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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 409898, 10 pages
http://dx.doi.org/10.1155/2011/409898
Review Article

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

1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
2Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 16 December 2010; Accepted 9 February 2011

Academic Editor: Giuseppe Marino

Copyright © 2011 Shih-sen Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, + is the set of nonnegative real numbers, and 𝐽𝐸2𝐸 is the normalized duality mapping defined by 𝐽(𝑥)=𝑓𝐸𝑥,𝑓=𝑥𝑓,𝑥=𝑓,𝑥𝐸.(1.1) Let 𝑇𝐶𝐶 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 {𝑥𝑛}𝐶, let 𝑊𝜔(𝑥𝑛) denote the weak 𝜔-limit set, that is, 𝑊𝜔𝑥𝑛=𝑥𝑧𝐶thereexistsasubsequence𝑛𝑖𝑥𝑛suchthat𝑥𝑛𝑖.𝑧(1.2)

Definition 1.1. (1) A mapping 𝑇𝐶𝐶 is said to be pseudocontraction [1], if for any 𝑥,𝑦𝐶, there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑥𝑦2.(1.3)
It is well known that [1] the condition (1.3) is equivalent to the following: []𝑥𝑦𝑥𝑦+𝑠(𝐼𝑇𝑥)(𝐼𝑇𝑦),(1.4) for all 𝑠>0 and all 𝑥,𝑦𝐶.
(2) 𝑇𝐶𝐶 is said to be strongly pseudocontractive, if there exists 𝑘(0,1) such that𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑘𝑥𝑦2,(1.5) for each 𝑥,𝑦𝐶 and for some 𝑗(𝑥𝑦)𝐽(𝑥𝑦).
(3) 𝑇𝐶𝐶 is said to be strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists 𝜆>0 such that𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑥𝑦2𝜆(𝐼𝑇)𝑥(𝐼𝑇)𝑦2,(1.6) for every 𝑥,𝑦𝐶 and for some 𝑗(𝑥𝑦)𝐽(𝑥𝑦).
In this case, we say 𝑇 is a 𝜆-strictly pseudocontractive mapping.
(4) 𝑇𝐶𝐶 is said to be 𝐿-Lipschitzian, if there exists 𝐿>0 such that 𝑇𝑥𝑇𝑦𝐿𝑥𝑦,𝑥,𝑦𝐶.(1.7)

Remark 1.2. It is easy to see that if 𝑇𝐶𝐶 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,𝑦𝐶,𝜆(𝐼𝑇)𝑥(𝐼𝑇)𝑦2(𝐼𝑇)𝑥(𝐼𝑇)𝑦,𝑗(𝑥𝑦)(𝐼𝑇)𝑥(𝐼𝑇)𝑦𝑥𝑦.(1.8) Simplifying it, we have 1(𝐼𝑇)𝑥(𝐼𝑇)𝑦𝜆𝑥𝑦,(1.9) that is, 𝑇𝑥𝑇𝑦1+𝜆𝜆𝑥𝑦,𝑥,𝑦𝐶.(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 𝑇𝐶𝐶 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 𝑇𝐶𝐶 be a Lipschitzian pseudocontraction mapping. For every given 𝑢𝐶 and 𝑠(0,1), define a mapping 𝑈𝑠𝐶𝐶 by 𝑈𝑠𝑥=𝑠𝑢+(1𝑠)𝑇𝑥,𝑥𝐶.(1.11) It is easy to see that 𝑈𝑠 is a continuous strongly pseudocontraction mapping. By using Lemma 1.3, there exists a unique fixed point 𝑥𝑠𝐶 of 𝑈𝑠 such that 𝑥𝑠=𝑠𝑢+(1𝑠)𝑇𝑥𝑠.(1.12)
The concept of pseudocontractive mappings is closely related to accretive operators. It is known that 𝑇 is pseudocontractive if and only if 𝐼𝑇 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=𝑇𝑢𝑢,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,𝑦𝐸 with 𝑥, 𝑦1 and 𝑥𝑦𝜀, 𝑥+𝑦2(1𝛿) holds. The modulus of convexity of 𝐸 is defined by𝛿𝐸(𝜀)=inf1𝑥+𝑦2[].𝑥,𝑦1,𝑥𝑦𝜀,𝜀0,2(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 𝛿𝐸. Then 𝛿𝐸[0,2][0,1] is continuous, increasing, 𝛿𝐸(0)=0, 𝛿𝐸(𝑡)>0 for 𝑡(0,2] and 𝑐𝑢+(1𝑐)𝑣12min{𝑐,1𝑐}𝛿𝐸(𝑢𝑣),(1.15) for all 𝑐[0,1], and 𝑢,𝑣𝐸 with 𝑢,𝑣1.
A Banach space 𝐸 is said to satisfy the Opial condition, if for any sequence {𝑥𝑛}𝐸 with 𝑥𝑛𝑥, then the following inequality holds: limsup𝑛𝑥𝑛𝑥<limsup𝑛𝑥𝑛,𝑦(1.16) for any 𝑦𝐸 with 𝑦𝑥.

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 𝑇𝐶𝐶 be a continuous pseudocontractive mapping. Then 𝐼𝑇 is demiclosed at zero, that is, for any sequence {𝑥𝑛}𝐸, if 𝑥𝑛𝑦 and (𝐼𝑇)𝑥𝑛0, then (𝐼𝑇)𝑦=0.

Lemma 1.7 (Chang [11]). Let 𝐽𝐸2𝐸 be the normalized duality mapping, then for any 𝑥,𝑦𝐸, 𝑥+𝑦2𝑥2+2𝑦,𝑗(𝑥+𝑦),𝑗(𝑥+𝑦)𝐽(𝑥+𝑦).(1.17)

Definition 1.8 (see [12]). Let {𝑇𝑛}𝐶𝐸 be a family of mappings with 𝑛=1𝐹(𝑇𝑛). We say {𝑇𝑛} satisfies the AKTT-condition, if for each bounded subset 𝐵 of 𝐶 the following holds: 𝑛=1sup𝑧𝐵𝑇𝑛+1𝑧𝑇𝑛𝑧<.(1.18)

Lemma 1.9 (see [12]). Suppose that the family of mappings {𝑇𝑛}𝐶𝐶 satisfies the AKTT-condition. Then for each 𝑦𝐶, {𝑇𝑛𝑦} converges strongly to a point in 𝐶. Moreover, let 𝑇𝐶𝐶 be the mapping defined by 𝑇𝑦=lim𝑛𝑇𝑛𝑦,𝑦𝐶.(1.19) Then, for each bounded subset 𝐵𝐶, lim𝑛sup𝑧𝐵𝑇𝑧𝑇𝑛𝑧=0.

2. Main Results

Theorem 2.1. Let 𝐸 be a uniformly convex Banach space with a modulus of convexity 𝛿𝐸, and 𝐶 be a nonempty closed convex subset of 𝐸. Let {𝑇𝑛}𝐶𝐶 be a family of 𝐿𝑛-Lipschitzian and pseudocontractive mappings with 𝐿=sup𝑛1𝐿𝑛< and =𝑛1𝐹(𝑇𝑛). Let {𝑥𝑛} be the sequence defined by 𝑥1𝑥𝐶,𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛,𝑛1,(2.1) where {𝛼𝑛} is a sequence in [0,1]. If the following conditions are satisfied: (i)limsup𝑛𝛼𝑛<1; (ii)there exists a compact subset 𝐾𝐸 such that 𝑛=1𝑇𝑛(𝐶)𝐾; (iii){𝑇𝑛} satisfies the AKTT-condition, and 𝐹(𝑇), where 𝑇𝐶𝐶 is the mapping defined by (1.19). Then 𝑥𝑛 converges strongly to some point 𝑝

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 the limit lim𝑛𝑥𝑛𝑝 exists. In fact, since {𝑇𝑛} is pseudocontractive, for each 𝑝, we have 𝑥𝑛𝑝2=𝑥𝑛𝑥𝑝,𝑗𝑛𝑝=𝛼𝑛𝑥𝑛1𝑥𝑝,𝑗𝑛𝑝+1𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑝,𝑗𝑛𝑝𝛼𝑛𝑥𝑛1𝑥𝑝𝑛+𝑝1𝛼𝑛𝑥𝑛𝑝2,𝑛1.(2.2) Simplifying, we have that 𝑥𝑛𝑥𝑝𝑛1𝑝,𝑛1.(2.3) Consequently, the limit lim𝑛𝑥𝑛𝑝 exists, and so the sequence {𝑥𝑛} is bounded. (II) Now, we prove that lim𝑛𝑥𝑛𝑇𝑛𝑥𝑛=0. In fact, by virtue of (2.1) and (1.4), we have 𝑥𝑛𝑥𝑝𝑛𝑝+1𝛼𝑛2𝛼𝑛𝑥𝑛𝑇𝑛𝑥𝑛=𝑥𝑛𝑝+1𝛼𝑛2𝑥𝑛1𝑇𝑛𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛𝑝+1𝛼𝑛2𝑥𝑛1𝑇𝑛𝑥𝑛=𝑥𝑛1+𝑥𝑛2=𝑥𝑝𝑛1𝑥𝑝𝑛1𝑝2𝑥𝑛1+𝑥𝑝𝑛𝑝2𝑥𝑛1.𝑝(2.4) Letting 𝑢=(𝑥𝑛1𝑝)/𝑥𝑛1𝑝 and 𝑣=(𝑥𝑛𝑝)/𝑥𝑛1𝑝, from (2.3), we know that 𝑢=1, 𝑣1. It follows from (2.4) and Lemma 1.5 that 𝑥𝑛𝑥𝑝𝑛1𝑝1𝛿𝐸𝑥𝑛1𝑥𝑛𝑥𝑛1.𝑝(2.5) Simplifying, we have that 𝑥𝑛1𝛿𝑝𝐸𝑥𝑛1𝑥𝑛𝑥𝑛1𝑥𝑝𝑛1𝑥𝑝𝑛.𝑝(2.6) This implies that 𝑛=1𝑥𝑛1𝛿𝑝𝐸𝑥𝑛1𝑥𝑛𝑥𝑛1𝑥𝑝0.𝑝(2.7) Letting lim𝑛𝑥𝑛𝑝=𝑟, if 𝑟=0, the conclusion of Theorem 2.1 is proved. If 𝑟>0, it follows from the property of modulus of convexity 𝛿𝐸 that 𝑥𝑛1𝑥𝑛0(𝑛). Therefore, from (2.1) and the condition (i), we have that 𝑥𝑛1𝑇𝑛𝑥𝑛=11𝛼𝑛𝑥𝑛𝑥𝑛10(as𝑛).(2.8) In view of (2.1) and (2.8), we have 𝑥𝑛𝑇𝑛𝑥𝑛=𝛼𝑛𝑥𝑛1𝑇𝑛𝑥𝑛0(as𝑛).(2.9)(III) Now, we prove that {𝑥𝑛} converges strongly to some point in . In fact, it follows from (2.9) and condition (ii) that there exists a subsequence {𝑥𝑛𝑖}{𝑥𝑛} such that 𝑥𝑛𝑖𝑇𝑛𝑖𝑥𝑛𝑖0 (as 𝑛𝑖), 𝑇𝑛𝑖𝑥𝑛𝑖𝑝 and 𝑥𝑛𝑖𝑝 (some point in 𝐶). Furthermore, by Lemma 1.9, we have 𝑇𝑛𝑖𝑝𝑇𝑝. consequently, we have 𝑝𝑇𝑝𝑝𝑥𝑛𝑖+𝑥𝑛𝑖𝑇𝑛𝑖𝑝+𝑇𝑛𝑖𝑝𝑇𝑝𝑝𝑥𝑛𝑖+𝑥𝑛𝑖𝑇𝑛𝑖𝑥𝑛𝑖+𝑇𝑛𝑖𝑥𝑛𝑖𝑇𝑛𝑖𝑝+𝑇𝑛𝑖𝑝𝑇𝑝(1+𝐿)𝑝𝑥𝑛𝑖+𝑥𝑛𝑖𝑇𝑛𝑖𝑥𝑛𝑖+𝑇𝑛𝑖𝑝𝑇𝑝0.(2.10) This implies that 𝑝=𝑇𝑝, that is, 𝑝𝐹(𝑇). Since 𝑥𝑛𝑖𝑝 and the limit lim𝑛𝑥𝑛𝑝 exists, we have 𝑥𝑛𝑝.
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 {𝑇𝑛}𝐶𝐶 be a family of 𝐿𝑛-Lipschitzian pseudocontractive mappings with 𝐿=sup𝑛1𝐿𝑛< and =𝑛1𝐹(𝑇𝑛). Let {𝑥𝑛} be the sequence defined by (2.1) and {𝛼𝑛} be a sequence in (0, 1). If the following conditions are satisfied: (i)limsup𝑛𝛼𝑛<1, (ii)for any bounded subset 𝐵 of 𝐶lim𝑛sup𝑧𝐵𝑇𝑚𝑇𝑛𝑧𝑇𝑛𝑧=0,foreach𝑚1.(2.11)Then the sequence {𝑥𝑛} converges weakly to some point 𝑢.

Proof. By the same method as given in the proof of Theorem 2.1, we can prove that the sequence {𝑥𝑛} is bounded and lim𝑛𝑥𝑛𝑝exists,foreach𝑝;lim𝑛𝑥𝑛𝑇𝑛𝑥𝑛=0.(2.12)
Now, we prove thatlim𝑛𝑇𝑚𝑥𝑛𝑥𝑛=0,foreach𝑚1.(2.13)
Indeed, for each 𝑚1, we have𝑇𝑚𝑥𝑛𝑥𝑛𝑇𝑚𝑥𝑛𝑇𝑚𝑇𝑛𝑥𝑛+𝑇𝑚𝑇𝑛𝑥𝑛𝑇𝑛𝑥𝑛+𝑇𝑛𝑥𝑛𝑥𝑛𝑇(1+𝐿)𝑛𝑥𝑛𝑥𝑛+𝑇𝑚𝑇𝑛𝑥𝑛𝑇𝑛𝑥𝑛𝑇(1+𝐿)𝑛𝑥𝑛𝑥𝑛+sup𝑧{𝑥𝑛}𝑇𝑚𝑇𝑛𝑧𝑇𝑛𝑧.(2.14) By (2.12) and condition (ii), we have lim𝑛𝑇𝑚𝑥𝑛𝑥𝑛=0,foreach𝑚1.(2.15) The conclusion of (2.13) is proved.
Finally, we prove that {𝑥𝑛} converges weakly to some point 𝑢.
In fact, since 𝐸 is uniformly convex, and so it is reflexive. Again since {𝑥𝑛}𝐶 is bounded, there exists a subsequence {𝑥𝑛𝑖}{𝑥𝑛} such that 𝑥𝑛𝑖𝑢. Hence from (2.13), for any 𝑚>1, we have𝑇𝑚𝑥𝑛𝑖𝑥𝑛𝑖0as𝑛𝑖.(2.16) By virtue of Lemma 1.6, 𝑢𝐹(𝑇𝑚), for all 𝑚1. This implies that 𝑢=𝑛𝐹𝑇𝑛𝑊𝜔𝑥𝑛.(2.17)
Next, we prove that 𝑊𝜔(𝑥𝑛) is a singleton. Let us suppose, to the contrary, that if there exists a subsequence {𝑥𝑛𝑗}{𝑥𝑛} such that 𝑥𝑛𝑗𝑞𝑊𝜔(𝑥𝑛) and 𝑞𝑢. By the same method as given above we can also prove that 𝑞=𝑛1𝐹(𝑇𝑛)𝑊𝜔(𝑥𝑛). Taking 𝑝=𝑢 and 𝑝=𝑞 in (2.12). We know that the following limitslim𝑛𝑥𝑛𝑢,lim𝑛𝑥𝑛𝑞(2.18) exist. Since 𝐸 satisfies the Opial condition, we have lim𝑛𝑥𝑛𝑢=limsup𝑛𝑖𝑥𝑛𝑖𝑢<limsup𝑛𝑖𝑥𝑛𝑖𝑞=lim𝑛𝑥𝑛𝑞=limsup𝑛𝑗𝑥𝑛𝑗𝑞<limsup𝑛𝑗𝑥𝑛𝑗𝑢=lim𝑛𝑥𝑛.𝑢(2.19) This is a contradiction, which shows that 𝑞=𝑢. Hence, 𝑊𝜔𝑥𝑛={𝑢}=𝑛1𝐹𝑇𝑛.(2.20) This implies that 𝑥𝑛𝑢.
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 𝑇𝐶𝐶 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 {𝑆𝑛}𝐶𝐶 be a family of 𝜆𝑛-strictly pseudocontractive mappings with =𝑛=1𝐹(𝑆𝑛) and 𝜆=inf𝑛1𝜆𝑛>0. For each 𝑛1 define a mapping 𝑇𝑛𝐶𝐶 by: 𝑇𝑛𝑥=𝑛𝑘=1𝛽𝑘𝑛𝑆𝑘𝑥,𝑥𝐶,𝑛1,(2.21) where {𝛽𝑘𝑛} is sequence of nonnegative real numbers satisfying the following conditions: (i)𝑛𝑘=1𝛽𝑘𝑛=1, for all 𝑛1; (ii)𝛽𝑘=lim𝑛𝛽𝑘𝑛>0, for all 𝑘1; (iii)𝑛=1𝑛𝑘=1|𝛽𝑘𝑛+1𝛽𝑘𝑛|<. Then, (1)each 𝑇𝑛, 𝑛1 is a 𝜆-strictly pseudocontractive mapping; (2){𝑇𝑛} satisfies the AKTT-condition; (3)if 𝑇𝐶𝐶 is the mapping defined by 𝑇𝑥=𝑘=1𝛽𝑘𝑆𝑘𝑥,𝑥𝐶.(2.22)Then 𝑇𝑥=lim𝑛𝑇𝑛𝑥 and 𝐹(𝑇)=𝑘=1𝐹(𝑇𝑛)==𝑛=1𝐹(𝑆𝑛).

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 {𝑆𝑛}𝐶𝐶 be a family of 𝜆𝑛-strictly pseudocontractive mappings with =𝑛=1𝐹(𝑆𝑛) and 𝜆=inf𝑛1𝜆𝑛>0. For each 𝑛1 define a mapping 𝑇𝑛𝐶𝐶 by 𝑇𝑛𝑥=𝑛𝑘=1𝛽𝑘𝑛𝑆𝑘𝑥,𝑥𝐶,𝑛1,(2.23) where {𝛽𝑘𝑛} is a sequence of nonnegative real numbers satisfying the following conditions: (i)𝑛𝑘=1𝛽𝑘𝑛=1, for all 𝑛1; (ii)𝛽𝑘=lim𝑛𝛽𝑘𝑛>0, for all 𝑘1; (iii)𝑛=1𝑛𝑘=1|𝛽𝑘𝑛+1𝛽𝑘𝑛|<.
Let {𝑥𝑛} be the sequence defined by 𝑥1𝑥𝐶,𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛,𝑛1,(2.24) where {𝛼𝑛} is a sequence in [0,1]. If the following conditions are satisfied: (i)limsup𝑛𝛼𝑛<1; (ii)there exists a compact subset 𝐾𝐸 such that 𝑛=1𝑆𝑛(𝐶)𝐾. Then, {𝑥𝑛} converges strongly to some point 𝑝.

Proof. Since {𝑆𝑛}𝐶𝐶 is a family of 𝜆𝑛-strictly pseudocontractive mappings with 𝜆=inf𝑛1𝜆𝑛>0. Therefore, {𝑆𝑛} is a family of 𝜆-strictly pseudocontractive mappings. By Remark 1.2, {𝑆𝑛} is a family of (1+𝜆)/𝜆-Lipschitzian and strictly pseudocontractive mappings. Hence, by Lemma 2.3, {𝑇𝑛} defined by (2.21) is a family of (1+𝜆)/𝜆-Lipschitzian, strictly pseudocontractive mappings with 𝑛=1𝐹(𝑇𝑛)==𝑛=1𝐹(𝑆𝑛) and it has also the following properties: (1){𝑇𝑛} satisfies the AKTT-condition; (2)if 𝑇𝐶𝐶 is the mapping defined by (2.22), then 𝑇𝑥=lim𝑛𝑇𝑛𝑥,𝑥𝐶 and 𝐹(𝑇)==𝑘=1𝐹(𝑆𝑛)=𝑛=1𝐹(𝑇𝑛). Hence, by Definition 1.1, {𝑇𝑛} is also a family of (1+𝜆)/𝜆-Lipschitzian and pseudocontractive mappings having the properties (1) and (2) and =𝑛=1𝐹(𝑇𝑛). Therefore, {𝑇𝑛} satisfies all the conditions in Theorem 2.1. By Theorem 2.1, the sequence {𝑥𝑛} converges strongly to some point 𝑝𝑘=1𝐹(𝑆𝑛)=𝑛=1𝐹(𝑇𝑛).
This completes the proof of Theorem 2.4.

Acknowledgment

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

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