Open Access

Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Fixed Point Theory and Applications20122012:118

DOI: 10.1186/1687-1812-2012-118

Received: 4 April 2012

Accepted: 2 July 2012

Published: 20 July 2012

Abstract

Let L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif be a uniformly convex modular function space with a strong Opial property. Let T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq2_HTML.gif be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.

MSC:47H09, 47H10.

Keywords

fixed point nonexpansive mapping fixed point iteration process Mann process Ishikawa process modular function space Orlicz space Opial property uniform convexity

1 Introduction

In 2008, Kirk and Xu [21] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq2_HTML.gif, i.e.,
T n ( x ) T n ( y ) α n ( x ) x y , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equa_HTML.gif

where lim sup n α n ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq3_HTML.gif, for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq4_HTML.gif. Their main result (Theorem 3.5) states that every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded and convex subset C of a uniformly convex Banach space X has a fixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization of nonexpansive mappings. The conditions on α n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq5_HTML.gif can be for instance expressed in terms of the derivatives of iterations of T for differentiable T. In 2009 these results were generalized by Hussain and Khamsi to metric spaces, [9].

In 2011, Khamsi and Kozlowski [18] extended their result proving the existence of fixed points of asymptotic pointwise ρ-nonexpansive mappings acting in modular function spaces. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise ρ-nonexpansive mapping. This paper aims at filling this gap.

Let us recall that modular function spaces are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others, see the book by Kozlowski [24] for an extensive list of examples and special cases. There exists an extensive literature on the topic of the fixed point theory in modular function spaces, see, e.g., [35, 8, 13, 14, 1720, 24] and the papers referenced there.

It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. There exists an extensive literature on the subject of iterative fixed point construction processes for asymptotically nonexpansive mappings in Hilbert, Banach and metric spaces, see, e.g., [1, 2, 6, 7, 9, 12, 16, 3036, 3842] and the works referred there. Kozlowski proved convergence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [25] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [26]. Recently, weak and strong convergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [28].

We would like to emphasize that all convergence theorems proved in this paper define constructive algorithms that can be actually implemented. When dealing with specific applications of these theorems, one should take into consideration how additional properties of the mappings, sets and modulars involved can influence the actual implementation of the algorithms defined in this paper.

The paper is organized as follows:
  1. (a)

    Section 2 provides necessary preliminary material on modular function spaces.

     
  2. (b)

    Section 3 introduces the asymptotic pointwise nonexpansive mappings and related notions.

     
  3. (c)

    Section 4 deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.

     
  4. (d)

    Section 5 utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.

     
  5. (e)

    Section 6 establishes the almost everywhere convergence theorem for generalized Ishikawa process.

     
  6. (f)

    Section 7 provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact set C.

     

2 Preliminaries

Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq6_HTML.gif be a δ-ring of subsets of Ω such that E A P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq7_HTML.gif for any E P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq8_HTML.gif and A Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq9_HTML.gif. Let us assume that there exists an increasing sequence of sets K n P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq10_HTML.gif such that Ω = K n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq11_HTML.gif. By we denote the linear space of all simple functions with supports from P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq6_HTML.gif. By M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq12_HTML.gif we will denote the space of all extended measurable functions, i.e., all functions f : Ω [ , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq13_HTML.gif such that there exists a sequence { g n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq14_HTML.gif, | g n | | f | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq15_HTML.gif and g n ( ω ) f ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq16_HTML.gif for all ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq17_HTML.gif. By 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq18_HTML.gif we denote the characteristic function of the set A.

Definition 2.1 Let ρ : M [ 0 , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq19_HTML.gif be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:
  1. (i)

    ρ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq20_HTML.gif;

     
  2. (ii)

    ρ is monotone, i.e., | f ( ω ) | | g ( ω ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq21_HTML.gif for all ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq17_HTML.gif implies ρ ( f ) ρ ( g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq22_HTML.gif, where f , g M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq23_HTML.gif;

     
  3. (iii)

    ρ is orthogonally subadditive, i.e., ρ ( f 1 A B ) ρ ( f 1 A ) + ρ ( f 1 B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq24_HTML.gif for any A , B Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq25_HTML.gif such that A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq26_HTML.gif, f M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq27_HTML.gif;

     
  4. (iv)

    ρ has the Fatou property, i.e., | f n ( ω ) | | f ( ω ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq28_HTML.gif for all ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq17_HTML.gif implies ρ ( f n ) ρ ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq29_HTML.gif, where f M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq27_HTML.gif;

     
  5. (v)

    ρ is order continuous in , i.e., g n E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq30_HTML.gif and | g n ( ω ) | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq31_HTML.gif implies ρ ( g n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq32_HTML.gif.

     
Similarly, as in the case of measure spaces, we say that a set A Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq33_HTML.gif is ρ-null if ρ ( g 1 A ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq34_HTML.gif for every g E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq35_HTML.gif. We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual, we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind we define
M ( Ω , Σ , P , ρ ) = { f M ; | f ( ω ) | < ρ -a.e. } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ1_HTML.gif
(2.1)

where each f M ( Ω , Σ , P , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq36_HTML.gif is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists we will write instead of M ( Ω , Σ , P , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq37_HTML.gif.

Definition 2.2 Let ρ be a regular function pseudomodular.
  1. (1)

    We say that ρ is a regular convex function semimodular if ρ ( α f ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq38_HTML.gif for every α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq39_HTML.gif implies f = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq40_HTML.gif ρ-a.e.;

     
  2. (2)

    We say that ρ is a regular convex function modular if ρ ( f ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq41_HTML.gif implies f = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq40_HTML.gif ρ-a.e.;

     

The class of all nonzero regular convex function modulars defined on Ω will be denoted by .

Let us denote ρ ( f , E ) = ρ ( f 1 E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq42_HTML.gif for f M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq43_HTML.gif, E Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq44_HTML.gif. It is easy to prove that ρ ( f , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq45_HTML.gif is a function pseudomodular in the sense of Def.2.1.1 in [24] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [2224].

Remark 2.1 We limit ourselves to convex function modulars in this paper. However, omitting convexity in Definition 2.1 or replacing it by s-convexity would lead to the definition of nonconvex or s-convex regular function pseudomodulars, semimodulars and modulars as in [24].

Definition 2.3[2224]

Let ρ be a convex function modular.
  1. (a)
    A modular function space is the vector space L ρ ( Ω , Σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq46_HTML.gif, or briefly L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif, defined by
    L ρ = { f M ; ρ ( λ f ) 0  as  λ 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equb_HTML.gif
     
  2. (b)
    The following formula defines a norm in L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif (frequently called Luxemurg norm):
    f ρ = inf { α > 0 ; ρ ( f / α ) 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equc_HTML.gif
     

In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.

Theorem 2.1[2224]

Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif.
  1. (1)

    L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif, f ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq48_HTML.gif is complete and the norm ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq49_HTML.gif is monotone w.r.t. the natural order in .

     
  2. (2)

    f n ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq50_HTML.gif if and only if ρ ( α f n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq51_HTML.gif for every α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq39_HTML.gif.

     
  3. (3)

    If ρ ( α f n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq51_HTML.gif for an α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq39_HTML.gif then there exists a subsequence { g n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq52_HTML.gif of { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif such that g n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq54_HTML.gif ρ-a.e.

     
  4. (4)

    If { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif converges uniformly to f on a set E P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq55_HTML.gif then ρ ( α ( f n f ) , E ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq56_HTML.gif for every α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq39_HTML.gif.

     
  5. (5)

    Let f n f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq57_HTML.gif ρ-a.e. There exists a nondecreasing sequence of sets H k P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq58_HTML.gif such that H k Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq59_HTML.gif and { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif converges uniformly to f on every H k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq60_HTML.gif (Egoroff theorem).

     
  6. (6)

    ρ ( f ) lim inf ρ ( f n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq61_HTML.gif whenever f n f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq62_HTML.gif ρ-a.e. (Note: this property is equivalent to the Fatou property.)

     
  7. (7)

    Defining L ρ 0 = { f L ρ ; ρ ( f , )  is order continuous } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq63_HTML.gif and E ρ = { f L ρ ; λ f L ρ 0  for every  λ > 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq64_HTML.gif we have:

     
  8. (a)

    L ρ L ρ 0 E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq65_HTML.gif,

     
  9. (b)

    E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq66_HTML.gif has the Lebesgue property, i.e., ρ ( α f , D k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq67_HTML.gif for α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq68_HTML.gif, f E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq69_HTML.gif and D k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq70_HTML.gif.

     
  10. (c)

    E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq66_HTML.gif is the closure of (in the sense of ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq71_HTML.gif).

     

The following definition plays an important role in the theory of modular function spaces.

Definition 2.4 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. We say that ρ has the Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif-property if
sup n ρ ( 2 f n , D k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equd_HTML.gif

whenever D k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq70_HTML.gif and sup n ρ ( f n , D k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq73_HTML.gif.

Theorem 2.2 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. The following conditions are equivalent:
  1. (a)

    ρ has Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif,

     
  2. (b)

    L ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq74_HTML.gif is a linear subspace of L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif,

     
  3. (c)

    L ρ = L ρ 0 = E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq75_HTML.gif,

     
  4. (d)

    if ρ ( f n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq76_HTML.gif, then ρ ( 2 f n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq77_HTML.gif,

     
  5. (e)

    if ρ ( α f n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq51_HTML.gif for an α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq68_HTML.gif, then f n ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq50_HTML.gif, i.e., the modular convergence is equivalent to the norm convergence.

     

We will also use another type of convergence which is situated between norm and modular convergence. It is defined, among other important terms, in the following definition.

Definition 2.5 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif.
  1. (a)

    We say that { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif is ρ-convergent to f and write f n f ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq78_HTML.gif if and only if ρ ( f n f ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq79_HTML.gif.

     
  2. (b)

    A sequence { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif where f n L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq80_HTML.gif is called ρ-Cauchy if ρ ( f n f m ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq81_HTML.gif as n , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq82_HTML.gif.

     
  3. (c)

    A set B L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq83_HTML.gif is called ρ-closed if for any sequence of f n B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq84_HTML.gif, the convergence f n f ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq85_HTML.gif implies that f belongs to B.

     
  4. (d)

    A set B L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq83_HTML.gif is called ρ-bounded if sup { ρ ( f g ) ; f B , g B } < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq86_HTML.gif.

     
  5. (e)

    A set B L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq83_HTML.gif is called strongly ρ-bounded if there exists β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq87_HTML.gif such that M β ( B ) = sup { ρ ( β ( f g ) ) ; f B , g B } < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq88_HTML.gif.

     
  6. (f)

    A set B L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq83_HTML.gif is called ρ-compact if for any { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq89_HTML.gif in C there exists a subsequence { f n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq90_HTML.gif and an f C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq91_HTML.gif such that ρ ( f n k f ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq92_HTML.gif.

     
  7. (g)

    A set C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq93_HTML.gif is called ρ-a.e. closed if for any { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif in C which ρ-a.e. converges to some f, then we must have f C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq91_HTML.gif.

     
  8. (h)

    A set C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq93_HTML.gif is called ρ-a.e. compact if for any { f n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq53_HTML.gif in C, there exists a subsequence { f n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq90_HTML.gif which ρ-a.e. converges to some f C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq91_HTML.gif.

     
  9. (i)
    Let f L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq94_HTML.gif and C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gif. The ρ-distance between f and C is defined as
    d ρ ( f , C ) = inf { ρ ( f g ) ; g C } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Eque_HTML.gif
     

Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, f n f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq57_HTML.gif does not imply in general λ f n λ f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq96_HTML.gif, λ > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq97_HTML.gif. Using Theorem 2.1, it is not difficult to prove the following:

Proposition 2.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif.
  1. (i)

    L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif is ρ-complete,

     
  2. (ii)

    ρ-balls B ρ ( x , r ) = { y L ρ ; ρ ( x y ) r } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq98_HTML.gif are ρ-closed and ρ-a.e. closed.

     

Let us compare different types of compactness introduced in Definition 2.5.

Proposition 2.2 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. The following relationships hold for sets C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq99_HTML.gif:
  1. (i)

    If C is ρ-compact, then C is ρ-a.e. compact.

     
  2. (ii)

    If C is ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq100_HTML.gif-compact, then C is ρ-compact.

     
  3. (iii)

    If ρ satisfies Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif, then ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq100_HTML.gif-compactness and ρ-compactness are equivalent in L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif.

     
Proof
  1. (i)

    follows from Theorem 2.1 part (3).

     
  2. (ii)

    follows from Theorem 2.1 part (2).

     
  3. (iii)

    follows from (2.2) and from Theorem 2.2 part (e).

     

 □

3 Asymptotic pointwise nonexpansive mappings

Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [18].

Definition 3.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif and let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gif be nonempty and ρ-closed. A mapping T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq101_HTML.gif is called an asymptotic pointwise mapping if there exists a sequence of mappings α n : C [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq102_HTML.gif such that
ρ ( T n ( f ) T n ( g ) ) α n ( f ) ρ ( f g ) for any  f , g L ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equf_HTML.gif
  1. (i)

    If α n ( f ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq103_HTML.gif for every f L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq94_HTML.gif and every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq104_HTML.gif, then T is called ρ-nonexpansive or shortly nonexpansive.

     
  2. (ii)

    If { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq105_HTML.gif converges pointwise to α : C [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq106_HTML.gif, then T is called asymptotic pointwise contraction.

     
  3. (iii)

    If lim sup n α n ( f ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq107_HTML.gif for any f L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq94_HTML.gif, then T is called asymptotic pointwise nonexpansive.

     
  4. (iv)

    If lim sup n α n ( f ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq108_HTML.gif for any f L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq94_HTML.gif with 0 < k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq109_HTML.gif, then T is called strongly asymptotic pointwise contraction.

     
Denoting a n ( x ) = max ( α n ( x ) , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq110_HTML.gif, we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if
ρ ( T n ( f ) T n ( g ) ) a n ( f ) ρ ( f g ) for all  f , g C , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ2_HTML.gif
(3.1)
lim n a n ( f ) = 1 , a n ( f ) 1 for all  f C ,  and  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ3_HTML.gif
(3.2)
Define b n ( f ) = a n ( f ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq111_HTML.gif. In view of (3.2), we have
lim n b n ( f ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ4_HTML.gif
(3.3)

The above notation will be consistently used throughout this paper.

By T ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq112_HTML.gif we will denote the class of all asymptotic pointwise nonexpansive mappings T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq2_HTML.gif.

In this paper, we will impose some restrictions on the behavior of a n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq113_HTML.gif and b n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq114_HTML.gif. This type of assumptions is typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, e.g., [25].

Definition 3.2 Define T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq115_HTML.gif as a class of all T T ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq116_HTML.gif such that
n = 1 b n ( x ) < for every  x C , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ5_HTML.gif
(3.4)
a n  is a bounded function for every  n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ6_HTML.gif
(3.5)

We recall the following concepts related to the modular uniform convexity introduced in [18]:

Definition 3.3 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. We define the following uniform convexity type properties of the function modular ρ: Let t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq117_HTML.gif, r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq118_HTML.gif, ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif. Define
D 1 ( r , ε ) = { ( f , g ) ; f , g L ρ , ρ ( f ) r , ρ ( g ) r , ρ ( f g ) ε r } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equg_HTML.gif
Let
δ 1 t ( r , ε ) = inf { 1 1 r ρ ( t f + ( 1 t ) g ) ; ( f , g ) D 1 ( r , ε ) } , if  D 1 ( r , ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equh_HTML.gif

and δ 1 ( r , ε ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq120_HTML.gif if D 1 ( r , ε ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq121_HTML.gif. We will use the following notational convention: δ 1 = δ 1 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq122_HTML.gif.

Definition 3.4 We say that ρ satisfies ( U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq123_HTML.gif if for every r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq118_HTML.gif, ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif, δ 1 ( r , ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq124_HTML.gif. Note that for every r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq118_HTML.gif, D 1 ( r , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq125_HTML.gif, for ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif small enough. We say that ρ satisfies ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif if for every s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq127_HTML.gif, ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq128_HTML.gif there exists η 1 ( s , ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq129_HTML.gif depending only on s and ε such that
δ 1 ( r , ε ) > η 1 ( s , ε ) > 0 for any  r > s . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equi_HTML.gif

We will need the following result whose proof is elementary. Note that for t = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq130_HTML.gif, this result follows directly from Definition 3.4.

Lemma 3.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gifand let t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq117_HTML.gif. Then for every s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq131_HTML.gif, ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gifthere exists η 1 t ( s , ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq132_HTML.gifdepending only on s and ε such that
δ 1 t ( r , ε ) > η 1 t ( s , ε ) > 0 for any  r > s . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equj_HTML.gif

The notion of bounded away sequences of real numbers will be used extensively throughout this paper.

Definition 3.5 A sequence { t n } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq133_HTML.gif is called bounded away from 0 if there exists 0 < a < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq134_HTML.gif such that t n a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq135_HTML.gif for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq136_HTML.gif. Similarly, { t n } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq133_HTML.gif is called bounded away from 1 if there exists 0 < b < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq137_HTML.gif such that t n b https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq138_HTML.gif for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq104_HTML.gif.

We will need the following generalization of Lemma 4.1 from [18] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see, e.g., [36].

Lemma 3.2 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gifand let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq139_HTML.gifbe bounded away from 0 and 1. If there exists R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq140_HTML.gifsuch that
lim sup n ρ ( f n ) R , lim sup n ρ ( g n ) R , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ7_HTML.gif
(3.6)
lim n ρ ( t n f n + ( 1 t n ) g n ) = R , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ8_HTML.gif
(3.7)
then
lim n ρ ( f n g n ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equk_HTML.gif
Proof Assume to the contrary that this is not the case and fix an arbitrary γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq141_HTML.gif. Passing to a subsequence if necessary, we may assume that there exists an ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif such that
ρ ( f n ) R + γ , ρ ( g n ) R + γ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ9_HTML.gif
(3.8)
while
ρ ( f n g n ) ( R + γ ) ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ10_HTML.gif
(3.9)
Since { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq142_HTML.gif is bounded away from 0 and 1 there exist 0 < a < b < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq143_HTML.gif such that a t n b https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq144_HTML.gif for all natural n. Passing to a subsequence if necessary, we can assume that t n t 0 [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq145_HTML.gif. For every t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq146_HTML.gif and f , g D 1 ( R + γ , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq147_HTML.gif, let us define λ f , g ( t ) = ρ ( t f + ( 1 t ) g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq148_HTML.gif. Observe that the function λ f , g : [ 0 , 1 ] [ 0 , R + γ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq149_HTML.gif is a convex function. Hence that the function
λ ( t ) = sup { λ f , g ( t ) : f , g D 1 ( R + γ , ε ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ11_HTML.gif
(3.10)
is also convex on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq150_HTML.gif, and consequently, it is a continuous function on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq151_HTML.gif. Noting that
δ 1 t ( R + γ , ε ) = 1 1 r λ ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ12_HTML.gif
(3.11)
we conclude that δ 1 t ( R + γ , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq152_HTML.gif is a continuous function of t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq153_HTML.gif. Hence
lim n δ 1 t n ( R + γ , ε ) = δ 1 t 0 ( R + γ , ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ13_HTML.gif
(3.12)
By (3.8) and (3.9)
δ 1 t n ( R + γ , ε ) 1 1 R + γ ρ ( t n f n + ( 1 t n ) g n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ14_HTML.gif
(3.13)
By (3.12) the left-hand side of (3.13) tends to δ 1 t 0 ( R + γ , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq154_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq155_HTML.gif while the right-hand side tends to γ R + γ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq156_HTML.gif in view of (3.7). Hence
δ 1 t 0 ( R + γ , ε ) γ R + γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ15_HTML.gif
(3.14)
By ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif and by Lemma 3.1, there exists η 1 t 0 ( R , ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq157_HTML.gif satisfying
0 < η 1 t 0 ( R , ε ) δ 1 t 0 ( R + γ , ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ16_HTML.gif
(3.15)
Combining (3.14) with (3.15) we get
0 < η 1 t 0 ( R , ε ) γ R + γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ17_HTML.gif
(3.16)

Letting γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq158_HTML.gif we get a contradiction which completes the proof. □

Let us introduce a notion of a ρ-type, a powerful technical tool which will be used in the proofs of our fixed point results.

Definition 3.6 Let K L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq159_HTML.gif be convex and ρ-bounded. A function τ : K [ 0 , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq160_HTML.gif is called a ρ-type (or shortly a type) if there exists a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq161_HTML.gif of elements of K such that for any z K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq162_HTML.gif there holds
τ ( z ) = lim sup n ρ ( y n z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equl_HTML.gif

Note that τ is convex provided ρ is convex. A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists. This will be the subject of Lemma 3.3 below. First, let us recall the definition of the Opial property and the strong Opial property in modular function spaces, [15, 17].

Definition 3.7 We say that L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif satisfies the ρ-a.e. Opial property if for every { f n } L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq163_HTML.gif which is ρ-a.e. convergent to 0 such that there exists a β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq87_HTML.gif for which
sup n { ρ ( β f n ) } < , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ18_HTML.gif
(3.17)
the following inequality holds for any g E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq164_HTML.gif not equal to 0
lim inf n ρ ( f n ) lim inf n ρ ( f n + g ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ19_HTML.gif
(3.18)
Definition 3.8 We say that L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gif satisfies the ρ-a.e. strong Opial property if for every { f n } L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq163_HTML.gif which is ρ-a.e. convergent to 0 such that there exists a β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq87_HTML.gif for which
sup n { ρ ( β f n ) } < , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ20_HTML.gif
(3.19)
the following equality holds for any g E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq164_HTML.gif
lim inf n ρ ( f n + g ) = lim inf n ρ ( f n ) + ρ ( g ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ21_HTML.gif
(3.20)

Remark 3.1 Note that the ρ-a.e. Strong Opial property implies ρ-a.e. Opial property [15].

Remark 3.2 Also, note that, by virtue of Theorem 2.1 in [15], every convex, orthogonally additive function modular ρ has the ρ-a.e. strong Opial property. Let us recall that ρ is called orthogonally additive if ρ ( f , A B ) = ρ ( f , A ) + ρ ( f , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq165_HTML.gif whenever A B = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq166_HTML.gif. Therefore, all Orlicz and Musielak-Orlicz spaces must have the strong Opial property.

Note that the Opial property in the norm sense does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for L p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq167_HTML.gif spaces for 1 p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq168_HTML.gif while the modular strong Opial property holds in L p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq167_HTML.gif for all p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq169_HTML.gif.

Lemma 3.3[27]

Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. Assume that L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq1_HTML.gifhas the ρ-a.e. strong Opial property. Let C E ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq170_HTML.gifbe a nonempty, strongly ρ-bounded and ρ-a.e. compact convex set. Then any ρ-type defined in C attains its minimum in C.

Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.

Theorem 3.1[18]

Assume ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifis ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif. Let C be a ρ-closed ρ-bounded convex nonempty subset. Then any T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq171_HTML.gifasymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed points Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq172_HTML.gifis ρ-closed.

4 Demiclosedness Principle

The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem 5.1. Our proof the Demiclosedness Principle uses the parallelogram inequality valid in the modular spaces with the ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif property (see Lemma 4.2 in [18]). We start with a technical result which will be used in the proof of Theorem 4.1.

Lemma 4.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a convex set, and let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif. If { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gifis a ρ-approximate fixed point sequence for T, that is, ρ ( T ( x k ) x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq175_HTML.gifas k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq176_HTML.gif, then for every fixed m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq177_HTML.gifthere holds
ρ ( T m ( x k ) x k m ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ22_HTML.gif
(4.1)

as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq176_HTML.gif.

Proof It follows from 3.5 that there exists a finite constant M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq178_HTML.gif such that
j = 1 m 1 sup { a j ( x ) ; x C } M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ23_HTML.gif
(4.2)
Using the convexity of ρ and the ρ-nonexpansiveness of T, we get
ρ ( T m ( x k ) x k m ) = ρ ( 1 m j = 0 m 1 ( T j + 1 ( x k ) T j ( x k ) ) ) 1 m j = 0 m 1 ρ ( T j + 1 ( x k ) T j ( x k ) ) ρ ( T ( x k ) x k ) ( j = 1 m 1 a j ( x n ) + 1 ) 1 m ( M + 1 ) ρ ( T ( x k ) x k ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ24_HTML.gif
(4.3)

as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq176_HTML.gif. □

Corollary 4.1 If, under the hypothesis of Lemma  4.1, ρ satisfies additionally the Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gifcondition, then ρ ( T m ( x k ) x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq179_HTML.gifas k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq176_HTML.gif.

The version of the Demiclosedness Principle used in this paper (Theorem 4.1) requires the uniform continuity of the function modular ρ in the sense of the following definition (see, e.g., [17]).

Definition 4.1 We say that ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif is uniformly continuous if to every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif and L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq180_HTML.gif, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq181_HTML.gif such that
| ρ ( g ) ρ ( g + h ) | ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ25_HTML.gif
(4.4)

provided ρ ( h ) < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq182_HTML.gif and ρ ( g ) L https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq183_HTML.gif.

Let us mention that the uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [37] or in sequence Orlicz spaces [11] the uniform continuity of the Orlicz modular is equivalent to the Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif-type condition.

Theorem 4.1 Demiclosedness Principle. Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. Assume that:
  1. (1)

    ρ is ( U C C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq184_HTML.gif,

     
  2. (2)

    ρ has strong Opial property,

     
  3. (3)

    ρ has Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif property and is uniformly continuous.

     

Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a nonempty, convex, strongly ρ-bounded and ρ-closed, and let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif. Let { x n } C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq185_HTML.gif, and x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq186_HTML.gif. If x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq187_HTML.gifρ-a.e. and ρ ( T ( x n ) x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq188_HTML.gif, then x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq189_HTML.gif.

Proof Let us recall that by definition of uniform continuity of ρ to every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif and L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq180_HTML.gif, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq181_HTML.gif such that
| ρ ( g ) ρ ( g + h ) | ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ26_HTML.gif
(4.5)
provided ρ ( h ) < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq182_HTML.gif and ρ ( g ) L https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq183_HTML.gif. Fix any m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq190_HTML.gif. Noting that ρ ( x n x ) M < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq191_HTML.gif due to the strong ρ-boundedness of C and that ρ ( T m ( x n ) x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq192_HTML.gif by Corollary (4.1), it follows from (4.5) with g = x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq193_HTML.gif and h = T m ( x n ) x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq194_HTML.gif that
| ρ ( x n x ) ρ ( x n x + T m ( x n ) x n ) | 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ27_HTML.gif
(4.6)
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq155_HTML.gif. Hence
lim sup n ρ ( x n x ) = lim sup n ρ ( T m ( x n ) x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ28_HTML.gif
(4.7)
Define the ρ-type φ by
φ ( x ) = lim sup n ρ ( x n x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ29_HTML.gif
(4.8)
By (4.7) we get
φ ( x ) = lim sup n ρ ( T m ( x n ) x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ30_HTML.gif
(4.9)
Hence, for every y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq195_HTML.gif there holds
φ ( T m ( y ) ) = lim sup n ρ ( T m ( x n ) T m ( y ) ) a m ( y ) lim sup n ρ ( x n y ) = a m ( y ) φ ( y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ31_HTML.gif
(4.10)
Using (4.10) with y = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq196_HTML.gif and by passing with m to infinity, we conclude that
lim sup m φ ( T m ( x ) ) φ ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ32_HTML.gif
(4.11)
Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since x n x ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq197_HTML.gif-a.e., it follows via the Opial property that for any y x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq198_HTML.gif
φ ( x ) = lim sup n ρ ( x n x ) < lim sup n ρ ( x n y ) = φ ( y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ33_HTML.gif
(4.12)
which implies that
φ ( x ) = inf { φ ( y ) : y C } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ34_HTML.gif
(4.13)
Combining (4.11) with (4.13), we have
φ ( x ) lim sup m φ ( T m ( x ) ) φ ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ35_HTML.gif
(4.14)
that is,
lim sup m φ ( T m ( x ) ) = φ ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ36_HTML.gif
(4.15)
We claim that
lim m ρ ( T m ( x ) x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ37_HTML.gif
(4.16)
Assume to the contrary that (4.16) does not hold, that is,
ρ ( T m ( x ) x )  does not tend to zero . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ38_HTML.gif
(4.17)
By Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif, it follows from (4.17) that ρ ( T m ( x ) x 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq199_HTML.gif does not tend to zero. By passing to a subsequence if necessary, we can assume that there exists 0 < t < M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq200_HTML.gif such that
ρ ( T m ( x ) x 2 ) > t > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ39_HTML.gif
(4.18)
for m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq177_HTML.gif, which implies that
ρ ( x n x ) + ρ ( x n T m ( x ) ) > t 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ40_HTML.gif
(4.19)
for every m , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq201_HTML.gif. Hence,
max { ρ ( x n x ) , ρ ( x n T m ( x ) ) } t 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ41_HTML.gif
(4.20)
for every m , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq201_HTML.gif. Applying the modular parallelogram inequality valid in ( U C C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq184_HTML.gif modular function spaces, see Lemma 4.2 in [18],
ρ 2 ( z + y 2 ) 1 2 ρ 2 ( z ) + 1 2 ρ 2 ( y ) Ψ ( r , s , 1 r ρ ( z y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ42_HTML.gif
(4.21)
where ρ ( z ) r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq202_HTML.gif, ρ ( y ) r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq203_HTML.gif and max { ρ ( z ) , ρ ( y ) } s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq204_HTML.gif for 0 < s < r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq205_HTML.gif, with r = M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq206_HTML.gif, s = t 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq207_HTML.gif, z = x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq208_HTML.gif, y = T m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq209_HTML.gif, we get
ρ 2 ( x n x + T m ( x ) 2 ) 1 2 ρ 2 ( x n x ) + 1 2 ρ 2 ( x n T m ( x ) ) Ψ ( M , t 4 , 1 M ρ ( x T m ( x ) ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ43_HTML.gif
(4.22)
Note that by (4.13)
φ 2 ( x ) φ 2 ( x + T m ( x ) 2 ) = lim sup n ρ 2 ( x n x + T m ( x ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ44_HTML.gif
(4.23)
Combining (4.22) with (4.23), we obtain
φ 2 ( x ) 1 2 lim sup n ρ 2 ( x n x ) + 1 2 lim sup n ρ 2 ( x n T m ( x ) ) Ψ ( M , t 4 , 1 M ρ ( x T m ( x ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ45_HTML.gif
(4.24)
which implies
0 Ψ ( M , t 4 , 1 M ρ ( x T m ( x ) ) ) 1 2 φ 2 ( T m ( x ) ) 1 2 φ 2 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ46_HTML.gif
(4.25)
Letting m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq210_HTML.gif and applying (4.15), we get
0 lim sup m Ψ ( M , t 4 , 1 M ρ ( x T m ( x ) ) ) 1 2 lim sup m φ 2 ( T m ( x ) ) 1 2 φ 2 ( x ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ47_HTML.gif
(4.26)

Using the properties of Ψ, we conclude that ρ ( x T m ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq211_HTML.gif tends to zero itself, which contradicts our assumption (4.17). Hence, ρ ( x T m ( x ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq212_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq210_HTML.gif. Clearly, then ρ ( x T m + 1 ( x ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq213_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq210_HTML.gif, that is, T m + 1 ( x ) x ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq214_HTML.gif while T m + 1 ( x ) T ( x ) ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq215_HTML.gif by ρ-continuity of T. By the uniqueness of the ρ-limit, we obtain T ( x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq216_HTML.gif, that is, x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq217_HTML.gif. □

5 Convergence of generalized Mann iteration process

The following elementary, easy to prove, lemma will be used in this paper.

Lemma 5.1[2]

Suppose { r k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq218_HTML.gifis a bounded sequence of real numbers and { d k , n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq219_HTML.gifis a doubly-index sequence of real numbers which satisfy
lim sup k lim sup n d k , n 0  and  r k + n r k + d k , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equm_HTML.gif

for each k , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq220_HTML.gif. Then { r k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq218_HTML.gifconverges to an r R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq221_HTML.gif.

Following Mann [29], let us start with the definition of the generalized Mann iteration process.

Definition 5.1 Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif and let { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif be an increasing sequence of natural numbers. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gif be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T, the sequence { t k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq224_HTML.gif, and the sequence { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif denoted by g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq225_HTML.gif is defined by the following iterative formula:
x k + 1 = t k T n k ( x k ) + ( 1 t k ) x k , where  x 1 C  is chosen arbitrarily . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ48_HTML.gif
(5.1)
Definition 5.2 We say that a generalized Mann iteration process g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq226_HTML.gif is well defined if
lim sup k a n k ( x k ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ49_HTML.gif
(5.2)

Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness, lim k a k ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq227_HTML.gif for every x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq228_HTML.gif. Hence we can always select a subsequence { a n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq229_HTML.gif such that (5.2) holds. In other words, by a suitable choice of { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif, we can always make g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq226_HTML.gif well defined.

The following result provides an important technique which will be used in this paper.

Lemma 5.2 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set. Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gifand let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gif. Assume that a sequence { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq231_HTML.gifis bounded away from 0 and 1. Let w be a fixed point of T and g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq225_HTML.gifbe a generalized Mann process. Then there exists r R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq232_HTML.gifsuch that
lim k ρ ( x k w ) = r . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ50_HTML.gif
(5.3)
Proof Let w F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq233_HTML.gif. Since
ρ ( x k + 1 w ) t k ρ ( T n k ( x k ) w ) + ( 1 t k ) ρ ( x k w ) = t k ρ ( T n k ( x k ) T n k ( w ) ) + ( 1 t k ) ρ ( x k w ) t k ( 1 + b n k ( w ) ) ρ ( x k w ) + ( 1 t k ) ρ ( x k w ) = t k b n k ( w ) ρ ( x k w ) + ρ ( x k w ) b n k ( w ) diam ρ ( C ) + ρ ( x k w ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equn_HTML.gif
it follows that for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq234_HTML.gif,
ρ ( x k + n w ) ρ ( x k w ) + diam ρ ( C ) i = k k + n 1 b n i ( w ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ51_HTML.gif
(5.4)

Denote r p = ρ ( x p w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq235_HTML.gif for every p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq236_HTML.gif and d k , n = diam ρ ( C ) i = k k + n 1 b n i ( w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq237_HTML.gif. Observe that since T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif, it follows that lim sup k lim sup n d k , n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq238_HTML.gif. By Lemma 5.1, there exists an r R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq239_HTML.gif such that lim k ρ ( x k w ) = r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq240_HTML.gif as claimed. □

The next result will be essential for proving the convergence theorems for iterative process.

Lemma 5.3 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set, and T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq241_HTML.gif. Assume that a sequence { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq139_HTML.gifis bounded away from 0 and 1. Let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gifand g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq242_HTML.gifbe a generalized Mann iteration process. Then
lim k ρ ( T n k ( x k ) x k ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ52_HTML.gif
(5.5)
and
lim k ρ ( x k + 1 x k ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ53_HTML.gif
(5.6)
Proof By Theorem 3.1, T has at least one fixed point w C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq243_HTML.gif. In view of Lemma 5.2, there exists r R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq232_HTML.gif such that
lim k ρ ( x k w ) = r . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ54_HTML.gif
(5.7)
Note that
lim sup k ρ ( T n k ( x k ) w ) = lim sup k ρ ( T n k ( x k ) T n k ( w ) ) lim sup k a n k ( w ) ρ ( x k w ) r , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ55_HTML.gif
(5.8)
and that
lim k ρ ( t k ( T n k ( x k ) w ) + ( 1 t k ) ( x k w ) ) = lim k ρ ( x k + 1 w ) = r . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ56_HTML.gif
(5.9)
Set f k = T n k ( x k ) w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq244_HTML.gif, g k = x k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq245_HTML.gif, and note that lim sup k ρ ( g k ) r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq246_HTML.gif by (5.7), and lim sup k ρ ( f k ) r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq247_HTML.gif by (5.8). Observe also that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ57_HTML.gif
(5.10)
Hence, it follows from Lemma 3.2 that
lim k ρ ( T n k ( x k ) x k ) = lim k ρ ( f k g k ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ58_HTML.gif
(5.11)
which by the construction of the sequence { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif is equivalent to
lim k ρ ( x k + 1 x k ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ59_HTML.gif
(5.12)

as claimed. □

In the next lemma, we prove that under suitable assumption the sequence { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.

Definition 5.3 A strictly increasing sequence { n i } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq248_HTML.gif is called quasi-periodic if the sequence { n i + 1 n i } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq249_HTML.gif is bounded, or equivalently, if there exists a number p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq250_HTML.gif such that any block of p consecutive natural numbers must contain a term of the sequence { n i } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq251_HTML.gif. The smallest of such numbers p will be called a quasi-period of { n i } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq251_HTML.gif.

Lemma 5.4 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gifsatisfying Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set, and T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq252_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gifbe bounded away from 0 and 1. Let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gifbe such that the generalized Mann process g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq225_HTML.gifis well defined. If, in addition, the set of indices J = { j ; n j + 1 = 1 + n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq253_HTML.gifis quasi-periodic, then { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gifis an approximate fixed point sequence, i.e.,
lim k ρ ( T ( x k ) x k ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ60_HTML.gif
(5.13)
Proof Let p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq236_HTML.gif be a quasi-period of J https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq254_HTML.gif. Observe that it is enough to prove that ρ ( T ( x k ) x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq175_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq255_HTML.gif through J https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq254_HTML.gif. Indeed, let us fix ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq119_HTML.gif. From ρ ( T ( x k ) x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq175_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq255_HTML.gif through J https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq254_HTML.gif it follows that
ρ ( T ( x k ) x k ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ61_HTML.gif
(5.14)
for sufficiently large k. By the quasi-periodicity of J https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq254_HTML.gif, to every positive integer k, there exists j k J https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq256_HTML.gif such that | k j k | p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq257_HTML.gif. Assume that k p j k k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq258_HTML.gif (the proof for the other case is identical). Since T is ρ-Lipschitzian with the constant M = sup { a 1 ( x ) ; x C } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq259_HTML.gif, there exist a 0 < δ < ε 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq260_HTML.gif such that
ρ ( T ( x ) T ( y ) ) < ε if  ρ ( x y ) < δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ62_HTML.gif
(5.15)
Note that by (5.6) and by Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif, ρ ( p ( x k + 1 x k ) ) < δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq261_HTML.gif for k sufficiently large. This implies that
ρ ( x k x j k ) 1 p ( ρ ( p ( x k x k 1 ) ) + + ρ ( p ( x j k + 1 x j k ) ) ) p δ p = δ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ63_HTML.gif
(5.16)
and therefore,
ρ ( x k T ( x k ) 3 ) 1 3 ρ ( x k x j k ) + 1 3 ρ ( x j k T ( x j k ) ) + 1 3 ρ ( T ( x j k ) T ( x k ) ) δ + ε 3 + ε 3 < ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ64_HTML.gif
(5.17)
which demonstrates that
ρ ( x k T ( x k ) 3 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ65_HTML.gif
(5.18)

as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq176_HTML.gif. By Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif again, we get ρ ( T ( x k ) x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq262_HTML.gif.

To prove that ρ ( T ( x k ) x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq175_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq255_HTML.gif through J https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq254_HTML.gif, observe that, since n k + 1 = n k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq263_HTML.gif for such k, there holds
ρ ( x k T ( x k ) 4 ) 1 4 ρ ( x k x k + 1 ) + 1 4 ρ ( x k + 1 T n k + 1 ( x k + 1 ) ) + 1 4 ρ ( T n k + 1 ( x k + 1 ) T n k + 1 ( x k ) ) + 1 4 ρ ( T T n k ( x k ) T ( x k ) ) 1 4 ρ ( x k x k + 1 ) + 1 4 ρ ( x k + 1 T n k + 1 ( x k + 1 ) ) + 1 4 a n k + 1 ( x k + 1 ) ρ ( x k x k + 1 ) + 1 4 M ρ ( T n k ( x k ) x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ66_HTML.gif
(5.19)

which tends to zero in view of (5.5), (5.6) and (5.2). □

The next theorem is the main result of this section.

Theorem 5.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. Assume that:
  1. (1)

    ρ is ( U C C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq184_HTML.gif,

     
  2. (2)

    ρ has Strong Opial Property,

     
  3. (3)

    ρ has Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif property and is uniformly continuous.

     

Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe nonempty, ρ-a.e. compact, convex, strongly ρ-bounded and ρ-closed, and let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq241_HTML.gif. Assume that a sequence { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq139_HTML.gifis bounded away from 0 and 1. Let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gifand g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq242_HTML.gifbe a well-defined generalized Mann iteration process. Assume, in addition, that the set of indices J = { j ; n j + 1 = 1 + n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq264_HTML.gifis quasi-periodic. Then there exists x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq189_HTML.gifsuch that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq187_HTML.gifρ-a.e.

Proof Observe that by Theorem 4.1 in [18], the set of fixed points F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq265_HTML.gif is nonempty, convex and ρ-closed. Note also that by Lemma 3.1 in [27], it follows from the strong Opial property of ρ that any ρ-type attains its minimum in C. By Lemma 5.4, the sequence { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif is an approximate fixed point sequence, that is,
ρ ( T ( x k ) x k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ67_HTML.gif
(5.20)
as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq176_HTML.gif. Consider y , z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq266_HTML.gif, two ρ-a.e. cluster points of { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif. There exits then { y k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq267_HTML.gif, { z k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq268_HTML.gif subsequences of { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif such that y k y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq269_HTML.gifρ-a.e. and z k z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq270_HTML.gifρ-a.e. By Theorem 4.1, y F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq271_HTML.gif and z F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq272_HTML.gif. By Lemma 5.2, there exist r y , r z R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq273_HTML.gif such that
r y = lim k ρ ( x k y ) , r z = lim k ρ ( x k z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ68_HTML.gif
(5.21)
We claim that y = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq274_HTML.gif. Assume to the contrary that y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq275_HTML.gif. Then, by the strong Opial property, we have
r y = lim inf k ρ ( y k y ) < lim inf k ρ ( y k z ) = lim inf k ρ ( z k z ) < lim inf k ρ ( z k y ) = r y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ69_HTML.gif
(5.22)

The contradiction implies that y = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq274_HTML.gif. Therefore, { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif has at most one ρ-a.e. cluster point. Since, C is ρ-a.e. compact it follows that the sequence { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif has exactly one ρ-a.e. cluster point, which means that ρ ( x k ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq276_HTML.gifρ-a.e. Using Theorem 4.1 again, we get x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq189_HTML.gif as claimed. □

Remark 5.2 It is easy to see that we can always construct a sequence { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif with the quasi-periodic properties specified in the assumptions of Theorem 5.1. When constructing concrete implementations of this algorithm, the difficulty will be to ensure that the constructed sequence { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif is not “too sparse” in the sense that the generalized Mann process g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq225_HTML.gif remains well defined. The similar quasi-periodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2, 25, 28].

6 Convergence of generalized Ishikawa iteration process

The two-step Ishikawa iteration process is a generalization of the one-step Mann process. The Ishikawa iteration process, [10], provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.

Definition 6.1 Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif and let { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif be an increasing sequence of natural numbers. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gif be bounded away from 0 and 1, and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gif be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T, the sequences { t k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq224_HTML.gif, { s k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq278_HTML.gif, and the sequence { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif denoted by g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq279_HTML.gif is defined by the following iterative formula:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ70_HTML.gif
(6.1)
Definition 6.2 We say that a generalized Ishikawa iteration process g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq280_HTML.gif is well defined if
lim sup k a n k ( x k ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ71_HTML.gif
(6.2)

Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness, lim k a k ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq227_HTML.gif for every x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq228_HTML.gif. Hence we can always select a subsequence { a n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq229_HTML.gif such that (6.2) holds. In other words, by a suitable choice of { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gif, we can always make g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq281_HTML.gif well defined.

Lemma 6.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set. Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gifand let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gifbe bounded away from 0 and 1, and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gifbe bounded away from 1. Let w F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq233_HTML.gifand g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq279_HTML.gifbe a generalized Ishikawa process. There exists then an r R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq232_HTML.gifsuch that lim k ρ ( x k w ) = r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq282_HTML.gif.

Proof Define T k : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq283_HTML.gif by
T k ( x ) = t k T n k ( s k T n k ( x ) + ( 1 s k ) x ) + ( 1 t k ) x , x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ72_HTML.gif
(6.3)
It is easy to see that x k + 1 = T k ( x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq284_HTML.gif and that F ( T ) F ( T k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq285_HTML.gif. Moreover, a straight calculation shows that each T k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq286_HTML.gif satisfies
ρ ( T k ( x ) T k ( y ) ) A k ( x ) ρ ( x y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ73_HTML.gif
(6.4)
where
A k ( x ) = 1 + t k a n k ( M k ( x ) ) ( 1 + s k a n k ( x ) s k ) t k , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ74_HTML.gif
(6.5)
and
M k ( x ) = s k T n k ( x ) + ( 1 s k ) x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ75_HTML.gif
(6.6)
Note that A k ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq287_HTML.gif, which follows directly from the fact that a n k ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq288_HTML.gif and from (6.5). Using (6.5) and the fact that M k ( w ) = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq289_HTML.gif, we have
B k ( w ) = A k ( w ) 1 = t k ( 1 + s k a n k ( w ) ) ( a n k ( w ) 1 ) ( 1 + a n k ( w ) ) b n k ( w ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ76_HTML.gif
(6.7)
Fix any M > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq290_HTML.gif. Since lim k a n k ( w ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq291_HTML.gif, it follows that there exists a k 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq292_HTML.gif such that for k > k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq293_HTML.gif, a n k ( w ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq294_HTML.gif. Therefore, using the same argument as in the proof of Lemma 5.2, we deduce that for k > k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq293_HTML.gif and n > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq295_HTML.gif
ρ ( x k + n w ) ρ ( x k w ) + diam ρ ( C ) i = k k + n 1 B i ( w ) ρ ( x k w ) + diam ρ ( C ) ( 1 + M ) i = k k + n 1 b n i ( w ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ77_HTML.gif
(6.8)

Arguing like in the proof of Lemma 5.2, we conclude that there exists an r R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq296_HTML.gif such that lim k ρ ( x k w ) = r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq297_HTML.gif. □

Lemma 6.2 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set. Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gifand let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gifbe bounded away from 0 and 1, and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gifbe bounded away from 1. Let g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq279_HTML.gifbe a generalized Ishikawa process. Define
y k = s k T n k ( x k ) + ( 1 s k ) x k . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ78_HTML.gif
(6.9)
Then
lim k ρ ( T n k ( y k ) x k ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ79_HTML.gif
(6.10)
or equivalently
lim k ρ ( x k + 1 x k ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ80_HTML.gif
(6.11)
Proof By Theorem 3.1, F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq298_HTML.gif. Let us fix w F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq233_HTML.gif. By Lemma 6.1, lim k ρ ( x k w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq299_HTML.gif exists. Let us denote it by r. Since w F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq233_HTML.gif, T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif, and lim k ρ ( x k w ) = r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq300_HTML.gif by Lemma 6.1, we have the following:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ81_HTML.gif
(6.12)
Note that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ82_HTML.gif
(6.13)

Applying Lemma 3.2 with u k = T n k ( y k ) w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq301_HTML.gif and v k = x k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq302_HTML.gif, we obtain the desired equality lim k ρ ( T n k ( y k ) x k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq303_HTML.gif, while (6.11) follows from (6.10) via the construction formulas for x k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq304_HTML.gif and y k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq305_HTML.gif. □

Lemma 6.3 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gifsatisfying Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set. Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq252_HTML.gifand let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq230_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq231_HTML.gifbe bounded away from 0 and 1, and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq306_HTML.gifbe bounded away from 1. Let g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq279_HTML.gifbe a well-defined generalized Ishikawa process. Then
lim k ρ ( T n k ( x k ) x k ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ83_HTML.gif
(6.14)
Proof Let y k = s k T n k ( x k ) + ( 1 s k ) x k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq307_HTML.gif. Hence
T n k ( x k ) x k = 1 1 s k ( T n k ( x k ) y k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ84_HTML.gif
(6.15)
Since { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gif is bounded away from 1, there exists 0 < s < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq308_HTML.gif such that s k s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq309_HTML.gif for every k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq310_HTML.gif. Hence,
ρ ( T n k ( x k ) x k ) = ρ ( 1 1 s k ( T n k ( x k ) y k ) ) ρ ( 1 1 s ( T n k ( x k ) y k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ85_HTML.gif
(6.16)

The right-hand side of this inequality tends to zero because ρ ( T n k ( x k ) y k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq311_HTML.gif by Lemma 6.2 and ρ satisfies Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif. □

Lemma 6.4 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifbe ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gifsatisfying Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe a ρ-closed, ρ-bounded and convex set, and T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq252_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gifbe bounded away from 0 and 1 and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gifbe bounded away from 1. Let { n k } N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq312_HTML.gifbe such that the generalized Ishikawa process g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq279_HTML.gifis well defined. If, in addition, the set J = { j ; n j + 1 = 1 + n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq253_HTML.gifis quasi-periodic, then { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq313_HTML.gifis an approximate fixed point sequence, i.e.,
lim k ρ ( T ( x k ) x k ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ86_HTML.gif
(6.17)

Proof The proof is analogous to that of Lemma 5.4 with (6.11) used instead of (5.6) and (6.14) replacing (5.5). □

Theorem 6.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gif. Assume that
  1. (1)

    ρ is ( U C C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq184_HTML.gif,

     
  2. (2)

    ρ has Strong Opial Property,

     
  3. (3)

    ρ has Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif property and is uniformly continuous.

     

Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq95_HTML.gifbe nonempty, ρ-a.e. compact, convex, strongly ρ-bounded and ρ-closed, and let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq241_HTML.gif. Let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gifbe bounded away from 0 and 1, and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gifbe bounded away from 1. Let { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gifbe such that the generalized Ishikawa process g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq279_HTML.gifis well defined. If, in addition, the set J = { j ; n j + 1 = 1 + n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq253_HTML.gifis quasi-periodic, then { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gifgenerated by g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq314_HTML.gifconverges ρ-a.e. to a fixed point x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq189_HTML.gif.

Proof The proof is analogous to that of Theorem 5.1 with Lemma 5.4 replaced by Lemma 6.4, and Lemma 5.2 replaced by Lemma 6.1. □

7 Strong convergence

It is interesting that, provided C is ρ-compact, both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property.

Theorem 7.1 Let ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq47_HTML.gifsatisfy conditions ( U U C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq126_HTML.gifand Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif. Let C L ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq99_HTML.gifbe a ρ-compact, ρ-bounded and convex set, and let T T r ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq173_HTML.gif. Let { t k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq223_HTML.gifbe bounded away from 0 and 1, and { s k } ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq277_HTML.gifbe bounded away from 1. Let { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq222_HTML.gifbe such that the generalized Mann process g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq225_HTML.gif (resp. Ishikawa process g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq314_HTML.gif) is well defined. Then there exists a fixed point x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq189_HTML.gifsuch that then { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gifgenerated by g M ( T , { t k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq225_HTML.gif (resp. g I ( T , { t k } , { s k } , { n k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq314_HTML.gif) converges strongly to a fixed point of T, that is
lim k ρ ( x k x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ87_HTML.gif
(7.1)
Proof By the ρ-compactness of C, we can select a subsequence { x p k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq315_HTML.gif of { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq174_HTML.gif such that there exists x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq186_HTML.gif with
lim k ρ ( T ( x p k ) x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ88_HTML.gif
(7.2)
Note that
ρ ( x p k x 2 ) 1 2 ρ ( x p k T ( x p k ) ) + 1 2 ρ ( T ( x p k x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ89_HTML.gif
(7.3)
which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif it follows from (7.3) that
ρ ( x p k x ) 0 as  k . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ90_HTML.gif
(7.4)
Observe that by the convexity of ρ and by ρ-nonexpansiveness of T, we have
ρ ( T ( x ) x 3 ) 1 3 ρ ( T ( x ) T ( x p k ) ) + 1 3 ρ ( T ( x p k ) x p k ) + 1 3 ρ ( x p k x ) 1 3 ρ ( x x p k ) + 1 3 ρ ( T ( x p k ) x p k ) + 1 3 ρ ( x p k x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ91_HTML.gif
(7.5)
which tends to zero by (7.4) and by Lemma 5.3 (resp. Lemma 6.4). Hence ρ ( T ( x ) x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq316_HTML.gif which implies that x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq189_HTML.gif. Applying Lemma 5.2 (resp. Lemma 6.1), we conclude that lim k ρ ( x k x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq317_HTML.gif exists. By (7.4) this limit must be equal to zero which implies that
lim k ρ ( x k x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_Equ92_HTML.gif
(7.6)

 □

Remark 7.1 Observe that in view of the Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_Article_278_IEq72_HTML.gif assumption, the ρ-compactness of the set C assumed in Theorem 7.1 is equivalent to the compactness in the sense of the norm defined by ρ.

Declarations

Acknowledgements

The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University
(2)
School of Mathematics and Statistics, University of New South Wales

References

  1. Bose SC: Weak convergence to the fixed point of an asymptotically nonexpansive. Proc. Am. Math. Soc. 1978, 68: 305–308. 10.1090/S0002-9939-1978-0493543-4View Article
  2. Bruck R, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, 65(2):169–179.MathSciNet
  3. Dominguez-Benavides T, Khamsi MA, Samadi S: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Anal. 2001, 46: 267–278. 10.1016/S0362-546X(00)00117-6MathSciNetView Article
  4. Dominguez-Benavides T, Khamsi MA, Samadi S: Asymptotically regular mappings in modular function spaces. Sci. Math. Jpn. 2001, 53: 295–304.MathSciNet
  5. Dominguez-Benavides T, Khamsi MA, Samadi S: Asymptotically nonexpansive mappings in modular function spaces. J. Math. Anal. Appl. 2002, 265(2):249–263. 10.1006/jmaa.2000.7275MathSciNetView Article
  6. Fukhar-ud-din H, Khan AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Comput. Math. Appl. 2009, 53: 1349–1360.MathSciNetView Article
  7. Gornicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carol. 1989, 30: 249–252.MathSciNet
  8. Hajji A, Hanebaly E: Perturbed integral equations in modular function spaces. Electron. J. Qual. Theory Differ. Equ. 2003, 20: 1–7. http://​www.​math.​u-szeged.​hu/​ejqtde/​MathSciNetView Article
  9. Hussain N, Khamsi MA: On asymptotic pointwise contractions in metric spaces. Nonlinear Anal. 2009, 71(10):4423–4429. 10.1016/j.na.2009.02.126MathSciNetView Article
  10. Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView Article
  11. Kaminska A: On uniform convexity of Orlicz spaces. Indag. Math. 1982, 44(1):27–36.MathSciNetView Article
  12. Khan AR: On modified Noor iterations for asymptotically nonexpansive mappings. Bull. Belg. Math. Soc. Simon Stevin 2010, 17: 127–140.MathSciNet
  13. Khamsi MA: Nonlinear semigroups in modular function spaces. Math. Jpn. 1992, 37(2):1–9.MathSciNet
  14. Khamsi MA: Fixed point theory in modular function spaces. Proceedings of the Workshop on Recent Advances on Metric Fixed Point Theory Held in Sevilla, September, 1995 1995, 31–35.
  15. Khamsi MA: A convexity property in Modular function spaces. Math. Jpn. 1996, 44(2):269–279.MathSciNet
  16. Khamsi MA: On asymptotically nonexpansive mappings in hyperconvex metric spaces. Proc. Am. Math. Soc. 2004, 132: 365–373. 10.1090/S0002-9939-03-07172-7MathSciNetView Article
  17. Khamsi MA, Kozlowski WM: On asymptotic pointwise contractions in modular function spaces. Nonlinear Anal. 2010, 73: 2957–2967. 10.1016/j.na.2010.06.061MathSciNetView Article
  18. Khamsi MA, Kozlowski WM: On asymptotic pointwise nonexpansive mappings in modular function spaces. J. Math. Anal. Appl. 2011, 380(2):697–708. 10.1016/j.jmaa.2011.03.031MathSciNetView Article
  19. Khamsi MA, Kozlowski WM, Reich S: Fixed point theory in modular function spaces. Nonlinear Anal. 1990, 14: 935–953. 10.1016/0362-546X(90)90111-SMathSciNetView Article
  20. Khamsi MA, Kozlowski WM, Shutao C: Some geometrical properties and fixed point theorems in Orlicz spaces. J. Math. Anal. Appl. 1991, 155(2):393–412. 10.1016/0022-247X(91)90009-OMathSciNetView Article
  21. Kirk WA, Xu HK: Asymptotic pointwise contractions. Nonlinear Anal. 2008, 69: 4706–4712. 10.1016/j.na.2007.11.023MathSciNetView Article
  22. Kozlowski WM: Notes on modular function spaces I. Ann. Soc. Math. Pol., 1 Comment. Math. 1988, 28: 91–104.MathSciNet
  23. Kozlowski WM: Notes on modular function spaces II. Ann. Soc. Math. Pol., 1 Comment. Math. 1988, 28: 105–120.MathSciNet
  24. Kozlowski WM Series of Monographs and Textbooks in Pure and Applied Mathematics 122. In Modular Function Spaces. Dekker, New York; 1988.
  25. Kozlowski WM: Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2011, 377: 43–52. 10.1016/j.jmaa.2010.10.026MathSciNetView Article
  26. Kozlowski WM: Common fixed points for semigroups of pointwise Lipschitzian mappings in Banach spaces. Bull. Aust. Math. Soc. 2011, 84: 353–361. 10.1017/S0004972711002668MathSciNetView Article
  27. Kozlowski WM: On the existence of common fixed points for semigroups of nonlinear mappings in modular function spaces. Ann. Soc. Math. Pol., 1 Comment. Math. 2011, 51(1):81–98.MathSciNet
  28. Kozlowski, WM, Sims, B: On the convergence of iteration processes for semigroups of nonlinear mappings in Banach spaces (to appear)
  29. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View Article
  30. Nanjaras B, Panyanak B: Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 268780
  31. Noor MA, Xu B: Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2002, 267: 444–453. 10.1006/jmaa.2001.7649MathSciNetView Article
  32. Passty GB: Construction of fixed points for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1982, 84: 212–216. 10.1090/S0002-9939-1982-0637171-7MathSciNetView Article
  33. Rhoades BE: Fixed point iterations for certain nonlinear mappings. J. Math. Anal. Appl. 1994, 183: 118–120. 10.1006/jmaa.1994.1135MathSciNetView Article
  34. Samanta SK: Fixed point theorems in a Banach space satisfying Opial’s condition. J. Indian Math. Soc. 1981, 45: 251–258.MathSciNet
  35. Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022-247X(91)90245-UMathSciNetView Article
  36. Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView Article
  37. Shutao C Dissertationes Mathematicae 356. Geometry of Orlicz Spaces 1996.
  38. Tan K-K, Xu H-K: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1992, 114: 399–404. 10.1090/S0002-9939-1992-1068133-2MathSciNetView Article
  39. Tan K-K, Xu H-K: A nonlinear ergodic theorem for asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1992, 45: 25–36. 10.1017/S0004972700036972MathSciNetView Article
  40. Tan K-K, Xu H-K: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309MathSciNetView Article
  41. Tan K-K, Xu H-K: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1994, 122: 733–739. 10.1090/S0002-9939-1994-1203993-5MathSciNetView Article
  42. Xu H-K: Existence and convergence for fixed points of asymptotically nonexpansive type. Nonlinear Anal. 1991, 16: 1139–1146. 10.1016/0362-546X(91)90201-BMathSciNetView Article

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© Dehaish and Kozlowski; licensee Springer 2012

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