On best proximity points for pseudocontractions in the intermediate sense for non-cyclic and cyclic self-mappings in metric spaces

Fixed Point Theory and Applications20132013:146

DOI: 10.1186/1687-1812-2013-146

Received: 17 September 2012

Accepted: 17 May 2013

Published: 5 June 2013

Abstract

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

1 Introduction

Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points. See, for instance, [14] and references therein. Results about the existence of a fixed point are discussed in those papers. On the other hand, important attention has been paid during the last decades to the study of the convergence properties of distances in cyclic contractive self-mappings on p subsets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq1_HTML.gif of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif , or a Banach space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq3_HTML.gif . The cyclic self-mappings under study have been of standard contractive or weakly contractive types and of Meir-Keeler type. The convergence of sequences to fixed points and best proximity points of the involved sets has been investigated in the last years. See, for instance, [520] and references therein. It has to be noticed that every nonexpansive mapping [21, 22] is a 0-strict pseudocontraction and also that strict pseudocontractions in the intermediate sense are asymptotically nonexpansive [2]. The uniqueness of the best proximity points to which all the sequences of iterations converge is proven in [6] for the extension of the contractive principle for cyclic self-mappings in either uniformly convex Banach spaces (then being strictly convex and reflexive [23]) or in reflexive Banach spaces [13]. The p subsets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq4_HTML.gif of the metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif , or the Banach space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq3_HTML.gif , where the cyclic self-mappings are defined, are supposed to be nonempty, convex and closed. If the involved subsets have nonempty intersections, then all best proximity points coincide, with a unique fixed point being allocated in the intersection of all the subsets, and framework can be simply given on complete metric spaces. The research in [6] is centered on the case of the 2-cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in [7] for Meir-Keeler cyclic contraction maps and, in general, with the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq5_HTML.gif -cyclic self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq6_HTML.gif defined on any number of subsets of the metric space with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq7_HTML.gif . Other recent research which has been performed in the field of cyclic maps is related to the introduction and discussion of the so-called cyclic representation of a set M, as the union of a set of nonempty sets as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq8_HTML.gif , with respect to an operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq9_HTML.gif [14]. Subsequently, cyclic representations have been used in [15] to investigate operators from M to M which are cyclic φ-contractions, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq10_HTML.gif is a given comparison function, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif is a metric space. The above cyclic representation has also been used in [16] to prove the existence of a fixed point for a self-mapping defined on a complete metric space which satisfies a cyclic weak φ-contraction. In [18], a characterization of best proximity points is studied for individual and pairs of non-self-mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq12_HTML.gif , where A and B are nonempty subsets of a metric space. The existence of common fixed points of self-mappings is investigated in [24] for a class of nonlinear integral equations, while fixed point theory is investigated in locally convex spaces and non-convex sets in [2528]. More recently, the existence and uniqueness of best proximity points of more general cyclic contractions have been investigated in [29, 30] and a study of best proximity points for generalized proximal contractions, a concept referred to non-self-mappings, has been proposed and reported in detail in [31]. Also, the study and characterization of best proximity points for cyclic weaker Meir-Keeler contractions have been performed in [32] and recent contributions on the study of best proximity and proximal points can be found in [3338] and references therein. In general, best proximity points do not fulfill the usual ‘best proximity’ condition http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq13_HTML.gif under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq15_HTML.gif . Furthermore, a class of cyclic φ-contractions, which contains the cyclic contraction maps as a subclass, has been proposed in [18] in order to investigate the convergence and existence results of best proximity points in reflexive Banach spaces completing previous related results in [6]. Also, the existence and uniqueness of best proximity points of cyclic φ-contractive self-mappings in reflexive Banach spaces have been investigated in [19]. This paper is devoted to the convergence properties and the existence of fixed points of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically pseudocontractive in the intermediate sense in the more general framework of metric spaces. The case of 2-cyclic pseudocontractive self-mappings is also considered. The combination of constants defining the contraction may be different on each of the subsets and only the product of all the constants is requested to be less than unity. It is assumed that the considered self-mapping can perform a number of iterations on each of the subsets before transferring its image to the next adjacent subset of the 2-cyclic self-mapping. The existence of a unique closed finite limiting sequence on any sequence of iterations from any initial point in the union of the subsets is proven if X is a uniformly convex Banach space and all the subsets of X are nonempty, convex and closed. Such a limiting sequence is of size http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq16_HTML.gif (with the inequality being strict if there is at least one iteration with image in the same subset as its domain), where p of its elements (all of them if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq17_HTML.gif ) are best proximity points between adjacent subsets. In the case that all the subsets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq18_HTML.gif intersect, the above limit sequence reduces to a unique fixed point allocated within the intersection of all such subsets.

2 Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces

If H is a real Hilbert space with an inner product http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq19_HTML.gif and a norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq20_HTML.gif and A is a nonempty closed convex subset of H, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq22_HTML.gif if
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ1_HTML.gif
(2.1)
for some sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq24_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif [14, 23]. Such a concept was firstly introduced in [1]. If (2.1) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq26_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq27_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically β-strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate sense if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq26_HTML.gif . If (2.1) is changed to the stronger condition
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ2_HTML.gif
(2.2)
then the above concepts translate into http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif being an asymptotically β-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one, respectively. Note that (2.1) is equivalent to
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ3_HTML.gif
(2.3)
or, equivalently,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ4_HTML.gif
(2.4)
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ5_HTML.gif
(2.5)
Note that the high-right-hand-side term http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq29_HTML.gif of (2.3) is expanded as follows for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq30_HTML.gif :
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ6_HTML.gif
(2.6)
The objective of this paper is to discuss the various pseudocontractive in the intermediate sense concepts in the framework of metric spaces endowed with a homogeneous and translation-invariant metric and also to generalize them to the β-parameter to eventually be replaced with a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq31_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq32_HTML.gif . Now, if instead of a real Hilbert space H endowed with an inner product http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq19_HTML.gif and a norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq20_HTML.gif , we deal with any generic Banach space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq33_HTML.gif , then its norm induces a homogeneous and translation invariant metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq34_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq35_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif so that (2.6) takes the form
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ7_HTML.gif
(2.7)
Define
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ8_HTML.gif
(2.8)
which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ9_HTML.gif
(2.9)

The following result holds related to the discussion (2.7)-(2.9) in metric spaces.

Theorem 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq37_HTML.gif be a metric space and consider a self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif . Assume that the following constraint holds:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ10_HTML.gif
(2.10)
with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ11_HTML.gif
(2.11)
for some parameterizing bounded real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq41_HTML.gif of general terms http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq44_HTML.gif satisfying the following constraints:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ12_HTML.gif
(2.12)
with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq45_HTML.gif and, furthermore, the following condition is satisfied:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ13_HTML.gif
(2.13)

if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq46_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq48_HTML.gif .

Then the following properties hold:
  1. (i)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq49_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq50_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically nonexpansive.

     
  2. (ii)
    Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif be complete, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq51_HTML.gif be, in addition, a translation-invariant homogeneous norm and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq52_HTML.gif , with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif being the metric-induced norm from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq54_HTML.gif , be a uniformly convex Banach space. Assume also that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is continuous. Then any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq55_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq56_HTML.gif is bounded and convergent to some point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq57_HTML.gif , being in general dependent on x, in some nonempty bounded, closed and convex subset C of A, where A is any nonempty bounded subset of X. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq58_HTML.gif is bounded; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq59_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq60_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq62_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq63_HTML.gif is a fixed point of the restricted self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif . Furthermore,
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ14_HTML.gif
    (2.14)
     

Proof Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif as follows:

(A) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq67_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq68_HTML.gif . Then one gets from (2.10)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ15_HTML.gif
(2.15)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ16_HTML.gif
(2.16)
which holds from (2.12)-(2.13) if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq70_HTML.gif since
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equa_HTML.gif
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq71_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif in (2.13) is equivalent to (2.16). Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq72_HTML.gif is ensured either with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq73_HTML.gif or with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq74_HTML.gif if
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ17_HTML.gif
(2.17)

However, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq75_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq76_HTML.gif has to be excluded because of the unboundedness or nonnegativity of the second right-hand-side term of (2.15).

(B) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq77_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq68_HTML.gif . Then one gets from (2.10)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ18_HTML.gif
(2.18)
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ19_HTML.gif
(2.19)
which holds from (2.12) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq78_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq79_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ20_HTML.gif
(2.20)
Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq80_HTML.gif together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq81_HTML.gif to be either http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq82_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq83_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq68_HTML.gif . It then follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq84_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq85_HTML.gif from (2.15)-(2.19) since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq87_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq85_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif equivalent to the translation-invariant homogeneous metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq88_HTML.gif . Such a norm exists since the metric is homogeneous and translation-invariant so that norm and metric are formally equivalent. Rename http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq89_HTML.gif and define a sequence of subsets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq90_HTML.gif of X. From Property (i), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq91_HTML.gif is bounded; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq92_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq93_HTML.gif is finite, since it is bounded for any finite http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif and, furthermore, it has a finite limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif . Thus, all the collections of subsets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq94_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq95_HTML.gif are bounded since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq96_HTML.gif is bounded. Define the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq97_HTML.gif which is nonempty bounded, closed and convex by construction. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq98_HTML.gif is complete, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq99_HTML.gif is a uniformly convex Banach space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif is asymptotically nonexpansive from Property (i), then it has a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq100_HTML.gif [1, 23]. Since the restricted self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif is also continuous, one gets from Property (i)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ21_HTML.gif
(2.21)

Then any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq101_HTML.gif is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in C; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif . This also implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq102_HTML.gif is bounded; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq60_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq104_HTML.gif . This implies also http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq105_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq106_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq107_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif which is then a fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif (otherwise, the above property http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq108_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq104_HTML.gif would be contradicted). Hence, Property (ii) is proven. □

First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex space which is also a complete metric space. Since the metric is homogeneous and translation-invariant, a norm can be induced by such a metric. Alternatively, the property could be established on any uniformly convex Banach space by taking a norm-induced metric which always exists. Conceptually similar arguments are used in later parallel results throughout the paper. Note that the proof of Theorem 2.1(i) has two parts: Case (A) refers to an asymptotically nonexpansive self-mapping which is contractive for any number of finite iteration steps and Case (B) refers to an asymptotically nonexpansive self-mapping which is allowed to be expansive for a finite number of iteration steps. It has to be pointed out concerning such a Theorem 2.1(ii) that the given conditions guarantee the existence of at least a fixed point but not its uniqueness. Therefore, the proof is outlined with the existence of a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq109_HTML.gif for any nonempty, bounded and closed subset A of X. Note that the set C, being in general dependent on the initial set A, is bounded, convex and closed by construction while any taken nonempty set of initial conditions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq110_HTML.gif is not required to be convex. However, the property that all the sequences converge to fixed points opens two potential possibilities depending on particular extra restrictions on the self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif , namely: (1) the fixed point is not unique so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq111_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif (and any A in X) so that some set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq113_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq114_HTML.gif contains more than one point. In other words, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq115_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif has not been proven although it is true that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq116_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif ; (2) there is only a fixed point in X. The following result extends Theorem 2.1 for a modification of the asymptotically nonexpansive condition (2.10).

Theorem 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq37_HTML.gif be a metric space and consider the self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif . Assume that the constraint below holds:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ22_HTML.gif
(2.22)
with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ23_HTML.gif
(2.23)
for some parameterizing real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq119_HTML.gif satisfying, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ24_HTML.gif
(2.24)
Then the following properties hold:
  1. (i)
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq120_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically nonexpansive, and then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq121_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif if
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ25_HTML.gif
    (2.25)
    and the following limit exists:
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ26_HTML.gif
    (2.26)
     
  2. (ii)

    Property (ii) of Theorem 2.1 if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif is complete and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq122_HTML.gif is a uniformly convex Banach space under the metric-induced norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif .

     

Sketch of the proof Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1 for Case (B). Using proving arguments similar to those used to prove Theorem 2.1, one proves Property (ii). □

The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions in the intermediate sense and the asymptotic strict contractions in the intermediate sense relies on Case (B) in the proof of Property (i) with the sequence of constants http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq123_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq124_HTML.gif ; as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif . The concepts of an asymptotic pseudocontraction and an asymptotic strict pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to (2.23) are revisited as follows in the context of metric spaces.

Definition 2.3 Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif is a complete metric space with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq34_HTML.gif being a homogeneous translation-invariant metric. Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense if
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ27_HTML.gif
(2.27)
for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq125_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif and some real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq128_HTML.gif being, in general, dependent on the initial points, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq130_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ28_HTML.gif
(2.28)

Definition 2.4 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq133_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq134_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq136_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif and the remaining conditions as in Definition 2.3 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq137_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq130_HTML.gif .

Definition 2.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically β-strictly contractive in the intermediate sense if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq140_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq141_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq143_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq144_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq48_HTML.gif , in Definition 2.3 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq130_HTML.gif .

Definition 2.6 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically contractive in the intermediate sense if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq147_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq149_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq150_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq151_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif in Definition 2.3 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq152_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq154_HTML.gif .

Remark 2.7 Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the convergence properties under the constraint (2.22), subject to (2.23), by noting the following:
  1. (1)

    If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq155_HTML.gif of asymptotically nonexpansive constants has a general term http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq156_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif , and it converges to a limit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq157_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq134_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif from (2.22) since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq159_HTML.gif from (2.27). Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is trivially asymptotically nonexpansive as expected.

     
  2. (2)

    If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq160_HTML.gif of asymptotically nonexpansive constants has the general term: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq161_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif , and it converges to a limit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq162_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq135_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is also trivially asymptotically nonexpansive as expected. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq163_HTML.gif , note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq164_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq165_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif , while http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq166_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq167_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq168_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif from (2.22)-(2.23).

     
  3. (3)

    If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically β-strictly contractive in the intermediate sense (Definition 2.5), then the sequence of asymptotically contractive constants is defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq169_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq170_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq171_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq172_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif , since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq173_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is an asymptotically strict contraction as expected since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq168_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif from (2.22)-(2.23). Note that the asymptotic convergence rate is arbitrarily fast as α and β are arbitrarily close to zero, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq174_HTML.gif becomes also arbitrarily close to zero, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq175_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq176_HTML.gif .

     
  4. (4)
    If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically contractive in the intermediate sense (Definition 2.6), then the sequence of asymptotically contractive constants is defined by
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equb_HTML.gif

    with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq151_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq177_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq178_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq179_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq180_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq181_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is an asymptotically strict contraction as expected since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq182_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif from (2.23). Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq183_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq185_HTML.gif . Note also that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq186_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq188_HTML.gif , while http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq189_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq190_HTML.gif . In the first case, the convergence to fixed points (see Theorem 2.8 below) is guaranteed to be asymptotically faster if the self-mapping is asymptotically β-strictly contractive in the intermediate sense than if it is just asymptotically contractive in the intermediate sense if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq191_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif . Note also that if the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq192_HTML.gif are identical in both cases, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq193_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq194_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq195_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq196_HTML.gif .

     
  5. (5)

    The above considerations could also be applied to Theorem 2.1 for the case http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq197_HTML.gif (Case (B) in the proof of Property (i)) being asymptotically nonexpansive for the asymptotically nonexpansive condition (2.10) subject to (2.11).

     

The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically contractive and pseudocontractive self-mappings in the intermediate sense. Therefore, it is assumed that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq198_HTML.gif .

Theorem 2.8 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif be a complete metric space endowed with a homogeneous translation-invariant metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq199_HTML.gif and consider the self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq200_HTML.gif is a uniformly convex Banach space endowed with a metric-induced norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif from the metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq54_HTML.gif . Assume that the asymptotically nonexpansive condition (2.22), subject to (2.23), holds for some parameterizing real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq201_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq203_HTML.gif satisfying, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq66_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ29_HTML.gif
(2.29)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq204_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq50_HTML.gif satisfying the conditions
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ30_HTML.gif
(2.30)
Furthermore, the following properties hold:
  1. (i)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq205_HTML.gif and any given nonempty, bounded and closed subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq110_HTML.gif of initial conditions if (2.29) hold with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq207_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq209_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif has a fixed point for any such set C if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is continuous.

     
  2. (ii)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif is asymptotically pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq210_HTML.gif and any given nonempty, bounded and closed subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq110_HTML.gif of initial conditions if (2.29) hold with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq211_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq212_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq214_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq64_HTML.gif has a fixed point for any such set C if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is continuous.

     
  3. (iii)

    If (2.29) hold with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq215_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq216_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq217_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq144_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically β-strictly contractive in the intermediate sense. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif has a unique fixed point.

     
  4. (iv)

    If (2.29) hold with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq218_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq220_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq221_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically strictly contractive in the intermediate sense. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif has a unique fixed point.

     

Proof (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the particular nonexpansive condition (2.22), subject to (2.23), so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq21_HTML.gif is asymptotically nonexpansive (see Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)). Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that the asymptotically nonexpansive self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is also a strict contraction, then continuous with a unique fixed point, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq222_HTML.gif (see Remark 2.7(3)) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq179_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq223_HTML.gif (see Remark 2.7(4)), respectively. (The above properties could also be got from Theorem 2.1 for Case (B) of the proof of Theorem 2.1(ii) - see Remark 2.7(5).) □

Example 2.9 Consider the time-varying pth order nonlinear discrete dynamic system
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ31_HTML.gif
(2.31)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq224_HTML.gif for some given nonempty bounded set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq225_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq226_HTML.gif is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq227_HTML.gif matrix sequence of elements http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq228_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq229_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq230_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq231_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq232_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq233_HTML.gif defines the state-sequence trajectory solution http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq234_HTML.gif . Equation (2.13) requires the consistency constraint http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq235_HTML.gif to calculate http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq236_HTML.gif . However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq237_HTML.gif in (2.31) with the initial condition http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq238_HTML.gif (and appropriate ad hoc re-definition of the mapping which generates the trajectory solution from given initial conditions) do not require such a consistency constraint. The dynamic system (2.31) is asymptotically linear if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq239_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq240_HTML.gif . Note that for the Euclidean distance (and norm), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq241_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq242_HTML.gif . Assume that the squared spectral norm of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq243_HTML.gif is upper-bounded by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq244_HTML.gif for some parameterizing scalar sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq245_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq246_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq192_HTML.gif which can be dependent, in a more general case, on the state http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq247_HTML.gif . This holds, for instance, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq248_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq249_HTML.gif is a real positive sequence satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq251_HTML.gif both being potentially dependent on the state as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the matrix is symmetric, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq252_HTML.gif can be taken exactly as the spectral radius of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq243_HTML.gif in such a case, i.e., it equalizes the absolute value of its dominant eigenvalue. We have to check the condition
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ32_HTML.gif
(2.32)

provided, for instance, that the distance is the Euclidean distance, induced by the Euclidean norm, then both being coincident, and provided also that we take the metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq253_HTML.gif which holds, in particular, if

(a) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq254_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq256_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq257_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq258_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq259_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq260_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq261_HTML.gif , as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq232_HTML.gif . This implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq262_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq263_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq232_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq264_HTML.gif is asymptotically nonexpansive being also an asymptotic strict β-pseudocontraction in the intermediate sense. This also implies that (2.31) is globally stable as it is proven as follows. Assume the contrary so that there is an infinite subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq265_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq266_HTML.gif which is unbounded, and then there is also an infinite subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq267_HTML.gif which is strictly increasing. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq268_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq269_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq232_HTML.gif , one has that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq270_HTML.gif , any given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq271_HTML.gif and some sufficiently large http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq272_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq273_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq274_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq275_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq277_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq278_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq279_HTML.gif . Now, take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq280_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq281_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq282_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq283_HTML.gif and any given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq284_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq285_HTML.gif , then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq286_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq287_HTML.gif for any given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq288_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq289_HTML.gif and some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq290_HTML.gif . Note that since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq267_HTML.gif is a strictly increasing real sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq291_HTML.gif implying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq292_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq293_HTML.gif , which leads to a contradiction to the inequality http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq294_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq295_HTML.gif for some sufficiently large http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq296_HTML.gif , then for some sufficiently large M, if such a strictly increasing sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq267_HTML.gif exists. Hence, there is no such sequence, and then no unbounded sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq265_HTML.gif for any initial condition in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq96_HTML.gif . As a result, for any initial condition in any given subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq225_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq297_HTML.gif (even if it is unbounded), any solution sequence of (2.31) is bounded, and then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable nonempty bounded closed sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq298_HTML.gif , which are not necessarily connected, such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq299_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq232_HTML.gif and any given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq300_HTML.gif . Assume that the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq96_HTML.gif of initial conditions is bounded, convex and closed and consider the collection of convex envelopes http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq301_HTML.gif , define constructively the closure convex set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq302_HTML.gif which is trivially bounded, convex and closed. Note that it is not guaranteed that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq303_HTML.gif is either open or closed since there is a union of infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the collection of sets is involved to ensure that A is convex since the union of convex sets is not necessarily convex (so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq304_HTML.gif is not guaranteed to be convex while A is convex). Consider now the self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq305_HTML.gif which defines exactly the same solution as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq233_HTML.gif for initial conditions in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq225_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq306_HTML.gif is identified with the restricted self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq307_HTML.gif from a nonempty bounded, convex and closed set to itself. Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq308_HTML.gif for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq309_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq310_HTML.gif are both continuous, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq305_HTML.gif is also continuous and has a fixed point in A from Theorem 2.8(i).

(b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above conclusions still hold with the modification http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq311_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq312_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq242_HTML.gif . From Remark 2.7(2), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq313_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq314_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq315_HTML.gif . Thus the convergence is guaranteed to be faster for an asymptotic β-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction in the intermediate sense with a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq316_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq317_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif with the remaining parameters and parametrical sequences being identical in both cases. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq318_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq319_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq320_HTML.gif are both continuous, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq321_HTML.gif is continuous and has a fixed point in A from Theorem 2.8(ii).

(c) If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically β-strictly contractive in the intermediate sense, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq322_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq232_HTML.gif so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).

(d) If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is asymptotically contractive in the intermediate sense, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq323_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq95_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq38_HTML.gif is an asymptotic strict contraction and has a unique fixed point from Theorem 2.8(iv).

Remark 2.10 Note that conditions like (2.32) can be tested on dynamic systems being different from (2.31) by redefining, in an appropriate way, the self-mapping which generates the solution sequence from given initial conditions. This allows to investigate the asymptotic properties of the self-mapping, the convergence of the solution to fixed points, then the system stability, etc. in a unified way for different dynamic systems. Close considerations can be discussed for different dynamic systems and convergence of the solutions generated by the different cyclic self-mappings defined on the union of several subsets to the best proximity points of each of the involved subsets.

3 Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate sense

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq324_HTML.gif be nonempty subsets of X. http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq325_HTML.gif is a cyclic self-mapping if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq327_HTML.gif . Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ33_HTML.gif
(3.1)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ34_HTML.gif
(3.2)
with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq328_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif , and that the asymptotically nonexpansive condition (2.22), subject to (2.23), is modified as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ35_HTML.gif
(3.3)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ36_HTML.gif
(3.4)

with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq328_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq47_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq329_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq330_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq331_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq332_HTML.gif and Theorems 2.1, 2.2 and 2.8 hold with the replacement http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq333_HTML.gif . Then if A and B are closed and convex, then there is a unique fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq325_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq334_HTML.gif . In the following, we consider the case that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq335_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq336_HTML.gif . The subsequent result based on Theorems 2.1, 2.2 and 2.8 holds.

Theorem 3.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif be a metric space and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq325_HTML.gif be a cyclic self-mapping, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq327_HTML.gif , where A and B are nonempty subsets of X. Define the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq337_HTML.gif of asymptotically nonexpansive iteration-dependent constants as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equc_HTML.gif
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq338_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif satisfies the constraint (3.1), subject to (3.2), and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ37_HTML.gif
(3.6)
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ38_HTML.gif
(3.7)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq340_HTML.gif ) and for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq341_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq342_HTML.gif ) provided that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif satisfies the constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq343_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq344_HTML.gif of general terms http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq345_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq346_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq154_HTML.gif fulfill the following constraints:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ39_HTML.gif
(3.8)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq347_HTML.gif and assuming that the following limits exist:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ40_HTML.gif
(3.9)
Then, the following properties hold:
  1. (i)
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif satisfies (3.3) subject to (3.4)-(3.9); http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq349_HTML.gif . Then
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equd_HTML.gif

    so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is a cyclic asymptotically nonexpansive self-mapping. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif is a best proximity point of A and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif is a best proximity point of B, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq351_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq352_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq353_HTML.gif , which are best proximity points of A and B (not being necessarily identical to x and y), respectively if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq354_HTML.gif is continuous.

     
  2. (ii)

    Property (i) also holds if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and (3.5b) provided that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq355_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq356_HTML.gif .

     
Proof The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ41_HTML.gif
(3.10)
and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ42_HTML.gif
(3.11)

since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq357_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq358_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq359_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq360_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq361_HTML.gif . Note that (3.8) implies that there is no division by zero in (3.11). Now, assume that (3.10) holds with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq362_HTML.gif . From (3.8) and (3.2), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq363_HTML.gif , equivalently, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq364_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq365_HTML.gif , which contradicts (3.5a) if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq76_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq362_HTML.gif in (3.5a) under (3.7) implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq366_HTML.gif and, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq367_HTML.gif from (3.6), there is no division by zero on the right-hand side of (3.10) if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq362_HTML.gif .

Also, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is continuous, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq368_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq369_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq370_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq371_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq372_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq358_HTML.gif . This proves Properties (i)-(ii). □

Remark 3.2 Note that Theorem 3.1 does not guarantee the convergence of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq373_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq374_HTML.gif to best proximity points if the initial points for the iterations http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif are not best proximity points if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is not contractive.

The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq375_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif subject to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq376_HTML.gif .

Theorem 3.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif be a metric space and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq325_HTML.gif be a cyclic self-mapping which satisfies the asymptotically nonexpansive constraint (3.1), subject to (3.2), where A and B are nonempty subsets of X. Let the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq377_HTML.gif of asymptotically nonexpansive iteration-dependent constants be defined by a general term http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq378_HTML.gif under the constraints http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq379_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq380_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq381_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq382_HTML.gif . Then the subsequent properties hold:
  1. (i)
    The following limits exist:
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ43_HTML.gif
    (3.12)
     
  2. (ii)
    Assume, furthermore, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq2_HTML.gif is complete, A and B are closed and convex and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq383_HTML.gif is translation-invariant and homogeneous and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq384_HTML.gif is uniformly convex where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif is the metric-induced norm. Then
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ44_HTML.gif
    (3.13)
     

http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq385_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq386_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq387_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq388_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq389_HTML.gif , where z and Tz are unique best proximity points in A and B, respectively. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq331_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq390_HTML.gif is the unique fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif .

Proof Note from (3.9), under (3.6) and (3.7), that there is no division by zero on the right-hand side of (3.10) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq391_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq392_HTML.gif . Then one has from (3.1)-(3.2), (3.5a), (3.6) and (3.7) that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ45_HTML.gif
(3.14)

There are several possible cases as follows.

Case A: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq393_HTML.gif is non-increasing. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq394_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq395_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq396_HTML.gif , one gets (3.12).

Case B: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq393_HTML.gif is non-decreasing. Then either http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq397_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq398_HTML.gif or it is unbounded. Then it has a subsequence which diverges, from which a strictly increasing subsequence can be taken. But this contradicts http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq399_HTML.gif following from (3.14) subject to the given parametrical constraints. Thus, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq400_HTML.gif is non-decreasing, it cannot have a strictly increasing subsequence so that it is bounded and has a finite limit as in Case A.

Case C: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq393_HTML.gif has an oscillating subsequence. It is proven that such a subsequence is finite. Assume not, then if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq401_HTML.gif , there is an integer sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq402_HTML.gif of general term subject to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq403_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Eque_HTML.gif
but the above expression is equivalent, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq404_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq405_HTML.gif which are in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq406_HTML.gif , but not jointly in either A or B, to
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equf_HTML.gif
which contradicts http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq407_HTML.gif since both sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq408_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq409_HTML.gif are bounded; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq395_HTML.gif . Then there is no infinite oscillating sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq410_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq411_HTML.gif so that there is a finite limit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq412_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq413_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq414_HTML.gif . Now, proceed by contradiction by assuming the existence of some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq415_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq416_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq48_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq417_HTML.gif . Thus, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq418_HTML.gif , there is some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq419_HTML.gif such that there are two consecutive nonzero elements of a nonzero real sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq420_HTML.gif , which can depend on x and y, which satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq421_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ46_HTML.gif
(3.15)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq417_HTML.gif . Otherwise, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq422_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq423_HTML.gif and any given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq424_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq425_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq426_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq427_HTML.gif . One gets, by combining (3.14) and (3.15), that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ47_HTML.gif
(3.16)

http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq428_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq429_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq430_HTML.gif , and some nonnegative real sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq431_HTML.gif which converges to zero since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq432_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq433_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq434_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq435_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq417_HTML.gif . The relations (3.16) contradict http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq436_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq437_HTML.gif is positive http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif (and it does not converge to zero) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq438_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq439_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif . Thus, one concludes that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq420_HTML.gif converges to zero, and then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq440_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq441_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq441_HTML.gif . This leads to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq442_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq443_HTML.gif by taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq444_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq342_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq341_HTML.gif . Property (i) has been proven.

Now, Property (ii) is proven. It is first proven that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq445_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq443_HTML.gif if the metric is translation-invariant and homogeneous so that it induces a norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif if A and B are nonempty, closed and convex subsets of X and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq446_HTML.gif is a uniformly convex Banach space. Assume not and take such a norm to yield http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq447_HTML.gif . Then if A is nonempty, closed and convex and B is nonempty and closed and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq448_HTML.gif . It is known that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq449_HTML.gif from Theorem 3.1(i) for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq450_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq451_HTML.gif is a uniformly convex Banach space for the metric-induced norm (being equivalent to the translation-invariant homogeneous metric), we have the following property for the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq452_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq453_HTML.gif satisfying for some strictly increasing nonnegative sequence of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq454_HTML.gif and any nonnegative sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq455_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq456_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq457_HTML.gif and any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq458_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ48_HTML.gif
(3.17)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ49_HTML.gif
(3.18)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ50_HTML.gif
(3.19)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq443_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif , which implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ51_HTML.gif
(3.20)
which has to be valid for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq459_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif . Now, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq460_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq461_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq462_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq463_HTML.gif , which is a contradiction to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq464_HTML.gif being strictly increasing, then contradicting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq465_HTML.gif being a uniformly convex Banach space, unless http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq466_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq467_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq468_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq469_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif , (3.15) for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq470_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq48_HTML.gif implies the existence of the first zero limit in (3.13). The existence of the second zero limit in (3.13) is proven in the same way since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq471_HTML.gif . Since those limits are zero, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq472_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq473_HTML.gif are Cauchy sequences in A converging to a best proximity point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq474_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq475_HTML.gif is necessarily the unique best proximity point in A since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq472_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq476_HTML.gif converge to the same point. Otherwise, the first limit of (3.13) would not exist if the sequences do not converge, then a contradiction holds to a proven result, and also Property (i) would not be true, since (3.12) would not hold, if the limit of the sequence would not be a best proximity point in A, then a contradiction holds to another proven result. In the same way, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq477_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq478_HTML.gif converge to a unique best proximity point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq479_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq480_HTML.gif . Now, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq481_HTML.gif . Assume not. Then since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq482_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq483_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq484_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq485_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq486_HTML.gif so that since A and B are convex,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equg_HTML.gif

which is a contradiction. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq481_HTML.gif is the unique best proximity of B. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq487_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq488_HTML.gif is the unique fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq489_HTML.gif which coincides with the unique best proximity point in A and B. □

Remark 3.4 Theorem 3.3 is known for strictly contractive cyclic self-mappings [20] satisfying the contractive condition (3.1) in the case that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq490_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq491_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq492_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq493_HTML.gif [57].

It is now assumed that the cyclic self-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq325_HTML.gif is asymptotically nonexpansive while not being strictly contractive for any finite number of iterations. The concepts of cyclic pseudocontractions and a strict contraction in the intermediate sense play an important role in the obtained results.

Theorem 3.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq3_HTML.gif be a uniformly convex Banach space endowed with a metric-induced norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq53_HTML.gif from a translation-invariant homogeneous metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq54_HTML.gif , where A and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq494_HTML.gif are nonempty, closed and convex subsets of X and assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif is a cyclic self-mapping. Define the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq495_HTML.gif of asymptotically nonexpansive iteration-dependent constants as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equh_HTML.gif
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq338_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif satisfies the constraint (3.1), subject to (3.2); and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ52_HTML.gif
(3.22)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq340_HTML.gif ) and for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq341_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq496_HTML.gif ) provided that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif satisfies the constraint (3.3), subject to (3.4), provided that the parameterizing bounded real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq497_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq498_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq499_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq500_HTML.gif of general terms http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq501_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq154_HTML.gif fulfill the following constraints:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ53_HTML.gif
(3.23)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq347_HTML.gif and assuming that the following limits exist:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ54_HTML.gif
(3.24)
Then the following properties hold:
  1. (i)
    If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif satisfies (3.3) subject to (3.20)-(3.24); http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq502_HTML.gif , then
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equi_HTML.gif

    so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is asymptotically nonexpansive. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq480_HTML.gif is a best proximity point of A and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif is a best proximity point of B, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq503_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq504_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq505_HTML.gif which are best proximity points of A and B (not being necessarily identical to x and y), respectively, if furthermore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is continuous.

     
  2. (ii)

    Property (i) also holds if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif satisfies (3.1) subject to (3.2), (3.22), (3.23)-(3.24) and (3.5b) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq506_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq507_HTML.gif .

     
  3. (iii)

    Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense so that (3.21a)-(3.21b) holds with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq508_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq509_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq510_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq511_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq136_HTML.gif , as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq512_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq354_HTML.gif is asymptotically nonexpansive and Property (i) holds.

     
  4. (iv)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is asymptotically pseudocontractive in the intermediate sense if (3.22) holds with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq513_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq514_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq515_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq516_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq209_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq381_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is asymptotically nonexpansive and Property (i) holds.

     
  5. (v)

    If the conditions of Property (iv) are modified as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq517_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq518_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq519_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq520_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq521_HTML.gif in (3.22), then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is asymptotically β-strictly contractive in the intermediate sense. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif has a unique best proximity point z in A and a unique best proximity point Tz in B to which the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq522_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq523_HTML.gif converge; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq341_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq524_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq525_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif .

     
  6. (vi)

    If (3.4) is modified by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq526_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq527_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq528_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq149_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq135_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq28_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is asymptotically-strictly contractive in the intermediate sense. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif has a unique best proximity point in A and a unique best proximity point in B to which the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq522_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq529_HTML.gif converge as in Property (v).

     
Proof The second condition of (2.18) now becomes under (3.1)-(3.2), or (3.3)-(3.4), and (3.23)-(3.24)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equj_HTML.gif

since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq359_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq530_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq48_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq531_HTML.gif . Also, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq348_HTML.gif is continuous, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq532_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq369_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq533_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq371_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq534_HTML.gif since A and B are closed and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq535_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq536_HTML.gif . This proves Properties (i)-(ii). To prove Property (iii), note that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq537_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense under (3.21a)-(3.21b)-(3.23) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq538_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq381_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq539_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif and (3.22) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq209_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq540_HTML.gif is asymptotically nonexpansive and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq541_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq25_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq542_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif are best proximity points. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq543_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq502_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq369_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq533_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq371_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq544_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif is continuous. Then Property (i) holds. Property (iv) is proven in a similar way as (iii) since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq339_HTML.gif is again asymptotically nonexpansive. Properties (v)-(vi) follow since in both cases http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq325_HTML.gif becomes a cyclic strictly contractive self-mapping for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq545_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq546_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq547_HTML.gif and some finite http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq548_HTML.gif in Theorem 3.3, Eq. (3.14). Thus, it is a direct proof that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq549_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq550_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq551_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq552_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq327_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq553_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq443_HTML.gif . Furthermore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq554_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq555_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq389_HTML.gif and there are unique best proximity points http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq475_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq556_HTML.gif . The convergence of the iterations to unique best proximity points follows using similar arguments as those used in the proof of Theorem 3.3(ii) based on the uniform convexity of the complete metric space and the fact that the subsets A and B are nonempty, convex and closed. □

Remark 3.6 Note that the existence of Theorem 3.5 of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq350_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq557_HTML.gif is guaranteed if A is nonempty, bounded, closed and convex and B is nonempty closed and convex is also guaranteed if A is compact and B is approximately compact with respect to A, i.e., if every sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq558_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq559_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq342_HTML.gif , has a convergent subsequence [6, 7, 31].

Example 3.7 Consider the time-varying scalar controlled discrete dynamic system:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ55_HTML.gif
(3.25)
under the feedback control sequence
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ56_HTML.gif
(3.26)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ57_HTML.gif
(3.27)
so that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ58_HTML.gif
(3.28)
where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq560_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq561_HTML.gif for some given nonempty bounded set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq96_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq562_HTML.gif is the control sequence. The above model can describe discrete-time dynamic systems under time-varying sampling periods or under a time-varying parameterization in general [39]. Assume that the suitable controlled solution (3.28) is of the form
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equk_HTML.gif
Then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ59_HTML.gif
(3.29)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ60_HTML.gif
(3.30)
The identities (3.30) allow the feedback generation of the control sequence (3.26) from its previous values and previous solution values as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ61_HTML.gif
(3.31)
for given parameterizing scalar sequences which can be dependent on the state http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq247_HTML.gif (see Example 2.9). We are now defining a cyclic self-map http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq563_HTML.gif so that the solution belongs alternately to positive (respectively, nonnegative) and negative (respectively, nonpositive) real intervals http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq564_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq565_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq566_HTML.gif (respectively, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq567_HTML.gif ), that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq568_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq569_HTML.gif . For such an objective, consider the scalar bounded sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq570_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq571_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq572_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq573_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq574_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq575_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq576_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq69_HTML.gif which satisfy
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ62_HTML.gif
(3.32a)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_Equ63_HTML.gif
(3.32b)
Note that by using the Euclidean distance and norm on R, it is possible to apply the theoretical formalism to the expressions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq577_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq578_HTML.gif to prove convergence to the best proximity points http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq579_HTML.gif to which the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq580_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq581_HTML.gif converge, respectively if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq582_HTML.gif and conversely if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq583_HTML.gif . Assume that:
  1. (1)

    The constraints (3.32a)-(3.32b) hold;

     
  2. (2)

    The parametrical constraints of the various parts (a) to (d) of Example 2.9 hold with the replacements and its appropriate replacements of the constraints http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq584_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq585_HTML.gif ;

     
  3. (3)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq245_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq246_HTML.gif are redefined for this example from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq586_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq587_HTML.gif , respectively, from (3.32a)-(3.32b).

     

From Theorem 3.5, the various properties of Example 2.9 hold also for this example if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq332_HTML.gif so that the cyclic self-map is such that it alternates the values of the solution sequence between http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq588_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq589_HTML.gif . The unique fixed point to which the solution converges is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq590_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq591_HTML.gif , then the corresponding results are modified by convergence to each of the unique best proximity points to which the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq580_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq592_HTML.gif converge; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_485_IEq593_HTML.gif .

Declarations

Acknowledgements

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651, and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. He is also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07 and to the referees for their useful comments.

Authors’ Affiliations

(1)
Institute of Research and Development of Processes, University of the Basque Country

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© De la Sen; licensee Springer. 2013

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