Abstract and Applied Analysis

Abstract and Applied Analysis / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 6150398 | https://doi.org/10.1155/2020/6150398

Thanomsak Laokul, "Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition", Abstract and Applied Analysis, vol. 2020, Article ID 6150398, 7 pages, 2020. https://doi.org/10.1155/2020/6150398

Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

Academic Editor: Kunquan Lan
Received24 Oct 2019
Accepted27 Mar 2020
Published18 Apr 2020

Abstract

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).

1. Introduction

Let be a Banach space. We denote by the family of nonempty closed bounded subsets of and the family of nonempty compact subsets of . For any nonempty subset of and , the distance from to is defined by

The radius of relative to is defined by

The diameter of is defined by

The set is said to be bounded if . The Hausdorff distance on is defined by

A multivalued mapping is said to be contractive if there exists a constant such that

If (5) is valid when , then is said to be nonexpansive. It is clear that every contractive mapping is nonexpansive, but, in general, the converse is not true. A point is said to be a fixed point of the multivalued mapping if . We denote by the set of all fixed points of . Additionally, if , then is called an endpoint (or a stationary point) of . We denote by the set of all endpoints of . We can see that for each mapping , . We may say in other words that the concept of endpoints seems to be more complicated than the concept of fixed points. Nevertheless, both concepts are coincident when is a single-valued mapping; in this case, .

Let be a multivalued nonexpansive mapping. Given and , let be defined by

By Nadler’s theorem [1], is a contraction and admits a fixed point , that is,

If is single-valued, then

The strong convergence of the net for a single-valued nonexpansive mapping , which is defined by (8), has been investigated by many authors; especially in 1967, Browder [2] established the great influence to the development of approximated fixed point theory. Some relevant definitions and concept will be given in the next section.

Theorem 1 (Browder [2]). Let be a nonempty bounded closed convex subset of a Hilbert space and be a nonexpansive mapping. Fix and let be defined by (8). Then, converges strongly as to the element of nearest to .

A natural question which arises is whether Browder’s convergence theorem can be extended to the multivalued case. Pietramala [3] (see also Jung [4]) gave the following example which shows that it is not true in general.

Example 1. Let be the square in the real plane and be defined by

It is easy to see that for any ,

This implies that is nonexpansive. It is also easy to see that the fixed point set of is the set . Let ; then, the mapping defined by (6) has the fixed point set . Let

Then, satisfies (7), but it does not converge. This example also shows that the sequence does not converge as to under the Hausdorff metric.

Pietramala [3] extended Browder’s convergence theorem to the multivalued mapping case with the endpoint condition. Acedo and Xu [5] showed the strong convergence of the net defined by (7) in a Hilbert space under the endpoint condition. Later on, it was extended to a Banach space with a sequentially continuous duality mapping by Kim and Jung [6]. Sahu [7] proved the strong convergence of in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Since then, the strong convergence of has been improved and many papers have appeared (see for instance [4, 8, 9]). Jung [4] provided the strong convergence of for a multivalued nonexpansive mapping in a uniformly convex or a reflexive Banach space having a uniformly Gâteaux differentiable norm. He also noted that the endpoint condition should be added in the main results of Sahu [7].

Theorem 2 (Jung [4]). Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , and a multivalued nonexpansive mapping. Suppose that is a nonexpansive retract of . Suppose that satisfies the endpoint condition and that for each and , the contraction defined by (6) has a fixed point . Then, has a fixed point if and only if remains bounded as , and in this case, converges strongly as to a fixed point of .

Very recently, Panyanak and Suantai [10] brought the concept of diametrically regular mappings which is more general than the concept of the endpoint condition to prove Browder’s theorem for multivalued nonexpansive mappings in Hadamard spaces.

Theorem 3 (Panyanak and Suantai [10]). Let be a nonempty closed convex subset of a Hadamard space and be a multivalued nonexpansive mapping. Fix and let be defined by (7). Suppose that is diametrically regular for . Then, has an endpoint if and only if is bounded as . In this case, converges strongly to the unique point in such that

In this paper, we will show that the result of Jung [4] in Theorem 2 can be approached without the endpoint condition by using the notion of diametrically regular mappings. Our results extend and complement many known results including those of [2, 4, 7, 1012].

2. Preliminaries and Lemmas

Throughout this paper, stands for the set of natural numbers and stands for the set of real numbers.

Let be a Banach space and let be a nonempty subset of . A multivalued mapping is said to be diametrically regular if there exists the net in such that . In this case, we call a diametrically regular net for . The mapping is said to satisfy the endpoint condition if for any . Notice that if is a nonempty closed convex subset of a uniformly convex Banach space and is a nonexpansive mapping, then is closed and convex (see [13]).

A Banach space is said to be uniformly convex if for each there exists such that for any the conditions , , and imply

It is well known that if is uniformly convex, then is reflexive and strictly convex (cf. [14]). The norm of is said to be Gâteaux differentiable (and is said to be smooth) if for each in its unit sphere . It is said to be uniformly Gâteaux differentiable if for each , this limit is attained uniformly for . We denote by the (normalized) duality mapping from into , i.e., where denotes the generalized duality pairing.

A subset of is said to be a retract if there exists a continuous mapping with . Any such mapping is a retraction of onto . If is nonexpansive, then is said to be a nonexpansive retract of (cf. [14, 15]).

Let be a linear continuous functional on and . We will sometimes write in place of the value . A linear continuous functional such that and for every is called a Banach limit. We know that if is a Banach limit, then for every . Let be a bounded sequence in . Then, the real-valued continuous function on defined by is convex.

The following lemma was given in [16].

Lemma 4 (see [16]). Let be a nonempty closed convex subset of a Banach space with a uniformly Gteaux differentiable norm and let be a bounded sequence in . Let be a Banach limit and . Then, if and only if for all .

The following result was essentially given by Reich [17] and also proved by Takahashi and Jeong [18].

Lemma 5 (see [18]). Let be a uniformly convex Banach space, a nonempty closed convex subset of , and a bounded sequence in . Then, the set consists of one point.

From now on, we will use the notation (resp., ) for a sequence converging weakly (resp., converging strongly) to a point . The following two lemmas are also needed.

Lemma 6 (see [19]). Let be a nonempty subset of a metric space , be a sequence in , and be a multivalued mapping. Then, if and only if and .

Lemma 7 (see [20]). Let be a nonempty closed convex subset of a uniformly convex Banach space and be a multivalued nonexpansive mapping. Then, the following implication holds:

3. Main Results

We begin this section by proving that the diametric regularity of a nonexpansive mapping with a nonempty fixed point set is weaker than the endpoint condition. Our proof follows the ideas of proof in Panyanak and Suantai [10].

Proposition 8. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , and a multivalued nonexpansive mapping with . Suppose that is a nonexpansive retract of and that for each and , the contraction defined by has a fixed point . Then, the following statement holds:
if satisfies the endpoint condition, then is a diametrically regular net for .

Proof. By Theorem 2, , where . Since satisfies the endpoint condition, . For we have

This implies that , and hence, is diametrically regular for .

The following example shows that the endpoint condition in Proposition 8 is necessary.

Example 2. Let , and be defined by

Then, is a multivalued nonexpansive mapping with and . Since for all , there is no net in which is diametrically regular for .

The main theorem is proved as follows.

Theorem 9. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , and a multivalued nonexpansive mapping. Suppose that is a nonexpansive retract of and that for each and , the contraction defined by has a fixed point . Suppose that is diametrically regular for . Then, has an endpoint if and only if is bounded as , and in this case, converges strongly as to an endpoint of .

Proof. Let . By the compactness of , for each there exists such that . Since is nonexpansive, This implies that and hence, . Therefore is bounded. Conversely, suppose that remains bounded as . Let be a sequence in converging to and put . For each , let be such that . Then, Since is bounded, we define by Since is continuous and convex, as . As is reflexive, attains its infimum over (cf. [21], p.79). Let be such that and let Then, is nonempty because . We also obtain that is a bounded closed convex subset of . Since is a nonexpansive retract of , we assume that is a nonexpansive retract of onto . Then, for any , we have and hence, . This implies that is a global minimum point over all of . Furthermore, is a unique element in by Lemma 5. Since is compact, for each , there exists for such that Let be a convergent subsequence of . Assume that . For sufficiently large, we obtain from (26) and (31) that Thus, . But is the unique global minimum element, . As we know from Lemma 5 that consists of one point, it follows that . We now show that . Take and choose so that . Thus, Since is diametrically regular for and (26), we obtain that Since is the global minimum point over all of , it implies that , and so, . On the other hand, for , we obtain from (24) that It follows that Hence, from (26) and (36), we obtain Thus, for , But by Lemma 4, we have for all . In particular, we have Combining (38) and (39), we get Therefore, there is a subsequence of which converges strongly to . To complete the proof, suppose that there is another subsequence of which converges strongly to (say) . Then, Since is diametrically regular for , we have that as . Thus, from (41) and Lemma 6, we get Since , and (42), we obtain from Lemma 7 that . It follows from (38) that Adding these two inequalities yields Thus, . This proves the strong convergence of .

Remark 10. Theorem 9 is an extension of Theorem 1 in [4] and Theorem 3.5 in [10].

It is well known that any closed convex set in Hilbert spaces is a nonexpansive retract with the proximity map as a nonexpansive retraction. Thus, we obtain the following result.

Corollary 11. Let be a real Hilbert space, a nonempty closed convex subset of , and a multivalued nonexpansive mapping. Suppose that for each and , the contraction defined by has a fixed point . Suppose that is diametrically regular for . Then, has an endpoint if and only if is bounded as , and in this case, converges strongly as to an endpoint of .

Remark 12. Corollary 11 is an extension of Corollary 1 in [4], and also, it is a non-self-mapping case of Theorem 3.6 in [10].

The following example given by Pietramala [3] (see also [10]) shows that the diametric regularity of in Theorem 9 and Corollary 11 is necessary.

Example 3. Let be the Euclidean space and . Let be defined by

Then, is a multivalued nonexpansive mapping with and . Fix . For each , let . It is easy to see that satisfies (7), and it is diametrically regular for . By Corollary 11, . However, if and , then satisfies (7) but it is not diametrically regular for . Moreover, .

Recall that the inward set of at is defined by

It is known that if is convex, then the closure of is for any , where

A mapping is said to satisfy (i) weak inwardness condition if for all and (ii) inwardness condition if for all . For details, we refer to [22].

The weak inwardness condition guarantees the existence of a fixed point of the contraction defined by (6); the weak inwardness condition for multivalued mappings is used. It has recently been shown by Xu [23] that if is a closed bounded convex subset of a uniformly convex Banach space and is a multivalued nonexpansive mapping satisfying the weak inwardness condition, then has a fixed point. Thus, we have the following results.

Corollary 13. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty bounded closed convex subset of , and a multivalued nonexpansive mapping. Suppose in addition that satisfies the weak in wardness condition. Fix and let be defined by (7). If is diametrically regular for , then converges strongly as to the unique point in .

Proof. Fix . For each , define the contraction by

As it is easily seen that also satisfies the week inwardness condition: for all , we have that has a fixed point denoted by . Also, we have that the fixed point set of is nonempty. Thus, the result follows from Theorem 9.

Corollary 14. Let be a real Hilbert space, a nonempty bounded closed convex subset of , and a multivalued nonexpansive mapping which satisfies the weak inwardness condition. Fix and let be defined by (7). If is diametrically regular for , then converges strongly as to an endpoint of .

Remark 15. Corollaries 13 and 14 are extensions of Corollaries 2 and 3 in [4], respectively.

Remark 16. In Theorem 2 of Jung [4], Browder’s convergence theorem for multivalued nonexpansive mappings in a reflexive Banach space with a uniformly Gâteaux differentiable norm can be also proved. In this paper, we relaxed the endpoint condition and we then applied Lemmas 6 and 7, which is a crucial tool, to prove Browder’s convergence theorem for multivalued nonexpansive mappings in uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm. Therefore, the following question should be of interest.

Without the endpoint condition, can the result of Jung [4] (Theorem 2) be approached by using the notion of diametrically regular mappings?

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Dr. Bancha Panyanak for drawing my attention to the subject and for many useful discussions. This research was supported by Chiang Mai University.

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Copyright © 2020 Thanomsak Laokul. 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.


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