#### Abstract

In this paper, we will introduce the concept of Suzuki type multivalued -contraction and we will prove some fixed point results in the setting of a metric space equipped with a binary relation. Our results generalize and extend various comparable results in the existing literature. Examples are provided to support the results proved here. As an application of our results, we obtain a homotopy result, proving the existence of a solution for a second-order differential equation and for a first-order fractional differential equation.

#### 1. Introduction and Preliminaries

Let be a metric space and be a mapping on . An element is called a fixed point of if it remains invariant under the action of ; that is, A mapping on a metric space is said to be a Banach contraction if holds for all , where . A Banach contraction mapping defined on a complete metric space has a unique fixed point. This result is known as Banach contraction principle. Several authors have extended and generalized Banach contraction principle in different directions.

Jleli and Samet [1] suggested a modification in the contraction condition and introduced a -contraction mapping. Consistent with [1], the following notations, definitions, and results will be needed in the sequel.

Suppose that where

() is nondecreasing;

( for each sequence , if and only if ;

() there exists and such that .

Example 1. Define for by Then, .

Let be a metric space and . A mapping is called a -contraction if for any , we have whenever and . Jleli and Samet [1] proved the following fixed point theorem in the framework of a generalized metric space in the sense of Branciar; i.e., the triangle inequality is replaced by the inequality , for all pairwise distinct points .

Theorem 2. Let be a complete generalized metric space in the sense of Branciari and be a -contraction. Then has a unique fixed point.

Hussain et al. [2] considered the following class of mappings: where

() is nondecreasing;

() if and only if ;

() for each sequence , if and only if ;

() there exists and such that ;

() .

Example 3. Let be defined by and , respectively, then and .

Hussain et al. [2] proved the following result.

Theorem 4. Let be a complete metric space and be a continuous mapping. Suppose there exist and with , such that for any , we have Then has a unique fixed point.

For other results in this direction, we refer to [3, 4] and references mentioned therein.

Let be a metric space and (respectively ) be the family of all nonempty closed and bounded (nonempty compact, respectively) subsets of . For and , define is the Hausdorff-Pompeiu metric on (or on ) induced by . Let be a given multivalued mapping. An element is called a fixed point of if A mapping is said to be a Nadler contraction if there exists such that for any Nadler [5] obtained the following multivalued version of Banach contraction principle.

Theorem 5. Let be a complete metric space and be a Nadler contraction. Then has at least one fixed point.

Later on, many researchers have obtained fixed point results for multivalued mappings satisfying generalized contraction type conditions. For example, recently, Hançer et al. [6] proved the following fixed point result for multivalued -contractions.

Theorem 6. Let be a complete metric space and be a multivalued mapping. Suppose that there exist and such that for any , provided that . Then has at least one fixed point.

Durmaz [7] introduced a new type of generalized multivalued -contraction and proved some interesting fixed point results (see also [8]). Kikkawa and Suzuki [9] refined Nadler’ result by proving the following theorem.

Theorem 7. Let be defined as . Let be a complete metric space and . Assume there exists such that Then has at least one fixed point.

We denote and define the following class of mappings, which was considered in [10].

Example 8. Let be defined by where are given by and . Obviously .

Example 9. Let be defined by where are defined by and for all . Note that .

Many results, dealing with existence of fixed points of mappings satisfying certain contraction type conditions in the framework of complete metric spaces endowed with a partial ordering, have appeared in the last decade. Ran and Reurings [11] proved an analogue of Banach’s fixed point theorem in a metric space endowed with partial ordering and gave an application of their results to solve matrix equations. Alam and Imdad [12] proved another variant of Banach’s fixed point theorem in a metric space equipped with a binary relation which generalized many comparable results, including Ran and Reuring’s result in [11]. Senapati and Dey [13] proved Banach’s fixed point theorem in metric spaces equipped with an arbitrary binary relation using -distance. They employed their results to prove the existence of solutions of nonlinear fractional differential equations and fractional thermostat model involving the Caputo fractional derivative. A very nice Ph.D. thesis was written on the same subject; see Dobrican [14].

Let us first recall the following definitions.

Definition 10. Let be a nonempty set and be a binary relation defined on . Then, is -related to if and only if .

We denote and .

Definition 11. Let be a nonempty set and a binary relation on . A sequence is called -preserving if

Definition 12. Let be a metric space. A binary relation defined on is called -self closed if whenever is an -preserving sequence and converges to , then there exists a subsequence of with either or for all .

Definition 13. Let be a metric space and a binary relation defined on . A mapping is -closed if for any , implies that for any and .

If is a multivalued map, then we set In particular, if is single-valued, then we denote Motivated by the results in [2, 10, 12], we introduce the concept of a Suzuki type multivalued -contraction mapping and present some fixed point results in metric spaces equipped with a binary relation. Our results extend and generalize several results given in [2, 1519]. We also provide applications of our results to homotopy theory proving the existence of a solution of second-order differential equations and first-order fractional differential equations.

#### 2. Multivalued Suzuki Type -Contraction

In this section, we obtain a fixed point result for multivalued Suzuki type -contraction in a metric space equipped with a binary relation .

Throughout this paper satisfies the following additional property: where .

Definition 14. Let be a metric space and a binary relation on . Assume that and . A mapping is a multivalued Suzuki type -contraction if for any with where with

Our first main result is the following.

Theorem 15. Let be a complete metric space, a binary relation on , and a multivalued Suzuki type -contraction. Suppose that following conditions hold: (1) is nonempty,(2) is -closed,(3) is -self closed or has closed graph. Then has at least one fixed point.

Proof. Since is nonempty, if we choose , then there exists some such that . If , the result follows. Assume that . As and we have Now and imply that Then by and , it follows that and hence Hence, we obtain and, in conclusion, we get that By we have Thus, We can choose such thatAs , , and is - closed, we have that . If , the result follows. Assume that . Also, Hence, By and , we have As and , we obtain that Hence, By , we have Hence, We can choose such that By (27), we get Since , and is -closed, we have .
Continuing this way, we can obtain a sequence such that and is -preserving. Obviously, we have , for all natural numbers . Hence,Letting , we have It follows from () that Now, we show that is a Cauchy sequence. If we set , then from (37), we obtainFurther, from (), there exist and such that Suppose . Let . Then there exists such that for all , we obtain that Hence, for all , we have This implies that If , then for there exists such that for all , we have which implies that Hence, for each case, we obtain that Thus, using (40) we have Therefore, So, there exists such that for all , we have which implies that Let . Then By the convergence of the series we get as . Hence, is Cauchy. Since is complete, there exists in such that . We show that has a fixed point. Assume on the contrary that does not have a fixed point. Then, for all natural numbers . As , we have which implies that Furthermore, gives thatThus which further implies thatHenceIf is -self closed, there exists a subsequence of such that either or . Assume that . If , then we have From (57), we obtain that a contradiction. Hence, for all .
By our assumption . Thus which implies thatConsequently, Also, From (61), it follows thatLetting in (64) we obtain that a contradiction. Hence, .
If has closed graph, since for each and , we get that .

If we take in Theorem 15, we obtain a Suzuki type generalization of the result in [6] in the framework of a complete metric space equipped with a binary relation .

Corollary 16. Let be a complete metric space, a binary relation on and . Assume that and . Suppose that there exists such that for any with , implies that If conditions (1)-(3) in Theorem 15 are satisfied, then has a fixed point.

If we take n, Theorem 15, then we have the following multivalued extension of Cirić result in [16].

Corollary 17. Let be a complete metric space, a binary relation on , and . Suppose that and there exist with , such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

Remark 18. Note that the conclusion of Corollary 17 can be written as

Notice that if we take in Theorem 15, using Remark 18, we obtain the following multivalued Suzuki type generalization of Chatterjea’s result in [15].

Corollary 19. Let be a complete metric space, a binary relation on , and . Suppose that and there exists such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied, then has a fixed point.

If we take in Theorem 15, using Remark 18, we obtain the following multivalued Kannan type result in [17].

Corollary 20. Let be a complete metric space, a binary relation on , and . Suppose that and there exist with such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

Taking in Theorem 15, we have a multivalued extension and generalization of Reich’s result in [20].

Corollary 21. Let be a complete metric space, a binary relation on , and . Suppose that and there exist with , such that for any with , the following implication is true implying that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

Similarly, if we take , then we obtain the following corollary.

Corollary 22. Let be a complete metric space, a binary relation on , and . Suppose that and there exists with , such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

We now give an example of a multivalued Suzuki type -contraction which is neither a multivalued Banach contraction nor a multivalued -contraction.

Example 23. Let Define the binary relation on as follows: Let be defined byDefine the mapping by Clearly, is -closed and is nonempty. Indeed, if , , then . Take an -preserving sequence such that converges to and for all . Then, , for all . Thus, or . Since both and are closed, either or .
Let and . We consider following cases: (1)If , then .(2)If , then clearly . For and we have which implies that All other cases are trivial. Thus all the conditions of Theorem 15 are satisfied. Moreover, has two fixed points and .
Note that, for , for any . Thus is not a Nadler’s contraction. Also, for any Thus, is not multivalued -contraction.
If is a single-valued map, then we have the following result.

Theorem 24. Let be a complete metric space, a binary relation on , and . Suppose that , and there exist with , such that for any with ,