Journal of Applied Mathematics

Volume 2013 (2013), Article ID 821737, 7 pages

http://dx.doi.org/10.1155/2013/821737

Research Article

Approximation Analysis for a Common Fixed Point of Finite Family of Mappings Which Are Asymptotically -Strict Pseudocontractive in the Intermediate Sense

H. Zegeye¹ and N. Shahzad²

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 16 February 2013; Accepted 18 April 2013

Academic Editor: Luigi Muglia

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 introduce an iterative process which converges strongly to a common fixed point of a finite family of uniformly continuous asymptotically -strict pseudocontractive mappings in the intermediate sense for . The projection of onto the intersection of closed convex sets and for each is not required. Moreover, the restriction that the interior of common fixed points is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

1. Introduction and Preliminaries

Let be a nonempty subset of a real Hilbert space . A mapping is called Lipschitzian if there exists such that , for all , . If , then is called nonexpansive, and if , is called contraction. is called uniformly -Lipschitzian if there exists such that , for all , and all . Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is uniformly -Lipschitzian with and hence Lipschitzian.

A mapping is said to be asymptotically nonexpansive if there exists a sequence with such that for all integers and all . is said to be asymptotically nonexpansive in the intermediate sense if there exist sequences with , such that for all integers and all .

A mapping is said to be asymptotically -strict pseudocontractive if there exist a constant and a sequence with , as , such that

The class of asymptotically -strict pseudocontractive mappings which includes the class of asymptotically nonexpansive, and hence the class of nonexpansive mappings was introduced by Liu [1] in 1996 (see, also [2]). Kim and Xu [3] proved that the fixed point set of asymptotically -strict pseudocontractions is closed and convex. Recall that a fixed point of a map is a set , and it is denoted by . In addition, it is noted in [3] that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .

A mapping is said to be an asymptotically -strict pseudocontractive in the intermediate sense if

where and such that , as . If we put

It follows that , as , and (2) is reduced to the following:

We note that the class of asymptotically -strict pseudocontractive mappings is properly contained in a class of asymptotically -strict pseudocontractive mapping in the intermediate sense (see examples in [4]). The class of asymptotically -strict pseudocontractive mappings in the intermediate sense was introduced by Sahu et al. [4]. They obtained a weak convergence theorem of modified Mann iterative processes for these class of mappings. In [4], Sahu et al. established the following classical result.

Theorem SXY1 (see [4]). Let be a real Hilbert space, and let be a nonempty closed and convex set. Let be a uniformly continuous and asymptotically -strict pseudocontractive mapping in the intermediate sense with sequences and such that is nonempty and . Assume that is a sequence in such that and . Let be a sequence defined by and

Then, converges weakly to a fixed point of .

But it is worth mentioning that the convergence obtained is a weak convergence. Furthermore, we observe from the proof of Theorem SXY1 that if, in addition, or has some compactness assumption, we obtain that the sequence given by (5) converges strongly to a fixed point of .

Attempts to modify the Mann iteration method (5) so that strong convergence is guaranteed, without compactness assumption on or , have recently been made. Sahu et al. [4] established the following hybrid Mann algorithm for approximating fixed points of asymptotically -strict pseudocontractive mappings in the intermediate sense.

Theorem SXY2 (see [4]). Let be a real Hilbert space, and let be a nonempty, closed, and convex set. Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequences and such that is nonempty and bounded. Assume that is a sequence in such that , for all . Let be a sequence in defined by and

where and . Then, converges strongly to .

Recently, Hu and Cai [5] studied the strong convergence of the modified Mann iteration process (5) for a finite family of asymptotically -strict pseudocontractive mappings in the intermediate sense. More precisely, they obtained the following theorem.

Theorem HC (see [5]). Let be a real Hilbert space, let be a nonempty, closed, and convex set. Let be uniformly continuous asymptotically -strict pseudocontractive mappings in the intermediate sense for some with sequences and such that and for , , , . Let , , and . Assume that is nonempty and bounded. Let be a sequence in such that for all . Let be a sequence defined by and

where , as , for , , where , a positive integer such that , as . Then, converges strongly to .

But we observe that the iterative algorithms (6) and (7) generate a sequence by projecting onto the intersection of closed convex sets and for each which is not easy to compute.

Attempts to remove projection mapping onto the intersection of closed convex sets and for each have recently been made. In [6], Zegeye et al. studied the strong convergence of the modified Mann iteration process for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense. More precisely, they proved the following theorem.

Theorem ZRS (see [6]). Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a uniformly -Lipschitzian and asymptotically -strict pseudocontractive mapping in the intermediate sense with sequences and . Assume that the interior of is nonempty. Let be a sequence defined by and

where and satisfy certain conditions. Then, converges strongly to a fixed point of .

But it is worth to mention that the assumption interior of is nonempty is severe restriction.

It is our purpose, in this paper, to construct an iteration scheme which converges strongly to a common fixed point of a finite family of uniformly continuous asymptotically -strict pseudocontractive mappings in the intermediate sense for , , , . The projection of onto the intersection of closed convex sets and for each is not required. Furthermore, the restriction that the interior of is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

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

Lemma 1. Let be a real Hilbert space. Then, for any given , , the following inequality holds:

Lemma 2 (see [7]). Let H be a real Hilbert space. Then, for all , and for , , , such that , the following equality holds:

Lemma 3 (see [8]). 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, .

Lemma 4 (see [9]). Let be a sequence of nonnegative real numbers satisfying the following relation:

where and satisfying the following conditions: , and . Then, .

Lemma 5 (see [4]). Let be a nonempty closed convex subset of a Hilbert space , and let be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then, is closed and convex.

Lemma 6 (see [4]). Let be a nonempty closed convex subset of a Hilbert space , and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Let be a sequence in such that and , as . Then, , as .

Lemma 7 (see [4]). Let be a nonempty closed convex subset of a Hilbert space , and be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then if is a sequence in such that converges weakly and , then .

Lemma 8 (see [10]). Let be a nonempty closed, convex subset of a Hilbert space and let be the metric projection mapping from onto . Given if and only if , for all .

2. Main Result

Theorem 9. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be uniformly continuous asymptotically -strict pseudocontractive mappings in the intermediate sense for some with sequences and , for , , , . Assume that is nonempty. Let be a sequence generated by

where such that , , , and satisfying for each and . Then, converges strongly to an element of .

Proof. Let . Let and . Then, from (13), Lemma 2, and asymptotically -strict pseudocontractiveness of , for each , we get that

since there exists such that and for all and for some satisfying . Thus, by induction,

which implies that , and hence and are bounded. Moreover, from (13), (14) and Lemma 1, we obtain that

for some and for all .

Now, following the method of proof of Lemma 3.2 of Maingé [8], we consider two cases.

Case  1. Suppose that there exists such that is nonincreasing for all . In this situation, is convergent. Then, from (17), we have that

for , , , . Moreover, from (13) and (19) and the fact that , we get that

as , and hence

Furthermore, from (19), (21), and Lemma 6, we obtain that

Let be a subsequence of such that    and converges weakly to . Then, from (21), we also get that converges weakly to . Moreover, since is uniformly continuous and , for all , we get that , as , for all . Therefore, by Lemma 7, we obtain that . Now, from Lemma 8, we have that

Then, from (18), (23), and Lemma 4, we obtain that , as . Consequently, .

Case  2. Suppose that there exists a subsequence of such that

for all . Then, by Lemma 3, there exist a nondecreasing sequence such that ,

and , for all . Then, from (17) and the fact that , we have

Then, we get that , as , for each , , , . Thus, as in Case 1, we obtain that and , as and

Now, from (18), we have that

This implies that

Then, using (25), inequality (29) implies that

In particular, since , we get

Furthermore, using (27) and the fact that , we obtain that , as . This together with (28) give , 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 9, we assume that , then we get the following corollary.

Corollary 10. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let be a sequence generated by

where such that , , , and for each . Then, converges strongly to an element of .

If in Theorem 9, we assume that each is asymptotically -strict pseudocontractive mapping, then we get the following corollary.

Corollary 11. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be asymptotically -strict pseudocontractive mappings for some with sequences , for , , , . Assume that is nonempty. Let be a sequence generated by

where such that , , , and satisfying for each and . Then, converges strongly to an element of .

Proof. Since every asymptotically -strict pseudocontractive mapping is uniformly continuous and asymptotically -strict pseudocontractive mapping in the intermediate sense with , for all and each , the conclusion follows from Theorem 9.

If in Theorem 9, we assume that each is asymptotically nonexpansive in the intermediate sense we obtain the following corollary.

Corollary 12. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be uniformly continuous asymptotically nonexpansive mappings in the intermediate sense with sequences and , for , , , . Assume that is nonempty. Let be a sequence generated by

where such that , ,     , and satisfying for each . Then, converges strongly to an element of .

Proof. Since every asymptotically nonexpansive mapping in the intermediate sense is asymptotically -strict pseudocontractive mapping in the intermediate sense with , for each , , , , the conclusion follows from Theorem 9.

If in Theorem 9, we assume that each is asymptotically nonexpansive, we obtain the following corollary.

Corollary 13. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be asymptotically nonexpansive mappings with sequences , for , , , . Assume that is nonempty. Let be a sequence generated by

where such that , , , and satisfying for each . Then, converges strongly to an element of .

Proof. Since every asymptotically nonexpansive mapping is uniformly continuous and asymptotically -strict pseudocontractive mapping with and , for all and , , , , the conclusion follows from Theorem 9.

Remark 14. Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem 9 extends Theorem SXY1, SXY2, HC, and Theorem ZRS in the sense that our convergence is either strong, does not require computation of closed convex sets and for each , or does not require the assumption that interior of set of fixed points is nonempty.

Remark 15. We also remark that Corollary 11 is more general than Theorem 3.1 of Kim and Xu [3] and Corollary 13 is more general than Theorem 2.2 of Kim and Xu [11] in the sense that our convergence is either strong, does not require computation of closed convex sets and for each , or does not require the assumption that interior of set of fixed points is nonempty.

Acknowledgment

The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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