Iterative Approximation of Endpoints for Multivalued Mappings in Banach Spaces

Abstract

The purpose of this paper is to introduce the modified Agarwal-O’Regan-Sahu iteration process (S-iteration) for finding endpoints of multivalued nonexpansive mappings in the setting of Banach spaces. Under suitable conditions, some weak and strong convergence results of the iterative sequence generated by the proposed process are proved. Our results especially improve and unify some recent results of Panyanak (J. Fixed Point Theory Appl. (2018)). At the end of the paper, we offer an example to illustrate the main results.

1. Introduction and Preliminaries

Throughout this paper, stands for the set of natural numbers, and stands for the set of real numbers. Let be a Banach space and C be a nonempty subset of X. For , set

We shall denote the set of all nonempty and compact subsets of C by . Set is known as the Hausdorff metric on the set . A multivalued mapping is said to be nonexpansive if

A point is said to be a fixed point of if and is said to be an endpoint (or a stationary point) of if . From now on, we will denote the set of all endpoints and the set of all fixed points of T by and , respectively. Note that, a multivalued mapping is said to satisfy the endpoint condition [1] if . Different iteration processes have been developed to approximate the fixed points of multivalued mappings. Keep in mind, Sastry and Babu [2] proved Mann and Ishikawa-type convergence results for multivalued nonexpansive mappings in the framework of Hilbert spaces. Panyanak [3] extended the results of Sastry and Babu [2] to the framework of uniformly convex Banach spaces. Actually, Panyanak [3] showed some results using Ishikawa-type iteration process without the endpoint condition. Song and Wang [4] showed that without the endpoint condition, their process was not well-defined. They reconstructed the process using the endpoint condition which made it well-defined. After this, Shahzad and Zegeye [5] introduced two types of Ishikawa iterations. Note that, their first type iteration also requires the endpoint condition.

For a multivalued mapping , if is an endpoint of T, then q is also a fixed point of T; but the converse is not always true (see the following example).

Example 1. Let . Define byCleary and .
For existence results of endpoints of multivalued mappings in the framework of Banach spaces, see [6–12]. Very recently, Panyanak [13] used an Ishikawa-type iteration process to approximate endpoints of multivalued nonexpansive mappings in the setting of Banach spaces.
Agarwal et al. [14] introduced an iteration process known as S-iteration process, which is independent of both Mann [15] and Ishikawa [16] iterations, for a single-valued mappings in Banach spaces:where . They proved that the rate of convergence of iteration process (4) is the same as Picard iteration process and faster than Mann [15] iteration process for the class of contraction mappings. Later, it was observed that this scheme also converges faster than Ishikawa [16] iteration process (see e.g., [17]; for more details and some recent literature of S-iteration process, see [18–24]).
Keeping above in mind, we introduce our iteration process as follows:where such that and such that .
In this way, we approximate endpoints of multivalued nonexpansive mappings by an iteration process which is independent of but faster than Ishikawa iteration process. Thus, our results improve and unify corresponding results of Panyanak [13] and references therein.

Definition 1. A Banach space X is said to be uniformly convex if for each , there is a such that for every ,

Definition 2. (see [25]). A Banach space X is said to have Opial’s property if for each sequence in X which weakly converges to and for every , it follows thatExamples of Banach spaces satisfying this condition are Hilbert spaces and all spaces ().

Definition 3. (see [13]). Let C be a nonempty subset of a Banach space X. A mapping is said to satisfy condition if there is a nondecreasing function with , for such thatfor each .

Definition 4. (see [13]). Let C be a nonempty subset of a Banach space X. A mapping is said to be semicompact if for every sequence in C such thatthere is a subsequence of such that for some . We see that, if C is compact then every multivalued mapping is semicompact.

Definition 5. Let C be a nonempty subset of a Banach space XA sequence in X is called Fejer-monotone with respect to C iffor each and .
The following important lemma is due to Xu [26].

Lemma 1. A Banach space X is uniformly convex if and only if for any number , and there is a strictly increasing and continuous function with such thatfor each with , , and .
The following lemma can be found in [13].

Lemma 2. For a multivalued mapping , the following statements hold.(a) x is a fixed point of T(b) x is an endpoint of T(c)If T is nonexpansive, then the mapping defined by is continuous

Lemma 3. (see [3]) Let , be two real sequences such that(a)(b)(c)Let be a sequence of nonnegative real numbers such that , then has a subsequence which converges to 0.

Lemma 4. (see [27]). Let C be a nonempty closed and convex subset of a uniformly convex Banach space and be a multivalued nonexpansive mapping. Then, the following is true:The following fact is needed which can be found in [28].

Proposition 1. Let C be a nonempty closed subset of a Banach space. Let be a Fejer-monotone sequence with respect to C. Then, converges (strongly) to the point of C if and only if .

2. Main Results

The following lemma is crucial.

Lemma 5. Let C be a nonempty closed convex subset of a uniformly convex Banach space X and be a multivalued nonexpansive mapping with . Let be a sequence defined by (5). Then, exists for each .

Proof. Let . For each , we havewhich implies thatHence, is a nonincreasing sequence, which implies exists for all .
First, we prove our weak convergence result.

Theorem 1. Let X be a uniformly convex Banach space with the Opial property, C be a nonempty closed convex subset of X, and be a multivalued nonexpansive mapping with . Let and be a sequence defined by (5). Then, converges weakly to an element of .
Proof. Fix . By Lemma 1., there exists a strictly increasing continuous function with such that

Thus,

It follows that

Thus, . But ψ is strictly increasing and continuous, we have . Hence,

To show that converges weakly to an element of , it suffices to show that has a unique weak subsequential limit in . For this purpose, we assume that there are subsequences and of such that and . By (18), . By Lemma 4, . Similarly, it can be shown that . Next, we prove . On the contrary suppose , then by Lemma 5 together withOpial’s property, we havewhich is a contradiction. So, . Hence proved.

Now, we approximate endpoints of the mapping T through strong convergence of the sequence defined by (5).

Theorem 2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X and be a multivalued nonexpansive mapping with . Let be such that and and let be a sequence defined by (5). If T is semicompact, then converges strongly to an element of .
Proof. In view of (17),

By Lemma 3, subsequences and of and exists, respectively, such that . Since ψ is strictly increasing and continuous, we have . Hence,

On the contrary, T is semicompact; we may choose by passing through a subsequence that for some . We need to show and . By Lemma 2 part (c), together with (21), we have

It follows from Lemma 2 part (b) that . By Lemma 5, exists, and hence p is the strong limit of .

In the next strong convergence result, we relax semicompactness of T with the help of condition (J).

Theorem 3. Let C be a nonempty closed convex subset of a uniformly convex Banach space X and be a multivalued nonexpansive mapping with . Let and be a sequence defined by (5). If T satisfies condition (J), then converges strongly to an element of .

Proof. Since T satisfies condition , by (18), we get that . Closeness of follows from the nonexpansiveness of T. In the view of Lemma 5, we have is Fejer-monotone with respect to . By Proposition 1, converges strongly to an element of .
Here is an example in support of the main theorems.

Example 2. Let and with absolute valued norm. Define a multivalued mapping by for each . Clearly T is semicompact and nonexpansive with . We prove that T satisfies condition with . When , we haveNext, we prove that defined by (5) strongly converges to 2.
Choose , for all . Let , then . Take such that;That is, . So,Now, . Choose such that . Similar calculation to the above gives . Hence,Continuing in this manner, , for all , and hence converges strongly to .

Remark 1. We see in Example 2. that, and , i.e., T does not satisfy the endpoint condition. Therefore, we cannot directly apply any result in [19, 22, 23].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The first and the fourth authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number RG-DES-2017-01-17).