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

  • Buthinah A Bin Dehaish1Email author and

    Affiliated with

    • WM Kozlowski2

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif be a uniformly convex modular function space with a strong Opial property. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq2_HTML.gif be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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: Primary 47H09; Secondary 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq2_HTML.gif , i.e.,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equa_HTML.gif

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq3_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq6_HTML.gif be a δ-ring of subsets of Ω such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq7_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq9_HTML.gif . Let us assume that there exists an increasing sequence of sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq10_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq11_HTML.gif . By ℰ we denote the linear space of all simple functions with supports from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq6_HTML.gif . By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq12_HTML.gif we will denote the space of all extended measurable functions, i.e., all functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq13_HTML.gif such that there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq16_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq17_HTML.gif . By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq18_HTML.gif we denote the characteristic function of the set A.

      Definition 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq19_HTML.gif be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq20_HTML.gif ;

         
      2. (ii)

        ρ is monotone, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq21_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq17_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq22_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq23_HTML.gif ;

         
      3. (iii)

        ρ is orthogonally subadditive, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq24_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq25_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq26_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq27_HTML.gif ;

         
      4. (iv)

        ρ has the Fatou property, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq28_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq17_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq29_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq27_HTML.gif ;

         
      5. (v)

        ρ is order continuous in ℰ, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq31_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq32_HTML.gif .

         
      Similarly, as in the case of measure spaces, we say that a set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq33_HTML.gif is ρ-null if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq34_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ1_HTML.gif
      (2.1)

      where each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 ℳ instead of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq38_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq39_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq40_HTML.gif ρ-a.e.;

         
      2. (2)

        We say that ρ is a regular convex function modular if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq41_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq40_HTML.gif ρ-a.e.;

         

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

      Let us denote http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq42_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq44_HTML.gif . It is easy to prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq46_HTML.gif , or briefly http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif , defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equb_HTML.gif
         
      2. (b)
        The following formula defines a norm in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif (frequently called Luxemurg norm):
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif .
      1. (1)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq48_HTML.gif is complete and the norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq49_HTML.gif is monotone w.r.t. the natural order in ℳ.

         
      2. (2)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq50_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq51_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq39_HTML.gif .

         
      3. (3)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq51_HTML.gif for an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq39_HTML.gif then there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq52_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq54_HTML.gif ρ-a.e.

         
      4. (4)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif converges uniformly to f on a set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq55_HTML.gif then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq56_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq39_HTML.gif .

         
      5. (5)

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq57_HTML.gif ρ-a.e. There exists a nondecreasing sequence of sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq58_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif converges uniformly to f on every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq60_HTML.gif (Egoroff theorem).

         
      6. (6)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq61_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq62_HTML.gif ρ-a.e. (Note: this property is equivalent to the Fatou property.)

         
      7. (7)
        Defining http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq64_HTML.gif we have:
        1. (a)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq65_HTML.gif ,

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq66_HTML.gif has the Lebesgue property, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq67_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq70_HTML.gif .

           
        3. (c)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq66_HTML.gif is the closure of ℰ (in the sense of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq71_HTML.gif ).

           
         

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

      Definition 2.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . We say that ρ has the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif -property if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equd_HTML.gif

      whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq73_HTML.gif .

      Theorem 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . The following conditions are equivalent:
      1. (a)

        ρ has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif ,

         
      2. (b)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq74_HTML.gif is a linear subspace of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif ,

         
      3. (c)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq75_HTML.gif ,

         
      4. (d)

        if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq76_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq77_HTML.gif ,

         
      5. (e)

        if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq51_HTML.gif for an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq68_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif .
      1. (a)

        We say that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif is ρ-convergent to f and write http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq78_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq79_HTML.gif .

         
      2. (b)

        A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq80_HTML.gif is called ρ-Cauchy if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq81_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq82_HTML.gif .

         
      3. (c)

        A set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq83_HTML.gif is called ρ-closed if for any sequence of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq84_HTML.gif , the convergence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq85_HTML.gif implies that f belongs to B.

         
      4. (d)

        A set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq83_HTML.gif is called ρ-bounded if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq86_HTML.gif .

         
      5. (e)

        A set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq83_HTML.gif is called strongly ρ-bounded if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq87_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq88_HTML.gif .

         
      6. (f)

        A set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq83_HTML.gif is called ρ-compact if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq89_HTML.gif in C there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq90_HTML.gif and an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq91_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq92_HTML.gif .

         
      7. (g)

        A set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq93_HTML.gif is called ρ-a.e. closed if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif in C which ρ-a.e. converges to some f, then we must have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq91_HTML.gif .

         
      8. (h)

        A set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq93_HTML.gif is called ρ-a.e. compact if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq53_HTML.gif in C, there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq90_HTML.gif which ρ-a.e. converges to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq91_HTML.gif .

         
      9. (i)
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif . The ρ-distance between f and C is defined as
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Eque_HTML.gif
         

      Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq57_HTML.gif does not imply in general http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq96_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq97_HTML.gif . Using Theorem 2.1, it is not difficult to prove the following:

      Proposition 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif .
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif is ρ-complete,

         
      2. (ii)

        ρ-balls http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . The following relationships hold for sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq99_HTML.gif :
      1. (i)

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

         
      2. (ii)

        If C is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq100_HTML.gif -compact, then C is ρ-compact.

         
      3. (iii)

        If ρ satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq100_HTML.gif -compactness and ρ-compactness are equivalent in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be nonempty and ρ-closed. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq101_HTML.gif is called an asymptotic pointwise mapping if there exists a sequence of mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq102_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equf_HTML.gif
      1. (i)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq103_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq94_HTML.gif and every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq104_HTML.gif , then T is called ρ-nonexpansive or shortly nonexpansive.

         
      2. (ii)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq105_HTML.gif converges pointwise to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq106_HTML.gif , then T is called asymptotic pointwise contraction.

         
      3. (iii)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq107_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq94_HTML.gif , then T is called asymptotic pointwise nonexpansive.

         
      4. (iv)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq108_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq94_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq109_HTML.gif , then T is called strongly asymptotic pointwise contraction.

         
      Denoting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq110_HTML.gif , we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ2_HTML.gif
      (3.1)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ3_HTML.gif
      (3.2)
      Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq111_HTML.gif . In view of (3.2), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ4_HTML.gif
      (3.3)

      The above notation will be consistently used throughout this paper.

      By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq112_HTML.gif we will denote the class of all asymptotic pointwise nonexpansive mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq2_HTML.gif .

      In this paper, we will impose some restrictions on the behavior of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq115_HTML.gif as a class of all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq116_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ5_HTML.gif
      (3.4)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ6_HTML.gif
      (3.5)

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

      Definition 3.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . We define the following uniform convexity type properties of the function modular ρ: Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif . Define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equg_HTML.gif
      Let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equh_HTML.gif

      and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq120_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq121_HTML.gif . We will use the following notational convention: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq122_HTML.gif .

      Definition 3.4 We say that ρ satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq123_HTML.gif if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq124_HTML.gif . Note that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq125_HTML.gif , for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif small enough. We say that ρ satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq128_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq129_HTML.gif depending only on s and ε such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equi_HTML.gif

      We will need the following result whose proof is elementary. Note that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq130_HTML.gif , this result follows directly from Definition 3.4.

      Lemma 3.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq117_HTML.gif . Then for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq132_HTML.gif depending only on s and ε such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq133_HTML.gif is called bounded away from 0 if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq134_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq135_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq136_HTML.gif . Similarly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq133_HTML.gif is called bounded away from 1 if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq137_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq138_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq139_HTML.gif be bounded away from 0 and 1. If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq140_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ7_HTML.gif
      (3.6)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ8_HTML.gif
      (3.7)
      then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equk_HTML.gif
      Proof Assume to the contrary that this is not the case and fix an arbitrary http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq141_HTML.gif . Passing to a subsequence if necessary, we may assume that there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ9_HTML.gif
      (3.8)
      while
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ10_HTML.gif
      (3.9)
      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq142_HTML.gif is bounded away from 0 and 1 there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq143_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq144_HTML.gif for all natural n. Passing to a subsequence if necessary, we can assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq145_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq147_HTML.gif , let us define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq148_HTML.gif . Observe that the function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq149_HTML.gif is a convex function. Hence that the function
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ11_HTML.gif
      (3.10)
      is also convex on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq150_HTML.gif , and consequently, it is a continuous function on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq151_HTML.gif . Noting that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ12_HTML.gif
      (3.11)
      we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq152_HTML.gif is a continuous function of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq153_HTML.gif . Hence
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ13_HTML.gif
      (3.12)
      By (3.8) and (3.9)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ14_HTML.gif
      (3.13)
      By (3.12) the left-hand side of (3.13) tends to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq154_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq155_HTML.gif while the right-hand side tends to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq156_HTML.gif in view of (3.7). Hence
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ15_HTML.gif
      (3.14)
      By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif and by Lemma 3.1, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq157_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ16_HTML.gif
      (3.15)
      Combining (3.14) with (3.15) we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ17_HTML.gif
      (3.16)

      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq159_HTML.gif be convex and ρ-bounded. A function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq160_HTML.gif is called a ρ-type (or shortly a type) if there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq161_HTML.gif of elements of K such that for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq162_HTML.gif there holds
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif satisfies the ρ-a.e. Opial property if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq163_HTML.gif which is ρ-a.e. convergent to 0 such that there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq87_HTML.gif for which
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ18_HTML.gif
      (3.17)
      the following inequality holds for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq164_HTML.gif not equal to 0
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ19_HTML.gif
      (3.18)
      Definition 3.8 We say that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif satisfies the ρ-a.e. strong Opial property if for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq163_HTML.gif which is ρ-a.e. convergent to 0 such that there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq87_HTML.gif for which
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ20_HTML.gif
      (3.19)
      the following equality holds for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq164_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq165_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq167_HTML.gif spaces for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq168_HTML.gif while the modular strong Opial property holds in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq167_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq169_HTML.gif .

      Lemma 3.3[27]

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq1_HTML.gif has the ρ-a.e. strong Opial property. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq170_HTML.gif be 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif . Let C be a ρ-closed ρ-bounded convex nonempty subset. Then any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq171_HTML.gif asymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed points http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq172_HTML.gif is 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a convex set, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif is a ρ-approximate fixed point sequence for T, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq175_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq176_HTML.gif , then for every fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq177_HTML.gif there holds
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ22_HTML.gif
      (4.1)

      as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq176_HTML.gif .

      Proof It follows from 3.5 that there exists a finite constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq178_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ23_HTML.gif
      (4.2)
      Using the convexity of ρ and the ρ-nonexpansiveness of T, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ24_HTML.gif
      (4.3)

      as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq176_HTML.gif . □

      Corollary 4.1 If, under the hypothesis of Lemma 4.1, ρ satisfies additionally the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif condition, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq179_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif is uniformly continuous if to every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq180_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq181_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ25_HTML.gif
      (4.4)

      provided http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif -type condition.

      Theorem 4.1 Demiclosedness Principle. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . Assume that:
      1. (1)

        ρ is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq184_HTML.gif ,

         
      2. (2)

        ρ has strong Opial property,

         
      3. (3)

        ρ has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif property and is uniformly continuous.

         

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a nonempty, convex, strongly ρ-bounded and ρ-closed, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq185_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq186_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq187_HTML.gif ρ-a.e. and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq188_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq189_HTML.gif .

      Proof Let us recall that by definition of uniform continuity of ρ to every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq180_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq181_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ26_HTML.gif
      (4.5)
      provided http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq183_HTML.gif . Fix any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq190_HTML.gif . Noting that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq191_HTML.gif due to the strong ρ-boundedness of C and that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq192_HTML.gif by Corollary (4.1), it follows from (4.5) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq193_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq194_HTML.gif that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ27_HTML.gif
      (4.6)
      as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq155_HTML.gif . Hence
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ28_HTML.gif
      (4.7)
      Define the ρ-type φ by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ29_HTML.gif
      (4.8)
      By (4.7) we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ30_HTML.gif
      (4.9)
      Hence, for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq195_HTML.gif there holds
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ31_HTML.gif
      (4.10)
      Using (4.10) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq196_HTML.gif and by passing with m to infinity, we conclude that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ32_HTML.gif
      (4.11)
      Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq197_HTML.gif -a.e., it follows via the Opial property that for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq198_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ33_HTML.gif
      (4.12)
      which implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ34_HTML.gif
      (4.13)
      Combining (4.11) with (4.13), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ35_HTML.gif
      (4.14)
      that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ36_HTML.gif
      (4.15)
      We claim that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ37_HTML.gif
      (4.16)
      Assume to the contrary that (4.16) does not hold, that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ38_HTML.gif
      (4.17)
      By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif , it follows from (4.17) that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq199_HTML.gif does not tend to zero. By passing to a subsequence if necessary, we can assume that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq200_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ39_HTML.gif
      (4.18)
      for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq177_HTML.gif , which implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ40_HTML.gif
      (4.19)
      for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq201_HTML.gif . Hence,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ41_HTML.gif
      (4.20)
      for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq201_HTML.gif . Applying the modular parallelogram inequality valid in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq184_HTML.gif modular function spaces, see Lemma 4.2 in [18],
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ42_HTML.gif
      (4.21)
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq202_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq204_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq205_HTML.gif , with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq207_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq209_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ43_HTML.gif
      (4.22)
      Note that by (4.13)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ44_HTML.gif
      (4.23)
      Combining (4.22) with (4.23), we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ45_HTML.gif
      (4.24)
      which implies
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ46_HTML.gif
      (4.25)
      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq210_HTML.gif and applying (4.15), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ47_HTML.gif
      (4.26)

      Using the properties of Ψ, we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq211_HTML.gif tends to zero itself, which contradicts our assumption (4.17). Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq212_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq210_HTML.gif . Clearly, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq213_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq210_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq214_HTML.gif while http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq215_HTML.gif by ρ-continuity of T. By the uniqueness of the ρ-limit, we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq216_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq218_HTML.gif is a bounded sequence of real numbers and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq219_HTML.gif is a doubly-index sequence of real numbers which satisfy
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equm_HTML.gif

      for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq220_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq218_HTML.gif converges to an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq221_HTML.gif .

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

      Definition 5.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif be an increasing sequence of natural numbers. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T, the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq224_HTML.gif , and the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq225_HTML.gif is defined by the following iterative formula:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ48_HTML.gif
      (5.1)
      Definition 5.2 We say that a generalized Mann iteration process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq226_HTML.gif is well defined if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ49_HTML.gif
      (5.2)

      Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq227_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq228_HTML.gif . Hence we can always select a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq229_HTML.gif such that (5.2) holds. In other words, by a suitable choice of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif , we can always make http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq226_HTML.gif well defined.

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

      Lemma 5.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif . Assume that a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq231_HTML.gif is bounded away from 0 and 1. Let w be a fixed point of T and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq225_HTML.gif be a generalized Mann process. Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq232_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ50_HTML.gif
      (5.3)
      Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq233_HTML.gif . Since
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equn_HTML.gif
      it follows that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq234_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ51_HTML.gif
      (5.4)

      Denote http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq235_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq236_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq237_HTML.gif . Observe that since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq238_HTML.gif . By Lemma 5.1, there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq239_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq240_HTML.gif as claimed. □

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

      Lemma 5.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq241_HTML.gif . Assume that a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq139_HTML.gif is bounded away from 0 and 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq242_HTML.gif be a generalized Mann iteration process. Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ52_HTML.gif
      (5.5)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ53_HTML.gif
      (5.6)
      Proof By Theorem 3.1, T has at least one fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq243_HTML.gif . In view of Lemma 5.2, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq232_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ54_HTML.gif
      (5.7)
      Note that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ55_HTML.gif
      (5.8)
      and that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ56_HTML.gif
      (5.9)
      Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq244_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq245_HTML.gif , and note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq246_HTML.gif by (5.7), and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq247_HTML.gif by (5.8). Observe also that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ57_HTML.gif
      (5.10)
      Hence, it follows from Lemma 3.2 that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ58_HTML.gif
      (5.11)
      which by the construction of the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif is equivalent to
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ59_HTML.gif
      (5.12)

      as claimed. □

      In the next lemma, we prove that under suitable assumption the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq248_HTML.gif is called quasi-periodic if the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq249_HTML.gif is bounded, or equivalently, if there exists a number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq250_HTML.gif such that any block of p consecutive natural numbers must contain a term of the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq251_HTML.gif . The smallest of such numbers p will be called a quasi-period of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq251_HTML.gif .

      Lemma 5.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq252_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif be such that the generalized Mann process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq225_HTML.gif is well defined. If, in addition, the set of indices http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq253_HTML.gif is quasi-periodic, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif is an approximate fixed point sequence, i.e.,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ60_HTML.gif
      (5.13)
      Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq236_HTML.gif be a quasi-period of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq254_HTML.gif . Observe that it is enough to prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq175_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq255_HTML.gif through http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq254_HTML.gif . Indeed, let us fix http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq119_HTML.gif . From http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq175_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq255_HTML.gif through http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq254_HTML.gif it follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ61_HTML.gif
      (5.14)
      for sufficiently large k. By the quasi-periodicity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq254_HTML.gif , to every positive integer k, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq256_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq257_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq258_HTML.gif (the proof for the other case is identical). Since T is ρ-Lipschitzian with the constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq259_HTML.gif , there exist a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq260_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ62_HTML.gif
      (5.15)
      Note that by (5.6) and by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq261_HTML.gif for k sufficiently large. This implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ63_HTML.gif
      (5.16)
      and therefore,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ64_HTML.gif
      (5.17)
      which demonstrates that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ65_HTML.gif
      (5.18)

      as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq176_HTML.gif . By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif again, we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq262_HTML.gif .

      To prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq175_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq255_HTML.gif through http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq254_HTML.gif , observe that, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq263_HTML.gif for such k, there holds
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . Assume that:
      1. (1)

        ρ is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq184_HTML.gif ,

         
      2. (2)

        ρ has Strong Opial Property,

         
      3. (3)

        ρ has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif property and is uniformly continuous.

         

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be nonempty, ρ-a.e. compact, convex, strongly ρ-bounded and ρ-closed, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq241_HTML.gif . Assume that a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq139_HTML.gif is bounded away from 0 and 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq242_HTML.gif be a well-defined generalized Mann iteration process. Assume, in addition, that the set of indices http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq264_HTML.gif is quasi-periodic. Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq189_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq187_HTML.gif ρ-a.e.

      Proof Observe that by Theorem 4.1 in [18], the set of fixed points http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif is an approximate fixed point sequence, that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ67_HTML.gif
      (5.20)
      as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq176_HTML.gif . Consider http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq266_HTML.gif , two ρ-a.e. cluster points of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif . There exits then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq267_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq268_HTML.gif subsequences of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq269_HTML.gif ρ-a.e. and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq270_HTML.gif ρ-a.e. By Theorem 4.1, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq271_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq272_HTML.gif . By Lemma 5.2, there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq273_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ68_HTML.gif
      (5.21)
      We claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq274_HTML.gif . Assume to the contrary that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq275_HTML.gif . Then, by the strong Opial property, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ69_HTML.gif
      (5.22)

      The contradiction implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq274_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif has exactly one ρ-a.e. cluster point, which means that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq276_HTML.gif ρ-a.e. Using Theorem 4.1 again, we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq189_HTML.gif as claimed. □

      Remark 5.2 It is easy to see that we can always construct a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif is not “too sparse” in the sense that the generalized Mann process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif be an increasing sequence of natural numbers. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T, the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq224_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq278_HTML.gif , and the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq279_HTML.gif is defined by the following iterative formula:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ70_HTML.gif
      (6.1)
      Definition 6.2 We say that a generalized Ishikawa iteration process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq280_HTML.gif is well defined if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ71_HTML.gif
      (6.2)

      Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq227_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq228_HTML.gif . Hence we can always select a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq229_HTML.gif such that (6.2) holds. In other words, by a suitable choice of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif , we can always make http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq281_HTML.gif well defined.

      Lemma 6.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif be bounded away from 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq233_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq279_HTML.gif be a generalized Ishikawa process. There exists then an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq232_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq282_HTML.gif .

      Proof Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq283_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ72_HTML.gif
      (6.3)
      It is easy to see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq284_HTML.gif and that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq285_HTML.gif . Moreover, a straight calculation shows that each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq286_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ73_HTML.gif
      (6.4)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ74_HTML.gif
      (6.5)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ75_HTML.gif
      (6.6)
      Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq287_HTML.gif , which follows directly from the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq288_HTML.gif and from (6.5). Using (6.5) and the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq289_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ76_HTML.gif
      (6.7)
      Fix any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq290_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq291_HTML.gif , it follows that there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq292_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq293_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq294_HTML.gif . Therefore, using the same argument as in the proof of Lemma 5.2, we deduce that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq293_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq295_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ77_HTML.gif
      (6.8)

      Arguing like in the proof of Lemma 5.2, we conclude that there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq296_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq297_HTML.gif . □

      Lemma 6.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif be bounded away from 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq279_HTML.gif be a generalized Ishikawa process. Define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ78_HTML.gif
      (6.9)
      Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ79_HTML.gif
      (6.10)
      or equivalently
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ80_HTML.gif
      (6.11)
      Proof By Theorem 3.1, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq298_HTML.gif . Let us fix http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq233_HTML.gif . By Lemma 6.1, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq299_HTML.gif exists. Let us denote it by r. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq233_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq300_HTML.gif by Lemma 6.1, we have the following:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ81_HTML.gif
      (6.12)
      Note that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ82_HTML.gif
      (6.13)

      Applying Lemma 3.2 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq301_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq302_HTML.gif , we obtain the desired equality http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq303_HTML.gif , while (6.11) follows from (6.10) via the construction formulas for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq304_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq305_HTML.gif . □

      Lemma 6.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq252_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq230_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq231_HTML.gif be bounded away from 0 and 1, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq306_HTML.gif be bounded away from 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq279_HTML.gif be a well-defined generalized Ishikawa process. Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ83_HTML.gif
      (6.14)
      Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq307_HTML.gif . Hence
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ84_HTML.gif
      (6.15)
      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif is bounded away from 1, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq308_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq309_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq310_HTML.gif . Hence,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ85_HTML.gif
      (6.16)

      The right-hand side of this inequality tends to zero because http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq311_HTML.gif by Lemma 6.2 and ρ satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif . □

      Lemma 6.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be a ρ-closed, ρ-bounded and convex set, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq252_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1 and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif be bounded away from 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq312_HTML.gif be such that the generalized Ishikawa process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq279_HTML.gif is well defined. If, in addition, the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq253_HTML.gif is quasi-periodic, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq313_HTML.gif is an approximate fixed point sequence, i.e.,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif . Assume that
      1. (1)

        ρ is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq184_HTML.gif ,

         
      2. (2)

        ρ has Strong Opial Property,

         
      3. (3)

        ρ has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif property and is uniformly continuous.

         

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq95_HTML.gif be nonempty, ρ-a.e. compact, convex, strongly ρ-bounded and ρ-closed, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq241_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif be bounded away from 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif be such that the generalized Ishikawa process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq279_HTML.gif is well defined. If, in addition, the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq253_HTML.gif is quasi-periodic, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif generated by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq314_HTML.gif converges ρ-a.e. to a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq47_HTML.gif satisfy conditions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq99_HTML.gif be a ρ-compact, ρ-bounded and convex set, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq173_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq223_HTML.gif be bounded away from 0 and 1, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq277_HTML.gif be bounded away from 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq222_HTML.gif be such that the generalized Mann process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq225_HTML.gif (resp. Ishikawa process http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq314_HTML.gif ) is well defined. Then there exists a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq189_HTML.gif such that then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif generated by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq225_HTML.gif (resp. http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq314_HTML.gif ) converges strongly to a fixed point of T, that is
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ87_HTML.gif
      (7.1)
      Proof By the ρ-compactness of C, we can select a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq315_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq174_HTML.gif such that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq186_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ88_HTML.gif
      (7.2)
      Note that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ89_HTML.gif
      (7.3)
      which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq72_HTML.gif it follows from (7.3) that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ90_HTML.gif
      (7.4)
      Observe that by the convexity of ρ and by ρ-nonexpansiveness of T, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ91_HTML.gif
      (7.5)
      which tends to zero by (7.4) and by Lemma 5.3 (resp. Lemma 6.4). Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq316_HTML.gif which implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq189_HTML.gif . Applying Lemma 5.2 (resp. Lemma 6.1), we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_IEq317_HTML.gif exists. By (7.4) this limit must be equal to zero which implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_278_Equ92_HTML.gif
      (7.6)

       □

      Remark 7.1 Observe that in view of the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-118/MediaObjects/13663_2012_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

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      © Dehaish and Kozlowski; licensee Springer 2012

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.