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Abstract and Applied Analysis

Volume 2011, Article ID 468716, 18 pages

http://dx.doi.org/10.1155/2011/468716

## On the Convergence of Implicit Iterative Processes for Asymptotically Pseudocontractive Mappings in the Intermediate Sense

^{1}School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China^{2}Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China^{3}Department of Mathematics, Kyungnam University, Masan 631-701, Republic of Korea

Received 20 November 2010; Accepted 9 February 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Xiaolong Qin 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

An implicit iterative process is considered. Strong and weak convergence theorems of common fixed points of a finite family of asymptotically pseudocontractive mappings in the intermediate sense are established in a real Hilbert space.

#### 1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm . Let be a nonempty closed convex subset of and a mapping. We denote by the fixed point of the mapping .

Recall that is said to be uniformly -lipschitz if there exists a positive constant such that is said to be nonexpansive if is said to be asymptotically nonexpansive if there exists a sequence with as such that The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. It is known that if is a nonempty bounded closed convex subset of a Hilbert space space , then every asymptotically nonexpansive mapping on has a fixed point. Since 1972, a host of authors have been studing strong and weak convergence problems of the iterative processes for such a class of mappings.

is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds: Putting we see that as . Then, (1.4) is reduced to the following: The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [2] (see also Bruck et al. [3]) as a generalization of the class of asymptotically nonexpansive mappings. It is known [4] that if is a nonempty bounded closed convex subset of a Hilbert space space , then every asymptotically nonexpansive mapping in the intermediate sense on has a fixed point.

is said to be strictly pseudocontractive if there exists a constant such that For such a case, is also said to be a -strict pseudocontraction. The class of strict pseudocontractions was introduced by Browder and Petryshyn [5] in 1967. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction.

is said to be an asymptotically strict pseudocontraction if there exist a sequence with as and a constant such that For such a case, is also said to be an asymptotically -strict pseudocontraction. The class of asymptotically strict pseudocontractions is introduced by Qihou [6] in 1996. It is clear that every asymptotically nonexpansive mapping is an asymptotical 0-strict pseudocontraction.

is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exists a sequence with as and a constant such that For such a case, is also said to be an asymptotically -strict pseudocontraction in the intermediate sense. Putting we see that as . Then, (1.9) is reduced to the following: The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu et al. [7] as a generalization of the class of asymptotically strict pseudocontractions, see [7] for more details. We also remark that if for each and in (1.9), then the class of asymptotically -strict pseudocontractions in the intermediate sense is reduced to the class of asymptotically nonexpansive mappings in the intermediate sense.

is said to be pseudocontractive if It is easy to see that (1.12) is equivalent to

is said to be asymptotically pseudocontractive if there exists a sequence with as such that It is easy to see that (1.14) is equivalent to We remark that the class of asymptotically pseudocontractive mappings was introduced by Schu [8] in 1991. For an asymptotically pseudocontractive mapping , Zhou [9] proved that if is also uniformly Lipschitz and uniformly asymptotically regular, then enjoys a nonempty fixed point set. Moreover, is closed and convex.

is said to be an asymptotically pseudocontractive mapping in the intermediate sense if there exists a sequence with as such that It is easy to see that (1.16) is equivalent to Put Then, (1.16) is reduced to the following It is easy to see that (1.19) is equivalent to The class of asymptotically pseudocontractive mappings in the intermediate sense which includes the class of asymptotically pseudocontractive mappings and the class of asymptotically strict pseudocontractions in the intermediate sense as special cases was introduced by Qin et al. [10].

In 2001, Xu and Ori [11], in the framework of Hilbert spaces, introduced the following implicit iteration process for a finite family of nonexpansive mappings with a real sequence in and an initial point : which can be written in the following compact form where (here the takes values in ).

They obtained the following weak convergence theorem.

Theorem XO. *Let be a real Hilbert space, a nonempty closed convex subset of , and be a finite family of nonexpansive mappings such that . Let be defined by (1.22). If is chosen so that as , then converges weakly to a common fixed point of the family of .*

Subsequently, fixed point problems based on implicit iterative processes have been considered by many authors, see [9, 12–23]. In 2004, Osilike [18] considered the implicit iterative process (1.22) for a finite family of strictly pseudocontractive mappings. To be more precise, he proved the following theorem.

Theorem O. *Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that . Let and let be a sequence in such that as . Then, the sequence defined by (1.22) converges weakly to a common fixed point of the mappings .*

In 2008, Qin et al. [20] considered the following implicit iterative process for a finite family of asymptotically strict pseudocontractions: where is an initial value, is a sequence in . Since for each , it can be written as , where , is a positive integer and as . Hence, the above table can be rewritten in the following compact form: A weak convergence theorem of the implicit iterative process (1.24) for a finite family of asymptotically strict pseudocontractions was established.

We remark that the implicit iterative process (1.24) has been used to study the class of asymptotically pseudocontractive mappings by Osilike and Akuchu [19]. They obtained strong convergence of the implicit iterative process (1.24), however, there is no weak convergence theorem.

In this paper, motivated by the above results, we reconsider the implicit iterative process (1.24) for asymptotically pseudocontractive mappings in the intermediate sense. Strong and weak convergence theorems of common fixed points of a finite family of asymptotically pseudocontractive mappings in the intermediate sense are established. The results presented in this paper mainly improve and extend the corresponding results announced in Chang et al. [24], Chidume and Shahzad [13], Górnicki [25], Osilike [18], Qin et al. [20], Xu and Ori [11], and Zhou and Chang [23].

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

Recall that a space is said to satisfy Opial's condition [26] if, for each sequence in , the convergence weakly implies that

Recall that a mapping is semicompact if any sequence in satisfying has a convergent subsequence.

Lemma 1.1 (see [27]). *In a real Hilbert space, the following inequality holds
*

Lemma 1.2 (see [28]). *Let , , and be three nonnegative sequences satisfying the following condition:
**
where is some nonnegative integer, and . Then, the limit exists.*

#### 2. Main Results

Theorem 2.1. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence such that for each , where is some positive integer. Let for each . Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: *(a)*, where , for all ; *(b)*, where . ** Then, converges weakly to some point in .*

*Proof. *First, we show that the sequence generated in the implicit iterative process (1.24) is well defined. Define mappings by
Notice that
From the restriction (a), we see that is a contraction for each . By Banach contraction principle, we see that there exists a unique fixed point such that
This shows that the implicit iterative process (1.24) is well defined for uniformly Lipschitz continuous and asymptotically pseudocontractive mappings in the intermediate sense. Let . In view of the assumption, we see that . Fixing , we see from Lemma 1.1 that
From the restriction (a), we see that there exists some such that
where . It follows that
In view of the restriction (b), we obtain from Lemma 1.2 that exists. Hence, the sequence is bounded. Reconsidering (2.4), we see from the restriction (a) that
This implies that
Notice that
It follows from (2.8) that
Observe that
In view of (2.8), and (2.10), we obtain that
Since for any positive integer , it can be written as , where . Observe that
Since for each , , on the other hand, we obtain from that . That is,
Notice that
Substituting (2.15) into (2.13), we arrive at
In view of (2.8), (2.10), and (2.12), we obtain from (2.16) that
Notice that
From (2.10) and (2.17), we arrive at
Notice that
It follows from (2.10) and (2.19) that
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that
Since the sequence is bounded, we see that there exists a subsequence such that converges weakly to a point . Choose and define for arbitrary but fixed . Notice that
It follows from (2.22) that
Note that
Since and (2.24), we arrive at

On the other hand, we have
Notice that
Substituting (2.26) and (2.27) into (2.28), we arrive at
Letting in (2.29), we see that for each . Since is uniformly -Lipschitz, we can obtain that for each . This means that .

Next we show that converges weakly to . Supposing the contrary, we see that there exists some subsequence of such that converges weakly to , where . Similarly, we can show . Notice that we have proved that exists for each . Assume that where is a nonnegative number. By virtue of the Opial property of , we see that
This is a contradiction. Hence . This completes the proof.

For the class of asymptotically pseudocontractive mappings, we have, from Theorem 2.1, the following results immediately.

Corollary 2.2. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions , where , for all . Then, converges weakly to some point in .*

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.1 the following results immediately.

Corollary 2.3. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each , where is some positive integer. Let for each . Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: *(a)*, where , for all ; *(b)*, where . ** Then converges weakly to some point in . *

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.1 the following results immediately.

Corollary 2.4. *Let be a nonempty closed convex subset of a Hilbert space . Let be an asymptotically nonexpansive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restriction , where , for all . Then, converges weakly to some point in .*

Next, we give strong convergence theorems with the help of semicompactness.

Theorem 2.5. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence such that for each , where is some positive integer. Let for each . Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: *(a)*, where , for all ; *(b)*, where . ** If one of is semicompact, then the sequence converges strongly to some point in .*

*Proof. *Without loss of generality, we may assume that is semicompact. From (2.22), we see that there exists a subsequence of that converges strongly to . For each , we get that
Since is Lipschitz continuous, we obtain from (2.22) that . In view of Theorem 2.1, we obtain that exists. Therefore, we can obtain the desired conclusion immediately.

For the class of asymptotically pseudocontractive mappings, we have from Theorem 2.5 the following results immediately.

Corollary 2.6. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: , where , for all . If one of is semicompact, then the sequence converges strongly to some point in .*

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.5 the following results immediately.

Corollary 2.7. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each , where is some positive integer. Let for each . Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: *(a)*, where , for all ; *(b)*, where . ** If one of is semicompact, then the sequence converges strongly to some point in . *

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.5 the following results immediately.

Corollary 2.8. *Let be a nonempty closed convex subset of a Hilbert space . Let be an asymptotically nonexpansive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restriction: , where , for all . If one of is semicompact, then the sequence converges strongly to some point in .*

In 2005, Chidume and Shahzad [13] introduced the following conception. Recall that a family with is said to satisfy Condition on if there is a nondecreasing function with and for all such that for all

Next, we give strong convergence theorems with the help of Condition .

Theorem 2.9. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence such that for each , where is some positive integer. Let for each . Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: *(a)*, where , for all ; *(b)*, where . ** If satisfies Condition , then the sequence converges strongly to some point in .*

*Proof. *In view of Condition , we obtain from (2.22) that , which implies . Next, we show that the sequence is Cauchy. In view of (2.6), for any positive integers , where , we obtain that
where . It follows that
It follows that is a Cauchy sequence in ,so converges strongly to some . Since is Lipschitz for each , we see that is closed. This in turn implies that . This completes the proof.

For the class of asymptotically pseudocontractive mappings, we have from Theorem 2.9 the following results immediately.

Corollary 2.10. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: , where , for all . If satisfies Condition , then the sequence converges strongly to some point in .*

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.9 the following results immediately.

Corollary 2.11. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each , where is some positive integer. Let for each . Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions: *(a)*, where , for all ; *(b)*, where . ** If satisfies Condition , then the sequence converges strongly to some point in . *

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.9 the following results immediately.

Corollary 2.12. *Let be a nonempty closed convex subset of a Hilbert space . Let be an asymptotically nonexpansive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restriction: , where , for all . If satisfies Condition , then the sequence converges strongly to some point in .*

Finally, we give the following strong convergence criteria.

Theorem 2.13. *, where , for all ; *(b)*, where . ** Then, the sequence converges strongly to some point in if and only if .*

*Proof. *The necessity is obvious. We only show the sufficiency. Assume that
In view of Lemma 1.2, we can obtain from (2.6) that . The desired results can be obtain from Theorem 2.9 immediately.

For the class of asymptotically pseudocontractive mappings, we have from Theorem 2.13 the following results immediately.

Corollary 2.14. *Let be a nonempty closed convex subset of a Hilbert space . Let be a uniformly -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restrictions , where , for all . Then, the sequence converges strongly to some point in if and only if .*

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.13 the following results immediately.

Corollary 2.15. *, where , for all ; *(b)*, where . ** Then, the sequence converges strongly to some point in if and only if .*

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.13 the following results immediately.

Corollary 2.16. *Let be a nonempty closed convex subset of a Hilbert space . Let be an asymptotically nonexpansive mapping with the sequence such that for each , where is some positive integer. Assume that the common fixed point set is nonempty. Let be a sequence generated in (1.24). Assume that the control sequence in satisfies the following restriction: , where , for all . Then, the sequence converges strongly to some point in if and only if .*

#### Acknowledgment

This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (2010-0016000).

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