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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 142759, 7 pages
http://dx.doi.org/10.1155/2013/142759
Research Article

Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptotically -Strict Pseudocontractive Mappings

1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 21 September 2012; Accepted 11 December 2012

Academic Editor: Cristina Marcelli

Copyright © 2013 H. Zegeye and N. Shahzad. 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

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptotically -strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let be a real normed linear space with dual . A gauge function is a continuous and strictly increasing function satisfying and , as . The generalized duality mapping from to associated with the gauge function (see, e.g., [1]) is defined by where denotes the duality pairing. In the case that , the duality mapping is called the normalized duality mapping.

Following Browder [2], we say that a Banach space has a weakly continuous duality mapping if there exists a gauge for which the duality mapping is single valued and weak-to-weak* sequentially continuous (i.e., if is a sequence in weakly convergent to a point , then the sequence converges weak* to ). It is known that has a weakly continuous duality mapping with a gauge function , for all .

Let be a nonempty subset of . A mapping is called asymptotically -strict pseudocontractive, with sequence , (see, e.g., [36]) if for all , there exist and a constant such that for all .

If denotes the identity operator, then (2) can be equivalently written as for all .

If , a real Hilbert space, it is shown by Osilike et al. [4] that (2) (and hence (3)) is equivalent to the inequality where . is called uniformly Lipschitz if there exists such that for all . It is shown in [4] that an asymptotically -strict pseudocontractive mapping is uniformly Lipschitz.

The class of asymptotically -strict pseudocontractive mappings was first introduced in Hilbert spaces by Liu [5]. He proved the following theorem.

Theorem Q (see [5]). Let be a closed convex and bounded subset of a Hilbert space . Let be completely continuous asymptotically -strict pseudocontractive mapping for some with sequence such that and . Let be a sequence generated by the modified Mann's iteration method: where is a real sequence satisfying for all and some . Then, converges strongly to a fixed point of  .

The iteration scheme (5) is called modified Mann’s iterative processes which was introduced by Schu [7, 8] and has been used by several authors (see, e.g., [35, 917]). We observe that Liu [5] proved strong convergence of scheme (5) to a fixed point of asymptotically -strict pseudocontractive mapping with additional assumption that is completely continuous, where is said to be completely continuous if for every bounded sequence , there exists a subsequence, say of such that the sequence converges strongly to some element of the range of .

In [12], Kim and Xu studied weak convergence theorem for the class of asymptotically -strict pseudocontractive mappings in the frame work of Hilbert spaces. In fact, they proved the following.

Theorem KX (see [12]). Let be a closed and convex subset of a Hilbert space . Let be an asymptotically -strict pseudocontractive mapping for some with sequence such that and . Let be a sequence generated by the modified Mann's iteration method: where is a real sequence satisfying , for all , and . Then, converges weakly to a fixed point of  .

In 2007, Osilike et al. [13] extended Theorem KX by proving weak convergence of scheme (6) to a fixed point of in the frame work of uniformly smooth Banach spaces which are also uniformly convex under suitable control conditions.

In 2011, Zhang and Xie [17] extended Theorem of Osilike et al. [13] to a more general real uniformly convex Banach space with Fréchet differentiable norm. In addition, they proved strong convergence of scheme (5) to a fixed point of asymptotically -strict pseudocontractive mapping provided that , where .

However, we observe that the convergence obtained above is either weak or requiring additional assumption like or is completely continuous. But the requirement that is not easy to verify, as is in general unknown, and there is also an example of asymptotically -strict pseudocontractive mapping which is not completely continuous as shown below.

An example of asymptotically -strict pseudocontractive mapping which is not completely continuous.

Example 1. Let and . Define by , where is a real sequence satisfying , , and . Then it is shown in [13] that is asymptotically -strict pseudocontractive mapping.

Now, we show that is not completely continuous. Let be a sequence in defined by , , . Then and is given by , , . Hence, since , as , there is no subsequence of such that converges strongly to a point in , as , as . Therefore, is not completely continuous.

Thus, one question is raised naturally: can we obtain a scheme that converges strongly to a fixed point of asymptotically -strict pseudocontractive mappings without those additional assumptions?

It is our purpose in this paper to provide an iterative scheme which converges strongly to a common fixed point of finite family of asymptotically -strict pseudocontractive mappings in Banach spaces. The assumption that or is completely continuous is not required.

2. Preliminaries

We need the following definitions from [18]. The Banach space is said to be uniformly convex if, given , there exists , such that, for all with and , . It is well known that , , and Sobolev spaces , (, are uniformly convex.

A Banach space is said to have a Fréchet differentiable norm if for all exists and is attained uniformly in . It is well known that uniformly smooth Banach spaces has a Fréchet differentiable norm.

In order to prove our results, we need the following lemmas.

Lemma 2 (see [19]). Let be a nonempty close convex subset of a real Banach space which has the Fréchet differentiable norm. For , let be defined for by Then, and

It is shown in [19] that if , a real Hilbert space, then , for . In our general setting, throughout this paper we assume that .

Lemma 3. Let be a real Banach space. Then the following inequality holds:

Lemma 4 (see [20]). Let be a uniformly convex Banach space and a closed ball of . Then, there exists a continuous strictly increasing convex function with such that for each and for , with .

Lemma 5 (see [21]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , and . Then, .

Lemma 6 (see [17]). Let be a nonempty closed convex subset of a real uniformly convex Banach space which has the Fréchet differentiable norm. Let be an asymptotically -strict pseudocontractive mapping with fixed point of , . Then is demiclosed at zero, that is, if and , as , then .

Lemma 7 (see [22]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

3. Main Results

We now prove our main theorem.

Theorem 8. Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from into . Let be asymptotically -strict pseudocontractive mappings for with sequences , for . Assume that is nonempty. Let be a sequence defined by and where , such that , for each , , satisfying , , , , for and (, , and constants), for , Then the sequence generated by (14) converges strongly to a common fixed point of .

Proof. Fix . Let and . Then, using Lemma 2 and (3) we have that
On the other hand using Lemma 4 we get that
Now substituting (16) into (15) we obtain that since for each . Then now, from (14) and (18) we get that where is a positive integer such that , for all , for some . Therefore, by induction, which implies that and hence are bounded.
Furthermore, from (14), Lemma 3, and (17) we get that for some .
Now, the rest of the proof is divided into two parts.
Case 1. Suppose that there exists such that is decreasing for all . Then we have that is convergent. Then from (21) and the assumptions on , , and we have that , as , which implies that for . Then from (14) we obtain that Again, from (23) we get that as . Thus, (24) and (25) imply that Therefore, since each , for , is uniformly -Lipschitzian and we have from (23), (26), and uniform continuity of that for each . Furthermore, the fact that is bounded and is reflexive implies that we can choose a subsequence of such that and Now, from (26) we get that and from Lemma 6 we have that , for each . Hence, . Therefore, putting in (29) and using the fact that is weakly sequentially continuous we immediately obtain that . Again, putting in inequality (22), we get that and, hence, it follows from (30) and Lemma 5 that , as . Consequently, .
Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 7, there exists a nondecreasing sequence such that , and for all . Then from (21) and following the method of Case 1, we get that for each . Thus, again following the method of Case 1, we obtain that and , as , for each and there exists such that Then now, putting in (22) we have that Since , (34) implies that Moreover, since , inequality (35) gives that Then, from (33) and the fact that , we obtain that , as . This together with (34) gives that , as . But , for all ; thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to an element of and the proof is complete.

If, in Theorem 8, we assume a single asymptotically -strict pseudocontractive mapping we get the following corollary.

Corollary 9. Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from into . Let be an asymptotically -strict pseudocontractive mapping for with sequences . Assume that is nonempty. Let be a sequence defined by and where , satisfying , , , and (, , and constants). Then the sequence generated by (37) converges strongly to a fixed point of .

Proof. Putting in (14), we get that and the scheme reduces to scheme (37) and following the method of proof of Theorem 8 we get that (see (21) and (22)) for some . Now, considering cases, as in the proof of Theorem 8, we obtain the required result.

Corollary 10. Let be a nonempty, closed, and convex subset of , . Let be asymptotically -strict pseudocontractive mappings for with sequences , for . Assume that is nonempty. Let be a sequence defined by and where , such that , for each , , satisfying , , , and (for , , and constants), for . Then the sequence generated by (39) converges strongly to a common fixed point of .

Proof. We note that , , spaces are uniformly convex which have Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from into (see, e.g., [18]). Thus, the result follows from Theorem 8.

Corollary 11. Let be a nonempty, closed, and convex subset of , . Let be an asymptotically -strict pseudocontractive mapping for some with sequences . Assume that is nonempty. Let be a sequence defined by and where , and (for , , and constants) satisfying , and . Then the sequence converges strongly to a fixed point of .

If in Theorem 8 we have that , a real Hilbert space, then is uniformly convex with Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping. Thus, we have the following corollary.

Corollary 12. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be asymptotically -strict pseudocontractive mappings for with sequences , for . Assume that is nonempty. Let be a sequence defined by and where , such that , for each , , satisfying , , , and (for , , and constants), for . Then the sequence generated by (41) converges strongly to a common fixed point of .

Corollary 13. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be an asymptotically -strict pseudocontractive mapping for some with sequences . Assume that is nonempty. Let be a sequence defined by and where , and (for , , and constants) satisfying , and . Then the sequence converges strongly to a fixed point of .

Remark 14. We note that Corollary 9 generalizes several recent results of this nature. Particularly, it extends Theorem KX of [12], Theorem 2 of Liu [5], and corresponding theorem of Schu [7] in the sense that our convergence is strong in more general Banach spaces possessing weakly sequentially continuous duality mappings without the requirement that be completely continuous.

Remark 15. Corollary 9 is an improvement of Theorem 3.2 of Osilike et al. [13] and Theorems 3.1 and 3.2 of Zhang and Xie [17] in the sense that our convergence is strong without the requirement that , provided that possesses weakly sequentially continuous duality mappings.

Acknowledgments

N. Shahzad gratefully acknowledges research support from the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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