Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces

Abstract

Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping. For the implicit iterates , , , . We proved that converges strongly to a fixed point of under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007).

1. Introduction

Let be a nonempty subset of a Banach space . We will denote by the family of all subsets of the family of nonempty closed and bounded subsets of the family of nonempty compact subsets of . Let symbolize the family of nonempty compact convex subsets of . Let be Hausdorff metric on ; that is,

(1.1)

where . A multivalued mapping is called nonexpansive (resp., contractive), if for any , there holds

(1.2)

A point is called a fixed point of if . In this paper, stands for the fixed point set of a mapping .

The fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings.

In 1968, Markin [1] firstly established the nonexpansive multivalued convergence results in Hilbert space. Banach's Contraction Principle was extended to a multivalued contraction in 1969. (Below is stated in a Banach space setting.)

Theorem 1.1 (see [2]).

Let be a nonempty closed subset of a Banach space and a multivalued contraction. Then has a fixed point.

In 1974, one breakthrough was achieved by Lim using Edelstein's method of asymptotic centers [3].

Theorem 1.2 (see Lim [3]).

Let be a nonempty closed bounded convex subset of a uniformly convex Banach space and a multivalued nonexpansive mapping. Then has a fixed point.

In 1990, Kirk and Massa [4] obtained another important result for multivalued nonexpansive mappings.

Theorem 1.3 (see Kirk and Massa [4]).

Let be a nonempty closed bounded convex subset of a Banach space and a multivalued nonexpansive mapping. Suppose that the asymptotic center in E of each bounded sequence of X is nonempty and compact. Then T has a fixed point.

In 1999, Sahu [5] obtained the strong convergence theorems of the nonexpansive type and nonself multivalued mappings for the following (1.3) iteration process:

(1.3)

where and . He proved that converges strongly to some fixed points of . Xu [6] extended Theorem 1.3 to a multivalued nonexpansive nonself mapping and obtained the fixed theorem in 2001. The recent fixed point results for nonexpansive mappings can be found in [7–12] and references therein.

Recently, Panyanak [13] studied the Mann iteration (1.4) and Ishikawa iterative processes (1.5) for as follows:

(1.4)

where , and fixed are such that ,

(1.5)

where , and fixed are such that and and proved the strong convergence theorems for multivalued nonexpansive mappings in Banach spaces.

In this paper, motivated by Panyanak [13] and the previous results, we will study the following iteration process (1.6). Let be a nonempty convex subset of ,

(1.6)

and we prove some strong convergence theorems of the sequence defined by (1.6) for nonexpansive multivalued mappings in Banach spaces. The results presented in this paper establish a new type iteration convergence theorems for multivalued nonexpansive mappings in Banach spaces and extend the corresponding results of Panyanak [13].

2. Preliminaries

Let be a real Banach space and let denote the normalized duality mapping from to defined by

(2.1)

where denotes the dual space of and denotes the generalized duality pair. It is well known that if is strictly convex, then is single valued. And if Banach space admits sequentially continuous duality mapping from weak topology to weak star topology, then, by [14, Lemma  1], we know that the duality mapping is also single valued. In this case, the duality mapping is also said to be weakly sequentially continuous; that is, if is a subject of with , then . By Theorem  1 of [14], we know that if admits a weakly sequentially continuous duality mapping, then satisfies Opial's condition, and is smooth; for the details, see [14]. In the sequel, we will denote the single-valued duality mapping by .

Throughout this paper, we write (resp., ) to indicate that the sequence weakly (resp., weak *) converges to , as usual will symbolize strong convergence. In order to show our main results, the following concepts and lemmas are needed.

A Banach space is called uniformly convex if for each there is a such that for with and holds. The modulus of convexity of is defined by

(2.2)

for all . is said to be uniformly convex if , and for all .

Lemma 2.1 (see [10]).

In Banach space , the following result (subdifferential inequality) is well known: for all , for all , for all ,

(2.3)

Definition 2.2.

A Banach space is said to satisfy Opial's condition if for any sequence in () implies

(2.4)

We know that Hilbert spaces, , and Banach space with weakly sequentially continuous duality mappings satisfy Opial's condition; for the details, see [14, 15].

Definition 2.3.

A multivalued mapping is said to satisfy Condition  if there is a nondecreasing function with for such that

(2.5)

Example of mappings that satisfy Condition   can be founded in [13].

3. Main Results

Now, we prove our results.

Theorem 3.1.

Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping, where and , the sequence is generated by (1.6).

Then,

(i)by the compactness of , there exists a subsequence of such that for some . In addition if then

(ii) is a fixed point of and the sequence converges strongly to .

Proof.

Part (i) is trivial. And part (ii) remains to be proved.

Due to the compactness of and boundness of , there exists a real number such that

(3.1)

It follows from (1.6), that

(3.2)

thus

(3.3)

therefore

(3.4)

so

(3.5)

Hence, is a fixed point of .

Next we show that exists.

For all , there exist , using Lemma 2.1, we obtain

(3.6)

so

(3.7)

If , then apparently holds.

Let , from (3.7), we have

(3.8)

We get that is a decreasing sequence, so

(3.9)

So the desired conclusion follows.

The proof is completed.

Remark 3.2.

The above result modified the gap in the proof of Theorem  3.1 in [13] by a new method; the gap discovered by Song and Wang [16] is as follows.

Panyanak [13] introduced the Ishikawa iterates (1.5) of a multivalued mapping . It is obvious that depends on and . For , we have

(3.10)

Clearly, if and , then the above inequalities cannot be assured. Indeed, from the monotone decreasing sequence of in the proof of (Theorem  3.1 [13]), we cannot obtain that is a decreasing sequence. Hence, the conclusion of Theorem  3.1 in [13] cannot be achieved.

Theorem 3.3.

Let be a Banach space satisfying Opial's condition and let be a nonempty weakly compact convex subset of . Suppose that is a multivalued nonexpansive mapping, where and , the sequence is generated by (1.6).

Then,

(i)by the weak compactness of , there exists a subsequence of such that for some . In addition if, then

(ii) is a fixed point of and the sequence converges weakly to .

Proof.

Part (i) is trivial. Now we prove part (ii).

It follows from (3.3) of Theorem 3.1 that

(3.11)

Since is weakly compact, from part (i), there exists a subsequence of such that

(3.12)

Suppose that does not belong to . By the compactness of , for any given , there exist such that and .

Thus , from

(3.13)

This is a contradiction by satisfying Opial's condition.

Hence, is a fixed point of .

It follows from (3.7) of Theorem 3.1 that

(3.14)

Next we show . Suppose not. There exists another subsequence of such that .

Then, we also obtain . From Opial's condition, we have

(3.15)

Which is a contradiction, so the conclusion of the theorem follows.

The proof is completed.

Corollary 3.4.

Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping from to , and let be a nonempty weakly compact convex subset of . Suppose that is a multivalued nonexpansive mapping, where and , the sequence is generated by (1.6).

Then,

(i)by the weak compactness of , there exists a subsequence of such that for some . In addition if, then

(ii) is a fixed point of and the sequence converges weakly to .

Proposition 3.5.

Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping. Then is a closed subset of .

Proof.

Suppose , such that , then we have

(3.16)

so

(3.17)

Hence, is a fixed point of .

Thus, is a closed subset of .

The proof is completed.

Theorem 3.6.

Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping satisfying Condition  , where and , then the sequence generated by (1.6) converges strongly to a fixed point.

Proof.

It follows from (3.3) of Theorem 3.1 that

(3.18)

The proof of remained part is omitted because it is similar to the proof of Theorem  3.8 in [13].