Abstract

In this research, under some appropriate conditions, we approximate stationary points of multivalued Suzuki mappings through the modified Agarwal-O’Regan-Sahu iteration process in the setting of 2-uniformly convex hyperbolic spaces. We also provide an illustrative numerical example. Our results improve and extend some recently announced results of the current literature.

1. Introduction

Let be a metric space and be a nonempty subset of . For , set

Let represent the set of all nonempty compact subsets of . The function defined by satisfies all the properties of the metric and is often called the Hausdorff metric on . Recall that a multivalued mapping is called the Suzuki mapping [1] if for each ,

The class of Suzuki mappings and its extensions in the setting of single-valued mappings are widely studied by many authors (see, e.g., [27] and references therein). Here, we will only focus on the multivalued version of Suzuki mappings. We can easily observe that if is nonexpansive, that is, for all , then is also a Suzuki mapping. Nevertheless, the following example shows that the converse of this statement may not hold in general.

Example 1. Let and be defined by

When and , then we have nothing to prove, because in this case, we have the following:

When and . In this case, is nonexpansive, and hence, is a Suzuki mapping.

By setting and , we have . Hence, is not nonexpansive.

An element in is called a stationary point (or called an endpoint) of whenever and is called a fixed point of whenever . Throughout the paper, the notations and will represent the set of all stationary points and the set of all fixed points of , respectively. Recall that a multivalued mapping is called quasinonexpansive provided that for each and . Existence of stationary points for different types of multivalued mappings is studied in [815]. The following statements hold: (i)(ii) if and only if (iii) if and only if

In 2005, Sastry and Babu [16] published a paper on the strong convergence of the fixed point for multivalued nonexpansive mappings using modified Mann and Ishikawa iterative processes in the setting of Hilbert spaces. In the year 2008, Panyanak [17] showed that the results of Sastry and Babu [16] can be extended to the slightly general context of uniformly convex Banach spaces. Song and Wang [18] improved the results of Panyanak [17]. For more details in this direction, see [19, 20] and others. In the year 2018, Panyanak [21] published a paper on the approximation of stationary points of multivalued nonexpansive mappings in the framework of Banach spaces using the modified Ishikawa iterative process. In 2019, Ullah et al. [22] quickly noted that the results of Panyanak [21] can be extended to the general context of CAT(0) spaces. In 2020, Laokul and Panyanak [23] used the Ishikawa iterative process for finding stationary points of multivalued Suzuki mappings in 2-uniformly convex hyperbolic spaces. Recently, Abdeljawad et al. [24] used the modified Agarwal-O’Regan-Sahu iterative process for finding stationary points of multivalued nonexpansive mappings in Banach spaces. Very recently, Ullah et al. [25] extended the results of Abdeljawad et al. [24] to the general context of 2-uniformly convex hyperbolic spaces. The modified Agarwal-O’Regan-Sahu iteration process reads as follows: where such that and such that . The purpose of this work is to prove, under some appropriate conditions, the strong and convergence results of stationary points for a wider class of multivalued nonexpansive mappings so-called multivalued Suzuki mappings using iterative process (6) in the general setting of 2-uniformly convex hyperbolic spaces. In this way, we improve and extend the corresponding results proved in [2125].

Now we recall some basic definitions and results, which will be used in the sequel.

Definition 2 [26]. A hyperbolic space is a metric space endowed with a function such that for all and , we have

If and , we use the notation for . It follows from that

The set is called convex if for any , one has .

Definition 3. A hyperbolic space is said to be uniformly convex if for every real number and , we can choose a such that for each with , , and , one has

A function providing such for given and is called a modulus of uniform convexity. The function is called monotone provided that it is nonincreasing in for each fixed .

Definition 4. Let be a uniformly convex hyperbolic space. For every and , set where the infimum is taken over each such that , and . We say that is -uniformly convex if

Remark 5. Notice that uniformly convex Banach space and CAT(0) spaces as well as CAT() spaces ( and for some are -uniformly convex hyperbolic spaces (see [23, 27, 28]).

Definition 6. Let be any bounded sequence in a complete 2-uniformly convex hyperbolic space and . The asymptotic radius of relative to is . Moreover, the asymptotic center of relative to is the set .

Definition 7. Let be a nonempty closed convex subset in a complete 2-uniformly convex hyperbolic space and . Let be any bounded sequence in . We say that -converges to if for each subsequence of . In this case, we write and call the of .

Now, we collect some basic facts about multivalued Suzuki mappings, which can be found in [2931].

Proposition 8. Let be a nonempty subset of a complete 2-uniformly convex hyperbolic space and . (i)If is a Suzuki mapping with a nonempty fixed point set, then is quasinonexpansive(ii)If is a Suzuki mapping, then the following holds:

The following facts are also needed.

Lemma 9 (see [17]). Let be such that and . Let be a sequence of nonnegative real numbers such that . Then, has a subsequence which converges to 0.

The following lemma is a characterization of 2-uniformly convex hyperbolic spaces.

Lemma 10 (see [23]). Let be a -uniformly convex hyperbolic space. Then, for each and .

Lemma 11 (see [23]). Let be a nonempty closed convex subset of a2-uniformly convex hyperbolic space and be a Suzuki mapping. Suppose that is a bounded sequence in such that and converges for every , then . Here, where the union is taken over all subsequences of . Furthermore, is a singleton.

2. Convergence Theorems in 2-Uniformly Convex Hyperbolic Spaces

Throughout the section, will stand for a complete 2-uniformly convex hyperbolic space with monotone modulus of uniform convexity.

The following lemma is crucial.

Lemma 12. Let be a nonempty closed convex subset of and be a Suzuki mapping with . Let be the sequence defined by (6). Then, exists for all .

Proof. Let . By Proposition 8(i), we have Hence, is a nonincreasing sequence, which implies exists for every .

First, we establish our -convergence theorem.

Theorem 13. Let be a nonempty closed convex subset of and be a Suzuki mapping with . Let and be the sequence defined by (6). Then, -converges to a stationary point of .

Proof. Fix . By Lemma 10, we have Thus, Since , it follows that Thus, and hence By Lemma 12, converges for all . By Lemma 11, is a singleton and is contained in . This shows that -converges to a point of .

Definition 14 (see [21]). Let be a nonempty subset of and . is said to satisfy condition if there is a nondecreasing function with the properties , for and for all . is called semicompact if for each sequence in satisfying one can find a strongly convergent subsequence of . Moreover, a sequence in is called a Fejér monotone with respect to provided that for each and .

The following facts are in [31].

Proposition 15. Let be a nonempty closed subset of and be a Fejér monotone sequence with respect to . Then, converges strongly to a point of if and only if .

The following theorem is based on the semicompactness of .

Theorem 16. Let be a nonempty closed convex subset of and be a Suzuki mapping with . Suppose be such that and . If is semicompact, then generated by (6) converges strongly to a stationary point of .

Proof. In view of (16), By Lemma 9, there exists subsequences, namely, and of and , respectively, such that . Hence, By the semicompactness of the mapping , one can find a strongly convergent subsequence of with the strong limit, say . We shall prove that . By Proposition 8(ii), we have Hence, . By Proposition 8(i), Now, we let and choose such that From (19) and (21), we have Hence, for all , that is, . Therefore, . By Lemma 12, exists. Hence, is the strong limit of .

Example 17. Let and be as in Example 1. Then, is a Suzuki mapping with . Semicompactness of follows from the compactness of . For each , we let and . The, and . Hence, by Theorem 16, the sequence of the modified Agarwal-O’Regan-Sahu iteration defined by (6) converges strongly to . However, we cannot directly apply any result in [22, 24, 25] because, in this situation, is not nonexpansive.

The following theorem requires condition .

Theorem 18. Let be a nonempty closed convex subset of and be a Suzuki mapping with . Suppose . If satisfies condition , then generated by (6) converges strongly to a stationary point of .

Proof. From (17), we have Since satisfies condition , we have So, from (23), we get Since the function is nondecreasing and and for every . Hence, Closeness of follows from the quasinonexpansiveness of . In the view of Lemma 12, we have which is a Fejér monotone with respect to . By Proposition 15, converges strongly to an element of .

Example 19. Let and be as in Example 1. Then, is a Suzuki mapping with and satisfies the condition . For each , we let . Then, by Theorem 18, the sequence of modified Agarwal-O’Regan-Sahu iteration defined by (6) converges strongly to . However, we cannot directly apply any result in [22, 24, 25] because, in this situation, is not nonexpansive.

3. Conclusions

Under some appropriate conditions, we have proven that the sequence of the modified Agarwal-O’Regan-Sahu iterative process defined by (6) converges to a stationary point of a multivalued Suzuki mapping. We have used a 2-uniformly convex hyperbolic space, which is more general than the uniformly convex Banach spaces, CAT(0) spaces, and some CAT() spaces. Moreover, the class of multivalued Suzuki mappings properly includes the class of multivalued nonexpansive mappings as shown by Example 1. Our iterative process is independent of but better than the Ishikawa iterative process. Hence, our presented results extend and improve the corresponding results in [2125].

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 authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/EU FEDER) and to the Basque Government for Grant IT1207-19.