Strong and Weak Convergence Theorems for an Infinite Family of Lipschitzian Pseudocontraction Mappings in Banach Spaces

Abstract

The purpose of this paper is to study the strong and weak convergence theorems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces.

1. Introduction and Preliminaries

Throughout this paper, we assume that 𝐸 is a real Banach space, 𝐸^∗ is the dual space of 𝐸, 𝐶 is a nonempty closed convex subset of 𝐸, ℛ⁺ is the set of nonnegative real numbers, and 𝐽∶𝐸→2^𝐸∗ is the normalized duality mapping defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖⋅‖𝑓‖,‖𝑥‖=‖𝑓‖,𝑥∈𝐸.(1.1) Let 𝑇∶𝐶→𝐶 be a mapping. We use 𝐹(𝑇) to denote the set of fixed points of 𝑇. We also use “→” to stand for strong convergence and “⇀” for weak convergence. For a given sequence {𝑥_𝑛}⊂𝐶, let 𝑊_𝜔(𝑥_𝑛) denote the weak 𝜔-limit set, that is, 𝑊_𝜔𝑥_𝑛=𝑧∈𝐶∶thereexistsasubsequence𝑥_𝑛𝑖⊂𝑥_𝑛suchthat𝑥_𝑛𝑖⇀𝑧.(1.2)

Definition 1.1. (1) A mapping 𝑇∶𝐶→𝐶 is said to be pseudocontraction [1], if for any 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖².(1.3)
It is well known that [1] the condition (1.3) is equivalent to the following: ‖𝑥−𝑦‖≤‖𝑥−𝑦+𝑠[(𝐼−𝑇𝑥)−(𝐼−𝑇𝑦)]‖,(1.4) for all 𝑠>0 and all 𝑥,𝑦∈𝐶.
(2) 𝑇∶𝐶→𝐶 is said to be strongly pseudocontractive, if there exists 𝑘∈(0,1) such that⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤𝑘‖𝑥−𝑦‖²,(1.5) for each 𝑥,𝑦∈𝐶 and for some 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦).
(3) 𝑇∶𝐶→𝐶 is said to be strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists 𝜆>0 such that⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖²−𝜆‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²,(1.6) for every 𝑥,𝑦∈𝐶 and for some 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦).
In this case, we say 𝑇 is a 𝜆-strictly pseudocontractive mapping.
(4) 𝑇∶𝐶→𝐶 is said to be 𝐿-Lipschitzian, if there exists 𝐿>0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.7)

Remark 1.2. It is easy to see that if 𝑇∶𝐶→𝐶 is a 𝜆-strictly pseudocontractive mapping, then it is a (1+𝜆)/𝜆-Lipschitzian mapping.
In fact, it follows from (1.6) that for any 𝑥,𝑦∈𝐶,𝜆‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²≤≤⟨‖((𝐼𝐼−−𝑇𝑇))𝑥𝑥−−((𝐼𝐼−−𝑇𝑇))𝑦𝑦,‖𝑗‖(𝑥𝑥−−𝑦𝑦)‖⟩.(1.8) Simplifying it, we have ‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖≤1𝜆‖𝑥−𝑦‖,(1.9) that is, ‖𝑇𝑥−𝑇𝑦‖≤1+𝜆𝜆‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.10)

Lemma 1.3 (see [2, Theorem 13.1] or [3]). Let 𝐸 be a real Banach space, 𝐶 be a nonempty closed convex subset of 𝐸, and 𝑇∶𝐶→𝐶 be a continuous strongly pseudocontractive mapping. Then 𝑇 has a unique fixed point in 𝐶.

Remark 1.4. Let 𝐸 be a real Banach space, 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝐶 be a Lipschitzian pseudocontraction mapping. For every given 𝑢∈𝐶 and 𝑠∈(0,1), define a mapping 𝑈_𝑠∶𝐶→𝐶 by 𝑈_𝑠𝑥=𝑠𝑢+(1−𝑠)𝑇𝑥,𝑥∈𝐶.(1.11) It is easy to see that 𝑈_𝑠 is a continuous strongly pseudocontraction mapping. By using Lemma 1.3, there exists a unique fixed point 𝑥_𝑠∈𝐶 of 𝑈_𝑠 such that 𝑥_𝑠=𝑠𝑢+(1−𝑠)𝑇𝑥_𝑠.(1.12)
The concept of pseudocontractive mappings is closely related to accretive operators. It is known that 𝑇 is pseudocontractive if and only if 𝐼−𝑇 is accretive, where 𝐼 is the identity mapping. The importance of accretive mappings is from their connection with theory of solutions for nonlinear evolution equations in Banach spaces. Many kinds of equations, for example, Heat, wave, or Schrödinger equations can be modeled in terms of an initial value problem:𝑑𝑢𝑑𝑡=𝑇𝑢−𝑢,𝑢(0)=𝑢⁰,(1.13) where 𝑇 is a pseudocontractive mapping in an appropriate Banach space.
In order to approximate a fixed point of Lipschitzian pseudocontractive mapping, in 1974, Ishikawa introduced a new iteration (it is called Ishikawa iteration). Since then, a question of whether or not the Ishikawa iteration can be replaced by the simpler Mann iteration has remained open. Recently Chidume and Mutangadura [4] solved this problem by constructing an example of a Lipschitzian pseudocontractive mapping with a unique fixed point for which every Mann-type iteration fails to converge.
Inspired by the implicit iteration introduced by Xu and Ori [5] for a finite family of nonexpansive mappings in a Hilbert space, Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9] proposed and studied convergence theorems for an implicit iteration process for a finite or infinite family of continuous pseudocontractive mappings.
The purpose of this paper is to study the strong and weak convergence problems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces. The results presented in this paper extend and improve some recent results of Xu and Ori [5], Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9].
For this purpose, we first recall some concepts and conclusions.
A Banach space 𝐸 is said to be uniformly convex, if for each 𝜀>0, there exists a 𝛿>0 such that for any 𝑥,𝑦∈𝐸 with ‖𝑥‖, ‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜀, ‖𝑥+𝑦‖≤2(1−𝛿) holds. The modulus of convexity of 𝐸 is defined by𝛿_𝐸(𝜀)=inf1−‖‖‖𝑥+𝑦2‖‖‖∶‖𝑥‖,‖𝑦‖≤1,‖𝑥−𝑦‖≥𝜀,∀𝜀∈[0,2].(1.14)
Concerning the modulus of convexity of 𝐸, Goebel and Kirk [10] proved the following result.

Lemma 1.5 (see [10, Lemma 10.1]). Let 𝐸 be a uniformly convex Banach space with a modulus of convexity 𝛿_𝐸. Then 𝛿_𝐸∶[0,2]→[0,1] is continuous, increasing, 𝛿_𝐸(0)=0, 𝛿_𝐸(𝑡)>0 for 𝑡∈(0,2] and ‖𝑐𝑢+(1−𝑐)𝑣‖≤1−2min{𝑐,1−𝑐}𝛿_𝐸(‖𝑢−𝑣‖),(1.15) for all 𝑐∈[0,1], and 𝑢,𝑣∈𝐸 with ‖𝑢‖,‖𝑣‖≤1.
A Banach space 𝐸 is said to satisfy the Opial condition, if for any sequence {𝑥_𝑛}⊂𝐸 with 𝑥_𝑛⇀𝑥, then the following inequality holds: limsup_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<limsup_𝑛→∞‖‖𝑥_𝑛−𝑦‖‖,(1.16) for any 𝑦∈𝐸 with 𝑦≠𝑥.

Lemma 1.6 (Zhou [8]). Let 𝐸 be a real reflexive Banach space with Opial condition. Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping. Then 𝐼−𝑇 is demiclosed at zero, that is, for any sequence {𝑥_𝑛}⊂𝐸, if 𝑥_𝑛⇀𝑦 and ‖(𝐼−𝑇)𝑥_𝑛‖→0, then (𝐼−𝑇)𝑦=0.

Lemma 1.7 (Chang [11]). Let 𝐽∶𝐸→2^𝐸∗ be the normalized duality mapping, then for any 𝑥,𝑦∈𝐸, ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,∀𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).(1.17)

Definition 1.8 (see [12]). Let {𝑇_𝑛}∶𝐶→𝐸 be a family of mappings with ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. We say {𝑇_𝑛} satisfies the AKTT-condition, if for each bounded subset 𝐵 of 𝐶 the following holds: ^∞_𝑛=1sup_𝑧∈𝐵‖‖𝑇_𝑛+1𝑧−𝑇_𝑛𝑧‖‖<∞.(1.18)

Lemma 1.9 (see [12]). Suppose that the family of mappings {𝑇_𝑛}∶𝐶→𝐶 satisfies the AKTT-condition. Then for each 𝑦∈𝐶, {𝑇_𝑛𝑦} converges strongly to a point in 𝐶. Moreover, let 𝑇∶𝐶→𝐶 be the mapping defined by 𝑇𝑦=lim_𝑛→∞𝑇_𝑛𝑦,∀𝑦∈𝐶.(1.19) Then, for each bounded subset 𝐵⊂𝐶, lim_𝑛→∞sup_𝑧∈𝐵‖𝑇𝑧−𝑇_𝑛𝑧‖=0.

2. Main Results

Theorem 2.1. Let 𝐸 be a uniformly convex Banach space with a modulus of convexity 𝛿_𝐸, and 𝐶 be a nonempty closed convex subset of 𝐸. Let {𝑇_𝑛}∶𝐶→𝐶 be a family of 𝐿_𝑛-Lipschitzian and pseudocontractive mappings with 𝐿∶=sup_𝑛≥1𝐿_𝑛<∞ and ℱ∶=⋂_𝑛≥1𝐹(𝑇_𝑛)≠∅. Let {𝑥_𝑛} be the sequence defined by 𝑥𝑥₁∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑛≥1,(2.1) where {𝛼_𝑛} is a sequence in [0,1]. If the following conditions are satisfied: (i)limsup_𝑛→∞𝛼_𝑛<1; (ii)there exists a compact subset 𝐾⊂𝐸 such that ⋃^∞𝑛=1𝑇_𝑛(𝐶)⊂𝐾; (iii){𝑇_𝑛} satisfies the AKTT-condition, and 𝐹(𝑇)⊂ℱ, where 𝑇∶𝐶→𝐶 is the mapping defined by (1.19). Then 𝑥_𝑛 converges strongly to some point 𝑝∈ℱ

Proof. First, we note that, by Remark 1.4, the method is well defined. So, we can divide the proof in three steps. (I) For each 𝑝∈ℱ the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. In fact, since {𝑇_𝑛} is pseudocontractive, for each 𝑝∈ℱ, we have ‖‖𝑥_𝑛−𝑝‖‖²==𝛼_𝑛𝑥_𝑛−𝑝,𝑗𝑥_𝑛−𝑝≤𝛼_𝑛⟨𝑥_𝑛−1−𝑝,𝑗𝑥_𝑛−𝑝⟩+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑝,𝑗𝑥_𝑛−𝑝‖‖𝑥_𝑛−1−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²,∀𝑛≥1.(2.2) Simplifying, we have that ‖‖𝑥_𝑛−𝑝‖‖≤‖‖𝑥_𝑛−1−𝑝‖‖,∀𝑛≥1.(2.3) Consequently, the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, and so the sequence {𝑥_𝑛} is bounded. (II) Now, we prove that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0. In fact, by virtue of (2.1) and (1.4), we have ‖‖𝑥_𝑛−𝑝‖‖≤‖‖‖𝑥_𝑛−𝑝+1−𝛼^𝑛2𝛼_𝑛𝑥_𝑛=‖‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖‖−𝑝+1−𝛼^𝑛2𝑥^𝑛−1=‖‖‖𝛼_𝑛−𝑇_𝑛𝑥_𝑛‖‖‖𝑥_𝑛−1+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑝+1−𝛼^𝑛2𝑥^𝑛−1=‖‖‖𝑥_𝑛−1−𝑇_𝑛𝑥_𝑛‖‖‖+𝑥_𝑛=‖‖𝑥2−𝑝_𝑛−1‖‖‖−𝑝‖‖⋅‖‖‖𝑥^𝑛−1−𝑝2‖‖𝑥_𝑛−1−𝑝‖‖+𝑥^𝑛−𝑝2‖‖𝑥_𝑛−1−𝑝‖‖‖‖‖.(2.4) Letting 𝑢=(𝑥_𝑛−1−𝑝)/‖𝑥_𝑛−1−𝑝‖ and 𝑣=(𝑥_𝑛−𝑝)/‖𝑥_𝑛−1−𝑝‖, from (2.3), we know that ‖𝑢‖=1, ‖𝑣‖≤1. It follows from (2.4) and Lemma 1.5 that ‖‖𝑥_𝑛−𝑝‖‖≤‖‖𝑥_𝑛−1−𝑝‖‖1−𝛿_𝐸‖‖𝑥^𝑛−1−𝑥_𝑛‖‖‖‖𝑥_𝑛−1−𝑝‖‖.(2.5) Simplifying, we have that ‖‖𝑥_𝑛−1−𝑝‖‖𝛿_𝐸‖‖𝑥^𝑛−1−𝑥_𝑛‖‖‖‖𝑥_𝑛−1−𝑝‖‖≤‖‖𝑥^𝑛−1−𝑝‖‖−‖‖𝑥_𝑛−𝑝‖‖.(2.6) This implies that ^∞_𝑛=1‖‖𝑥_𝑛−1−𝑝‖‖𝛿_𝐸‖‖𝑥^𝑛−1−𝑥_𝑛‖‖‖‖𝑥_𝑛−1−𝑝‖‖≤‖‖𝑥⁰−𝑝‖‖.(2.7) Letting lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑟, if 𝑟=0, the conclusion of Theorem 2.1 is proved. If 𝑟>0, it follows from the property of modulus of convexity 𝛿_𝐸 that ‖𝑥_𝑛−1−𝑥_𝑛‖→0(𝑛→∞). Therefore, from (2.1) and the condition (i), we have that ‖‖𝑥_𝑛−1−𝑇_𝑛𝑥_𝑛‖‖=11−𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖⟶0(as𝑛⟶∞).(2.8) In view of (2.1) and (2.8), we have ‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖=𝛼_𝑛‖‖𝑥_𝑛−1−𝑇_𝑛𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(2.9)(III) Now, we prove that {𝑥_𝑛} converges strongly to some point in ℱ. In fact, it follows from (2.9) and condition (ii) that there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that ‖𝑥_𝑛𝑖−𝑇_𝑛𝑖𝑥_𝑛𝑖‖→0 (as 𝑛_𝑖→∞), 𝑇_𝑛𝑖𝑥_𝑛𝑖→𝑝 and 𝑥_𝑛𝑖→𝑝 (some point in 𝐶). Furthermore, by Lemma 1.9, we have 𝑇_𝑛𝑖𝑝→𝑇𝑝. consequently, we have ‖𝑝−𝑇𝑝‖≤≤‖‖‖‖𝑝𝑝−−𝑥𝑥_{𝑛𝑖𝑛𝑖}‖‖+‖‖𝑥_𝑛𝑖−𝑇_𝑛𝑖𝑝‖‖+‖‖𝑇_𝑛𝑖𝑝−𝑇𝑝‖‖≤(1+𝐿)‖‖‖‖𝑝+−‖‖𝑥𝑥_{𝑛𝑖𝑛𝑖}−𝑇_𝑛𝑖𝑥_𝑛𝑖‖‖+‖‖𝑇_𝑛𝑖𝑥_𝑛𝑖−𝑇_𝑛𝑖𝑝‖‖+‖‖𝑇_𝑛𝑖𝑝−𝑇𝑝‖‖‖‖+‖‖𝑥_𝑛𝑖−𝑇_𝑛𝑖𝑥_𝑛𝑖‖‖+‖‖𝑇_𝑛𝑖𝑝−𝑇𝑝‖‖⟶0.(2.10) This implies that 𝑝=𝑇𝑝, that is, 𝑝∈𝐹(𝑇)⊂ℱ. Since 𝑥_𝑛𝑖→𝑝 and the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, we have 𝑥_𝑛→𝑝.
This completes the proof of Theorem 2.1.

Theorem 2.2. Let 𝐸 be a uniformly convex Banach space satisfying the Opial condition. Let 𝐶 be a nonempty closed convex subset of 𝐸 and {𝑇_𝑛}∶𝐶→𝐶 be a family of 𝐿_𝑛-Lipschitzian pseudocontractive mappings with 𝐿∶=sup_𝑛≥1𝐿_𝑛<∞ and ℱ∶=⋂_𝑛≥1𝐹(𝑇_𝑛)≠∅. Let {𝑥_𝑛} be the sequence defined by (2.1) and {𝛼_𝑛} be a sequence in (0, 1). If the following conditions are satisfied: (i)limsup_𝑛→∞𝛼_𝑛<1, (ii)for any bounded subset 𝐵 of 𝐶lim_𝑛→∞sup_𝑧∈𝐵‖‖𝑇_𝑚𝑇_𝑛𝑧−𝑇_𝑛𝑧‖‖=0,foreach𝑚≥1.(2.11)Then the sequence {𝑥_𝑛} converges weakly to some point 𝑢∈ℱ.

Proof. By the same method as given in the proof of Theorem 2.1, we can prove that the sequence {𝑥_𝑛} is bounded and lim_𝑛→∞‖‖𝑥_𝑛l−im𝑝_𝑛→∞‖‖exists,foreach𝑝∈ℱ;‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖=0.(2.12)
Now, we prove thatlim_𝑛→∞‖‖𝑇_𝑚𝑥_𝑛−𝑥_𝑛‖‖=0,foreach𝑚≥1.(2.13)
Indeed, for each 𝑚≥1, we have‖‖𝑇_𝑚𝑥_𝑛−𝑥_𝑛‖‖≤≤(‖‖1𝑇+_𝑚𝑥𝐿_𝑛)−‖‖𝑇𝑇_𝑚𝑛𝑇_𝑛𝑥_𝑛‖‖+‖‖𝑇_𝑚𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖‖≤(1+𝐿)‖‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖‖+‖‖𝑇_𝑚𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖𝑥_𝑛−𝑥_𝑛‖‖+sup_{𝑧∈{𝑥𝑛}}‖‖𝑇_𝑚𝑇_𝑛𝑧−𝑇_𝑛𝑧‖‖.(2.14) By (2.12) and condition (ii), we have lim_𝑛→∞‖‖𝑇_𝑚𝑥_𝑛−𝑥_𝑛‖‖=0,foreach𝑚≥1.(2.15) The conclusion of (2.13) is proved.
Finally, we prove that {𝑥_𝑛} converges weakly to some point 𝑢∈ℱ.
In fact, since 𝐸 is uniformly convex, and so it is reflexive. Again since {𝑥_𝑛}⊂𝐶 is bounded, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑢. Hence from (2.13), for any 𝑚>1, we have‖‖𝑇_𝑚𝑥_𝑛𝑖−𝑥_𝑛𝑖‖‖⟶0as𝑛_𝑖⟶∞.(2.16) By virtue of Lemma 1.6, 𝑢∈𝐹(𝑇_𝑚), for all 𝑚≥1. This implies that 𝑢∈ℱ∶=_𝑛≥𝐹𝑇_𝑛∩𝑊_𝜔𝑥_𝑛.(2.17)
Next, we prove that 𝑊_𝜔(𝑥_𝑛) is a singleton. Let us suppose, to the contrary, that if there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑞∈𝑊_𝜔(𝑥_𝑛) and 𝑞≠𝑢. By the same method as given above we can also prove that 𝑞∈ℱ∶=⋂_𝑛≥1𝐹(𝑇_𝑛)∩𝑊_𝜔(𝑥_𝑛). Taking 𝑝=𝑢 and 𝑝=𝑞 in (2.12). We know that the following limitslim_𝑛→∞‖‖𝑥_𝑛−𝑢‖‖,lim_𝑛→∞‖‖𝑥_𝑛−𝑞‖‖(2.18) exist. Since 𝐸 satisfies the Opial condition, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑢‖‖==lliimm_𝑛→∞sup_{𝑛𝑖→∞}‖‖𝑥_𝑛𝑖−𝑢‖‖<limsup_{𝑛𝑖→∞}‖‖𝑥_𝑛𝑖−𝑞‖‖<limsu‖‖p_{𝑛𝑗→∞}𝑥_𝑛−𝑞‖‖=limsup_{𝑛𝑗→∞}‖‖𝑥_𝑛𝑗−𝑞‖‖‖‖𝑥_𝑛𝑗−𝑢‖‖=lim_𝑛→∞‖‖𝑥_𝑛−𝑢‖‖.(2.19) This is a contradiction, which shows that 𝑞=𝑢. Hence, 𝑊_𝜔𝑥_𝑛={𝑢}⊂ℱ∶=_𝑛≥1𝐹𝑇_𝑛.(2.20) This implies that 𝑥_𝑛⇀𝑢.
This completes the proof of Theorem 2.2.

In the next lemma, we propose a sequence of mappings that satisfy condition (iii) in Theorem 2.1. Moreover, we apply this lemma to obtain a corollary of our main Theorem 2.1.

Let 𝐸 be a Banach space and 𝐶 be a nonempty closed convex subset of 𝐸. From Definition 1.1(3), we know that if 𝑇∶𝐶→𝐶 is a 𝜆-strictly pseudocontractive mapping, then it is a ((1+𝜆)/𝜆)-Lipschitzian pseudocontractive mapping.

On the other hand, by the same proof as given in [12] we can prove the following result.

Lemma 2.3 (see [12] or [9]). Let 𝐸 be a smooth Banach space, 𝐶 be a closed convex subset of 𝐸. Let {𝑆_𝑛}∶𝐶→𝐶 be a family of 𝜆_𝑛-strictly pseudocontractive mappings with ℱ∶=⋂^∞𝑛=1𝐹(𝑆_𝑛)≠∅ and 𝜆∶=inf_𝑛≥1𝜆_𝑛>0. For each 𝑛≥1 define a mapping 𝑇_𝑛∶𝐶→𝐶 by: 𝑇_𝑛𝑥=^𝑛_𝑘=1𝛽^𝑘𝑛𝑆_𝑘𝑥,𝑥∈𝐶,𝑛≥1,(2.21) where {𝛽^𝑘𝑛} is sequence of nonnegative real numbers satisfying the following conditions: (i)∑^𝑛𝑘=1𝛽^𝑘𝑛=1, for all 𝑛≥1; (ii)𝛽^𝑘∶=lim_𝑛→∞𝛽^𝑘𝑛>0, for all 𝑘≥1; (iii)∑^∞𝑛=1∑^𝑛𝑘=1|𝛽^𝑘𝑛+1−𝛽^𝑘𝑛|<∞. Then, (1)each 𝑇_𝑛, 𝑛≥1 is a 𝜆-strictly pseudocontractive mapping; (2){𝑇_𝑛} satisfies the AKTT-condition; (3)if 𝑇∶𝐶→𝐶 is the mapping defined by 𝑇𝑥=^∞_𝑘=1𝛽^𝑘𝑆_𝑘𝑥,𝑥∈𝐶.(2.22)Then 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 and 𝐹(𝑇)=⋂^∞𝑘=1𝐹(𝑇_𝑛)=ℱ∶=⋂^∞𝑛=1𝐹(𝑆_𝑛).

The following result can be obtained from Theorem 2.1 and Lemma 2.3 immediately.

Theorem 2.4. Let 𝐸 be a uniformly convex Banach space, 𝐶 be a nonempty closed convex subset of 𝐸. Let {𝑆_𝑛}∶𝐶→𝐶 be a family of 𝜆_𝑛-strictly pseudocontractive mappings with ℱ∶=⋂^∞𝑛=1𝐹(𝑆_𝑛)≠∅ and 𝜆∶=inf_𝑛≥1𝜆_𝑛>0. For each 𝑛≥1 define a mapping 𝑇_𝑛∶𝐶→𝐶 by 𝑇_𝑛𝑥=^𝑛_𝑘=1𝛽^𝑘𝑛𝑆_𝑘𝑥,𝑥∈𝐶,𝑛≥1,(2.23) where {𝛽^𝑘𝑛} is a sequence of nonnegative real numbers satisfying the following conditions: (i)∑^𝑛𝑘=1𝛽^𝑘𝑛=1, for all 𝑛≥1; (ii)𝛽^𝑘∶=lim_𝑛→∞𝛽^𝑘𝑛>0, for all 𝑘≥1; (iii)∑^∞𝑛=1∑^𝑛𝑘=1|𝛽^𝑘𝑛+1−𝛽^𝑘𝑛|<∞.
Let {𝑥_𝑛} be the sequence defined by 𝑥𝑥₁∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑛≥1,(2.24) where {𝛼_𝑛} is a sequence in [0,1]. If the following conditions are satisfied: (i)limsup_𝑛→∞𝛼_𝑛<1; (ii)there exists a compact subset 𝐾⊂𝐸 such that ⋃^∞𝑛=1𝑆_𝑛(𝐶)⊂𝐾. Then, {𝑥_𝑛} converges strongly to some point 𝑝∈ℱ.

Proof. Since {𝑆_𝑛}∶𝐶→𝐶 is a family of 𝜆_𝑛-strictly pseudocontractive mappings with 𝜆∶=inf_𝑛≥1𝜆_𝑛>0. Therefore, {𝑆_𝑛} is a family of 𝜆-strictly pseudocontractive mappings. By Remark 1.2, {𝑆_𝑛} is a family of (1+𝜆)/𝜆-Lipschitzian and strictly pseudocontractive mappings. Hence, by Lemma 2.3, {𝑇_𝑛} defined by (2.21) is a family of (1+𝜆)/𝜆-Lipschitzian, strictly pseudocontractive mappings with ⋂^∞𝑛=1𝐹(𝑇_𝑛)=ℱ∶=⋂^∞𝑛=1𝐹(𝑆_𝑛)≠∅ and it has also the following properties: (1){𝑇_𝑛} satisfies the AKTT-condition; (2)if 𝑇∶𝐶→𝐶 is the mapping defined by (2.22), then 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥,𝑥∈𝐶 and 𝐹(𝑇)=ℱ∶=⋂^∞𝑘=1𝐹(𝑆_𝑛)=⋂^∞𝑛=1𝐹(𝑇_𝑛). Hence, by Definition 1.1, {𝑇_𝑛} is also a family of (1+𝜆)/𝜆-Lipschitzian and pseudocontractive mappings having the properties (1) and (2) and ℱ∶=⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Therefore, {𝑇_𝑛} satisfies all the conditions in Theorem 2.1. By Theorem 2.1, the sequence {𝑥_𝑛} converges strongly to some point 𝑝∈ℱ∶⋂^∞𝑘=1𝐹(𝑆_𝑛)=⋂^∞𝑛=1𝐹(𝑇_𝑛).
This completes the proof of Theorem 2.4.

Acknowledgment

This paper was supported by the Natural Science Foundation of Yibin University (no. 2009Z01).