Convergence theorems for mixed type asymptotically nonexpansive mappings

  • Weiping Guo1,

    Affiliated with

    • Yeol Je Cho2Email author and

      Affiliated with

      • Wei Guo3

        Affiliated with

        Fixed Point Theory and Applications20122012:224

        DOI: 10.1186/1687-1812-2012-224

        Received: 27 April 2012

        Accepted: 16 November 2012

        Published: 11 December 2012

        Abstract

        In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

        Keywords

        mixed type asymptotically nonexpansive mapping strong and weak convergence common fixed point uniformly convex Banach space

        1 Introduction

        Let K be a nonempty subset of a real normed linear space E. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq1_HTML.gif is said to be asymptotically nonexpansive if there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq2_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq3_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ1_HTML.gif
        (1.1)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif .

        In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, and proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.

        Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [216]), which extend and improve the result of Goebel and Kirk in several ways.

        Recently, Chidume et al.[10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.

        Definition 1.1[10]

        Let K be a nonempty subset of a real normed linear space E. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq6_HTML.gif be a nonexpansive retraction of E onto K. A nonself-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq7_HTML.gif is said to be asymptotically nonexpansive if there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq8_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq9_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq10_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ2_HTML.gif
        (1.2)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif .

        Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.

        In 2003, also, Chidume et al.[10] studied the following iteration scheme:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ3_HTML.gif
        (1.3)

        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq11_HTML.gif is a sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq12_HTML.gif and P is a nonexpansive retraction of E onto K, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.

        In 2006, Wang [11] generalized the iteration process (1.3) as follows:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ4_HTML.gif
        (1.4)

        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq13_HTML.gif are two asymptotically nonexpansive nonself-mappings and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq14_HTML.gif are real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq15_HTML.gif , and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).

        The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.

        2 Preliminaries

        Let E be a real Banach space, K be a nonempty closed convex subset of E and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq6_HTML.gif be a nonexpansive retraction of E onto K. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq16_HTML.gif be two asymptotically nonexpansive self-mappings and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq17_HTML.gif be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ5_HTML.gif
        (2.1)

        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq14_HTML.gif are two sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq15_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).

        We denote the set of common fixed points of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq22_HTML.gif and denote the distance between a point z and a set A in E by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq23_HTML.gif .

        Now, we recall some well-known concepts and results.

        Let E be a real Banach space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq24_HTML.gif be the dual space of E and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq25_HTML.gif be the normalized duality mapping defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equa_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq26_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq27_HTML.gif denotes duality pairing between E and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq24_HTML.gif . A single-valued normalized duality mapping is denoted by j.

        A subset K of a real Banach space E is called a retract of E[10] if there exists a continuous mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq6_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq28_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq29_HTML.gif . Every closed convex subset of a uniformly convex Banach space is a retract. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq30_HTML.gif is called a retraction if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq31_HTML.gif . It follows that if a mapping P is a retraction, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq32_HTML.gif for all y in the range of P.

        A Banach space E is said to satisfy Opial’s condition[17] if, for any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif of E, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq34_HTML.gif weakly as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq10_HTML.gif implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equb_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq35_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq36_HTML.gif .

        A Banach space E is said to have a Fréchet differentiable norm[18] if, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq37_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equc_HTML.gif

        exists and is attained uniformly in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq38_HTML.gif .

        A Banach space E is said to have the Kadec-Klee property[19] if for every sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif in E, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq34_HTML.gif weakly and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq39_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq34_HTML.gif strongly.

        Let K be a nonempty closed subset of a real Banach space E. A nonself-mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq7_HTML.gif is said to be semi-compact[11] if, for any sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif in K such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq40_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq10_HTML.gif , there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq41_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq41_HTML.gif converges strongly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq42_HTML.gif .

        Lemma 2.1[15]

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq45_HTML.gif be three nonnegative sequences satisfying the following condition:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equd_HTML.gif

        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq46_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq47_HTML.gif is some nonnegative integer, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq49_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq50_HTML.gif exists.

        Lemma 2.2[8]

        Let E be a real uniformly convex Banach space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq51_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif . Also, suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq52_HTML.gif are two sequences of E such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Eque_HTML.gif

        hold for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq53_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq54_HTML.gif .

        Lemma 2.3[10]

        Let E be a real uniformly convex Banach space, K be a nonempty closed convex subset of E and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq7_HTML.gif be an asymptotically nonexpansive mapping with a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq9_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq10_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq55_HTML.gif is demiclosed at zero, i.e., if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq34_HTML.gif weakly and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq56_HTML.gif strongly, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq57_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq58_HTML.gif is the set of fixed points of T.

        Lemma 2.4[16]

        Let X be a uniformly convex Banach space and C be a convex subset of X. Then there exists a strictly increasing continuous convex function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq59_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq60_HTML.gif such that, for each mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq61_HTML.gif with a Lipschitz constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq62_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equf_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq64_HTML.gif .

        Lemma 2.5[16]

        Let X be a uniformly convex Banach space such that its dual space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq65_HTML.gif has the Kadec-Klee property. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is a bounded sequence and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq66_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equg_HTML.gif

        exists for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq67_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq68_HTML.gif denotes the set of all weak subsequential limits of  http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq69_HTML.gif .

        3 Strong convergence theorems

        In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

        Lemma 3.1 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq70_HTML.gif be two asymptotically nonexpansive self-mappings with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq72_HTML.gif be two asymptotically nonexpansive nonself-mappings with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq73_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq75_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif , respectively, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq77_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif be the sequence defined by (2.1), where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq14_HTML.gif are two real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq15_HTML.gif . Then
        1. (1)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq78_HTML.gif exists for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif ;

           
        2. (2)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq80_HTML.gif exists.

           
        Proof (1) Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq81_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif , it follows from (2.1) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ6_HTML.gif
        (3.1)
        and so
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ7_HTML.gif
        (3.2)

        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq75_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq82_HTML.gif . It follows from Lemma 2.1 that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq78_HTML.gif exists.

        (2) Taking the infimum over all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif in (3.2), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equh_HTML.gif

        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif . It follows from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq82_HTML.gif and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

        Lemma 3.2 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq70_HTML.gif be two asymptotically nonexpansive self-mappings with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq72_HTML.gif be two asymptotically nonexpansive nonself-mappings with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq73_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq75_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif , respectively, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq77_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif be the sequence defined by (2.1) and the following conditions hold:
        1. (a)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq14_HTML.gif are two real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq83_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq84_HTML.gif ;

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq85_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif .

           

        Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq86_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif .

        Proof Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq81_HTML.gif . For any given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq78_HTML.gif exists by Lemma 3.1. Now, we assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq87_HTML.gif . It follows from (3.2) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq82_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equi_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equj_HTML.gif
        Taking lim sup on both sides in (3.1), we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq88_HTML.gif and so
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equk_HTML.gif
        Using Lemma 2.2, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ8_HTML.gif
        (3.3)
        By the condition (b), it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equl_HTML.gif
        and so, from (3.3), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ9_HTML.gif
        (3.4)
        Since
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equm_HTML.gif
        Taking lim inf on both sides in the inequality above, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equn_HTML.gif
        by (3.4) and so
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equo_HTML.gif
        Using (3.1), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equp_HTML.gif
        In addition, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equq_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equr_HTML.gif
        It follows from Lemma 2.2 that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ10_HTML.gif
        (3.5)
        Now, we prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equs_HTML.gif
        Indeed, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq89_HTML.gif by the condition (b). It follows from (3.5) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ11_HTML.gif
        (3.6)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq6_HTML.gif is a nonexpansive retraction of E onto K, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equt_HTML.gif
        and so
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ12_HTML.gif
        (3.7)
        Furthermore, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equu_HTML.gif
        Thus it follows from (3.5), (3.6) and (3.7) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ13_HTML.gif
        (3.8)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq91_HTML.gif by the condition (b) and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equv_HTML.gif
        Using (3.3) and (3.8), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ14_HTML.gif
        (3.9)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ15_HTML.gif
        (3.10)
        It follows from
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equw_HTML.gif
        and (3.3) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ16_HTML.gif
        (3.11)
        In addition, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equx_HTML.gif
        Using (3.3) and (3.11), we obtain that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ17_HTML.gif
        (3.12)
        Thus, using (3.9), (3.10) and the inequality
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equy_HTML.gif
        we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq92_HTML.gif . It follows from (3.6) and the inequality
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equz_HTML.gif
        that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ18_HTML.gif
        (3.13)
        Since
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equaa_HTML.gif
        from (3.8), (3.11) and (3.13), it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ19_HTML.gif
        (3.14)
        Again, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq93_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq94_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif are two asymptotically nonexpansive nonself-mappings, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ20_HTML.gif
        (3.15)
        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif . It follows from (3.12), (3.14) and (3.15) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ21_HTML.gif
        (3.16)
        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif . Moreover, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equab_HTML.gif
        Using (3.4), (3.8) and (3.12), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ22_HTML.gif
        (3.17)
        In addition, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equac_HTML.gif
        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq95_HTML.gif . Thus it follows from (3.6), (3.10), (3.16) and (3.17) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equad_HTML.gif
        Finally, we prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equae_HTML.gif
        In fact, by the condition (b), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equaf_HTML.gif
        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq95_HTML.gif . Thus it follows from (3.5), (3.6), (3.9) and (3.10) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equag_HTML.gif

        This completes the proof. □

        Now, we find two mappings, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq97_HTML.gif , satisfying the condition (b) in Lemma 3.2 as follows.

        Example 3.1[20]

        Let ℝ be the real line with the usual norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq98_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq99_HTML.gif . Define two mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq100_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equah_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equai_HTML.gif
        Now, we show that T is nonexpansive. In fact, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq101_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq102_HTML.gif , then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equaj_HTML.gif
        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq104_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq105_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq106_HTML.gif , then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equak_HTML.gif

        This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq107_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif . Similarly, we can show that S is an asymptotically nonexpansive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq108_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif .

        Next, we show that two mappings S, T satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

        Case 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq101_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equal_HTML.gif
        Case 2. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq102_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equam_HTML.gif
        Case 3. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq105_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq109_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equan_HTML.gif
        Case 4. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq104_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equao_HTML.gif

        Therefore, the condition (b) in Lemma 3.2 is satisfied.

        Theorem 3.1 Under the assumptions of Lemma 3.2, if one of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif is completely continuous, then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif defined by (2.1) converges strongly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif .

        Proof Without loss of generality, we can assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif is completely continuous. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is bounded by Lemma 3.1, there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq110_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq111_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq110_HTML.gif converges strongly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq112_HTML.gif . Moreover, we know that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equap_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equaq_HTML.gif
        by Lemma 3.2, which imply that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equar_HTML.gif
        as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq113_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq114_HTML.gif . Thus, by the continuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equas_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equat_HTML.gif

        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif . Thus it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq115_HTML.gif . Furthermore, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq116_HTML.gif exists by Lemma 3.1, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq117_HTML.gif . This completes the proof. □

        Theorem 3.2 Under the assumptions of Lemma 3.2, if one of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif is semi-compact, then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif defined by (2.1) converges strongly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif .

        Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq86_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq95_HTML.gif by Lemma 3.2 and one of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif is semi-compact, there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq41_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq41_HTML.gif converges strongly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq118_HTML.gif . Moreover, by the continuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq120_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif . Thus it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq115_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq116_HTML.gif exists by Lemma 3.1, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq117_HTML.gif . This completes the proof. □

        Theorem 3.3 Under the assumptions of Lemma 3.2, if there exists a nondecreasing function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq121_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq123_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq124_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equau_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq29_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq22_HTML.gif , then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif defined by (2.1) converges strongly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif .

        Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq86_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq95_HTML.gif by Lemma 3.2, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq125_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq121_HTML.gif is a nondecreasing function satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq123_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq124_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq80_HTML.gif exists by Lemma 3.1, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq126_HTML.gif .

        Now, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is a Cauchy sequence in K. In fact, from (3.2), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equav_HTML.gif
        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif . For any m, n, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq128_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equaw_HTML.gif
        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq129_HTML.gif . Thus, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equax_HTML.gif
        Taking the infimum over all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equay_HTML.gif

        Thus it follows from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq126_HTML.gif that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is a Cauchy sequence. Since K is a closed subset of E, the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges strongly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq118_HTML.gif . It is easy to prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq132_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq133_HTML.gif are all closed and so F is a closed subset of K. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq134_HTML.gif , the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges strongly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif . This completes the proof. □

        4 Weak convergence theorems

        In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.

        Lemma 4.1 Under the assumptions of Lemma 3.1, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq135_HTML.gif , the limit
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equaz_HTML.gif

        exists for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq136_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is the sequence defined by (2.1).

        Proof Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq137_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq138_HTML.gif and, from Lemma 3.1, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq139_HTML.gif exists. Thus it remains to prove Lemma 4.1 for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq140_HTML.gif .

        Define the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq141_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equba_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq29_HTML.gif . It is easy to prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ23_HTML.gif
        (4.1)
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq127_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq142_HTML.gif , it follows from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq144_HTML.gif that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq145_HTML.gif . Setting
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ24_HTML.gif
        (4.2)
        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq146_HTML.gif , from (4.1) and (4.2), it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbb_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq148_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq79_HTML.gif . Let
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equ25_HTML.gif
        (4.3)
        Then, using (4.3) and Lemma 2.4, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbc_HTML.gif
        It follows from Lemma 3.1 and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq149_HTML.gif that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq150_HTML.gif uniformly for all m. Observe that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbd_HTML.gif

        Thus we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq151_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq152_HTML.gif exists for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq140_HTML.gif . This completes the proof. □

        Lemma 4.2 Under the assumptions of Lemma 3.1, if E has a Fréchet differentiable norm, then, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq153_HTML.gif , the limit
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Eqube_HTML.gif

        exists, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is the sequence defined by (2.1). Furthermore, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq68_HTML.gif denotes the set of all weak subsequential limits of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq154_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq156_HTML.gif .

        Proof This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □

        Theorem 4.1 Under the assumptions of Lemma 3.2, if E has a Fréchet differentiable norm, then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif defined by (2.1) converges weakly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif .

        Proof Since E is a uniformly convex Banach space and the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif is bounded by Lemma 3.1, there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq157_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif which converges weakly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq158_HTML.gif . By Lemma 3.2, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbf_HTML.gif

        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq76_HTML.gif . It follows from Lemma 2.3 that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq159_HTML.gif .

        Now, we prove that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges weakly to q. Suppose that there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq160_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq160_HTML.gif converges weakly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq161_HTML.gif . Then, by the same method given above, we can also prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq162_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq163_HTML.gif . It follows from Lemma 4.2 that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbg_HTML.gif

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq164_HTML.gif , which shows that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges weakly to q. This completes the proof. □

        Theorem 4.2 Under the assumptions of Lemma 3.2, if the dual space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq24_HTML.gif of E has the Kadec-Klee property, then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif defined by (2.1) converges weakly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif .

        Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq157_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif which converges weakly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq159_HTML.gif .

        Now, we prove that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges weakly to q. Suppose that there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq160_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq160_HTML.gif converges weakly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq118_HTML.gif . Then, as for q, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq134_HTML.gif . It follows from Lemma 4.1 that the limit
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbh_HTML.gif

        exists for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq136_HTML.gif . Again, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq165_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq166_HTML.gif by Lemma 2.5. This shows that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges weakly to q. This completes the proof. □

        Theorem 4.3 Under the assumptions of Lemma 3.2, if E satisfies Opial’s condition, then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif defined by (2.1) converges weakly to a common fixed point of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq21_HTML.gif .

        Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq157_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif which converges weakly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq159_HTML.gif .

        Now, we prove that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges weakly to q. Suppose that there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq160_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq160_HTML.gif converges weakly to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq167_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq168_HTML.gif . Then, as for q, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq169_HTML.gif . Using Lemma 3.1, we have the following two limits exist:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbi_HTML.gif
        Thus, by Opial’s condition, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_Equbj_HTML.gif

        which is a contradiction and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq170_HTML.gif . This shows that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_329_IEq33_HTML.gif converges weakly to q. This completes the proof. □

        Declarations

        Acknowledgements

        The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

        Authors’ Affiliations

        (1)
        School of Mathematics and Physics, Suzhou University of Science and Technology
        (2)
        Department of Mathematics Education and the RINS College of Education, Gyeongsang National University
        (3)
        Department of Aerospace Engineering and Mechanics, University of Minnesota

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