Journal of Mathematics

Journal of Mathematics / 2021 / Article
Special Issue

Recent Advances in Fixed Point Theory in Abstract Spaces

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 3886659 | https://doi.org/10.1155/2021/3886659

Shagun Sharma, Sumit Chandok, "Existence of Best Proximity Point with an Application to Nonlinear Integral Equations", Journal of Mathematics, vol. 2021, Article ID 3886659, 7 pages, 2021. https://doi.org/10.1155/2021/3886659

Existence of Best Proximity Point with an Application to Nonlinear Integral Equations

Academic Editor: Huseyin Isik
Received10 Jul 2021
Accepted17 Aug 2021
Published08 Oct 2021

Abstract

Using the idea of modified -proximal admissible mappings, we derive some new best proximity point results for -contraction mappings in metric spaces. We also provide some illustrations to back up our work. As a result of our findings, several fixed-point results for such mappings are also found. We obtain the existence of a solution for nonlinear integral equations as an application.

1. Introduction and Preliminaries

Problems originating in several disciplines of mathematical analysis, such as obtaining the existence of a solution for integral and differential equations, are solved using fixed-point theory. The study of fixed points for nonself-mapping, on the contrary, is also fascinating. A nonself-contraction does not necessarily have a fixed point for two given nonempty closed subsets and of a complete metric space . In this case, it is important to identify a point such that is minimum. Essentially, ifwhere is the minimum value and hence is an approximate solution of equation with least possible error, such a solution is known as the best proximity point of mapping .

Various academics have discovered a number of best proximity point theorems and associated fixed-point results in metric or normed linear spaces (see [17] and references cited therein). Inspired by the work of Geraghty [8] and Kutbi and Sintunavarat [9], we introduced a new class of nonself-contractive mappings known as -contraction. The goal of this work is to use the modified -proximal admissible mappings concept to obtain some best proximity point outcomes for -contraction mappings in metric space. On a metric space enriched with an arbitrary binary relation, some optimal proximity point results are proved. For such a class of mappings, we also obtain certain fixed-point findings. As an application of our obtained results, we find the solution of a nonlinear integral equation.

The set of natural numbers and the set of real numbers are denoted by and , respectively, throughout this work.

Let and be two nonempty subsets of , a metric space. Define

Definition 1. Let be a pair of nonempty subsets of a metric space with . Then, the pair is said to have ₤-property (see [10]) if and only if, for any and ,Many authors have refined the concept of -admissible mappings developed by Samet [10] for proximal mappings (see [11, 12]).

Definition 2. Let and be two nonempty subsets of a metric space . A mapping is called modified -proximal admissible if there exists a mapping such that :for all .
Let be a metric space and be a binary relation over . Denoteis the symmetric, transitive relation attached to . Clearly,

Definition 3. Let and be two nonempty subsets of metric space . A mapping is called modified proximal comparative mapping; if ,for all .

Definition 4. Let be a metric space. A mapping is called transitive [13] if it satisfies the following condition:for all .

2. Main Results

To start with, we have the following definition:

Definition 5. Let and be two nonempty subsets of a metric space . A mapping is an contraction mapping if there exist two functions and for which and for all such thatfor all .

Theorem 1. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let be transitive and be continuous mapping satisfying the following assertions:(i) is contraction(ii) and satisfy the ₤-property(iii) is modified -proximal admissibleFurthermore, suppose that there exists and such that and . Then, the mapping has a best proximity point.

Proof. By assumption, there exists and such that and .
Since , then . By definition of , there exists such that . Since is modified -proximal admissible, and , we obtainAgain, ; then, . By definition of , there exists such that . Since is modified -proximal admissible, and , we haveContinuing in this fashion, we can construct a sequence such thatAs a pair satisfies ₤-property, we haveFor , we havefor all . Therefore, the sequence is strictly decreasing andfor some .
Next, we claim that . Assume, on the contrary, that . On taking limit as in equation (14), we obtain thatwhich is a contradiction. Therefore, .
Next, we show that is a Cauchy sequence. Using triangle inequality, (12) and (13), we have, for all and ,It implies thatOn taking limit as in (18), we obtain thatTherefore, is a Cauchy sequence in . Since is a closed subset of a complete metric , we obtainfor some . Since is continuous,for some . By (12), we obtainHence, is a best proximity point of .
The following hypotheses can be used to substitute ’s continuity hypothesis in Theorem 1.
() If is a sequence in such that , for all and as , then there exists a subsequence of such that , for all .

Theorem 2. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let be transitive and be a mapping satisfying the following assertions:(i) is contraction(ii) and satisft the ₤-property(iii) is modified -proximal admissibleFurthermore, suppose that () holds and there exists and such that and . Then, the mapping has a best proximity point.

Proof. Following the proof of Theorem 1, the sequence is Cauchy and converges to some in .
From our assumption and using (12), there exists a subsequence of such that , for all .
Now, we shall show that has a best proximity point. Using triangle inequality and (12), we obtainAgain, by triangle inequality and equation (23), we haveTaking in (24), we haveThis shows that has a best proximity point.

3. Best Proximity Points on a Metric Space Endowed with an Arbitrary Binary Relation

Using modified proximal comparative mapping, we establish some optimal proximity point findings on a metric space equipped with an arbitrary binary relation in this section.

Theorem 3. Let and be two nonempty closed subsets of a complete metric space such that is nonempty and be a binary relation over . Suppose that is a continuous mapping satisfying the following assertions:(i)There exists such that, for all , ,(ii) and satisfy the ₤-property.(iii) is modified proximal comparative mapping.If there exists and such that , , then the mapping has a best proximity point.

Proof. Define a mapping bySuppose that such thatHold for some . By definition of , we obtainCondition (iii) implies that , which gives us from the definition of that . This shows is a modified -proximal admissible mapping.
Furthermore, using assumption, we haveCondition (i) implies thatThus, all the conditions of Theorem 1 are satisfied; then, has a best proximity point.
We may replace the continuity hypothesis of in Theorem 3 by the following hypothesis.
() If is a sequence in such that for all and as , then there exists a subsequence of such that , for all .

Theorem 4. Let and be two nonempty closed subsets of a complete metric space such that is nonempty and be a binary relation over . Suppose that is a mapping satisfying the following assertions:(i)There exists such that, for all , ,(ii) and satisfy the ₤-property.(iii) is modified proximal comparative mapping.Furthermore, we assume that holds and there exists and such that , . Then, the mapping has a best proximity point.

Proof. By evaluating the mapping defined in Theorem 3 and noting that condition implies condition , the result is derived from Theorem 2.

In this section, we obtain some results on the existence of fixed points as a result of our findings.

If we take in Theorem 1, we obtain the following result:

Corollary 1 (see Theorem 10 in [9]). Let be a complete metric space. Let be transitive and be continuous mapping satisfying the following assertions:(i) is contraction(ii) is admissible(iii)There exists such that Then, the mapping has a fixed point.
By considering , for all and , for all in Corollary 1, we get the famous Banach contraction theorem.

Corollary 2 (see [14]). Let be a complete metric space and be a mapping satisfying the following assertion:for all . Then, the mapping has a fixed point.
If we take in Theorem 2, we obtain the following result.

Corollary 3 (see Theorem 12 in [9]). Let be a complete metric space. Let be transitive and be a mapping satisfying the following assertions:(i) is contraction(ii) is admissible(iii)There exists such that (iv)() holdsThen, the mapping has a fixed point.
If we take in Theorems 3 and 4, we obtain the following fixed-point results:

Corollary 4. Suppose that is a continuous mapping on a metric space with a binary relation over , satisfyingfor , , where . If there exists a such that , then the mapping has a fixed point.

Corollary 5. Suppose that is a self-mapping on a metric space with a binary relation over , satisfyingfor , , where . If there exists such that and holds, then the mapping has a fixed point.

5. Examples

We give several illustrations that support our findings in this section.

Example 1. Consider with metricfor all .
Suppose and , such that .
Define byfor all .
Define bythen is transitive. If and in , for , thenAlso, and and if and only if . Then, . This shows is a modified -proximal admissible and is continuous. Since and , then , for each .
Next, we prove that is contraction. Let , for all . Take and in , where . ConsiderIt implies thatfor all . Hence, is a contraction. All conditions of Theorem 1 are satisfied and has a best proximity point .

Example 2. Consider with metricfor all . Suppose thatThen, . Define byfor all .
Define byfor all . Hence, is transitive. If , then and if and only if . Therefore, . This shows is a modified -proximal admissible and is continuous. Since and , then , for each .
Next, we prove that is contraction.
Let . Take in ; then,It implies thatfor all . Therefore, is a contraction. All conditions of Theorem 1 are satisfied and has a best proximity point .

6. Application to Integral Equations

In this section, we obtain the solution of integral equation as an application of our obtained results.

If we take in Theorem 1, we obtain the solution of nonlinear integral equation.

Theorem 5. Let be the set of all continuous functions on closed interval , with metric defined byfor all . Consider the nonlinear integral equation:where , , and for each . Suppose that the following statements hold:(i) is continuous on and is integrable with respect to on (ii), for all , where , for all (iii)For all , , and , for all (iv)For all and , ; (v)There exists such that and , for all Then, nonlinear integral equation (49) has a solution in .

Proof. Define a mapping byfor all and for all . Then, is a continuous mapping.
Define a mapping byWe shall show that is a modified -proximal admissible mapping. Indeed, for such that , we have , for all . It follows from condition (iii) that .
Therefore, , and hence, is a modified -proximal admissible mapping and is transitive. Now, we claim that is a contraction. Let , for all . Consider