Convergence of Three-Step Iterations Scheme for Nonself Asymptotically Nonexpansive Mappings

Weak and strong convergence theorems of three-step iterations are established for nonself asymptotically nonexpansive mappings in uniformly convex Banach space. The results obtained in this paper extend and improve the recent ones announced by Suantai (2005), Khan and Hussain (2008), Nilsrakoo and Saejung (2006), and many others.

Suppose that is a real uniformly convex Banach space, is a nonempty closed convex subset of . Let be a self-mapping of .

A mapping is called nonexpansiveprovided

for all .

is called asymptotically nonexpansive mapping if there exists a sequence with such that

for all and .

The class of asymptotically nonexpansive maps which is an important generalization of the class nonexpansive maps was introduced by Goebel and Kirk [1]. They proved that every asymptotically nonexpansive self-mapping of a nonempty closed convex bounded subset of a uniformly convex Banach space has a fixed point.

is called uniformly-Lipschitzian if there exists a constant such that , the following inequality holds:

for all .

Asymptotically nonexpansive self-mappings using Ishikawa iterative and the Mann iterative processes have been studied extensively by various authors to approximate fixed points of asymptotically nonexpansive mappings (see [2, 12]). Noor [3] introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces. Glowinski and Le Tallec [4] applied a three-step iterative process for finding the approximate solutions of liquid crystal theory, and eigenvalue computation. It has been shown in [1] that the three-step iterative scheme gives better numerical results than the two-step and one-step approximate iterations. Xu and Noor [5] introduced and studied a three-step scheme to approximate fixed point of asymptotically nonexpansive mappings in a Banach space. Very recently, Nilsrakoo and Saejung [6] and Suantai [7] defined new three-step iterations which are extensions of Noor iterations and gave some weak and strong convergence theorems of the modified Noor iterations for asymptotically nonexpansive mappings in Banach space. It is clear that the modified Noor iterations include Mann iterations [8], Ishikawa iterations [9], and original Noor iterations [3] as special cases. Consequently, results obtained in this paper can be considered as a refinement and improvement of the previously known results

where , , , , , , , and in satisfy certain conditions.

If , then (1.4) reduces to the modified Noor iterations defined by Suantai [7] as follows:

where , , , , , and in satisfy certain conditions.

If , then (1.4) reduces to Noor iterations defined by Xu and Noor [5] as follows:

If , then (1.4) reduces to modified Ishikawa iterations as follows:

If , then (1.4) reduces to Mann iterative process as follows:

Let be a real normed space and be a nonempty subset of . A subset of is called a retract of if there exists a continuous map such that for all . Every closed convex subset of a uniformly convex Banach space is a rectract. A map is called a retraction if . In particular, a subset is called a nonexpansive retract of if there exists a nonexpansive retraction such that for all .

Iterative techniques for converging fixed points of nonexpansive nonself-mappings have been studied by many authors (see, e.g., Khan and Hussain [10], Wang [11]). Evidently, we can obtain the corresponding nonself-versions of (1.5)(1.7). We will obtain the weak and strong convergence theorems using (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. Very recently, Suantai [7] introduced iterative process and used it for the weak and strong convergence of fixed points of self-mappings in a uniformly convex Banach space. As remarked earlier, Suantai [7] has established weak and strong convergence criteria for asymptotically nonexpansive self-mappings, while Chidume et al. [12] studied the Mann iterative process for the case of nonself-mappings. Our results will thus improve and generalize corresponding results of Suantai [7] and others for nonself-mappings and those of Chidume et al. [12] in the sense that our iterative process contains the one used by them. The concept of nonself asymptotically nonexpansive mappings was introduced by Chidume et al. [12] as the generalization of asymptotically nonexpansive self-mappings and obtained some strong and weak convergence theorems for such mappings given (1.9) as follows: for

where and for some .

A nonself-mapping is called asymptotically nonexpansive if there exists a sequence with such that

for all , and . is called uniformly-Lipschitzianif there exists constant such that

for all , and . From the above definition, it is obvious that nonself asymptotically nonexpansive mappings are uniformly -Lipschitzian.

Now, we give the following nonself-version of (1.4):

for

, where , , , , , , , and in satisfy certain conditions.

The aim of this paper is to prove the weak and strong convergence of the three-step iterative sequence for nonself asymptotically nonexpansive mappings in a real uniformly convex Banach space. The results presented in this paper improve and generalize some recent papers by Suantai [7], Khan and Hussain [10], Nilsrakoo and Saejung [6], and many others.

Throughout this paper, we assume that is a real Banach space, is a nonempty closed convex subset of , and is the set of fixed points of mapping . A Banach space is said to be uniformly convex if the modulus of convexity of is as follows:

for all (i.e., is a function

Recall that a Banach space is said to satisfy Opial's condition [13] if, for each sequence in , the condition weakly as and for all with implies that

Lemma 2.1 (see [12]).

Let be a uniformly convex Banach space, a nonempty closed convex subset of and a nonself asymptotically nonexpansive mapping with a sequence and , then is demiclosed at zero.

Lemma 2.2 (see [12]).

Let be a real uniformly convex Banach space, a nonempty closed subset of with as a sunny nonexpansive retraction and a mapping satisfying weakly inward condition, then .

Lemma 2.3 (see [14]).

Let , , and be sequences of nonnegative real sequences satisfying the following conditions: , , where and , then exists.

Lemma 2.4 (see [6]).

Let be a uniformly convex Banach space and , then there exists a continuous strictly increasing convex function with such that

for all , and with .

Lemma 2.5 (See [7], Lemma ).

Let be a Banach space which satisfies Opial's condition and let be a sequence in . Let be such that and . If , are the subsequences of which converge weakly to , respectively, then .

In this section, we prove theorems of weak and strong of the three-step iterative scheme given in (1.12) to a fixed point for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. In order to prove our main results the followings lemmas are needed.

Lemma 3.1.

If and are sequences in such that and is sequence of real numbers with for all and , then there exists a positive integer and such that for all .

Proof.

By , there exists a positive integer and such that

Let with . From , then there exists a positive integer such that

from which we have . Put , then we have for all .

Lemma 3.2.

Let be a real Banach space and a nonempty closed and convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12), then we have, for any , exists.

Proof.

Consider

Thus, we have

Since and from Lemma 2.3, it follows that exits.

Lemma 3.3.

Let be a real uniformly convex Banach space and a nonempty closed and convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12), then one has the following conclusions.

If , then

If either or and , then

If the following conditions

,

either and or and are satisfied, then

Proof.

Let . By Lemma 3.2, we know that exits for any . Then the sequence is bounded. It follows that the sequences and are also bounded. Since is a nonself asymptotically nonexpansive mapping, then the sequences , , and are also bounded. Therefore, there exists such that , , , , , . By Lemma 2.4 and (1.12), we have

Let .

Therefore, the assumption implies that .

Thus, we have

From the last inequality, we have

By condition

there exists a positive integer and such that and for all then it follows from (3.7) that

for all . Thus, for , we write

Letting , we have , so that

From is continuous strictly increasing with and (1), then we have

By using a similar method for inequalities (3.8) and (3.10), we have

Next, to prove

we assume that and ,

By Lemma 3.1, there exists a positive integer and such that for all . This together with (3.18) implies that for ,

It follows from (3.15) and (3.16) that

This completes the proof.

Next, we show that .

Lemma 3.4.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12) with the following restrictions:

and ,

,

then .

Proof.

We first consider

We note that every asymptotically nonexpansive mapping is uniformly -Lipschitzian. Also note that

In addition,

We denote as the identity maps from into itself. Thus, by above inequality, we write

which implies that

In the next result, we prove our first strong convergence theorem as follows.

Theorem 3.5.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12) with the following restrictions:

and ,

.

If, in addition, is either completely continuous or demicompact, then converges strongly to a fixed point of .

Proof.

By Lemma 3.2, is bounded. It follows by our assumption that is completely continuous, there exists a subsequence of such that as . Therefore, by Lemma 3.4, we have which implies that as . Again by Lemma 3.4, we have

It folows that . Moreover, since exists, then , that is, converges strongly to a fixed point of .

We assume that is demicompact. Then, using the same ideas and argument, we also prove that converges strongly to a fixed point of .

Finally, we prove the weak convergence of the iterative scheme (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.

Theorem 3.6.

Let be a real uniformly convex Banach space satisfying Opial's condition and a nonempty closed convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12) with the following restrictions:

and ,

,

then converges weakly to a fixed point of .

Proof.

Let . Then as in Lemma 3.2, exists. We prove that has a unique weak subsequential limit in . We assume that and are weak limits of the subsequences , , or , respectively. By Lemma 3.4, and is demiclosed by Lemma 2.1, and in the same way, . Therefore, we have . It follows from Lemma 2.5 that . Thus, converges weakly to an element of This completes the proof.

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