Open Access

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 A i X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq1_HTML.gif of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif, or a Banach space ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 A i X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq4_HTML.gif of the metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif, or the Banach space ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 p ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq5_HTML.gif-cyclic self-mapping T : i p ¯ A i i p ¯ A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq6_HTML.gif defined on any number of subsets of the metric space with p ¯ : = { 1 , 2 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 M = i = 1 m M i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq8_HTML.gif, with respect to an operator f : M M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 φ : R 0 + R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq10_HTML.gif is a given comparison function, M X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq11_HTML.gif and ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 S , T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 x = S x = T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq13_HTML.gif under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq14_HTML.gif and d ( x , S x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 q p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 q = p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq17_HTML.gif) are best proximity points between adjacent subsets. In the case that all the subsets A i X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq19_HTML.gif and a norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq20_HTML.gif and A is a nonempty closed convex subset of H, then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some β [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq22_HTML.gif if
lim sup n sup x , y A ( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ1_HTML.gif
(2.1)
for some sequence { α n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq23_HTML.gif, α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq24_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif [14, 23]. Such a concept was firstly introduced in [1]. If (2.1) holds for β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq26_HTML.gif, then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if α n α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq27_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif, then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically β-strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate sense if β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq26_HTML.gif. If (2.1) is changed to the stronger condition
( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) 0 ; x , y A , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ2_HTML.gif
(2.2)
then the above concepts translate into T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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
T n x T n y 2 α n x y 2 + β ( I T n ) x ( I T n ) y 2 + ξ n ; x , y A , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ3_HTML.gif
(2.3)
or, equivalently,
T n x T n y , x y 1 2 β [ ( α n + β ) x y 2 + ( β 1 ) T n x T n y 2 + ξ n ] ; x , y A , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ4_HTML.gif
(2.4)
where
ξ n : = max { 0 , sup x , y A ( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) } ; n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ5_HTML.gif
(2.5)
Note that the high-right-hand-side term ( I T n ) x ( I T n ) y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq29_HTML.gif of (2.3) is expanded as follows for any x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq30_HTML.gif:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 { β n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq31_HTML.gif in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq32_HTML.gif. Now, if instead of a real Hilbert space H endowed with an inner product , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq19_HTML.gif and a norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq20_HTML.gif, we deal with any generic Banach space ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq33_HTML.gif, then its norm induces a homogeneous and translation invariant metric d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq34_HTML.gif defined by d ( x , y ) = d ( x y , 0 ) = x y 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq35_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif so that (2.6) takes the form
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ7_HTML.gif
(2.7)
Define
μ n ( x , y ) : = min ( ρ [ 1 , 1 ] : d 2 ( x y , T n x T n y ) d 2 ( x , y ) + d 2 ( T n x , T n y ) + 2 ρ d ( x , y ) d ( T n x , T n y ) ) ; x , y A , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ8_HTML.gif
(2.8)
which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that
{ 1 } { ρ R : ( I T n ) x ( I T n ) y 2 d 2 ( x , y ) + d 2 ( T n x , T n y ) + 2 ρ d ( x , y ) d ( T n x , T n y ) } ( ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq37_HTML.gif be a metric space and consider a self-mapping T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif. Assume that the following constraint holds:
d 2 ( T n x , T n y ) α n ( x , y ) d 2 ( x , y ) + β n ( x , y ) ( d 2 ( x , y ) + d 2 ( T n x , T n y ) ) + 2 μ n ( x , y ) β n ( x , y ) d ( x , y ) d ( T n x , T n y ) + ξ n ( x , y ) ; x , y X , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ10_HTML.gif
(2.10)
with
ξ n = ξ n ( x , y ) : = max ( 0 , ( 1 β n ( x , y ) ) d 2 ( T n x , T n y ) ( α n ( x , y ) + β n ( x , y ) ) d 2 ( x , y ) 2 μ n ( x , y ) β n ( x , y ) d ( x , y ) d ( T n x , T n y ) ) 0 ; x , y X as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ11_HTML.gif
(2.11)
for some parameterizing bounded real sequences { α n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq39_HTML.gif, { β n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq40_HTML.gif and { μ n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq41_HTML.gif of general terms α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq42_HTML.gif, β n = β n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq43_HTML.gif, μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq44_HTML.gif satisfying the following constraints:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ12_HTML.gif
(2.12)
with lim sup n [ β n ( x , y ) max ( 1 , 1 + 2 μ n ( x , y ) ) ] < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq45_HTML.gif and, furthermore, the following condition is satisfied:
( μ n ( x , y ) 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) ) 0 ; x , y X as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ13_HTML.gif
(2.13)

if and only if α n + 2 β n ( 1 + μ n ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq46_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq47_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq48_HTML.gif.

Then the following properties hold:
  1. (i)

    lim n d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq49_HTML.gif for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq50_HTML.gif so that T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically nonexpansive.

     
  2. (ii)
    Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif be complete, d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq51_HTML.gif be, in addition, a translation-invariant homogeneous norm and let ( X , ) ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq52_HTML.gif, with https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq53_HTML.gif being the metric-induced norm from d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq54_HTML.gif, be a uniformly convex Banach space. Assume also that T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is continuous. Then any sequence { T n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq55_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq56_HTML.gif is bounded and convergent to some point z x = z x ( x ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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, d ( T n x , T n + m x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq58_HTML.gif is bounded; n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq59_HTML.gif, lim n d ( T n x , T n + m x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq60_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq62_HTML.gif and z x = z x ( x ) = T z x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq63_HTML.gif is a fixed point of the restricted self-mapping T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif. Furthermore,
    lim n ( d 2 ( T n + 1 x , T n + 1 y ) d 2 ( T n x , T n y ) ) = 0 ; x , y A . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ14_HTML.gif
    (2.14)
     

Proof Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq65_HTML.gif and n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif as follows:

(A) d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq67_HTML.gif for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq50_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq68_HTML.gif. Then one gets from (2.10)
d 2 ( T n x , T n y ) ( α n + β n ) d 2 ( x , y ) + β n d 2 ( T n x , T n y ) + 2 μ n β n d 2 ( x , y ) + ξ n d ( T n x , T n y ) k a n d 2 ( x , y ) + ξ n 1 β n ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ15_HTML.gif
(2.15)
x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif, where
k a n = k a n ( x , y ) = α n + β n ( 1 + 2 μ n ) 1 β n 1 ; x , y X  as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ16_HTML.gif
(2.16)
which holds from (2.12)-(2.13) if lim sup n β n ( x , y ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq70_HTML.gif since
( μ n ( x , y ) 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) ) 0 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equa_HTML.gif
x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq71_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif in (2.13) is equivalent to (2.16). Note that 0 k a n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq72_HTML.gif is ensured either with min ( α n + β n ( 1 + 2 μ n ) , 1 β n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq73_HTML.gif or with max ( α n + β n ( 1 + 2 μ n ) , 1 β n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq74_HTML.gif if
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ17_HTML.gif
(2.17)

However, β n > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq75_HTML.gif with ξ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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) d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq77_HTML.gif for some x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq50_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq68_HTML.gif. Then one gets from (2.10)
d ( T n x , T n y ) 2 ( α n + β n ) d 2 ( x , y ) + β n d 2 ( T n x , T n y ) + 2 μ n β n d 2 ( T n x , T n y ) + ξ n d ( T n x , T n y ) 2 k b n d 2 ( x , y ) + ξ n 1 β n ( 1 + 2 μ n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ18_HTML.gif
(2.18)
where
k b n = k b n ( x , y ) = α n + β n 1 β n ( 1 + 2 μ n ) 1 as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ19_HTML.gif
(2.19)
which holds from (2.12) and k b n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq78_HTML.gif if lim sup n [ β n ( x , y ) max ( 1 , 1 + 2 μ n ( x , y ) ) ] < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq79_HTML.gif, and
μ n ( x , y ) [ 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) , 1 β n ( x , y ) 2 β n ( x , y ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 ξ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq80_HTML.gif together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking k n = k n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq81_HTML.gif to be either k a n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq82_HTML.gif or k b n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq83_HTML.gif for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq68_HTML.gif. It then follows that lim sup n ( d ( T n x , T n y ) d ( x , y ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq84_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq85_HTML.gif from (2.15)-(2.19) since 0 k n = k n ( x , y ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq86_HTML.gif and k n ( x , y ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq87_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq85_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif. Thus, T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq53_HTML.gif equivalent to the translation-invariant homogeneous metric d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 A 0 A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq89_HTML.gif and define a sequence of subsets A j : = { T j x : x A 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq90_HTML.gif of X. From Property (i), { d ( T n x , T n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq91_HTML.gif is bounded; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq92_HTML.gif if d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq93_HTML.gif is finite, since it is bounded for any finite n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif and, furthermore, it has a finite limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif. Thus, all the collections of subsets i = 1 k A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq94_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq95_HTML.gif are bounded since A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq96_HTML.gif is bounded. Define the set C = C ( A 0 ) : = cl [ convex ( i = 1 A k ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq97_HTML.gif which is nonempty bounded, closed and convex by construction. Since ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq98_HTML.gif is complete, ( X , ) ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq99_HTML.gif is a uniformly convex Banach space and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif is asymptotically nonexpansive from Property (i), then it has a fixed point z = T z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq100_HTML.gif [1, 23]. Since the restricted self-mapping T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif is also continuous, one gets from Property (i)
lim n d ( T n x , T n z ) = lim n d ( T n x , z ) = d ( lim n T n x , z ) d ( x , z ) < ; x A . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ21_HTML.gif
(2.21)

Then any sequence { T n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq101_HTML.gif is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in C; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif. This also implies d ( T n x , T n + m x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq102_HTML.gif is bounded; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif, n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq103_HTML.gif and lim n d ( T n x , T n + m x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq60_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq104_HTML.gif. This implies also T n x z x ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq105_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq106_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif such that z x ( x ) = T z x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq107_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif which is then a fixed point of T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif (otherwise, the above property lim n d ( T n x , T n + m x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq108_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 z Fix ( T | C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 A X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif, namely: (1) the fixed point is not unique so that z x z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq111_HTML.gif for any x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq112_HTML.gif (and any A in X) so that some set Fix ( T | C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq113_HTML.gif for some C = C ( A ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq114_HTML.gif contains more than one point. In other words, d 2 ( T n x , T n y ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq115_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif has not been proven although it is true that lim n ( d 2 ( T n + 1 x , T n + 1 y ) d 2 ( T n x , T n y ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq116_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq37_HTML.gif be a metric space and consider the self-mapping T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif. Assume that the constraint below holds:
d 2 ( T n x , T n y ) α n ( x , y ) d 2 ( x , y ) + β n ( x , y ) ( d 2 ( x , y ) + d 2 ( T n x , T n y ) ) + 2 μ n ( x , y ) β n ( x , y ) d 2 ( T n x , T n y ) + ξ n ( x , y ) ; x , y X , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ22_HTML.gif
(2.22)
with
ξ n = ξ n ( x , y ) : = max ( 0 , ( 1 β n ( x , y ) ) d 2 ( T n x , T n y ) ( α n ( x , y ) + β n ( x , y ) ) d 2 ( x , y ) 2 μ n ( x , y ) β n ( x , y ) d 2 ( T n x , T n y ) ) 0 ; x , y X as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ23_HTML.gif
(2.23)
for some parameterizing real sequences α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq117_HTML.gif, β n = β n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq118_HTML.gif and μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq119_HTML.gif satisfying, for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif,
{ α n ( x , y ) } [ 0 , ) , { μ n ( x , y ) } [ 1 , 1 β n ( x , y ) 2 β n ( x , y ) ) , { β n ( x , y ) } [ 0 , 1 ] ; x , y X , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ24_HTML.gif
(2.24)

Then the following properties hold:

(i) lim n d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq120_HTML.gif so that T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically nonexpansive, and then lim n d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq121_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq47_HTML.gif if
α n ( x , y ) + β n ( x , y ) 1 β n ( x , y ) ( 1 + 2 μ n ( x , y ) ) 1 μ n ( x , y ) [ 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) , 1 β n ( x , y ) 2 β n ( x , y ) ) ; x , y X , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ25_HTML.gif
(2.25)
and the following limit exists:
α n ( x , y ) + 2 β n ( x , y ) ( 1 + μ n ( x , y ) ) 1 ; x , y X as n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ26_HTML.gif
(2.26)

(ii) Property (ii) of Theorem  2.1 if ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif is complete and ( X , ) ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq122_HTML.gif is a uniformly convex Banach space under the metric-induced norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 k n ( x , y ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq123_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq47_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif and k n ( x , y ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq124_HTML.gif; as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif, x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif is a complete metric space with d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq34_HTML.gif being a homogeneous translation-invariant metric. Thus, T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense if
lim sup n ( ( 1 β n ( 1 + 2 μ n ) ) d 2 ( T n x , T n y ) ( α n + β n ) d 2 ( x , y ) ) 0 ; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ27_HTML.gif
(2.27)
for β n = β [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq125_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq126_HTML.gif and some real sequences { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq127_HTML.gif, { μ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq128_HTML.gif being, in general, dependent on the initial points, i.e., α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq129_HTML.gif, μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq130_HTML.gif and
{ μ n } [ 1 , 1 β 2 β ) and { α n } [ 1 , ) ; n N , α n 1  and  μ n 1  as  n ; x , y A , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ28_HTML.gif
(2.28)

Definition 2.4 T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with { μ n } [ 1 , 1 β n 2 β n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq131_HTML.gif, { β n } [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq132_HTML.gif, { α n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq133_HTML.gif, α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq134_HTML.gif, β n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq135_HTML.gif, μ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq136_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif and the remaining conditions as in Definition 2.3 with α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq137_HTML.gif, β n = β n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq138_HTML.gif and μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq130_HTML.gif.

Definition 2.5 T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically β-strictly contractive in the intermediate sense if α n [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq139_HTML.gif, β n = β [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq140_HTML.gif, μ n [ 1 , 1 β 2 β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq141_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq142_HTML.gif, μ n μ [ 1 , 1 β 2 β min ( 1 , 1 α + β ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq143_HTML.gif, α n α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq144_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq48_HTML.gif, in Definition 2.3 with α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq145_HTML.gif, μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq130_HTML.gif.

Definition 2.6 T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically contractive in the intermediate sense if α n [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq139_HTML.gif, { β n } [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq146_HTML.gif, μ n [ 1 , 1 β n 2 β n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq147_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq148_HTML.gif, μ n μ [ 1 , 1 + α 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq149_HTML.gif, α n α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq150_HTML.gif, and β n β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq151_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif in Definition 2.3 with α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq152_HTML.gif, β n = β n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq153_HTML.gif and μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real sequence { k 1 n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq155_HTML.gif of asymptotically nonexpansive constants has a general term k 1 n : = ( α n + β 1 β ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β 1 + β ) 1 / 2 , ) [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq156_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq126_HTML.gif, and it converges to a limit k 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq157_HTML.gif since ξ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq158_HTML.gif and α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq134_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif from (2.22) since μ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq159_HTML.gif from (2.27). Then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is trivially asymptotically nonexpansive as expected.

     
  2. (2)

    If T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), then the sequence { k 2 n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq160_HTML.gif of asymptotically nonexpansive constants has the general term: k 2 n : = ( α n + β n 1 β n ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β n 1 + β n ) 1 / 2 , ) [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq161_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq126_HTML.gif, and it converges to a limit k 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq162_HTML.gif since α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq24_HTML.gif, β n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq135_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif. Then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is also trivially asymptotically nonexpansive as expected. Since α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq163_HTML.gif, note that β n > β k 2 n > k 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq164_HTML.gif and β n < β k 2 n < k 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq165_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif, while k 1 n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq166_HTML.gif, k 2 n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq167_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif since ξ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq168_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif from (2.22)-(2.23).

     
  3. (3)

    If T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 k 3 n : = ( α n + β 1 β ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β 1 + β ) 1 / 2 , ) [ ( β 1 + β ) 1 / 2 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq169_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif and k 3 n k 3 = ( α + β 1 β ( 1 + 2 μ ) ) 1 / 2 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq170_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif for any μ [ 1 , 1 α 2 β 2 β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq171_HTML.gif such that μ n μ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq172_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif, since α + 2 β ( 1 + μ ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq173_HTML.gif. Then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is an asymptotically strict contraction as expected since ξ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq168_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 k 3 = O ( α + β ) = o ( α + β ) = o ( max ( α , β ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq174_HTML.gif becomes also arbitrarily close to zero, and k 3 2 K ( α + β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq175_HTML.gif with K = K ( β , μ ) = 1 1 β ( 1 + 2 μ ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq176_HTML.gif.

     
  4. (4)
    If T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is asymptotically contractive in the intermediate sense (Definition 2.6), then the sequence of asymptotically contractive constants is defined by
    k 4 n : = ( α n + β n 1 β n ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β n 1 + β n ) 1 / 2 , ) [ ( β n 1 + β n ) 1 / 2 , ) ; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equb_HTML.gif
     
with β n β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq151_HTML.gif and k 4 n k 4 = ( 1 + α 2 | μ | ) 1 / 2 [ 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq177_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif for some μ [ 1 , 1 + α 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq178_HTML.gif since μ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq179_HTML.gif with | μ | > 1 + α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq180_HTML.gif so that k 4 [ ( 1 + α 2 ) 1 / 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq181_HTML.gif. Then T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq21_HTML.gif is an asymptotically strict contraction as expected since ξ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq182_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif from (2.23). Note that k 3 = k 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq183_HTML.gif if μ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq179_HTML.gif and | μ | = 1 + α 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq184_HTML.gif and k 4 2 1 2 + o ( α ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq185_HTML.gif. Note also that k 3 < k 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq186_HTML.gif if μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq187_HTML.gif and | μ | < 1 + α 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq188_HTML.gif, while k 3 k 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq189_HTML.gif if μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq187_HTML.gif and | μ | 1 + α 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 β n > β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq191_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif. Note also that if the sequences { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq127_HTML.gif and { μ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq192_HTML.gif are identical in both cases, then k 3 n < k 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq193_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq194_HTML.gif such that β n > β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq191_HTML.gif and k 3 n k 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq195_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif such that β n β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq196_HTML.gif.
  1. (5)

    The above considerations could also be applied to Theorem 2.1 for the case d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 { α n ( x , y ) } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq198_HTML.gif.

Theorem 2.8 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif be a complete metric space endowed with a homogeneous translation-invariant metric d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq199_HTML.gif and consider the self-mapping T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif. Assume that ( X , ) ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq200_HTML.gif is a uniformly convex Banach space endowed with a metric-induced norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq53_HTML.gif from the metric d : X × X R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq54_HTML.gif. Assume that the asymptotically nonexpansive condition (2.22), subject to (2.23), holds for some parameterizing real sequences α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq201_HTML.gif, β n = β n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq202_HTML.gif and μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq203_HTML.gif satisfying, for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq66_HTML.gif,
{ α n ( x , y ) } [ 1 , ) , { μ n ( x , y ) } [ 1 , 1 β n ( x , y ) 2 β n ( x , y ) ) , { β n ( x , y ) } [ 0 , β ) [ 0 , 1 ] ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ29_HTML.gif
(2.29)
x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq47_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif. Then lim n d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq204_HTML.gif for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq50_HTML.gif satisfying the conditions
α n ( x , y ) + β n ( x , y ) 1 β n ( x , y ) ( 1 + 2 μ n ( x , y ) d ( x , y ) ) 1 ; α n ( x , y ) + 2 β n ( x , y ) ( 1 + μ n ( x , y ) ) 1 ; x , y X as n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ30_HTML.gif
(2.30)

Furthermore, the following properties hold:

(i) T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif is asymptotically β-strictly pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set C = C ( A ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq205_HTML.gif and any given nonempty, bounded and closed subset A X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq110_HTML.gif of initial conditions if (2.29) hold with 0 β n = β < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq206_HTML.gif, { μ n } [ 1 , 1 β 2 β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq207_HTML.gif, { α n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq208_HTML.gif, α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq24_HTML.gif and μ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq209_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif. Also, T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif has a fixed point for any such set C if T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is continuous.

(ii) T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif is asymptotically pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set C = C ( A ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq210_HTML.gif and any given nonempty, bounded and closed subset A X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq110_HTML.gif of initial conditions if (2.29) hold with { β n } [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq211_HTML.gif, { μ n } [ 1 , 1 β n 2 β n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq212_HTML.gif, { α n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq213_HTML.gif, β n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq135_HTML.gif, α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq134_HTML.gif and μ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq214_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif; x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq36_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif. Also, T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq64_HTML.gif has a fixed point for any such set C if T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is continuous.

(iii) If (2.29) hold with α n [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq139_HTML.gif, β n = β [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq215_HTML.gif, μ n [ 1 , 1 β 2 β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq216_HTML.gif, μ n μ [ 1 , 1 α 2 β 2 β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq217_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif and α n α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq144_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif, then T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically β-strictly contractive in the intermediate sense. Also, T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif has a unique fixed point.

(iv) If (2.29) hold with α n [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq139_HTML.gif, { β n } [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq218_HTML.gif, μ n [ 1 , 1 β n 2 β n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq219_HTML.gif, μ n μ [ 1 , 1 + α 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq220_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq126_HTML.gif, β n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq135_HTML.gif and α n α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq221_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif, then T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically strictly contractive in the intermediate sense. Also, T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is also a strict contraction, then continuous with a unique fixed point, since α + 2 β ( 1 + μ ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq222_HTML.gif (see Remark 2.7(3)) and μ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq179_HTML.gif with | μ | > 1 + α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 p th order nonlinear discrete dynamic system
x k + n = T n x k x k + 1 + F n ( x k ) ( x k + 1 x k ) + η n ( x k ) ; x 1 A 1 R p , k N ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ31_HTML.gif
(2.31)
k , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq224_HTML.gif for some given nonempty bounded set A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq225_HTML.gif, where { F n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq226_HTML.gif is a R p × p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq227_HTML.gif matrix sequence of elements F n k : R p R p × p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq228_HTML.gif with F n k = F n ( x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq229_HTML.gif and η n k : R p R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq230_HTML.gif with η n k = η n ( x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq231_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq232_HTML.gif, and T : R p R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq233_HTML.gif defines the state-sequence trajectory solution { x k ( x 1 , x 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq234_HTML.gif. Equation (2.13) requires the consistency constraint F 1 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq235_HTML.gif to calculate x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq236_HTML.gif. However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing x k + 1 x k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq237_HTML.gif in (2.31) with the initial condition x 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 η n ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq239_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; x R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq240_HTML.gif. Note that for the Euclidean distance (and norm), d ( x k + n , x k + 1 ) F n k d ( x k , x k + 1 ) + η n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq241_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq242_HTML.gif. Assume that the squared spectral norm of F n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq243_HTML.gif is upper-bounded by k n k 2 = k n 2 ( x k ) = α n k + β n k 1 β n k ( 1 + 2 μ n k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq244_HTML.gif for some parameterizing scalar sequences { α n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq245_HTML.gif, { β n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq246_HTML.gif and { μ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq192_HTML.gif which can be dependent, in a more general case, on the state x k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq247_HTML.gif. This holds, for instance, if F n k = F n ( x k ) = 1 a n k ( α n k + β n k 1 β n k ( 1 + 2 μ n k ) ) 1 / 2 P n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq248_HTML.gif, where { a n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq249_HTML.gif is a real positive sequence satisfying a n k P n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq250_HTML.gif and P n k ( = P n ( x k ) ) : N × R p R p × p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 k n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq252_HTML.gif can be taken exactly as the spectral radius of F n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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
lim sup n ( ( 1 + 2 μ n k β n k ) d 2 ( T n x k , T n y k ) ( α n k + β n k ) d 2 ( x k , y k ) ) 0 ; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 ( R p , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq253_HTML.gif which holds, in particular, if
  1. (a)

    { | η n k | } R 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq254_HTML.gif, { α n k } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq255_HTML.gif, β n k = β [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq256_HTML.gif, { μ n k } [ 1 , 1 β 2 β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq257_HTML.gif; n , k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq258_HTML.gif, α n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq259_HTML.gif and η n k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq260_HTML.gif, μ n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq261_HTML.gif, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq232_HTML.gif. This implies that k 1 n k 2 = α n k + β 1 β n k ( 1 + 2 μ n k ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq262_HTML.gif; n , k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq258_HTML.gif and k 1 n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq263_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq232_HTML.gif. Thus, T : R p R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 L u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq265_HTML.gif of { x n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq266_HTML.gif which is unbounded, and then there is also an infinite subsequence L u a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq267_HTML.gif which is strictly increasing. Since η n k = η n ( x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq268_HTML.gif and k 1 n k = k 1 n ( x k ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq269_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq232_HTML.gif, one has that for x 1 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq270_HTML.gif, any given k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq271_HTML.gif and some sufficiently large m 01 = m 01 ( x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq272_HTML.gif, m 02 = m 02 ( x k ) N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq273_HTML.gif, ε 1 = ε 1 ( m 01 ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq274_HTML.gif, ε 2 = ε 2 ( m 02 ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq275_HTML.gif such that k m 1 = k m 1 ( x k ) 1 + ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq276_HTML.gif and η m 2 ( x k ) ε 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq277_HTML.gif; m 1 m 01 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq278_HTML.gif, m 2 m 02 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq279_HTML.gif. Now, take m 0 = max ( m 01 , m 02 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq280_HTML.gif and ε = max ( ε 1 , ε 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq281_HTML.gif. Then x k + m / x k 1 + ε + ε / x k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq282_HTML.gif; m ( N ) m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq283_HTML.gif and any given k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq284_HTML.gif. If x k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq285_HTML.gif, then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take x k ( 0 ) , x k + m L u a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq286_HTML.gif such that x k + m x k M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq287_HTML.gif for any given M R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq288_HTML.gif, m ( N ) m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq289_HTML.gif and some m ¯ = m ¯ ( M ) m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq290_HTML.gif. Note that since L u a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq267_HTML.gif is a strictly increasing real sequence { M ( m ¯ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq291_HTML.gif implying M ( m ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq292_HTML.gif as m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq293_HTML.gif, which leads to a contradiction to the inequality M 1 + ε ( 1 + 1 x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq294_HTML.gif for ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq295_HTML.gif for some sufficiently large m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq296_HTML.gif, then for some sufficiently large M, if such a strictly increasing sequence L u a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq267_HTML.gif exists. Hence, there is no such sequence, and then no unbounded sequence L u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq265_HTML.gif for any initial condition in A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq96_HTML.gif. As a result, for any initial condition in any given subset A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq225_HTML.gif of R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 { A i R p : i N } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq298_HTML.gif, which are not necessarily connected, such that x k A k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq299_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq232_HTML.gif and any given x 0 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq300_HTML.gif. Assume that the set A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq96_HTML.gif of initial conditions is bounded, convex and closed and consider the collection of convex envelopes { convex A i R p : i N } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq301_HTML.gif, define constructively the closure convex set C ( A 0 ) = cl ( convex ( i = 1 convex A i ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq302_HTML.gif which is trivially bounded, convex and closed. Note that it is not guaranteed that i = 1 convex A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 i = 1 convex A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq304_HTML.gif is not guaranteed to be convex while A is convex). Consider now the self-mapping T ¯ : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq305_HTML.gif which defines exactly the same solution as T : R p R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq233_HTML.gif for initial conditions in A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq225_HTML.gif so that T ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq306_HTML.gif is identified with the restricted self-mapping T : R p | C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq307_HTML.gif from a nonempty bounded, convex and closed set to itself. Note that ( R p , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq308_HTML.gif for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then F n k : A R p R p × p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq309_HTML.gif and η n k : A R p R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq310_HTML.gif are both continuous, then T ¯ : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq305_HTML.gif is also continuous and has a fixed point in A from Theorem 2.8(i).

     
  2. (b)

    If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above conclusions still hold with the modification k 2 n k 2 = α n k + β n k 1 β n k ( 1 + 2 μ n k ) ( 1 ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq311_HTML.gif and μ n k α n k β n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq312_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq242_HTML.gif. From Remark 2.7(2), β n k > β k 2 n k > k 1 n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq313_HTML.gif and β n k < β k 2 n k < k 1 n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq314_HTML.gif for any n , k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 { β n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq316_HTML.gif such that β n k > β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq317_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif with the remaining parameters and parametrical sequences being identical in both cases. If F n k : A R p R p × p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq318_HTML.gif and η n k : A R p R p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq319_HTML.gif; n , k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq320_HTML.gif are both continuous, then T ¯ : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq321_HTML.gif is continuous and has a fixed point in A from Theorem 2.8(ii).

     
  3. (c)

    If T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically β-strictly contractive in the intermediate sense, then k 3 n k = α n k + β n k 1 β n k ( 1 + 2 μ n k ) k 3 = o ( max ( α , β ) ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq322_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq232_HTML.gif so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).

     
  4. (d)

    If T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq38_HTML.gif is asymptotically contractive in the intermediate sense, k 4 n k = ( α n k + β n k 1 β n k ( 1 + 2 μ n k ) ) 1 / 2 k 4 1 2 + o ( α ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq323_HTML.gif; k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq95_HTML.gif. Thus, T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 A , B X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq324_HTML.gif be nonempty subsets of X. T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq325_HTML.gif is a cyclic self-mapping if T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq326_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq327_HTML.gif. Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ33_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ34_HTML.gif
(3.2)
with ( ξ n γ n ( x , y ) D 2 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq328_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq47_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif, and that the asymptotically nonexpansive condition (2.22), subject to (2.23), is modified as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ35_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ36_HTML.gif
(3.4)

with ( ξ n γ n ( x , y ) D 2 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq328_HTML.gif; x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq47_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif, where { γ n ( x , y ) } [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq329_HTML.gif and D = dist ( A , B ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq330_HTML.gif. If A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq331_HTML.gif, then D = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq332_HTML.gif and Theorems 2.1, 2.2 and 2.8 hold with the replacement A A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq333_HTML.gif. Then if A and B are closed and convex, then there is a unique fixed point of T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq325_HTML.gif in A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq334_HTML.gif. In the following, we consider the case that A B = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq335_HTML.gif so that D > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq336_HTML.gif. The subsequent result based on Theorems 2.1, 2.2 and 2.8 holds.

Theorem 3.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif be a metric space and let T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq325_HTML.gif be a cyclic self-mapping, i.e., T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq326_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq327_HTML.gif, where A and B are nonempty subsets of X. Define the sequence { k n } n N [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq337_HTML.gif of asymptotically nonexpansive iteration-dependent constants as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equc_HTML.gif
( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq338_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif provided that T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq339_HTML.gif satisfies the constraint (3.1), subject to (3.2), and
[ ( d ( T n x , T n y ) d ( x , y ) β n = 1 ) ( γ n = 0 ) , ( x , y ) ( A × B ) ( B × A ) , n N ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ37_HTML.gif
(3.6)
and
k n = k n ( x , y ) = α n + β n 1 β n ( 1 + 2 μ n ) 1 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ38_HTML.gif
(3.7)
n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif for x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq112_HTML.gif ( y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq340_HTML.gif) and for x B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq341_HTML.gif ( y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq342_HTML.gif) provided that T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq339_HTML.gif satisfies the constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences { α n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq343_HTML.gif, { β n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq40_HTML.gif, { μ n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq41_HTML.gif and { γ n ( x , y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq344_HTML.gif of general terms α n = α n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq345_HTML.gif, β n = β n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq346_HTML.gif and μ n = μ n ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq154_HTML.gif fulfill the following constraints:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ39_HTML.gif
(3.8)
γ n = γ n ( x , y ) max ( 0 , 1 k n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq347_HTML.gif and assuming that the following limits exist:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ40_HTML.gif
(3.9)
Then, the following properties hold:
  1. (i)
    T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq348_HTML.gif satisfies (3.3) subject to (3.4)-(3.9); ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq349_HTML.gif. Then
    lim n d ( T n x , T n y ) [ D , d ( x , y ) ] ; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equd_HTML.gif
     
so that T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq348_HTML.gif is a cyclic asymptotically nonexpansive self-mapping. If x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq112_HTML.gif is a best proximity point of A and y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq350_HTML.gif is a best proximity point of B, then lim n d ( T n x , T n y ) = D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq351_HTML.gif and T 2 n x z x = z ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq352_HTML.gif and T 2 n y z y = z ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq353_HTML.gif, which are best proximity points of A and B (not being necessarily identical to x and y), respectively if T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq354_HTML.gif is continuous.
  1. (ii)

    Property (i) also holds if T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq348_HTML.gif satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and (3.5b) provided that d ( T n x , T n y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq355_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq356_HTML.gif.

     
Proof The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ41_HTML.gif
(3.10)
and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)
d ( T n x , T n y ) 2 k b n d 2 ( x , y ) + ξ n + γ n D 2 1 β n ( 1 + 2 μ n ) lim n d ( T n x , T n y ) [ D , d ( x , y ) ] ; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ42_HTML.gif
(3.11)

since T n x , T n x A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq357_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif since T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq326_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq358_HTML.gif, and k n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq359_HTML.gif and γ n ( x , y ) ( 1 k n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq360_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 β n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq362_HTML.gif. From (3.8) and (3.2), μ n [ 1 + α n 2 , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq363_HTML.gif, equivalently, | μ n | 1 + α n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq364_HTML.gif and d ( T n x , T n y ) > α n + 1 2 | μ n | d 2 ( x , y ) d 2 ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq365_HTML.gif, which contradicts (3.5a) if ξ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq76_HTML.gif so that β n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq362_HTML.gif in (3.5a) under (3.7) implies that ξ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq366_HTML.gif and, since γ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq367_HTML.gif from (3.6), there is no division by zero on the right-hand side of (3.10) if β n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq362_HTML.gif.

Also, if T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq348_HTML.gif is continuous, then lim n d ( T 2 n x , T 2 n y ) = d ( lim n T 2 n x , lim n T 2 n y ) = D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq368_HTML.gif so that T 2 n x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq369_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif, lim n T 2 n x cl A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq370_HTML.gif, T 2 n y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq371_HTML.gif and lim n T 2 n x cl B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq372_HTML.gif since T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq326_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq358_HTML.gif. This proves Properties (i)-(ii). □

Remark 3.2 Note that Theorem 3.1 does not guarantee the convergence of { T 2 n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq373_HTML.gif and { T 2 n y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq374_HTML.gif to best proximity points if the initial points for the iterations x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq112_HTML.gif and y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq350_HTML.gif are not best proximity points if T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq348_HTML.gif is not contractive.

The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with k n = α n + β n ( 1 + 2 μ n ) 1 β n < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq375_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq126_HTML.gif subject to lim n k n = k c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq376_HTML.gif.

Theorem 3.3 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif be a metric space and let T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 { k n } n N [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq377_HTML.gif of asymptotically nonexpansive iteration-dependent constants be defined by a general term k n ( x , y ) = k n : = α n + β n ( 1 + 2 μ n ) 1 β n [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq378_HTML.gif under the constraints γ n ( x , y ) = γ n : = δ j ( 1 k n ) ( 1 β n ) = o ( 1 β n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq379_HTML.gif, β n 1 μ n 1 + α n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq380_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq381_HTML.gif and lim n k n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq382_HTML.gif. Then the subsequent properties hold:

(i) The following limits exist:
lim n d ( T n x , T n y ) = D ; ( x , y ) ( A × B ) ( B × A ) ; lim n d ( T n x , T n + 1 x ) = D ; x A B . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ43_HTML.gif
(3.12)
(ii) Assume, furthermore, that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq2_HTML.gif is complete, A and B are closed and convex and d : X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq383_HTML.gif is translation-invariant and homogeneous and ( X , d ) ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq384_HTML.gif is uniformly convex where https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq53_HTML.gif is the metric-induced norm. Then
lim n d ( T 2 n x , T 2 n + 2 x ) = lim n d ( T 2 n + 1 x , T 2 n + 3 x ) = 0 ; x A B , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ44_HTML.gif
(3.13)

{ T 2 n x } z A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq385_HTML.gif, { T 2 n + 1 x } T z B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq386_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif, and { T 2 n y } T z B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq387_HTML.gif, { T 2 n + 1 x } z A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq388_HTML.gif; x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq61_HTML.gif, y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq389_HTML.gif, where z and Tz are unique best proximity points in A and B, respectively. If A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq331_HTML.gif, then z = T z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq390_HTML.gif is the unique fixed point of T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 ξ n = γ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq391_HTML.gif if β n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq392_HTML.gif. Then one has from (3.1)-(3.2), (3.5a), (3.6) and (3.7) that
d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) k j n d 2 ( T j n x , T j n y ) + ( 1 k j n ) D 2 + ξ j n 1 β j n ; j , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ45_HTML.gif
(3.14)

There are several possible cases as follows.

Case A: { d ( T j n x , T j n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq393_HTML.gif is non-increasing. Then d ( T j n x , T j n y ) g = g ( x , y ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq394_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq395_HTML.gif. Since { k n } [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq396_HTML.gif, one gets (3.12).

Case B: { d ( T j n x , T j n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq393_HTML.gif is non-decreasing. Then either { d ( T j n x , T j n y ) } d ( T j n x , T j n y ) g = g ( x , y ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq397_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq399_HTML.gif following from (3.14) subject to the given parametrical constraints. Thus, if { d ( T j n x , T j n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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: { d ( T j n x , T j n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq393_HTML.gif has an oscillating subsequence. It is proven that such a subsequence is finite. Assume not, then if lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq401_HTML.gif, there is an integer sequence { p n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq402_HTML.gif of general term subject to p n ( n , 2 n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq403_HTML.gif such that
lim sup n ( d 2 ( T ( j + 1 ) n + p n x , T ( j + 1 ) n + p n y ) d 2 ( T j n + p n x , T j n + p n y ) ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Eque_HTML.gif
but the above expression is equivalent, for x p n = T p n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq404_HTML.gif and y p n = T p n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq405_HTML.gif which are in A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq406_HTML.gif, but not jointly in either A or B, to
lim sup n ( d 2 ( T ( j + 1 ) n x p n , T ( j + 1 ) n y p n ) d 2 ( T j n + p n x p n , T j n + p n y p n ) ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equf_HTML.gif
which contradicts lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq407_HTML.gif since both sequences { T j n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq408_HTML.gif and { T j n y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq409_HTML.gif are bounded; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq395_HTML.gif. Then there is no infinite oscillating sequence { d ( T j n x , T j n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq410_HTML.gif for some ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq411_HTML.gif so that there is a finite limit g = g ( x , y ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq412_HTML.gif of { d ( T j n x , T j n y ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq413_HTML.gif, ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq414_HTML.gif. Now, proceed by contradiction by assuming the existence of some ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq415_HTML.gif such that d ( T j n x , T j n y ) g = g ( x , y ) = D + ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq416_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq48_HTML.gif; j N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq417_HTML.gif. Thus, for any j , n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq418_HTML.gif, there is some n ( n 0 ) N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq419_HTML.gif such that there are two consecutive nonzero elements of a nonzero real sequence { ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq420_HTML.gif, which can depend on x and y, which satisfy ε n + 1 ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq421_HTML.gif and
d ( T j n x , T j n y ) = D + ε n + 1 d ( T j n x , T j n y ) = D + ε n ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ46_HTML.gif
(3.15)
j N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq417_HTML.gif. Otherwise, if ε n + 1 < ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq422_HTML.gif for any n ( N ) n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq423_HTML.gif and any given j , n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq424_HTML.gif and ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq425_HTML.gif, then d ( T j n x , T j n y ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq426_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq427_HTML.gif. One gets, by combining (3.14) and (3.15), that
( D + ε n ) 2 ( D + ε n + 1 ) 2 ( D + ε n ) 2 = d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) ( D + ε n ) 2 k j n d 2 ( T j n x , T j n y ) + ( 1 k j n ) D 2 + ξ j n 1 β j n ( D + ε n ) 2 = k j n ( D + ε n ) 2 + ( 1 k j n ) D 2 + ξ j n 1 β j n ( D + ε n ) 2 = k j n ( D + ε n ) 2 + ( 1 + δ j n + ρ j n ) ( 1 k j n ) D 2 ( D + ε n ) 2 ( 1 + δ j n + ρ j n ) D 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ47_HTML.gif
(3.16)

j , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq428_HTML.gif since k j n < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq429_HTML.gif; j , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq430_HTML.gif, and some nonnegative real sequence { ρ j n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq431_HTML.gif which converges to zero since ξ j n γ j n D 2 δ j ( 1 k j n ) ( 1 β j n ) D 2 = o ( 1 β j n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq432_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif; j N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq433_HTML.gif for any ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq434_HTML.gif so that ξ j n 1 β j n = 1 k j n γ j n ξ j n δ j n ( n N ) ( 1 k j n ) δ j n D 2 + ρ j n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq435_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq25_HTML.gif; j N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq417_HTML.gif. The relations (3.16) contradict lim sup n ( D + ε n ) 2 ( 1 + δ j n + ρ j n ) D 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq436_HTML.gif since { ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq437_HTML.gif is positive n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq126_HTML.gif (and it does not converge to zero) and ρ j n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq438_HTML.gif, δ j n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq439_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq28_HTML.gif. Thus, one concludes that { ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq420_HTML.gif converges to zero, and then lim n d ( T n x , T n y ) = D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq440_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq441_HTML.gif; ( x , y ) ( A × B ) ( B × A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq441_HTML.gif. This leads to lim n d ( T n x , T n + 1 x ) = D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq442_HTML.gif; x A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq443_HTML.gif by taking y = T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq444_HTML.gif with y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq350_HTML.gif if x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq112_HTML.gif and y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq342_HTML.gif if x B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq341_HTML.gif. Property (i) has been proven.

Now, Property (ii) is proven. It is first proven that lim n d ( T 2 n x , T 2 n + 2 x ) = lim n d ( T 2 n + 1 x , T 2 n + 3 x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq445_HTML.gif; x A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq443_HTML.gif if the metric is translation-invariant and homogeneous so that it induces a norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq53_HTML.gif if A and B are nonempty, closed and convex subsets of X and ( X , ) ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq446_HTML.gif is a uniformly convex Banach space. Assume not and take such a norm to yield d ( T 2 n x , T 2 n + 2 x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq447_HTML.gif. Then if A is nonempty, closed and convex and B is nonempty and closed and x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq112_HTML.gif, then { T 2 n x } , { T 2 n + 2 x } A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq448_HTML.gif. It is known that d ( T 2 n x , T 2 n + 2 x ) d x d ( x , T 2 x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq449_HTML.gif from Theorem 3.1(i) for y = T 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq450_HTML.gif. Since ( X , ) ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_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 { T 2 n x } , { T 2 n + 2 x } A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq452_HTML.gif and { p n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq453_HTML.gif satisfying for some strictly increasing nonnegative sequence of functions { δ n : [ 0 , r 2 n R 2 n ] R 0 + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq454_HTML.gif and any nonnegative sequences { r 2 n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq455_HTML.gif and { R 2 n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq456_HTML.gif satisfying r 2 n R 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq457_HTML.gif and any sequence { p 2 n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq458_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ48_HTML.gif
(3.17)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ49_HTML.gif
(3.18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ50_HTML.gif
(3.19)
x A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq443_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif, which implies that
max ( r 2 n , 2 [ p 2 n ( 1 δ ( r 2 n R 2 n ) ) R 2 n ] ) T 2 n x + T 2 n + 2 x 2 [ ( 1 δ ( r 2 n R 2 n ) ) R 2 n + p 2 n ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_Equ51_HTML.gif
(3.20)
which has to be valid for R 2 n = p 2 n 1 δ ( 2 ) ( 1 δ ( 2 ) ) p 2 n 1 δ ( 2 ) + p 2 n = 2 p 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq459_HTML.gif; n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-146/MediaObjects/13663_2012_Article_485_IEq69_HTML.gif. Now, for p 2 n ( X )