Interpolative Prešić Type Contractions and Related Results

Alansari, Monairah; Ali, Muhammad Usman

doi:https://doi.org/10.1155/2022/6911475

Journal of Function Spaces

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Fixed-Point Theory and Applications to Probabilistic Functional Analysis

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Research Article | Open Access

Volume 2022 | Article ID 6911475 | https://doi.org/10.1155/2022/6911475

Interpolative Prešić Type Contractions and Related Results

Monairah Alansari¹and Muhammad Usman Ali²

Academic Editor: Antonio Francisco Roldán López de Hierro

Received19 Nov 2021

Accepted16 Jan 2022

Published21 Feb 2022

Abstract

In this article, we will extend the notion of interpolative Kannan contraction by introducing the notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions for mappings defined on product spaces. Through these notions, we will derive some results to ensure the existence of fixed points and best proximity points for such mappings.

1. Introduction and Preliminaries

The Banach contraction principle is the most significant and basic result of metric fixed point theory. Through this result, we can obtain a unique fixed point of a self-map , provided that is a contraction map on a complete metric space . This result motivated Prešić to study about the existence of fixed points of the operators defined on product spaces, that is, , for any fixed . As an outcome of this motivation, Prešić [1] presented the following noteworthy extension of the Banach contraction principle.

Theorem 1 (see [1]). Suppose that be a complete metric space and be a map, for any fixed , satisfying the following inequality:

for every , where are the nonnegative real numbers with . Then, there exists a unique point of that satisfies the equation .

This result is used to discuss the existence of equilibrium points for the ^th-order nonlinear difference equation of the formwhere is a continuous map. Note that a point is known as an equilibrium point of (2) if . Such a point is also known as a fixed point of . Some well-known generalizations of this work have been studied by several authors, for example, [2–5].

Kannan and Chatterjea made a vital contribution in the development of this field through the fixed point results derived in [6, 7], respectively. Recently, Karapınar [8] modified the Kannan contraction by introducing interpolative Kannan contraction, stated as, a map is called an interpolative Kannan contraction [8] iffor all with and , where and .

After that, many existing contraction-type conditions have been generalized in the sense of interpolative Kannan contraction, for example, Karapınar et al. [9] studied interpolative Reich-Rus-Ćirić type contraction in partial metric spaces, Aydi et al. [10] studied interpolative Ćirić-Reich-Rus type contractions in Branciari metric spaces, Mohammadi et al. [11] extended the concept of -contractions by interpolative Ćirić-Reich-Rus type F-contractions, Karapınar et al. [12] studied interpolative Hardy–Rogers type contractions, Debnath and Sen [13] studied set-valued interpolative Hardy-Rogers and set-valued Reich-Rus-Ćirić-type contractions, Sarwar et al. [14] presented rational type interpolative contractions, Khan et al. [15] worked on interpolative -type -contractions, Altun and Tasdemir [16] presented interpolative proximal contractions for nonself mappings, Fulga and Yesilkaya [17] studied interpolative Suzuki-type contractions, Karapınar et al. [18] defined -interpolative contractions, and Alansari and Ali [19] studied multivalued interpolative Reich-Rus-Ćirić-type contractions.

Gaba and Karapınar [20] extended the notion of interpolative Kannan contraction through exponential powers, stated as, a map is called an -interpolative Kannan contraction, iffor all with and , where with and . Readers can find other similar generalizations in [21].

Consider and be nonvoid subsets of a metric space . It is well-known that a fixed point of a map is a solution of . If , then fixed point of does not exist, that is, for all . In this situation, we try to find , such that attain the minimum value in some sense. It is obvious that the smallest value that can be obtained by for any will be greater or equal to , that is, distance between and . A point is said to be a best proximity point of , if . The existence of such points of nonself maps has been discussed by several researchers in different ways, for example, Caballero et al. [22] studied the existence of best proximity points for nonself maps satisfying Geraghty contraction and -property in metric spaces, Bilgili et al. [23], Aydi et al. [24], and Pitea [25] extended the work of Caballero et al. [22] by introducing generalized Geraghty contraction, -Geraghty contraction and generalized almost -Geraghty contraction for nonself maps, Basha and Shahzad [26] and Basha [27] defined proximal-type contractions to study the existence of best proximity points, Jleli and Samet [28] defined --proximal contraction to ensure the existence of best proximity points, Jleli et al. [29] and Aydi et al. [30] defined generalized --proximal contractions to extend the work of Jleli and Samet [28], Abkar and Gabeleh [31] and Kumam et al. [32] studied the existence of best proximity points for multivalued nonself maps in metric spaces, Ali et al. [33] defined implicit proximal contractions, Sahin et al. [34] defined proximal nonunique contraction, and Ali et al. [35] studied the existence of best proximity points for Prešić type nonself operators satisfying proximal type contractions.

This article aims to present the notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions for mappings defined on product spaces. Through these notions, we will study the existence of fixed points and best proximity points for such mappings.

2. Main Results

We begin this section with the following definition.

Definition 1. A map is called an interpolative Prešić type-I contraction, if for each , we getwhere is a map, with , , and .
The following theorem is used to study the existence of fixed points for the above map.

Theorem 2. Consider an interpolative Prešić type-I contraction map on a complete metric space . Also, consider that(i)If , then .(ii)There exist two elements with .(iii)For every sequence in with for some natural number and , we have .Then, there exists at least one point of that satisfies the equation .

Proof. Hypothesis (ii) assures that there are two points, say, and in withBy using these two points, we can define a sequence with . From hypothesis (i), it can be concluded that . Hence,By (5), we getthat is,By (9), we obtainSince , thus, by (10), we getHence, by (11), we conclude thatBy triangle inequality and (12), for each with , we obtainIn view of the above inequality and the convergence of , we say that the sequence is a Cauchy in . By the completeness of , we get a point , such that . Also, by (iii), we get , since and .
Here, the claim is . If the claim is wrong, then by (5), for each , we getBy triangle inequality and (14), we obtainLetting in (15), we get . Hence, the claim is true, that is, .

Example 1. Consider equipped with a metric for each . Define and byandThen, one can easily verify that the axioms of Theorem 2 are satisfied. Hence, there is at least one element , such that .
In the following, we present interpolative Prešić type-II contraction map and related fixed point result.

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

Theorem 3. Consider an interpolative Prešić type-II contraction map on a complete metric space . Also, consider that(i)If , then .(ii)There exist two elements with .(iii)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

Proof. In view of the hypothesis (ii), we getFor some and in . Through these two points, we can construct a sequence with . Also, hypothesis (i) implies that . Hence,By (18), we getBy performing some calculations, we getHence, it can be seen that is a Cauchy sequence in with . Also, by (iii), we get . Suppose that . Then, by (18), for each , we getBy triangle inequality and (23), we obtainLetting in (24), we get . Hence, our supposition is wrong and .

Example 2. Consider equipped with a metric for each . Define and byandThen, one can easily verify that the axioms of Theorem 3 are satisfied. Hence, there is at least one element , such that .
Some consequences of the above results can be obtained in the form of the following listed corollaries. The following corollary is obtained from Theorem 2 by considering .

Corollary 1. Consider a map on a complete metric space , such that for each , we getwhere is a map, with , and . Also, consider that(i)If , then (ii)There exist two elements with (iii)For every sequence in with for some natural number and , we have

Then, there exists at least one point of that satisfies the equation .

The following corollary is a special case of Theorem 3 which can be obtained by considering .

Corollary 2. Consider a map on a complete metric space , such that for each with , we getwhere is a map, with and . Also, consider that(i)If , then (ii)There exist two elements with (iii)For every sequence in with for some natural number and , we have

Then, there exists at least one point of that satisfies the equation .

By defining for each in Theorem 2 or Theorem 3, we get the following.

Corollary 3. Consider a map on a complete metric space that satisfiesfor each , where with and . Then, there exists at least one point of that satisfies the equation .

From the above corollary, we can also obtain the following result.

Corollary 4. Consider a map on a complete metric space that satisfiesfor each , where with and . Then, there exists at least one point of that satisfies the equation .

In the following, we will study about the interpolative Prešić type proximal contractions and related results.

Let be a metric space and be nonvoid subsets of . We will use the following notations.

Note that a point is known as a best proximity point of if . The collection of all such points for is denoted by .

Definition 3. A map is called an interpolative Prešić type-I proximal contraction, if for each with , we getwhere is a map, with and .
The following theorem is used to ensure the existence of best proximity points for the above defined maps.

Theorem 4. Consider an interpolative Prešić type-I proximal contraction map on a metric space . Also, consider that(i)If and , then .(ii)There exist elements with and .(iii).(iv) is nonempty and complete with respect to .(v)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

Proof. From hypothesis (ii), we have , , and in with and . Hypothesis (iii) implies that , and there is satisfying . Thus, from hypothesis (i), we get . Hence, by using hypotheses (i) and (ii) repeatedly, we conclude that and for all .
Since for each and for each , then, by (32), we getNow, by following the proof of Theorem 2, we say that is a Cauchy sequence in . Since is complete, we have a point , such that . Also, by (v), we get , since and . Clearly, , and there is with . Here, the claim is . Suppose it is wrong, then by (33), for each , we getLetting in (34), we obtain , and it contradicts our assumption. Hence, our claim is true, that is, . Therefore, .
In the following, we present the notion of interpolative Prešić type-II proximal contraction.

Definition 4. A map is called an interpolative Prešić type-II proximal contraction, if for each with and , we getwhere is a map, with and .
The existence of best proximity points for above map can be ensured by the result given.

Theorem 5. Consider an interpolative Prešić type-II proximal contraction map on a metric space . Also, consider that(i)If and , then .(ii)There exist elements with and .(iii).(iv) is nonempty and complete with respect to .(v)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

Proof. The proof can be derived on the same steps as the proof of Theorem 4 is done.

2.1. Results for Extended Interpolative Prešić Type Maps

This subsection presents the extensions of the above listed results. Theorems 6 and 7 can be considered as an extended version of Theorem 2 and Theorem 3, respectively.

Theorem 6. Consider an extended interpolative Prešić type-I contraction map , for any fixed , on a complete metric space , that is, for each , , we getwhere is a map, with , , and . Also, consider that(i)If , then .(ii)There exist elements , such that(iii)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

Proof. Hypothesis (ii) implies the existence of elements in withThrough these points, we can define a sequence with for all . Hence, by considering hypothesis (i), it can be concluded that . Then, we say thatBy (36), we getthat is,By (41), we obtainSince , thus, by (42), we getHence, by (43), we getthat is,From triangle inequality and (45), for each with , we obtainAbove inequality yields that is a Cauchy sequence in a complete space . Hence, we get a point with . Also, by (iii), we get , since and .
Here, the claim is . If the claim is wrong, then by (36), for each , we getBy triangle inequality and (47), for each , we obtainLetting in(48), we get . Hence, the claim is true, that is, .

Theorem 7. Consider an extended interpolative Prešić type-II contraction map , for any fixed , on a complete metric space , that is, for each , with , we getwhere is a map, with , , and . Also, consider that(i)If , then .(ii)There exist elements , such that(iii)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

Proof. The proof can be obtained on the same steps as the proofs of Theorems 6 and 2 are done.
The following theorems can be considered as an extended form of Theorems 3 and 5, respectively.

Theorem 8. Consider an extended interpolative Prešić type-I proximal contraction map , for any fixed , on a metric space , that is, for each , with

we getwhere is a map, with , , , and are the nonvoid subsets of . Also, consider that(i)If and , then .(ii)There exist elements with and(iii).(iv) is nonempty and complete with respect to .(v)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

Proof. By hypothesis (ii), we get , in with andHypothesis (iii) implies that , and there is satisfyingThen, from hypothesis (i), we get . Repeated use of hypotheses (i), (ii), and (iii) yields and for all . AsandandThen, by (52), we getFollowing the proof of Theorem 6, we say that is Cauchy in , and by the completeness of , we get a point with . Also, by (v), we get , since and . Clearly, , and there is with . Here, the claim is . Suppose it is wrong, then by (52), for each , we getLetting in (60), we obtain , and it contradicts our assumption. Hence, our claim is true, that is, . Therefore, .

Theorem 9. Consider an extended interpolative Prešić type-II proximal contraction map , for any fixed , on a metric space , that is, for each , withandwe getwhere is a map, with , and . Also, consider that(i)If and , then .(ii)There exist elements with and(iii).(iv) is nonempty and complete with respect to .(v)For every sequence in with for some natural number and , we have .

Then, there exists at least one point of that satisfies the equation .

The proof of the above theorem can be derived by viewing the proof of Theorem 8.

3. Conclusion

This article provides a few results dealing with fixed points and best proximity points of the mappings defined on product spaces. The notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions are introduced in the context of metric spaces to discuss the existence of fixed points and best proximity points of such maps, respectively. These notions are derived by considering the concept of interpolative Kannan contraction.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under grant number KEP-5-130-42. The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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Copyright © 2022 Monairah Alansari and Muhammad Usman Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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