Quadruple Best Proximity Points with Applications to Functional and Integral Equations

Hammad, Hasanen A.; Rashwan, Rashwan A.; Nafea, A.; Jarad, Fahd

doi:https://doi.org/10.1155/2022/1849891

Advances in Mathematical Physics

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

Research Article | Open Access

Volume 2022 | Article ID 1849891 | https://doi.org/10.1155/2022/1849891

Quadruple Best Proximity Points with Applications to Functional and Integral Equations

Hasanen A. Hammad,^1,2Rashwan A. Rashwan,³A. Nafea,²and Fahd Jarad^4,5,6

Academic Editor: Sergey Shmarev

Received23 Feb 2022

Accepted04 Apr 2022

Published06 May 2022

Abstract

This manuscript is devoted to obtaining a quadruple best proximity point for a cyclic contraction mapping in the setting of ordinary metric spaces. The validity of the theoretical results is also discussed in uniformly convex Banach spaces. Furthermore, some examples are given to strengthen our study. Also, under suitable conditions, some quadruple fixed point results are presented. Finally, as applications, the existence and uniqueness of a solution to a system of functional and integral equations are obtained to promote our paper.

1. Introduction and Preliminaries

Fixed point (FP) theory has many applications not only in the nonlinear analysis and its trends, from solutions of differential and integral equations, functional equations arising from dynamical programming, topology, and a dynamical system, but also in economics, game theory, biological sciences, computer sciences, and chemistry, etc. [1–4].

The FP technique became more attractive and elegant when Banach [5] introduced his principle, which is stated as follows: a mapping defined on a complete metric space (MS) has a unique FP if is a contraction, i.e., . He used this method for studying the existence of solutions for some integral equations.

The FP technique was extended to a coupled and tripled FP by Bhaskar and Lakshmikantham [6] and Berinde and Borcut [7], respectively. Many researchers have worked in these directions and obtained exciting results and life applications that serve the scientific communities, which in turn has led to the fixed points being brilliant and pioneering in the field of functional analysis until the present time. For more details, see [8–20].

Not only did the matter stop here, but Karapinar and Sadarangani [21] were able to generalize the triple point to the quadruple and generalized the previous results on this scale in terms of theories and applications. After that, this trend spreads to others; see, for instance, [22–25].

In 1978, Pathak and B. Fisher [26] were able to merge the state and decision space to clarify the importance of the FP methodology in finding the solution to the following functional system: where and are the state space and the decision space, respectively; denotes the transformation of the process; and refers to the optimal return function with the initial state . The above system is called a functional equation arising from dynamical programming which is commonly used in modeling and optimization problems. To clarify the participation of fixed and coincidence points and to delve deeper into this trend, we guide the reader to read these papers [27–30].

On the other hand, the importance of the FP technique lies in the fact that it presents a unified process and an important tool in solving equations that do not have to be linear. In the case of that is, a contraction mapping does not possess a FP; it became necessary to search a pointthat makesthe minimum, meaning the pointis in close proximity to

Assume that are nonempty closed subsets of a complete MS and is a given mapping. A point is called the best proximity point (BPP, for short) if , where is described as

It should be noted that if , then a BPP reduces to a usual FP.

The initial paper concerned with the BPP was presented by Fan [31]. This direction is very interesting in optimization, so many researchers have discussed and developed this direction in several applications; see, [32–36].

Now, we need the definitions below.

Definition 1 (see [37]). A Banach space is said to be (i)strictly convex, if(ii)uniformly convex, if for any there is so that

Clearly, a uniformly convex Banach space (UCBS) is strictly convex, but the opposite does not hold.

Definition 2 (see [21]). Assume that is a MS and is a subset of . A point is called a quadruple fixed point (QFP, for short) of the map if

Our paper is arranged as follows: In Section 2, some new definitions and supporting examples are presented. Also, the convergence of quadruple best proximity (QBP, for short) points for a pair of cyclic contraction mappings without and with property are obtained in the context of metric spaces (MSs). Moreover, quadruple fixed point (QFP) results for cyclic contraction mappings are established, and an example for supporting the above results is discussed in Section 3. In Section 4, the existence of a solution for quadruple functional equations arising in dynamical programming is discussed, and an example for supporting the results is presented. Ultimately, in Section 5, the existence of solutions for a system of quadruple integral equations is given, and an example is obtained to strengthen this contribution.

2. Main Results

This part is devoted to present the convergence of QBP points for a pair of cyclic contraction mappings in the setting of ordinary MSs.

We begin this part with the definitions below.

Definition 3. Let and be two nonempty closed subsets of a MS . The pair is called to satisfy the property , if there exist and such that as then

Example 1. Suppose that and are two nonempty subsets of a MS with . Then, satisfies the property

Example 2. Assume that , and are nonempty subsets of a MS with , and . If the pair satisfies the property then the pair satisfies also the property

Example 3. Suppose that and are nonempty subsets of a UCBS, then the pair satisfies the property if one of the hypotheses below holds: (i) is convex(ii) is convex relatively compact

Definition 4. Assume that is a MS and , are two nonempty subsets of We say that the pair has the property if has the property and the stipulation below is fulfilled:
If the sequences in and in so that the following statements are fulfilled:
(†₁) and
(†₂) so that then there is so that

Example 4. For nonempty subsets and of a MS , assume that . Then, the pair posses the property

Example 5. Assume that all requirements of Example 2 hold. If the pair has the property then the pair has the property too

Example 6. Assume that are two nonempty subsets of a UCBS and is convex. Then, verifies the property

Definition 5. Assume that is a MS and , are nonempty closed subsets of Also assume is a given mapping. We say that a quadruple is a QBP point of if, Clearly, when in Definition 5, then a QBP point reduces to a QFP.

Definition 6. Assume that is a MS and , are nonempty closed subsets of We say that the mappings and are cyclic contractions if there is so that the inequality below holds: for all and

Notice that, if the pair is a cyclic contraction, then the pair is a cyclic contraction too.

Example 7. Consider equipped with the distance . Let and Obviously, Describe two mappings and as for all and respectively. For each and and let we have

This leads to the pair as a cyclic contraction with

Example 8. Consider endowed with for all and suppose that

Clearly, . Define and by respectively. Then, we obtain

Also, if then one can write for any . In addition, let then it follows from (17) and (19) that

Thus, the pair is a cyclic contraction.

The lemma below is very important in the sequel.

Lemma 7. Assume that is a MS and are nonempty closed subsets of Let and be two cyclic contraction mappings. If and the sequences in are defined as follows: for all then we get

Proof. Consider, for each Using (11), we have By mathematical induction, we obtain for each that Passing we find that Again, for each by induction, one can write this yields after passing Analogously, we have This finishes the required proof.

Lemma 8. Assume that is a MS and , are nonempty closed subsets of so that and satisfy the property UC. Let the mappings and be cyclic contractions. If and the sequences in are defined as (22), then for each there is so that

Proof. According to Lemma 7, we get Because fulfills the property UC, then we have Also, verifies the property UC, we have Assuming (31) is not true. Then, for each with there is so that Hence, we can select the smallest integer with fulfilling (35). Therefore, Thus, we obtain As one can write Applying the triangle inequality, we get Applying (11), we obtain that It follows that Letting we have which is a contradiction since . This implies that (31) is fulfilled, and this finishes the proof.

Lemma 9. Assume that is a MS and , are nonempty closed subsets of so that and satisfy the property . Let the mappings and be cyclic contractions. If and for all the sequences in are defined by (22). Then, , and are Cauchy sequences.

Proof. Based on Lemma 7, one can get As satisfies the property UC, then Similarly, since verifies the property UC, then .
Now, we claim that, , so that Assume that (44) is not true. Then, , , and such that Therefore, we can select a smallest integer with fulfilling (45). Hence, one can get Setting we have Using Lemma 8, we can write for all Applying the triangle inequality, we get Letting we obtain that This is a contradiction. This achieves the inequality (44). It follows from (44), and the property of that is a Cauchy sequence. By the same manner, we can show that , and are Cauchy sequences. This finishes the proof.

Now, via the property we shall discuss the existence and convergence of QBP points.

Theorem 10. Assume that and are nonempty closed subsets of so that the property are satisfied on and . Let the mappings and be cyclic contractions. If and for all the sequences in are described as (22) Then, has a QBP point and has a QBP point . Moreover, we have In addition, if and , then

Proof. Based on Lemma 7, we conclude that From Lemma 9, we find that , and are Cauchy sequences. Thus, there are so that , , , and . Hence, we have Passing in (53) we find that By the same method, we have Now, consider Passing we obtain Analogously, we can obtain Therefore, is a QBP point of
Analogously, we can prove that there are so that , , , and . Moreover, we get Hence, is a QBP point of
Ultimately, let and , then we claim that (52) holds. For each , one can write Letting we get Also, , we get Passing one can obtain Similarly, we obtain It follows from (61), (63), (64), and (66) that This leads to Since then we have According to (66) and (68), we have This finishes the proof.

It should be noted that every pair of nonempty closed subsets and of a so that is convex fulfills the property , then we can state the result below.

Corollary 11. Assume that is a and , are nonempty closed subsets of . Assume also and are cyclic contraction mappings. If and the sequences in are defined as (22), for each , then has a QBP point and has a QBP point . Moreover, we get In addition, if and , then

The following example supports Corollary 11.

Example 9. Let be a equipped with the usual norm. Take and Obviously, Describe two mappings and as for each and respectively. For all , , and fixed we have Thus, the mappings and are cyclic contractions with Because and are closed convex, the pairs and justify the property . Therefore, all requirements of Corollary 11 are fulfilled. Thus, has a QBP point and has a QBP point. We note that a point is a unique QBP point of and a point is a unique QBP point of . Therefore, we obtain

In a compact subset of a MS, we can obtain the QBP point result as follows.

Theorem 12. Assume that is a MS and , are nonempty compact subsets of Assume also and are cyclic mappings. If and the sequences in are defined in (22), for each Then has a QBP point and has a QBP point . Moreover, we get In addition, if and , then

Proof. Since and (22) holds for each we get The compactness of illustrates that the sequences , and have the convergent subsequences , and , respectively, so that Now, we have Applying Lemma 7, we find that Taking in (79) we get By the same manner, one can obtain Notice that As we have Analogously, we have Thus, has a QBP point . By the same argument, since is compact, we can also claim that has a QBP point . To prove we can follow the same approach used in the proof of Theorem 10.

3. Quadruple Fixed Point Technique

This part is devoted to present new QFP consequences in the sense of cyclic contraction mappings.

Theorem 13. Assume that is a MS and , are nonempty closed subsets of Let the mappings and be cyclic contractions. If and the sequences in are described as (22), for each If , then has a QFP point and has a QFP point . Moreover, we get In addition, if and , then and have a common QFP in

Proof. Because we find that the pairs and justify the property . Using Theorem 10, we see that has a QBP point , that is, and has a QBP point , that is, From (88) and since , we obtain This means that is a QFP of
Again, from (89) and since , we get This means that is a QFP of
Now, let and . From Theorem 10, one can write Since we have It follows that Therefore, the quadruple is a common QFP of and . This is enough to end the proof.

Example 10. Consider equipped with the usual norm. Take and Describe two mappings and as for each and respectively. Then, and are cyclic contractions with Moreover, for each and , we get Therefore, all postulates of Theorem 13 are justified. Then, and have a common QFP .

Putting in the above theorem, we have the result below.

Corollary 14. Assume that is a complete MS and is a closed subset of . Assume also that and are cyclic contraction mappings so that if and the sequences in are defined as (22), for each then has a QFP and has a QFP . Also, we obtain Moreover, if and , then and have a common QFP in .

The following corollary is very important in the application part. We get this result by placing in Corollary 14.

Corollary 15. Let be a complete MS and be a nonempty closed subset of . Assume also the mapping verifying for all , and . Then, has a unique QFP .

4. Solving a System of Functional Equations

Here, we apply Corollary 15 to discuss the existence of the solution for the following quadruple functional equations: where and are a state and decision space, respectively; ; ; and .

Assume that is the set of all bounded real-valued functions on . Consider

Moreover, define a distance on in the form of

Obviously, the pair is a complete MS.

In the theorem below, we will discuss the existence of the solution for the system (99).

Theorem 16. Assume that the following postulates are fulfilled:
(a₁) the functions and are bounded
(a₂) for each , and we have Then, system (99) has a unique bounded solution ().

Proof. Describe an operator on the space by for each and The existence solution for system (99) is equivalent to find a QFP of the operator
Clearly, the mapping is well-defined (because the functions and are bounded).
Hence, by the postulate , we obtain This implies that the contractive stipulation of Corollary 15 holds with . Then, the mapping has a unique QFP, which is a of problem (99)

The example below justifies Theorem 16.

Example 11. Consider a quadruple system of functional equations below: for all

Clearly, system (105) is comparable to system (99) with and Clearly, the postulate of Theorem 16 is fulfilled. To achieve the postulate we have

Therefore, the postulate of Theorem 16 is fulfilled. Thus, problem (105) has a in

5. Solving a System of Integral Equations

The existence of solutions for a system of quadruple integral equations is presented here by using the results of Corollary 15.

Consider the following problem: where with .

Suppose that is endowed with

Moreover, define a distance on in the form of

Hence, is a complete MS.

Theorem 17. Suppose that the following hypotheses hold:
The functions and are continuous so that For all , , , , , , , , we get

Then, problem (107) has a unique solution on .

Proof. Define the mapping by The existence solution of (107) corresponds to finding a QFP of
Assume that , , , , , , , , we get Hence, the stipulation of Corollary 15 holds with . Therefore, has a QFP, which in turn is considered the unique solution to (107)

Example 12. Consider a system of quadruple integral equations below: for all

Problem (114) is another shape of problem (107) with and

Obviously, the hypothesis of Theorem 17 holds. For the hypothesis , one can write

Hence, the hypothesis of Theorem 17 is justified with . Therefore, the mapping has a unique QFP which in turn is considered the unique solution to (114).

6. Conclusion

One of the central problems in approximation theory is to determine points that minimize the distance to a given point or subset. The best approximation has always attracted analysts because it carries enough potential to be extended especially with the functional analytic approach in nonlinear analysis. The best proximity point has many applications such as obtaining the existence of a unique solution for a variational inequality problem, integral and differential equations, and many other directions. The fixed point method is considered one of the distinguished methods for obtaining these points under cyclic contraction mappings, due to its smoothness and clarity. So, in this paper, the existence of a quadruple best proximity point for a cyclic contraction mapping is introduced in ordinary metric space. The validity of theoretical results in a uniformly convex Banach space was also discussed. Moreover, several examples are given to strengthen the theoretical results. Finally, our paper has been provided with applications on the existence and uniqueness of the solution to a system of functional and integral equations.

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 and significantly in writing this article.

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