Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi--Nonexpansive Mapping in Banach Space

Abstract

We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi--nonexpansive mapping and an asymptotically quasi-nonexpansive mapping , defined on a nonempty closed convex subset of a Banach space.

1. Introduction

Let be a nonempty subset of a real normed linear space and let be a mapping. Denote by the set of fixed points of , that is, . Throughout this paper, we always assume that . Now let us recall some known definitions.

Definition 1.1.

A mapping is said to be

(i)nonexpansive, if for all ;

(ii)asymptotically nonexpansive, if there exists a sequence with such that for all and ;

(iii)quasi-nonexpansive, if for all ;

(iv)asymptotically quasi-nonexpansive, if there exists a sequence with such that for all and .

Note that from the above definitions, it follows that a nonexpansive mapping must be asymptotically nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, but the converse does not hold (see [1]).

If is a closed nonempty subset of a Banach space and is nonexpansive, then it is known that may not have a fixed point (unlike the case if is a strict contraction), and even when it has, the sequence defined by (the so-called Picard sequence) may fail to converge to such a fixed point.

In [2, 3] Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the studies of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicians (see for more details [1, 4]).

In [5] Diaz and Metcalf studied quasi-nonexpansive mappings in Banach spaces. Ghosh and Debnath [6] established a necessary and sufficient condition for convergence of the Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subset of a Banach space. The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping were studied extensively by Goebel and Kirk [7], Liu [8], Wittmann [9], Reich [10], Gornicki [11], Schu [12] Shioji and Takahashi [13], and Tan and Xu [14] in the settings of Hilbert spaces and uniformly convex Banach spaces.

There are many methods for approximating fixed points of a nonexpansive mapping. Xu and Ori [15] introduced implicit iteration process to approximate a common fixed point of a finite family of nonexpansive mappings in a Hilbert space. Recently, Sun [16] has extended an implicit iteration process for a finite family of nonexpansive mappings, due to Xu and Ori, to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces. In [17] it has been studied the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces, which extends and improves the mentioned papers (see also [18, 19] for applications and other methods of implicit iteration processes).

There are many concepts which generalize a notion of nonexpansive mapping. One of such concepts is -nonexpansivity of a mapping ([20]). Let us recall some notions.

Definition 1.2.

Let , be two mappings of a nonempty subset of a real normed linear space . Then is said to be

(i)- nonexpansive, if for all ;

(ii)asymptotically - nonexpansive, if there exists a sequence with such that for all and ;

(iii)asymptotically quasi - nonexpansive mapping, if there exists a sequence with such that for all and

Remark 1.3.

If then an asymptotically -nonexpansive mapping is asymptotically quasi--nonexpansive. But, there exists a nonlinear continuous asymptotically quasi -nonexpansive mappings which is asymptotically -nonexpansive.

In [21] a weakly convergence theorem for -asymptotically quasi-nonexpansive mapping defined in Hilbert space was proved. In [22] strong convergence of Mann iterations of -nonexpansive mapping has been proved. Best approximation properties of -nonexpansive mappings were investigated in [20]. In [23] the weak convergence of three-step Noor iterative scheme for an -nonexpansive mapping in a Banach space has been established. Recently, in [24] the weak and strong convergence of implicit iteration process to a common fixed point of a finite family of -asymptotically nonexpansive mappings were studied. Assume that the family consists of one -asymptotically nonexpansive mapping . Now let us consider an iteration method used in [24], for , which is defined by

(1.1)

where and are two sequences in . From this formula one can easily see that the employed method, indeed, is not implicit iterative processes. The used process is some kind of modified Ishikawa iteration.

Therefore, in this paper we will extend of the implicit iterative process, defined in [16], to -asymptotically quasi-nonexpansive mapping defined on a uniformly convex Banach space. Namely, let be a nonempty convex subset of a real Banach space and be an asymptotically quasi -nonexpansive mapping, and let be an asymptotically quasi-nonexpansive mapping. Then for given two sequences and in we will consider the following iteration scheme:

(1.2)

In this paper we will prove the weak and strong convergences of the implicit iterative process (1.2) to a common fixed point of and . All results presented here generalize and extend the corresponding main results of [15–17] in a case of one mapping.

2. Preliminaries

Throughout this paper, we always assume that is a real Banach space. We denote by and the set of fixed points and the domain of a mapping respectively. Recall that a Banach space is said to satisfy Opial condition [25], if for each sequence in converging weakly to implies that

(2.1)

for all with It is well known that (see [26]) inequality (2.1) is equivalent to

(2.2)

Definition 2.1.

Let be a closed subset of a real Banach space and let be a mapping.

(i)A mapping is said to be semiclosed (demiclosed) at zero, if for each bounded sequence in the conditions converges weakly to and converges strongly to imply

(ii)A mapping is said to be semicompact, if for any bounded sequence in such that then there exists a subsequence such that strongly.

(iii) is called a uniformly -Lipschitzian mapping, if there exists a constant such that for all and

The following lemmas play an important role in proving our main results.

Lemma 2.2 (see [12]).

Let be a uniformly convex Banach space and let be two constants with Suppose that is a sequence in and and are two sequences in such that

(2.3)

holds some Then

Lemma 2.3 (see [14]).

Let and be two sequences of nonnegative real numbers with If one of the following conditions is satisfied:

(i)

(ii)

then the limit exists.

3. Main Results

In this section we will prove our main results. To formulate one, we need some auxiliary results.

Lemma 3.1.

Let be a real Banach space and let be a nonempty closed convex subset of Let be an asymptotically quasi -nonexpansive mapping with a sequence and be an asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in which satisfy the following conditions:

(i)

(ii)

If is the implicit iterative sequence defined by (1.2), then for each the limit exists.

Proof.

Since for any given it follows from (1.2) that

(3.1)

Again from (1.2) we derive that

(3.2)

which means

(3.3)

Then from (3.3) one finds

(3.4)

and so

(3.5)

By condition (ii) we have and therefore

(3.6)

Hence from (3.5) we obtain

(3.7)

By putting the last inequality can be rewritten as follows:

(3.8)

From condition (i) we find

(3.9)

Denoting in (3.8) one gets

(3.10)

and Lemma 2.3 implies the existence of the limit . This means the limit

(3.11)

exists, where is a constant. This completes the proof.

Now we prove the following result.

Theorem 3.2.

Let be a real Banach space and let be a nonempty closed convex subset of Let be a uniformly -Lipschitzian asymptotically quasi--nonexpansive mapping with a sequence and let be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in which satisfy the following conditions:

(i)

(ii)

Then the implicitly iterative sequence defined by (1.2) converges strongly to a common fixed point in if and only if

(3.12)

Proof.

The necessity of condition (3.12) is obvious. Let us proof the sufficiency part of theorem.

Since are uniformly -Lipschitzian mappings, so and are continuous mappings. Therefore the sets and are closed. Hence is a nonempty closed set.

For any given we have (see (3.8))

(3.13)

here as before with Hence, one finds

(3.14)

From (3.14) due to Lemma 2.3 we obtain the existence of the limit . By condition (3.12), one gets

(3.15)

Let us prove that the sequence converges to a common fixed point of and In fact, due to for all and from (3.13), we obtain

(3.16)

Hence, for any positive integers from (3.16) with we find

(3.17)

which means that

(3.18)

for all , where

Since then for any given there exists a positive integer number such that

(3.19)

Therefore there exists such that

(3.20)

Consequently, for all from (3.18) we derive

(3.21)

which means that the strong convergence of the sequence is a common fixed point of and This proves the required assertion.

We need one more auxiliary result.

Proposition 3.3.

Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of Let be a uniformly -Lipschitzian asymptotically quasi--nonexpansive mapping with a sequence and let be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in which satisfy the following conditions:

(i)

(ii)

(iii)

Then the implicitly iterative sequence defined by (1.2) satisfies the following:

(3.22)

Proof.

First, we will prove that

(3.23)

According to Lemma 3.1 for any we have . It follows from (1.2) that

(3.24)

By means of asymptotically quasi--nonexpansivity of and asymptotically quasi-nonexpansivity of from (3.3) we get

(3.25)

Now using

(3.26)

with (3.25) and applying Lemma 2.2 to (3.24) one finds

(3.27)

Now from (1.2) and (3.27) we infer that

(3.28)

On the other hand, we have

(3.29)

which implies

(3.30)

The last inequality with (3.3) yields that

(3.31)

Then (3.27) and (3.24) with the Squeeze theorem imply that

(3.32)

Again from (1.2) we can see that

(3.33)

From (3.11) one finds

(3.34)

Now applying Lemma 2.2 to (3.33) we obtain

(3.35)

Consider

(3.36)

Then from (3.27), (3.28), and (3.35) we get

(3.37)

Finally, from

(3.38)

with (3.28) and (3.37) we obtain

(3.39)

Analogously, one has

(3.40)

which with (3.28) and (3.35) implies

(3.41)

Now we are ready to formulate one of main results concerning weak convergence of the sequence .

Theorem 3.4.

Let be a real uniformly convex Banach space satisfying Opial condition and let be a nonempty closed convex subset of Let be an identity mapping, let be a uniformly -Lipschitzian asymptotically quasi--nonexpansive mapping with a sequence and, be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in satisfying the following conditions:

(i)

(ii)

(iii)

If the mappings and are semiclosed at zero, then the implicitly iterative sequence defined by (1.2) converges weakly to a common fixed point of and

Proof.

Let , then according to Lemma 3.1 the sequence converges. This provides that is a bounded sequence. Since is uniformly convex, then every bounded subset of is weakly compact. Since is a bounded sequence in then there exists a subsequence such that converges weakly to Hence from (3.39) and (3.41) it follows that

(3.42)

Since the mappings and are semiclosed at zero, therefore, we find and which means

Finally, let us prove that converges weakly to In fact, suppose the contrary, that is, there exists some subsequence such that converges weakly to and . Then by the same method as given above, we can also prove that

Taking and and using the same argument given in the proof of (3.11), we can prove that the limits and exist, and we have

(3.43)

where are two nonnegative numbers. By virtue of the Opial condition of , one finds

(3.44)

This is a contradiction. Hence This implies that converges weakly to This completes the proof of Theorem 3.4.

Now we formulate next result concerning strong convergence of the sequence .

Theorem 3.5.

Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of Let be a uniformly -Lipschitzian asymptotically quasi--nonexpansive mapping with a sequence and be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in satisfying the following conditions:

(i)

(ii)

(iii)

If at least one mapping of the mappings and is semicompact, then the implicitly iterative sequence defined by (1.2) converges strongly to a common fixed point of and

Proof.

Without any loss of generality, we may assume that is semicompact. This with (3.39) means that there exists a subsequence such that strongly and Since are continuous, then from (3.39) and (3.41) we find

(3.45)

This shows that According to Lemma 3.1 the limit exists. Then

(3.46)

which means that converges to This completes the proof.

Note that all results presented here generalize and extend the corresponding main results of [15–17] in a case of one mapping.