Fixed Point Results for Multivalued Mappings with Applications

Khan, Arshad; Sarwar, Muhammad; Khan, Farhan; Alsamir, Habes; Hammad, Hasanen A.

doi:https://doi.org/10.1155/2021/9921728

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Unique and Non-Unique Fixed Points and their Applications

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Volume 2021 | Article ID 9921728 | https://doi.org/10.1155/2021/9921728

Fixed Point Results for Multivalued Mappings with Applications

Arshad Khan,¹Muhammad Sarwar,¹Farhan Khan,¹Habes Alsamir,²and Hasanen A. Hammad³

Academic Editor: Santosh Kumar

Received17 Mar 2021

Revised08 Jul 2021

Accepted07 Aug 2021

Published24 Aug 2021

Abstract

In this manuscript, using the concept of multivalued contractions, some new Banach- and Caristi-type fixed point results are established in the context of metric spaces. For the reliability of the presented results, some examples and applications to Volterra integral type inclusion are also studied. The established results unify and generalize some existing results from the literature.

1. Introduction and Preliminaries

Volterra integral equations appear in different scientific applications such as in the spread of epidemics, semiconductor devices, and population dynamics. Also, the dynamics of multispan uniform continuous beams subjected to a moving load is one of the best applications of Volterra integral equations. A universal method for finding sorption uptake curves of fluid multicomponent mixtures in porous solid at variable and constant concentration of mixture components on the basis of the Volterra integral equation has been proposed. The fixed point theory for multivalued mappings is a serious subject of set-valued analysis. In Banach spaces, several well-known fixed point theorems of single-valued mappings such as Banach and Schauder have been extended to multivalued mappings. There are a lot of applications for multivalued mappings such as optimal control theory, differential inclusions, game theory, and many branches in physics.

The Banach contraction principle is used in a variety of fields of mathematics. This technique has many applications in studying the existence of solutions for nonlinear Volterra integral equations, and nonlinear integrodifferential equations in Banach spaces.

Recently, it has been widely spread. For example, in abstract spaces, the Fredholm integral equation introduced by Fredholm [1] and the solutions of Fredholm- and Volterra-type integral equations have been discussed analytically by Rezan et al. [2–4], Rus [5], Aydi et al. [6], Karapinar et al. [7], and Hammad and De la Sen [8, 9] and numerically by Panda et al. [10, 11] and Berenguer et al. [12].

Throughout this paper, the symbols , , , and refer to the family of all nonempty, nonempty closed, nonempty closed bounded, and nonempty compact subset of , respectively.

For multivalued contraction mappings, we have the following.

Definition 1 (see [13]). Suppose that is a metric space. Denote by , the set of all nonempty closed bounded subsets of . The Pompeiu-Hausdorff metric induced by the distance is formulated as follows: for all where Also, the pair is called a generalized Hausdorff distance induced by

In 1969, the stipulation of Banach in single-valued mappings was modified to multivalued mappings by Nadler [13] as follows.

Theorem 2. Let be a complete metric space (CMS) and be a multivalued mapping on so that is a nonempty closed, bounded subset of If for each there is so that then, has a fixed point in

In the literature, via abstract spaces, some authors obtained nice fixed point results for contractive mappings under certain conditions, for example, Hussain et al. [14] prove the existence of several fixed point results in ordinary and partially order metric spaces by studying the notion of Geraghty-type contractive mapping via simulation function along with -class function. In integral type, Branciari [15] introduced some common fixed point results under general contractive conditions.

In Meir-Keeler type, Agarwal et al. [16] obtained some exciting fixed point results. In Menger probabilistic metric spaces, Chauhan et al. [17] discussed a hybrid coincidence and common fixed point theorem under a strict contractive condition with an application. Via the notion of -admissible mapping, the existence of fixed point theorems under -distance mappings with an application is presented by Kutbi and Sintunavarat [18], and the others concerned with studying the notion of multivalued mapping and its contributions in fixed point theory such as Nadler [13], Ali and Kamran [19], Aubin and Siegel [20], Covitz and Nadler [21], Hot [22], and Ali et al. [23].

Du and Karapinar [24] introduce the concept of a Caristi-type cyclic map and present a new convergence theorem and a best proximity point theorem for Caristi-type cyclic maps. Petrusel and Sîntmarian [25] obtained a new result in the link of single-valued and multivalued Caristi-type mappings. Hussain et al. [26] introduce the notion of Suzuki-type multivalued contraction with simulation functions and then set up some new fixed point and data dependence results for these types of contraction mappings. Karapinar [27] used lower semicontinuous mappings to generalize Caristi-Kirk’s fixed point theorem on partial metric spaces. Abdeljawad and Karapinar [28] generalize Cristi-Kirik’s fixed point theorem to Cone metric spaces using Cone-valued lower semicontinuous maps. The relation between Caristi’s result and its restriction to the function verifying Caristi’s stipulations with continuous real functions is explained by Jachymski [29]. Khojasteh et al. [30] introduce the idea that many known fixed point theorems can easily be derived from the Caristi theorem. Also, Karapinar et al. [31] proposed a new fixed point theorem which is inspired from both Caristi and Banach.

In 2013, Ali et al. [32] initiated the idea of generalized -admissible mappings. Via this concept, fixed point consequences to generalized Mizoguchi’s fixed point theorem are derived. For single-valued mappings, Caristi [33] introduced an important theorem in fixed point field and called it the “Caristi fixed point theorem.” This theorem was generalized to multivalued mappings in Banach spaces by Feng and Liu [34] as follows:

Theorem 3. Assume that is a CMS and is a multivalued mapping so that is a closed subset of if there is a constant so that for all and where then, there is a fixed point of in with and is lower semicontinuous.

Theorem 4. Assume that is a CMS and is a multivalued mapping so that is a nonempty subset of Let be a lower semicontinuous and bounded from below function and be a nondecreasing, continuous, and subadditive function so that If for each there is an element verifying then, the mapping has a fixed point on

Recently, Isik et al. [35] generalized Banach contraction and Caristi’s fixed point theorem for a single-valued map as the following:

Theorem 5. Let be a CMS and be a self-continuous mapping, if there is the mapping so that , and then, there is a unique fixed point of .

Theorem 6. Assume that be a CMS and is a self-mapping. Let be a set of mappings so that the hypotheses in the following hold: (1) is continuous and strictly increasing(2)For each sequence if and only if (3)For all If the function is lower semicontinuous and bounded from as follows so that then, has a fixed point.

Definition 7 (see [36]). A function is called subadditive if for every

Definition 8 (see [13]). Suppose that be a multivalued mapping, a point is called a fixed point of if

Definition 9 (see [37]). Let be a metric space and be nonempty subsets of . Then, a set-valued mapping is called a set-valued cyclic map if and , where .

The lemma in the following is very useful in the sequel.

Lemma 10 (Zorn’s lemma). Let be a partially order set. If every totally ordered subset of has an upper bound then contains a maximal element.

Lemma 11 (see [10]). Assume that and are closed and bounded subset of and let For each positive then there is so that Moreover, if is a compact then there is so that

The focus of this work is extending Theorems 3, 4, 5, and 6 for multivalued mappings via generalized contractive conditions. An example and application for the existence of solution of Volterra integral inclusion are also given.

2. Fixed Point Results

We begin this section with the first main result.

Theorem 12. Let be a complete metric space and be a multivalued mapping. If there is a nondecreasing mapping so that and for all , there is , so that where and is lower semicontinuous; then, has a fixed point in .

Proof. Let be an arbitrary element in Since is nonempty for any we can build a sequence with for each so that Since , then, we can write Because the mapping is nondecreasing, so by (9), we have This implies that the sequence is nonincreasing. Since is bounded as follows, then there is so that as
Now, for with we get Passing the limit in the above inequality as , one can write this proves that is a Cauchy sequence. The completeness of leads to there is so that Since is lower semicontinuous, decreasing, is nondecreasing and by (9), we obtain that or, equivalently, this implies therefore, converges to zero. Again, because is lower semicontinuous, then, we have Hence, also, the closed property of implies that ☐

Remark 13. Theorem 12 is more general than Theorem 3, because if verifies the stipulation of Theorem 3, then for each there is so that this insinuate that this equivalent to yields Setting , we have

Moreover, is lower semicontinuous; therefore, has a fixed point by Theorem 12.

If we put for all in Theorem 12, we get the result as follows.

Corollary 14. Suppose that is a CMS and is a multivalued mapping. If for any there is so that then, there exists a fixed point of

Theorem 15. Let be a CMS and be a multivalued mapping, where is a compact subset of . If there exists a nondecreasing mapping so that , and then, there is a fixed point of in

Proof. Consider is an arbitrary point of According to Lemma 11, construct a sequence with for all so that this implies that is a nonincreasing sequence, as is bounded from the following so there is so that
To show that is a Cauchy sequence, let with and applying Lemma 11, we have As we get Thus, is a Cauchy sequence. The completeness of leads to there is so that Thus, the sequence converges to . Since for each this implies that This finishes the proof.☐

Example 1. Let It is obvious that the pair is complete metric space. Define a multivalued mapping by

Now, we verify that the multivalued mapping satisfied the condition of Theorem 15 with Note that iff or so we have the following two cases.

Case 1. If and , we have That is, which implies that

Case 2. If , then, we have that is, therefore,

Hence, the condition of Theorem 15 is satisfied. Also, is a fixed point of

Example 2. Suppose that for all . It is obvious that the pair is a CMS. Define a multivalued mapping by then, we have it follows from (37) and (38) that

Hence, satisfies the condition of Theorem 15 with

Remark 16. Theorem 15 upgrades the Nadler fixed point result in a finite-dimensional space. Indeed, if holds condition of Nadler’s theorem, then for we have or, equivalently, yields

Taking we get moreover, for each but is a finite-dimensional space; therefore, is compact and hence, by Theorem 15, there is a fixed point of

Let be the set of all mappings having the same stipulations (1)–(3) of Theorem 6. Note: by property (2), of Theorem 6, we have if and only if

Example 3. All the functions in the following belong to (i)(ii)(iii)The lemmas in the following help us to supplement the theoretical results.

Lemma 17. Assume that is a CMS and is a multivalued mapping. Let be a function defined in Theorem 6 and Define the relation “” on so that then, the relation “” is a partial order on and is a partial order space.

Proof. Since iff , then, we have this shows that
If and , then because ; thus, Moreover, iff ; therefore, ; this implies that
Finally, if and then it follows from that From the above results, we conclude that “” is a partial order on ☐

Lemma 18. Let the pair be a CMS and be a multivalued mapping. Consider be a function defined in Theorem 6, and is a nondecreasing, continuous, and subadditive function that ensures that Define the relation “” on by then, the relation “” is a partial on and is a partial order space.

Proof. Since iff and then, we get this implies that
If and then Since thus Moreover, iff and ; therefore, i.e.,
Again, if and then it follows from , and is nondecreasing and subadditive that and this completes the required. So, “” is a partial order on ☐

Now, we can state and prove the next main theorems.

Theorem 19. Let be a CMS, be a multivalued mapping, be a function defined as Theorem 6, and If for all , there is so that then, the mapping has a fixed point in

Proof. Since is a partial order space, then we need to prove only that has a maximal element. Suppose that is increasing sequence in , that is, for with then from (54), we have is decreasing, since is bounded as follows then holds. Assume that is increasing sequence in that ensures Now, for with , then and
Applying (54), we get As in the above inequality, we have Since is continuous so, we obtain Hence, is a Cauchy sequence in Since is complete then there exists so that
as Thus, is an upper bound for for each Now, we want to show that is also an upper bound for Suppose that with for each Then, by (54), we obtain that for each Also, from (54), we have Since then, by taking the limit as we get Hence, for each there is so that also yields for each Thus, is an upper bound for By Zorn’s lemma, has a maximal element The condition implies that this shows that Since is maximal, therefore, Therefore, This ends the proof.☐

Theorem 20. Let be a CMS, be a multivalued mapping, be a function defined as Theorem 6, , and be a nondecreasing continuous subadditive function such that . If for any , there is so that then, there is a fixed point of in .

Proof. By the same manner of the proof of Theorem 19, we can easily show that there is a maximal point of partial order space , and by hypothesis, there is that ensures This implies that As is a maximal element of therefore, ; hence, This completes the proof.☐

3. Supportive Application

Here, we use Theorem 15 with to determine the existence of a solution to the Volterra-type integral inclusion of the following form: where , refers to the class of nonempty compact and convex subset of and is a continuous function.

Let be the space of all continuous real-valued functions on Define the distance for all It is clear that the pair is a metric space.

We shall consider Problem (63) under the hypotheses as follows: (1) and are continuous functions, as well as is compact and convex(2)There is a continuous function so thatfor each and (3)We have

Now, our main theorem in this part becomes valid for showing.

Theorem 21. Under the assumptions (1)–(3), Problem (63) has a solution on

Proof. Define the multivalued mapping by ☐

The unique solution of Problem (63) is equivalent to find a fixed point of in

It is obvious that is compact because for any is lower semicontinuous. For clarification, let for each Thus, by Michael selection theorem [38], there is a continuous function so that It follows that this implies that is nonempty. Hence, it is compact.

Proof. Suppose that such that Then, for each and Now, by condition (1), there exist such that Let us define the multivalued operator by Since is lower semicontinuous, then there exist such that for all . Thus, for any By condition (3), we can write This implies that Now, by exchanging the rule of and we have yielding is equivalent to Therefore, Therefore, all the conditions of Theorem 15 are fulfilled. Therefore, the operator has a fixed point which is a solution to Problem (63).☐

Data Availability

No data were used.

Conflicts of Interest

The authors declare that they have no competing interests concerning the publication of this article.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.

Acknowledgments

The fourth author Habes Aslamir extends his appreciation to the College of Business Administration-Finance Department, Dar Al Uloom University for funding this work.

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