A New Approximation Method for Common Fixed Points of a Finite Family of Generalized Asymptotically Quasinonexpansive Mappings in Banach Spaces

Abstract

We introduce a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasinonexpansive mappings. Several strong and weak convergence theorems of the proposed iteration in Banach spaces are established. The main results obtianed in this paper generalize and refine many known results in the current literature.

1. Introduction

Let 𝐶 be a convex subset of a Banach space 𝑋, and let {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a family of self-mappings of 𝐶. Suppose that 𝛼_𝑖𝑛∈[0,1], for all 𝑛=1,2,3,… and 𝑖=1,2,…,𝑘.

For 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by the following algorithm:𝑥_𝑛+1=(1−𝛼_𝑘𝑛)𝑥_𝑛+𝛼_𝑘𝑛𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛},𝑦_{(𝑘−1)𝑛}=(1−𝛼_{(𝑘−1)𝑛})𝑥_𝑛+𝛼_{(𝑘−1)𝑛}𝑇^𝑛_𝑘−1𝑦_{(𝑘−2)𝑛},𝑦_{(𝑘−2)𝑛}=(1−𝛼_{(𝑘−2)𝑛})𝑥_𝑛+𝛼_{(𝑘−2)𝑛}𝑇^𝑛_𝑘−2𝑦_{(𝑘−3)𝑛},⋮𝑦_2𝑛=(1−𝛼_2𝑛)𝑥_𝑛+𝛼_2𝑛𝑇^𝑛₂𝑦_1𝑛,𝑦_1𝑛=(1−𝛼_1𝑛)𝑥_𝑛+𝛼_1𝑛𝑇^𝑛₁𝑦_0𝑛,(1.1) where 𝑦_0𝑛=𝑥_𝑛 for all 𝑛. The iterative process (1.1) for a finite family of mappings introduced by Khan et al. [1], and the iterative process is the generalized form of the modified Mann (one-step) iterative process by Schu [2], the modified Ishikawa (two-step) iterative process by Tan and Xu [3], and the three-step iterative process by Xu and Noor [4].

Common fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities. Many researchers [1, 5–19] are interested in studying approximation method for finding common fixed points of nonlinear mapping. Also, approximation methods for finding fixed points for nonexpansive mappings can be seen in [12–16, 20, 21].

In 2003, Sun [17] studied an implicit iterative scheme initiated by Xu and Ori [22] for a finite family of asymptotically quasinonexpansive mappings. Shahzand and Udomene [18], in 2006, proved some convergence theorems for the modified Ishikawa iterative process of two asymptotically quasinonexpence mappings to a common fixed point. Nammanee et al. [23] introduced a three-step iteration scheme for asymptotically nonexpansive mappings and proved weak and strong convergence theorems of that iteration scheme under some control conditions. In 2007, Fukhar-ud-din and Khan [24] studied a new three-step iteration scheme for approximating a common fixed point of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Shahzad and Zegeye [19] introduced a new concept of generalized asymptotially nonexpansive mappings and proved some strong convergence theorems for fixed points of finite family of this class. Recently, Khan et al. [1] introduced the iterative sequence (1.1) for a finite family of asymptotically quasinonexpansive mappings in Banach spaces.

Motivated by Khan et al. [1], we introduce a new iterative scheme for finding a common fixed point of a finite family of generalized asymptotically quasinonexpansive mappings as follows:

For 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by𝑥_𝑛+1=(1−𝛼_𝑘𝑛)𝑦_{(𝑘−1)𝑛}+𝛼_𝑘𝑛𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛},𝑦_{(𝑘−1)𝑛}=(1−𝛼_{(𝑘−1)𝑛})𝑦_{(𝑘−2)𝑛}+𝛼_{(𝑘−1)𝑛}𝑇^𝑛_𝑘−1𝑦_{(𝑘−2)𝑛},𝑦_{(𝑘−2)𝑛}=(1−𝛼_{(𝑘−2)𝑛})𝑦_{(𝑘−3)𝑛}+𝛼_{(𝑘−2)𝑛}𝑇^𝑛_𝑘−2𝑦_{(𝑘−3)𝑛},⋮𝑦_2𝑛=(1−𝛼_2𝑛)𝑦_1𝑛+𝛼_2𝑛𝑇^𝑛₂𝑦_1𝑛,𝑦_1𝑛=(1−𝛼_1𝑛)𝑦_0𝑛+𝛼_1𝑛𝑇^𝑛₁𝑦_0𝑛,(1.2) where 𝑦_0𝑛=𝑥_𝑛 for all 𝑛.

The aim of this paper is to obtain strong and weak convergence results for the iterative process (1.2) of a finite family of generalized asymptotically quasinonexpansive mappings in Banach spaces.

2. Preliminaries

In this section, we give some definitions and lemmas used in the main results.

Let 𝐶 be a nonempty subset of a real Banach space 𝑋, and let 𝑇 be a self-mapping of 𝐶. The fixed point set of 𝑇 is denoted by 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

Then let 𝑇 is called(i)nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶; (ii)quasinonexpansive if 𝐹(𝑇)≠∅ and ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖, for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇); (iii)asymptotically nonexpansive if there exists a sequence {𝑟_𝑛} in [0,∞) with lim_𝑛→∞𝑟_𝑛=0 and ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(1+𝑟_𝑛)‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶 and 𝑛=1,2,3,…; (iv)asymptotically quasinonexpansive if 𝐹(𝑇)≠∅ and there exists a sequence {𝑟_𝑛} in [0,∞) with lim_𝑛→∞𝑟_𝑛=0 and ‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑟_𝑛)‖𝑥−𝑝‖, for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇) and 𝑛=1,2,3,…; (v)generalized quasinonexpansive if 𝐹(𝑇)≠∅ and there exists a sequence {𝑠_𝑛} in [0,∞) with 𝑠_𝑛→0 as 𝑛→∞ such that ‖𝑇^𝑛𝑥−𝑝‖≤‖𝑥−𝑝‖+𝑠_𝑛, for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇) and 𝑛=1,2,3,…;(vi)generalized asymptotically quasinonexpansive [19] if 𝐹(𝑇)≠∅ and there exist two sequences {𝑟_𝑛} and {𝑠_𝑛} in [0,∞) with 𝑟_𝑛→0 and 𝑠_𝑛→0 as 𝑛→∞ such that ‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑟_𝑛)‖𝑥−𝑝‖+𝑠_𝑛, for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇) and 𝑛=1,2,3,…;(vii)uniformly 𝐿-Lipschitzian if there exists constant 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶 and 𝑛=1,2,3,…; (viii)(𝐿−𝛾) uniform Lipschitz if there are constants 𝐿>0 and 𝛾>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖^𝛾, for all 𝑥,𝑦∈𝐶 and 𝑛=1,2,3,…. (ix)semicompact if for a sequence {𝑥_𝑛} in 𝐶 with lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖→𝑝∈𝐶.

From the definition of these mappings, it can be seen that (i)a quasinonexpansive mapping is generalized quasinonexpansive;(ii)an asymptotically quasinonexpansive mapping is generalized asymptotically quasinonexpansive;(iii)a generalized quasinonexpansive mapping is generalized asymptotically quasinonexpansive;(iv)a uniformly 𝐿-Lipschitzian mapping is (𝐿−1) uniform Lipschitz.

The map 𝑇∶𝐶→𝑋 is said to be demiclosed at 0 if for each sequence {𝑥_𝑛} in 𝐶 converging weakly to 𝑥∈𝐶 and 𝑇(𝑥_𝑛) converging strongly to 0, we get 𝑇𝑥=0.

A Banach space 𝑋 is said to have Opial's property if for each sequence {𝑥_𝑛} converging weakly to 𝑥∈𝐶 and 𝑥≠𝑦, we have the condition liminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖.(2.1)

Condition 2 (A^′′). Let 𝐶 be a subset of a normed space 𝑋. A family of self-mappings {𝑇_𝑖∶𝑖=1,2,…,𝑘} of 𝐶 is said to have Condition (A^′′) if there exists a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑟)>0 for all 𝑟∈(0,∞) such that ‖𝑥−𝑇_𝑖𝑥‖≥𝑓(𝑑(𝑥,𝐹)) for some 1≤𝑖≤𝑘 and for all 𝑥∈𝐶 where 𝑑(𝑥,𝐹)=inf{‖𝑥−𝑝‖∶𝑝∈𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)}.

The condition (A^′′) defined above by the authors is the generalization of the condition (A) [25] when 𝑘=1 and condition (A^′) [26] for 𝑘=2.

The following lemmas are needed for proving our main results.

Lemma 2.1 (cf. [17, Lemma 2.2]). Let the sequences {𝑎_𝑛}, {𝛿_𝑛} and {𝑐_𝑛} of real numbers satisfy: 𝑎_𝑛+1≤(1+𝛿_𝑛)𝑎_𝑛+𝑐_𝑛,where𝑎_𝑛≥0,𝛿_𝑛≥0,𝑐_𝑛≥0∀𝑛=1,2,3,…(2.2) and ∑^∞_𝑛=1,𝛿_𝑛<∞, ∑^∞_𝑛=1,𝑐_𝑛<∞. (i)lim_𝑛→∞𝑎_𝑛 exists;(ii)if liminf_𝑛→∞𝑎_𝑛=0, then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.2 (see [2, Lemma 1.3]). Let 𝑋 be a uniformly convex Banach space. Assume that 0<𝑏≤𝑡_𝑛≤𝑐<1, 𝑛=1,2,3,…. Let the sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 be such that limsup_𝑛→∞‖𝑥_𝑛‖≤𝑎, limsup_𝑛→∞‖𝑦_𝑛‖≤𝑎 and lim_𝑛→∞‖𝑡_𝑛𝑥_𝑛+(1−𝑡_𝑛)𝑦_𝑛‖=𝑎, where 𝑎≥0. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

3. Convergence in Banach Spaces

The aim of this section is to establish the strong convergence of the iterative scheme (1.2) to converge to a common fixed point of a finite family of asymptotically quasinonexpansive mappings in a Banach space under some appropriate conditions.

Lemma 3.1. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋, and {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a finite family of generalized asymtotically quasinonexpansive self-mappings of 𝐶, that is, ‖𝑇^𝑛_𝑖𝑥−𝑝_𝑖‖≤(1+𝑟_in)‖𝑥−𝑝_𝑖‖+𝑠_in, for all 𝑥∈𝐶 with the sequence {𝑟_in},{𝑠_in}⊂[0,∞) and 𝑝_𝑖∈𝐹(𝑇_𝑖), 𝑖=1,2,…,𝑘. Suppose that 𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)≠∅, 𝑥₁∈𝐶, and the iterative sequence {𝑥_𝑛} is defined by (1.2). Let 𝑟_𝑛=max_{1≤𝑖≤𝑘}{𝑟_in} and 𝑠_𝑛=max_{1≤𝑖≤𝑘}{𝑠_in} Then for 𝑝∈𝐹, we get the following:(i)‖𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛‖≤(2+𝑟_𝑛)‖𝑥_𝑛−𝑝‖+𝑠_𝑛,forall𝑖=1,2,…,𝑘;(ii)‖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖≤(2+𝑟_𝑛)‖𝑦_{(𝑖−1)𝑛}−𝑝‖+𝑠_𝑛,forall𝑖=1,2,…,𝑘;(iii)‖𝑦_𝑘𝑛−𝑝‖≤(1+𝑟_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑠_𝑛∑^𝑘_𝑖=1(1+𝑟_𝑛)^𝑘−𝑖; (iv)‖𝑥_𝑛+1−𝑝‖≤(1+𝑟_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑠_𝑛∑^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)}; (v)‖𝑥_𝑛+1−𝑝‖≤(1+𝛿_𝑛)‖𝑥_𝑛−𝑝‖+𝑐_𝑛,forall𝑛∈ℕ,where 𝑐_𝑛=𝑠_𝑛∑^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)} and 𝛿_𝑛=(^𝑘₁)𝑟_𝑛+(^𝑘₂)𝑟²_𝑛+⋯+(^𝑘_𝑘)𝑟^𝑘_𝑛.(vi) If ∑^∞_𝑛=1𝑟_𝑛<∞ and ∑^∞_𝑛=1𝑠_𝑛<∞, then lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.

Proof. Let 𝑝∈𝐹. (i)For 𝑖=1,2,3,…,𝑘, we have ‖𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛‖≤‖𝑥_𝑛−𝑝‖+‖𝑇^𝑛_𝑖𝑥_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖+(1+𝑟_𝑛)‖𝑥_𝑛−𝑝‖+𝑠_𝑖𝑛,≤(2+𝑟_𝑛)‖𝑥_𝑛−𝑝‖+𝑠_𝑛.(3.1)(ii)Similarly to part (i), we have ‖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖≤(2+𝑟_𝑛)‖𝑦_{(𝑖−1)𝑛}−𝑝‖+𝑠_𝑛,∀𝑖=1,2,…,𝑘.(3.2)(iii)By part (i) and 0≤𝛼_𝑖𝑛≤1, we have ‖𝑦_1𝑛−𝑝‖=‖(1−𝛼_1𝑛)(𝑥_𝑛−𝑝)+𝛼_1𝑛(𝑇^𝑛₁𝑥_𝑛−𝑝)‖≤(1−𝛼_1𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛‖𝑇^𝑛₁𝑥_𝑛−𝑝‖≤(1−𝛼_1𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛(1+𝑟_1𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛𝑠_1𝑛≤(1+𝑟_𝑛)‖𝑥_𝑛−𝑝‖+𝑠_𝑛,‖𝑦_2𝑛−𝑝‖=‖(1−𝛼_2𝑛)(𝑦_1𝑛−𝑝)+𝛼_2𝑛(𝑇^𝑛₂𝑦_1𝑛−𝑝)‖≤(1−𝛼_2𝑛)‖𝑦_1𝑛−𝑝‖+𝛼_2𝑛‖𝑇^𝑛₂𝑦_1𝑛−𝑝‖≤(1−𝛼_2𝑛)‖𝑦_1𝑛−𝑝‖+𝛼_2𝑛(1+𝑟_2𝑛)‖𝑦_1𝑛−𝑝‖+𝛼_2𝑛𝑠_2𝑛≤(1+𝑟_𝑛)‖𝑦_1𝑛−𝑝‖+𝑠_𝑛≤(1+𝑟_𝑛)(1+𝑟_𝑛)‖𝑥_𝑛−𝑝‖+𝑠_𝑛(1+𝑟_𝑛)+𝑠_𝑛≤(1+𝑟_𝑛)²‖𝑥_𝑛−𝑝‖+𝑠_𝑛((1+𝑟_𝑛)+1),‖𝑦_3𝑛−𝑝‖=‖(1−𝛼_3𝑛)(𝑦_2𝑛−𝑝)+𝛼_3𝑛(𝑇^𝑛₃𝑦_2𝑛−𝑝)‖≤(1−𝛼_3𝑛)‖𝑦_2𝑛−𝑝‖+𝛼_3𝑛‖𝑇^𝑛₃𝑦_2𝑛−𝑝‖≤(1−𝛼_3𝑛)‖𝑦_2𝑛−𝑝‖+𝛼_3𝑛(1+𝑟_3𝑛)‖𝑦_2𝑛−𝑝‖+𝛼_3𝑛𝑠_3𝑛≤(1+𝑟_𝑛)‖𝑦_2𝑛−𝑝‖+𝑠_𝑛≤(1+𝑟_𝑛)³‖𝑥_𝑛−𝑝‖+𝑠_𝑛((1+𝑟_𝑛)²+(1+𝑟_𝑛⋮)+1),‖𝑦_𝑘𝑛−𝑝‖≤(1+𝑟_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑠_𝑛^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)}.(3.3)(iv)By part (ii) and part (iii), we get ‖𝑥_𝑛+1−𝑝‖=‖(1−𝛼_𝑘𝑛)(𝑦_{(𝑘−1)𝑛}−𝑝)+𝛼_𝑘𝑛(𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝)‖≤(1−𝛼_𝑘𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝛼_𝑘𝑛‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝‖≤(1−𝛼_𝑘𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝛼_𝑘𝑛(1+𝑟_𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝛼_𝑘𝑛𝑠_𝑘𝑛≤(1+𝑟_𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝑠_𝑛≤(1+𝑟_𝑛)(1+𝑟_𝑛)^𝑘−1‖𝑥_𝑛−𝑝‖+(1+𝑟_𝑛)𝑠_𝑛^𝑘−1_𝑖=1(1+𝑟_𝑛)^{(𝑘−1−𝑖)}+𝑠_𝑛=(1+𝑟_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑠_𝑛^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)}.(3.4)(v) Put 𝛿_𝑛=(^𝑘₁)𝑟_𝑛+(^𝑘₂)𝑟²_𝑛+⋯+(^𝑘_𝑘)𝑟^𝑘_𝑛 and 𝑐_𝑛=𝑠_𝑛∑^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)}. Then (v) is directly obtained by (iv).(vi)By (𝑣), we have ‖𝑥_𝑛+1−𝑝‖≤(1+𝛿_𝑛)‖𝑥_𝑛−𝑝‖+𝑐_𝑛 for all 𝑛∈𝑁, where 𝑐_𝑛=𝑠_𝑛∑^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)} and 𝛿_𝑛=(^𝑘₁)𝑟_𝑛+(^𝑘₂)𝑟²_𝑛+⋯+(^𝑘_𝑘)𝑟^𝑘_𝑛. From Σ^∞_𝑛=1𝑟_𝑛<∞ and Σ^∞_𝑛=1𝑠_𝑛<∞, it follows that Σ^∞_𝑛=1𝛿_𝑛<∞ and Σ^∞_𝑛=1𝑐_𝑛<∞. By Lemma 2.1, we get lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋, and {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a finite family of generalized asymtotically quasinonexpansive self-mappings of 𝐶, that is, ‖𝑇^𝑛_𝑖𝑥−𝑝_𝑖‖≤(1+𝑟_in)‖𝑥−𝑝_𝑖‖+𝑠_in, for all 𝑥∈𝐶 and 𝑝_𝑖∈𝐹(𝑇_𝑖),𝑖=1,2,…,𝑘. Suppose that 𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)≠∅ is closed, 𝑥₁∈𝐶 and the iterative sequence {𝑥_𝑛} is defined by (1.2). Assume that ∑^∞_𝑛=1𝑟_𝑛<∞ and ∑^∞_𝑛=1𝑠_𝑛<∞, where 𝑟_𝑛=max_{1≤𝑖≤𝑘}{𝑟_in} and 𝑠_𝑛=max_{1≤𝑖≤𝑘}{𝑠_in}. Then {𝑥_𝑛} converges strongly to a common fixed point of the family of mappings if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, where 𝑑(𝑥,𝐹)=inf_𝑝∈𝐹‖𝑥−𝑝‖.

Proof. The necessity is obvious and then we prove only the sufficiency. Let 𝑝∈𝐹. Since 1+𝑡≤𝑒^𝑡 for 𝑡≥0, we obtain (1+𝑡)^𝑘≤𝑒^𝑘𝑡, for 𝑘=1,2,…. Thus by Lemma 3.1(iv) and (v), for positive integers 𝑚 and 𝑛, we have ‖𝑥_𝑛+𝑚−𝑝‖≤(1+𝑟_{(𝑛+𝑚−1)})^𝑘‖𝑥_{(𝑛+𝑚−1)}−𝑝‖+𝑐_{(𝑛+𝑚−1)}≤exp{𝑘𝑟_{(𝑛+𝑚−1)}}‖𝑥_{(𝑛+𝑚−1)}−𝑝‖+𝑐_{(𝑛+𝑚−1)}≤⋯≤exp{𝑘^{𝑛+𝑚−1}_𝑖=𝑛𝑟_𝑖}‖𝑥_𝑛−𝑝‖+^{𝑛+𝑚−1}_𝑖=𝑛𝑐_𝑖≤exp{𝑘^∞_𝑖=1𝑟_𝑖}‖𝑥_𝑛−𝑝‖+^∞_𝑖=𝑛𝑐_𝑖=𝑀‖𝑥_𝑛−𝑝‖+^∞_𝑖=𝑛𝑐_𝑖,(3.5) where 𝑀=exp{𝑘∑^∞_𝑖=1𝑟_𝑖}.
By Lemma 3.1(v), we have‖𝑥_𝑛+1−𝑝‖≤(1+𝛿_𝑛)‖𝑥_𝑛−𝑝‖+𝑐_𝑛,∀𝑝∈𝐹,(3.6) where 𝛿_𝑛=(^𝑘₁)𝑟_𝑛+(^𝑘₂)𝑟²_𝑛+⋯+(^𝑘_𝑘)𝑟^𝑘_𝑛 and 𝑐_𝑛=𝑠_𝑛∑^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)}.
It follows that 𝑑(𝑥_𝑛+1,𝐹)≤(1+𝛿_𝑛)𝑑(𝑥_𝑛,𝐹).(3.7) From the given condition liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 and Lemma 2.1, we get lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.(3.8)
Next, we show that {𝑥_𝑛} is a Cauchy sequence in 𝐶. By (3.8) and ∑^∞_𝑛=1𝑐_𝑛<∞, we get that for any 𝜖>0, there exists a positive integer 𝑛₀ such that, for all 𝑛≥𝑛₀,𝑑(𝑥_𝑛𝜖,𝐹)<,3𝑀^∞_𝑛=𝑛0𝑐_𝑛<𝜖3.(3.9)
From 𝑑(𝑥_𝑛,𝐹)<𝜖/3𝑀, there exists 𝑝₀∈𝐹 such that‖𝑥_𝑛0−𝑝₀𝜖‖<.3𝑀(3.10)
For any positive integer 𝑚, by (3.5), (3.9), and (3.10), we have‖𝑥_𝑛0+𝑚−𝑥_𝑛0‖≤‖𝑥_𝑛0+𝑚−𝑝₀‖+‖𝑥_𝑛0−𝑝₀‖≤𝑀‖𝑥_𝑛0−𝑝₀‖+^∞_𝑖=𝑛0𝑐_𝑖+‖𝑥_𝑛0−𝑝₀‖𝜖<𝑀+𝜖3𝑀3𝜖+𝑀3𝑀=𝜖.(3.11) Thus, {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, 𝑥_𝑛→𝑞∈𝑋. Actually, 𝑞∈𝐶 because {𝑥_𝑛}⊂𝐶 and 𝐶 is a closed subset of 𝑋. Next we show that 𝑞∈𝐹. Since 𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖) is closed, by the continuity of 𝑑(𝑥,𝐹) with 𝑑(𝑥_𝑛,𝐹)→0 and 𝑥_𝑛→𝑞 as 𝑛→∞, we get 𝑑(𝑞,𝐹)=0 and then 𝑞∈𝐹. Therefore, the proof is complete.

Since any asymptotically quasinonexpansive mapping is generalized asymptotically quasinonexpansive, the next corollary is obtained immediately from Theorem 3.2.

Corollary 3.3 (see [5, Theorem 3.2]). Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋, and {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a finite family of asymptotically quasinonexpansive self-mappings of 𝐶, that is, ‖𝑇^𝑛_𝑖𝑥−𝑝_𝑖‖≤(1+𝑟_in)‖𝑥−𝑝_𝑖‖, for all 𝑥∈𝐶 and 𝑝_𝑖∈𝐹(𝑇_𝑖), 𝑖=1,2,…,𝑘. 𝑥₁∈𝐶 and the iterative sequence {𝑥_𝑛} be defined by (1.2). Then {𝑥_𝑛} converges strongly to a common fixed point of the family of mappings if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, where 𝑑(𝑥,𝐹)=inf_𝑝∈𝐹‖𝑥−𝑝‖.

4. Convergence in Uniformly Convex Banach Spaces

In this section, strong and weak convergence results for the iterative process (1.2) on uniformly convex Banach spaces are proved without using the condition liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 appearing in Section 3.

Theorem 4.1. Let 𝐶 be a nonempty closed convex subset of an uniformly convex real Banach space 𝑋. Let {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a finite family of uniformly (𝐿−𝛾_𝑖) Lipschitzian and generalized asymptotically quasinonexpansive self-mappings of 𝐶, that is, ‖𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦‖≤𝐿‖𝑥−𝑦‖^𝛾𝑖 and ‖𝑇^𝑛_𝑖𝑥−𝑝_𝑖‖≤(1+𝑟_in)‖𝑥−𝑝_𝑖‖+𝑠_in, for all 𝑥,𝑦∈𝐶 and 𝑝_𝑖∈𝐹(𝑇_𝑖), 𝑖=1,2,…,𝑘. Suppose that {𝑇_𝑖∶𝑖=1,2,…,𝑘} satisfies condition (A^′′) and 𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝑥₁∈𝐶 and the iterative sequence {𝑥_𝑛} be defined by (1.2) with {𝛼_in}^𝑛_𝑖=1⊂[𝑎,𝑏], where 0<𝑎<𝑏<1. Assume that ∑^∞_𝑛=1𝑟_𝑛<∞,∑^∞_𝑛=1𝑠_𝑛<∞ where 𝑟_𝑛=max_{1≤𝑖≤𝑘}{𝑟_in} and 𝑠_𝑛=max_{1≤𝑖≤𝑘}{𝑠_in}. Then {𝑥_𝑛} converges strongly to a common fixed point of the family of mappings.

Proof. Let 𝑝∈𝐹. By Lemma 3.1(vi), we get that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Then there is a real number 𝑐≥0 such that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑐.(4.1)
By Lemma 3.1(iii), we have‖𝑦_𝑘𝑛−𝑝‖≤(1+𝑟_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑠_𝑛^𝑘_𝑖=1(1+𝑟_𝑛)^{(𝑘−𝑖)},∀𝑝∈𝐹.(4.2)
By taking limsup on both sides of the above inequality, we getlimsup_𝑛→∞‖𝑦_𝑖𝑛−𝑝‖≤𝑐,for𝑖=1,2,…,𝑘−1.(4.3) Since ‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑝‖≤(1+𝑟_𝑛)‖𝑦_{(𝑖−1)𝑛}−𝑝‖+𝑠_𝑛 and (4.3), we obtain limsup_𝑛→∞‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑝‖≤𝑐,for𝑖=1,2,…,𝑘.(4.4) Since lim_𝑛→∞‖𝑥_𝑛+1−𝑝‖=𝑐, we have lim_𝑛→∞‖(1−𝛼_𝑘𝑛)(𝑦_{(𝑘−1)𝑛}−𝑝)+𝛼_𝑘𝑛(𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝)‖=𝑐.(4.5) Using (4.1), (4.4), and Lemma 2.2, we conclude that lim_𝑛→∞‖𝑦_{(𝑘−1)𝑛}−𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}‖=0.(4.6) We assume that lim_𝑛→∞‖𝑦_{(𝑗−1)𝑛}−𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}‖=0,forsome2≤𝑗≤𝑘.(4.7) It follows from (4.5) and (4.7) that 𝑐≤liminf_𝑛→∞‖𝑦_{(𝑗−1)𝑛}−𝑝‖,for2≤𝑗≤𝑘.(4.8) By Lemma 3.1(iv), (1.2), and (4.8), we get lim_𝑛→∞‖(1−𝛼_{(𝑗−1)𝑛})(𝑦_{(𝑗−2)𝑛}−𝑝)+𝛼_{(𝑗−1)𝑛}(𝑇^𝑛_𝑗−1𝑦_{(𝑗−2)𝑛}−𝑝)‖=lim_𝑛→∞‖𝑦_{(𝑗−1)𝑛}−𝑝‖=𝑐.(4.9) Using (4.3), (4.4) and Lemma 2.2, we conclude that lim_𝑛→∞‖𝑦_{(𝑗−2)𝑛}−𝑇^𝑛_𝑗−1𝑦_{(𝑗−2)𝑛}‖=0.(4.10) Therefore, by mathematical induction, we obtain lim_𝑛→∞‖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖=0,for𝑖=1,2,…,𝑘.(4.11)
From (1.2), we have‖𝑦_𝑖𝑛−𝑦_{(𝑖−1)𝑛}‖=𝛼_𝑖𝑛‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑦_{(𝑗−1)𝑛}‖,for𝑖=1,2,…,𝑘−1.(4.12) By (4.11), we obtain that ‖𝑦_𝑖𝑛−𝑦_{(𝑖−1)𝑛}‖→0as𝑛→∞,for𝑖=1,2,…,𝑘−1.(4.13) From ‖𝑥_𝑛−𝑦_𝑖𝑛‖≤‖𝑥_𝑛−𝑦_1𝑛‖+‖𝑦_1𝑛−𝑦_2𝑛‖+⋯+‖𝑦_{(𝑖−1)𝑛}−𝑦_𝑖𝑛‖,(4.14) for 𝑖=1,2,…,𝑘−1, it follows by (4.13) that ‖𝑥_𝑛−𝑦_𝑖𝑛‖→0as𝑛→∞,for𝑖=1,2,…,𝑘−1.(4.15) From (4.11), when 𝑖=1, we get lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛‖=0. For 2≤𝑖≤𝑘, we have ‖𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛‖≤‖𝑥_𝑛−𝑦_{(𝑖−1)𝑛}‖+‖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖+‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑥_𝑛‖≤‖𝑥_𝑛−𝑦_{(𝑖−1)𝑛}‖+‖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖+𝐿‖𝑦_{(𝑖−1)𝑛}−𝑥_𝑛‖^𝛾𝑖.(4.16)
From (4.11) and (4.15), we conclude thatlim_𝑛→∞𝛾_𝑖𝑛=0,for𝑖=1,2,…,𝑘,(4.17) where 𝛾_𝑖𝑛=‖𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛‖. From (1.2), we have ‖𝑥_𝑛+1−𝑥_𝑛‖≤(1−𝛼_𝑘𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖+𝛼_𝑘𝑛‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖≤(1−𝛼_𝑘𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖+𝛼_𝑘𝑛(‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑦_{(𝑘−1)𝑛}‖+‖𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖)=‖𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖+𝛼_𝑘𝑛‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑦_{(𝑘−1)𝑛}‖.(4.18) From (4.11) and (4.15), lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.(4.19) For 𝑖=1,2,…,𝑘, we have ‖𝑥_𝑛+1−𝑇_𝑖𝑥_𝑛+1‖≤‖𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1‖+‖𝑇_𝑖𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1‖≤𝛾_𝑖(𝑛+1)+𝐿‖𝑥_𝑛+1−𝑇^𝑛_𝑖𝑥_𝑛+1‖^𝛾𝑖≤𝛾_𝑖(𝑛+1)+𝐿(‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛‖+‖𝑇^𝑛_𝑖𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛+1‖)^𝛾𝑖≤𝛾_𝑖(𝑛+1)+𝐿(‖𝑥_𝑛+1−𝑥_𝑛‖+𝛾_𝑖𝑛+𝐿‖𝑥_𝑛−𝑥_𝑛+1‖^𝛾𝑖)^𝛾𝑖.(4.20) Using (4.17) and (4.19), we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑇_𝑖𝑥_𝑛+1‖=0,for𝑖=1,2,…,𝑘.(4.21) Therefore, by using condition (A^′′), there exists a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑟)>0 for all 𝑟∈(0,∞) such that lim_𝑛→∞𝑓(𝑑(𝑥_𝑛,𝐹))≤lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑗𝑥_𝑛‖=0,(4.22) for some 1≤𝑗≤𝑘, that is lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.(4.23) By Theorem 3.2, we conclude that {𝑥_𝑛} converges strongly to a point 𝑝∈𝐹.

Lemma 4.2. Let 𝐶 be a nonempty closed convex subset of an uniformly convex real Banach space 𝑋, and {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a family of (𝐿−𝛾_𝑖) uniform Lipschitz and generalized asymtotically quasinonexpansive self-mappings of 𝐶, that is, ‖𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦‖≤𝐿‖𝑥−𝑦‖^𝛾𝑖 and ‖𝑇^𝑛_𝑖𝑥−𝑝_𝑖‖≤(1+𝑟_𝑖𝑛)‖𝑥−𝑝_𝑖‖+𝑠_𝑖𝑛, for all 𝑥,𝑦∈𝐶 and 𝑝_𝑖∈𝐹(𝑇_𝑖), 𝑖=1,2,…,𝑘. Suppose that 𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝑥₁∈𝐶 and the iterative sequence {𝑥_𝑛} be defined by (1.2) with {𝛼_𝑖𝑛}^𝑛_𝑖=1⊂[𝑎,𝑏], where 0<𝑎<𝑏<1. Assume that ∑^∞_𝑛=1𝑟_𝑛<∞, ∑^∞_𝑛=1𝑠_𝑛<∞ where 𝑟_𝑛=max_{1≤𝑖≤𝑘}{𝑟_𝑖𝑛} and 𝑠_𝑛=max_{1≤𝑖≤𝑘}{𝑠_𝑖𝑛}. Then, (i)lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖=0, for all 𝑖=1,2,,…,𝑘; (ii)lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0, for all 𝑖=1,2,,…,𝑘.

Proof. (i) Let 𝑝∈𝐹. By Lemma 3.1(vi), we obtain that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists and we then suppose that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑐.(4.24) By (4.24) and Lemma 3.1(iii), we have limsup_𝑛→∞‖𝑦_𝑖𝑛−𝑝‖≤𝑐,for𝑖=1,2,…,𝑘−1.(4.25)
By (1.2), we have ‖𝑥_𝑛+1−𝑝‖≤(1−𝛼_𝑘𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝛼_𝑘𝑛‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝‖≤(1−𝛼_𝑘𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝛼_𝑘𝑛(1+𝑟_𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝛼_𝑘𝑛𝑠_𝑘𝑛≤(1+𝑟_𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝑠_𝑛=(1+𝑟_𝑛)‖(1−𝛼_{(𝑘−1)𝑛})(𝑦_{(𝑘−2)𝑛}−𝑝)+𝛼_{(𝑘−1)𝑛}(𝑇^𝑛_𝑘−1𝑦_{(𝑘−2)𝑛}−𝑝)‖+𝑠_𝑛≤(1+𝑟_𝑛)[(1−𝛼_{(𝑘−1)𝑛})‖𝑦_{(𝑘−2)𝑛}−𝑝‖+𝛼_{(𝑘−1)𝑛}‖𝑇^𝑛_𝑘−1𝑦_{(𝑘−2)𝑛}−𝑝‖]+𝑠_𝑛≤(1+𝑟_𝑛)[(1−𝛼_{(𝑘−1)𝑛})‖𝑦_{(𝑘−2)𝑛}−𝑝‖+𝛼_{(𝑘−1)𝑛}(1+𝑟_𝑛)‖𝑦_{(𝑘−2)𝑛}−𝑝‖+𝛼_{(𝑘−1)𝑛}𝑠_𝑛]+𝑠_𝑛≤(1+𝑟_𝑛)²‖𝑦_{(𝑘−2)𝑛}−𝑝‖+𝑠_𝑛(1+𝑟_𝑛)+𝑠_𝑛⋮≤(1+𝑟_𝑛)^𝑘−𝑖‖𝑦_𝑖𝑛−𝑝‖+𝑠_𝑛^𝑘−1_𝑖=1(1+𝑟_𝑛)^𝑘−𝑖,(4.26) for 𝑖=1,2,…,𝑘−1. It follows that 𝑐≤liminf_𝑛→∞‖𝑦_𝑖𝑛−𝑝‖,for𝑖=1,2,…,𝑘−1.(4.27) From (4.25) and (4.27), we obtain lim_𝑛→∞‖𝑦_𝑖𝑛−𝑝‖=𝑐,for𝑖=1,2,…,𝑘−1,(4.28) and then lim_𝑛→∞‖(1−𝛼_𝑖𝑛)(𝑦_{(𝑖−1)𝑛}−𝑝)+𝛼_𝑖𝑛(𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑝)‖=𝑐,(4.29) for 𝑖=1,2,…,𝑘−1.
Since ‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑝‖≤(1+𝑟_𝑛)‖𝑦_{(𝑖−1)𝑛}−𝑝‖+𝑠_𝑛, for 𝑖=1,2,…,𝑘−1, we havelimsup_𝑛→∞‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑝‖≤𝑐,for𝑖=1,2,…,𝑘−1.(4.30) From (4.25), (4.29), (4.30) and Lemma 2.2, we obtain lim_𝑛→∞‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑦_{(𝑖−1)𝑛}‖=0,for𝑖=1,2,…,𝑘−1.(4.31) Now we want to show that (4.31) is also true for 𝑖=𝑘.
Since ‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝‖≤(1+𝑟_𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+𝑠_𝑛, lim_𝑛→∞𝑟_𝑛=0 and lim_𝑛→∞𝑠_𝑛=0, it follows by (4.28),limsup_𝑛→∞‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝‖≤𝑐.(4.32) We also have lim_𝑛→∞‖(1−𝛼_𝑘𝑛)(𝑦_{(𝑘−1)𝑛}−𝑝)+𝛼_𝑘𝑛(𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝)‖=lim_𝑛→∞‖𝑥_𝑛+1−𝑝‖=𝑐.(4.33) Hence, by (4.25), (4.32), and Lemma 2.2, we obtain lim_𝑛→∞‖𝑦_{(𝑘−1)𝑛}−𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}‖=0.(4.34) Then, (4.31) and (4.34) give us lim_𝑛→∞‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑦_{(𝑖−1)𝑛}‖=0,for𝑖=1,2,…,𝑘.(4.35) From ‖𝑥_𝑛−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖≤‖𝑥_𝑛−𝑦_{(𝑖−1)𝑛}‖+‖𝑦_{(𝑖−1)𝑛}−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖,(4.36) it implies by (4.15) and (4.35) that lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖=0,(4.37) for 𝑖=1,2,3,…,𝑘.
(ii) From part (i), for 𝑖=1, we havelim_𝑛→∞‖𝑇^𝑛₁𝑥_𝑛−𝑥_𝑛‖=0.(4.38) For 𝑖=2,3,4,…,𝑘, we get ‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖≤‖𝑇^𝑛_𝑖𝑥_𝑛−𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}‖+‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑥_𝑛‖≤𝐿‖𝑥_𝑛−𝑦_{(𝑖−1)𝑛}‖^𝛾𝑖+‖𝑇^𝑛_𝑖𝑦_{(𝑖−1)𝑛}−𝑥_𝑛‖.(4.39) By part (i) and (4.15), lim_𝑛→∞‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖=0,for𝑖=1,2,…,𝑘.(4.40) For 1≤𝑖≤𝑘, we obtain ‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1‖+‖𝑇_𝑖^𝑛+1𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛‖+‖𝑇_𝑖^𝑛+1𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1‖+𝐿‖𝑥_𝑛+1−𝑥_𝑛‖^𝛾𝑖+𝐿‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖^𝛾𝑖.(4.41) From (4.19) and (4.40), we then have lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0,for𝑖=1,2,…,𝑘.(4.42)

Theorem 4.3. Under the hypotheses of Lemma 4.2, assume that 𝑇^𝑚_𝑗 is semicompact for some positive integers 𝑚 and 1≤𝑗≤𝑘. Then {𝑥_𝑛} converges strongly to a common fixed point of the family {𝑇_𝑖∶𝑖=1,2,…,𝑘}.

Proof. Suppose that 𝑇^𝑚_𝑗 is semicompact for some positive integers 𝑚≥1 and 1≤𝑗≤𝑘. We have ‖𝑇^𝑚_𝑗𝑥_𝑛−𝑥_𝑛‖≤‖𝑇^𝑚_𝑗𝑥_𝑛−𝑇_𝑗^𝑚−1𝑥_𝑛‖+‖𝑇_𝑗^𝑚−1𝑥_𝑛−𝑇_𝑗^𝑚−2𝑥_𝑛‖+⋯+‖𝑇²_𝑗𝑥_𝑛−𝑇_𝑗𝑥_𝑛‖+‖𝑇_𝑗𝑥_𝑛−𝑥_𝑛‖≤(𝑚−1)𝐿‖𝑇_𝑗𝑥_𝑛−𝑥_𝑛‖^𝛾𝑗+‖𝑇_𝑗𝑥_𝑛−𝑥_𝑛‖.(4.43) Then, by Lemma 4.2(ii), we get ‖𝑇^𝑚_𝑗𝑥_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. Since {𝑥_𝑛} is bounded and 𝑇^𝑚_𝑗 is semicompact, there exists a subsequence {𝑥_𝑛𝑙} of {𝑥_𝑛} such that 𝑥_𝑛𝑙→𝑞∈𝐶 as 𝑙→∞.
By continuity of 𝑇_𝑖 and Lemma 4.2(ii), we obtain ‖𝑞−𝑇_𝑗𝑞‖=lim_𝑙→∞‖𝑥_𝑛𝑙−𝑇_𝑗𝑥_𝑛𝑙‖=0,∀𝑗=1,2,…,𝑘.(4.44) Therefore, 𝑞∈𝐹 and then Theorem 3.2 implies that {𝑥_𝑛} converges strongly to a common fixed point 𝑞 of the family {𝑇_𝑖∶𝑖=1,2,…,𝑘}.

We note that in practical Theorem 4.3 is very useful in the case that one of 𝑇_𝑖, 𝑖=1,2,3,…,𝑘, is semicompact.

Theorem 4.4. Let 𝐶 be a nonempty closed convex subset of an uniformly convex real Banach space 𝑋 satisfying the Opial property, and {𝑇_𝑖∶𝑖=1,2,…,𝑘} be a family of (𝐿−𝛾_𝑖) uniform Lipschitz and generalized asymtotically quasinonexpansive self-mappings of 𝐶, that is, ‖𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦‖≤𝐿‖𝑥−𝑦‖^𝛾𝑖 and ‖𝑇^𝑛_𝑖𝑥−𝑝_𝑖‖≤(1+𝑟_𝑖𝑛)‖𝑥−𝑝_𝑖‖+𝑠_𝑖𝑛, for all 𝑥,𝑦∈𝐶 and 𝑝_𝑖∈𝐹(𝑇_𝑖),𝑖=1,2,…,𝑘. Suppose that 𝐹=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝑥₁∈𝐶 and the iterative sequence {𝑥_𝑛} be defined by (1.2) with {𝛼_𝑖𝑛}^𝑛_𝑖=1⊂[𝑎,𝑏], where 0<𝑎<𝑏<1. Assume that ∑^∞_𝑛=1𝑟_𝑛<∞, where 𝑟_𝑛=max_{1≤𝑖≤𝑘}{𝑟_𝑖𝑛}. If 𝐼−𝑇_𝑖,𝑖=1,2,…,𝑘, is demiclosed at 0, then {𝑥_𝑛} converges weakly to a common fixed point of the family of mappings.

Proof. Let 𝑝∈𝐹. By Lemma 3.1(v), we get lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Then we follow the proof of Theorem 3.2 by Khan et al. [1] until we can conclude that {𝑥_𝑛} converges weakly to a common fixed point 𝑝∈𝐹.

Acknowledgments

This paper was supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Ph. D. Program Thai Doctoral degree from the Office of the Higher Education Commission, the Graduate School of Chiang Mai University, and the Thailand Research Fund.