- Research Article
- Open Access

# On the Convergence of an Implicit Iterative Process for Generalized Asymptotically Quasi-Nonexpansive Mappings

- Xiaolong Qin
^{1}, - ShinMin Kang
^{2}and - RaviP Agarwal
^{3, 4}Email author

**2010**:714860

https://doi.org/10.1155/2010/714860

© Xiaolong Qin et al. 2010

**Received:**27 June 2010**Accepted:**10 November 2010**Published:**10 November 2010

## Abstract

The purpose of this paper is to introduce and consider a general implicit iterative process which includes Schu's explicit iterative processes and Sun's implicit iterative processes as special cases for a finite family of generalized asymptotically quasi-nonexpansive mappings. Strong convergence of the purposed iterative process is obtained in the framework of real Banach spaces.

## Keywords

- Banach Space
- Nonexpansive Mapping
- Real Banach Space
- Nondecreasing Function
- Convex Banach Space

## 1. Introduction and Preliminaries

*uniformly convex*if for any there exists such that for any ,

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Let be a nonempty closed and convex subset of a Banach space . Let be a mapping. Denote by the fixed point set of .

A nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive; however, the inverse may be not true. See the following example [1].

Example 1.1.

Then is quasi-nonexpansive but not nonexpansive.

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence . The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972. It is known that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive mapping on has a fixed point. Further, the set of fixed points of is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

*asymptotically quasi-nonexpansive*if , and there exists a positive sequence with as such that

*asymptotically nonexpansive in the intermediate sense*if it is continuous and the following inequality holds:

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [3] (see also Bruck et al. [4]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if is a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point; see [5] more details.

*asymptotically quasi-nonexpansive in the intermediate sense*if it is continuous, , and the following inequality holds:

*generalized asymptotically nonexpansive*if there exist two positive sequences with and with as such that

It is easy to see that the class of generalized asymptotically nonexpansive includes the class of asymptotically nonexpansive as a special case.

*generalized asymptotically quasi-nonexpansive*if , and there exist two positive sequences with and with as such that

The class of generalized asymptotically quasi-nonexpansive was considered by Shahzad and Zegeye [6]; see [6, 7] for more details.

where is a sequence in the interval and : is an asymptotically nonexpansive mapping.

In 1991, Schu [8] obtained the following results.

Theorem Schu 1.

Let be a uniformly convex Banach space, closed bounded and convex, and asymptotically nonexpansive with sequence for which and is bounded away. Let be a sequence generated in (1.13). Then .

Theorem Schu 2.

Let be a uniformly convex Banach space, closed bounded and convex, and asymptotically nonexpansive with sequence for which and is bounded away. Let be a sequence generated in (1.13). Suppose that is compact for some positive integer . Then the sequence converges strongly to some fixed point of .

Theorem Schu 3.

Then the sequence converges strongly to some fixed point of .

We remark that the implicit iterative process (1.16) was first considered by Sun [9]; see [9] for more details.

Shahzad and Zegeye [6] obtained the following results.

Theorem SZ 1.

Let be a real uniformly convex Banach space and be a nonempty closed convex subset of . Let , where , be uniformly Lipschitz, generalized asymptotically quasi-nonexpansive self-mappings of with , such that and for all . Suppose that and there exists one member in which is either semicompact or satisfies condition . Let for some . From arbitrary , define the sequence by (1.16). Then converges strongly to a common fixed point of the mappings .

Theorem SZ 2.

Let be a real uniformly convex Banach space and a nonemptyclosed convex subset of . Let , where , be generalized asymptotically quasi-nonexpansive self-mappings of with , such that and for all . Suppose that is closed. Let for some . From arbitrary , define the sequence by (1.16). Then converges strongly to a common fixed point of the mappings if and only if .

We remark that our implicit iterative process (1.18) which includes the explicit iterative process (1.13) and the implicit iterative process (1.16) as special cases is general.

The purpose of this paper is to study the convergence of the implicit iteration process (1.18) for two finite families of generalized asymptotically quasi-nonexpansive mappings. Strong convergence theorems are obtained in the framework of real Banach spaces. The results presented in this paper improve and extend the corresponding results in Shahzad and Zegeye [6], Sun [9], Chang et al. [10], Chidume and Shahzad [11], Guo and Cho [12], Kim et al. [13], Qin et al. [14], Thianwan and Suantai [15], Xu and Ori [16], and Zhou and Chang [17].

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

Lemma 1.2 (see [18]).

where is some positive integer. If and , then exists.

Lemma 1.3 (see [19]).

## 2. Main Results

Lemma 2.1.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each and a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

Proof.

From the restriction (a), we see that is a contraction for each . From Banach contraction mapping principle, we can prove that the sequence generated in (1.18) is well defined.

This completes the proof.

Recall that a mapping
is said to be *semicompact* if for any bounded sequence
in
such that
as
, then there exists a subsequence
such that
.

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

Theorem 2.2.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

If one of or one of is semicompact, then the sequence converges strongly to some point in .

Proof.

Since is Lipshcitz continuous, we obtain from (2.27) that . This means that . In view of Lemma 2.1, we obtain that exists. Therefore, we can obtain the desired conclusion immediately.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.3.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

If one of is semicompact, then the sequence converges strongly to some point in .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.4.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , for all ;

If one of is semicompact, then the sequence converges strongly to some point in .

*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 2.5.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

If and satisfy Condition , then the sequence converges strongly to some point in .

Proof.

It follows that is a Cauchy sequence in and so converges strongly to some . Since and are Lipschitz for each , we see that is closed. This in turn implies that . This completes the proof.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.6.

Let be a real uniformly convex uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and where . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

If satisfies Condition , then the sequence converges strongly to some point in .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.7.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , and , for all ;

If satisfies Condition , then the sequence converges strongly to some point in .

Finally, we give a strong convergence theorem criterion.

Theorem 2.8.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

Then converges strongly to some point in if and only if .

Proof.

In view of Theorem 2.5, we can conclude the desired conclusion easily.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.9.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and where . Let , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

Then converges strongly to some point in if and only if .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.10.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

## Authors’ Affiliations

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