On Interpolative Prešić-Type Set-Valued Contractions

Abstract

This study aims to present the notions of interpolative Prešić-type set-valued contractions for the set-valued operators defined on product spaces. With the help of these notions, we have studied the existence of fixed points for such set-valued operators. An application of the obtained results is also discussed with the help of graph theory.

1. Introduction and Preliminaries

Banach [1] initiated the study of the existence of fixed points for self-maps defined on a metric space. This study was further strengthened by Kannan and Chatterjea through their fixed point results derived in [2, 3], respectively. Following this study, Nadler [4] proposed a result to ensure the existence of fixed points for set-valued maps. Prešić [5] extended Banach contraction principle to the maps defined on product spaces, that is, , for any fixed . Afterwards, this result was extended by Ćirić and Prešić [6]. The results of Prešić [5] and Ćirić and Prešić [6] are presented below.

Theorem 1. (see [5]). Let , for any fixed , be a map on complete metric space and satisfiesfor each , where with . Then, there exists a unique fixed point of , that is, .

Theorem 2. (see [6]). Let , for any fixed , be a map on complete metric space and satisfiesfor each , where . Then, there exists a unique fixed point of , that is, .
Karapınar [7] presented interpolative Kannan contraction by following the Kannan contraction as follows.
A map is called an interpolative Kannan contraction [7] iffor each with and , where and .
The above work of Karapınar [7] is adopted by several researchers; for example, the notions of interpolative Ćirić–Reich–Rus type contractions in Branciari metric spaces, and partial metric spaces are defined by Aydi et al. [8] and Karapınar et al. [9], the notions of interpolative type F-contractions are defined by Mohammadi et al. [10] and Alansari and Ali [11], the notions of interpolative Hardy-Rogers type contractions, and set-valued interpolative Hardy–Rogers type contractions are defined by Karapınar et al. [12] and Debnath and Sen [13], the notion of interpolative Suzuki-type contraction is discussed by Fulga and Yesilkaya [14], and the notion of interpolative proximal contraction is discussed by Altun and Tasdemir [15].
Gaba and Karapınar [16] redefined the notion of interpolative Kannan contraction through modifying exponential powers in a following way.

A map is called an -interpolative Kannan contraction iffor each with and , where with and .

Recently, Alansari and Ali [17] defined the notion of extended interpolative Prešić-type contraction map as follows.

A map , for any fixed , on a metric space is called extended interpolative Prešić type contraction if, for each , , we havewhere is a map, with , , and .

The purpose of this study is to extend and redefine the concepts of interpolative Prešić-type contractions by introducing interpolative Prešić-type set-valued contractions for set-valued maps. We will also present a few fixed-point results to study the existence of fixed points of such maps.

The literature of metric fixed-point theory contains several interesting results that are the generalizations of Banach fixed-point theorem, for example, the study of common fixed-point results for two or more maps [18] and the study of the existence of fixed points for the maps defined on generalized metric spaces, such as -metric space [19], partial metric space [20], dislocated quasi-metric [21], hypergraphical metric space [22], and soft metric space [23, 24].

Before the next section, we recall the Pompeiu–Hausdorff distance. The Pompeiu–Hausdorff distance is a map defined bywhere and represents the collection of all nonvoid closed and bounded subsets of .

2. Main Results

We begin this section with the following definition.

Definition 1. A map is said to be an interpolative Prešić type-I set-valued contraction if, for all , the following inequality exhibitswhere is a map, with , , and .
With the help of below stated result, we will study the existence of fixed points for the above map.

Theorem 3. Let be an interpolative Prešić type-I set-valued contraction map on a complete metric space . Also, consider that(i)If , then , for all (ii)There exist with , for all (iii)For each sequence in with , for some natural number , and , we have

Then, there exists an element of with .

Proof. By hypothesis (ii), we get two points in , say and , withLet ; then, by (7), we obtainThat is,Clearly, ; thus, by (10), we obtainAs , thus, by (11), we obtainThe fact yields the existence of some satisfying the inequality . Thus, by the last two inequalities, we obtainAs , , and , by hypothesis (i), we get . Thus, we say that . Again, by considering (7), we obtainThat is,As , thus, by (15), we obtainAs , then there is some such that . Thus, we obtainContinuing in that way, we reach to a sequence with the facts for all andandFor simplicity, we use for each . We will show with induction that for each , where and . Trivially, and . Suppose thatThen,Thus, , for each . Now, by considering this fact and the triangle inequality, for each with , we obtainHence, the convergence of , as , and the above inequality yields that is a Cauchy sequence in . Now, the completeness of yields the existence of a point such that . By hypothesis (iii), we get , as and . Now, we claim . If it is wrong, then, by (7), for each , we obtainThat is,By triangle inequality and (24), we obtainHence, by applying the limit as in (25), we get . This shows that the claim is true and .

Example 1. Let denote the set of all real numbers with a usual metric for each . Define maps and byThe hypotheses of Theorem 3 can be verified on the above defined maps. Hence, there exists an element of with .
We now present an interpolative Prešić type-II set-valued contraction map along with fixed-point result.

Definition 2. A map is called an interpolative Prešić type-II set-valued contraction if, for each with , we obtainwhere is a map, with , , and .

Theorem 4. Let be an interpolative Prešić type-II set-valued contraction map on a complete metric space . Also, consider that(i)If , then , for all (ii)There exist with for all (iii)For each sequence in with , for some natural number , and , we have

Then, there exists an element of with .

Proof. Hypothesis (ii) makes sure the existence of two points in , say and , that satisfies the following:By defining one value of as in the above inequality, we reach toThus, by (27), we obtainThat is,Since , thus, by (31), we obtainFrom the above inequality and by the fact , there exists some such thatSince and , , by hypothesis (i), we get . By proceeding the proof on the above steps, we reach to a sequence of the form for all andandBy viewing the above inequality and the proof of the above theorem, we conclude that is a Cauchy sequence in , and there exists a point with . From hypothesis (iii), we have for each . This implies . Suppose that . Then, by (27), for each , we obtainBy triangle inequality and (36), we obtainThus, by taking the limit in (37), we get . This shows that the supposition is wrong and .

2.1. Results for Extended Interpolative Prešić Type Set-Valued Operators

This section presents the extensions of the above listed results. Theorems 5 and 6 can be considered as an extended version of Theorems 3 and 4, respectively.

Theorem 5. Let , for any fixed , be an extended interpolative Prešić type-I set-valued contraction map on a complete metric space , that is, for every , , we havewhere is a map, with , , and . Also, consider that(i)If , then for all , .(ii)There are satisfying(iii)For each sequence in with , for some natural number , and , we have .Then, there exists an element of with .

Proof. Hypothesis (ii) says that there are points in satisfying the condition:Thus, for , we obtainThen, by (38), we obtainThat is,Since , thus, by (43), we obtainSince , then (44) givesAs , thus, there exists of a formHypothesis (i) implies that , since and , . By the repeated application of hypothesis (i) and (38), we reach to a sequence with the facts for all andandFor simplicity, take for each ; from (48), we obtainNow, we prove by induction that for each , where and . Trivially, , for each . Suppose that for each for some given , as induction hypothesis. Then, by (49), we obtainHence, it is shown by induction that , for each . This fact along with triangle inequality yield thatfor each with . Hence, the above inequality and the convergence of ensure that is a Cauchy sequence in . Now, the completeness of yields the existence of a point with . By hypothesis (iii), we get , as and . Now, we claim that . Suppose it is wrong, then, by (38), for each , we obtainThat is,By triangle inequality and (53), we obtainAfter applying the limit as in (54), we get . Hence, the claim is true and .

Theorem 6. Let , for any fixed , be an extended interpolative Prešić type-II set-valued contraction map on a complete metric space ; that is, for every , with , we havewhere is a map, with , , and . Also, consider that(i)If , then , for all and .(ii)There are satisfying(iii)For each sequence in with , for some natural number , and , we have