Application of Some Fixed-Point Theorems in Orthogonal Extended S-Metric Spaces

Rezaei, Mohammad Mahdi; Sedghi, Shaban; Parvaneh, Vahid

doi:https://doi.org/10.1155/2021/3040469

Journal of Mathematics

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

Research Article | Open Access

Volume 2021 | Article ID 3040469 | https://doi.org/10.1155/2021/3040469

Application of Some Fixed-Point Theorems in Orthogonal Extended S-Metric Spaces

Mohammad Mahdi Rezaei,¹Shaban Sedghi,¹and Vahid Parvaneh²

Academic Editor: Xiaolong Qin

Received29 Apr 2021

Accepted07 Oct 2021

Published29 Oct 2021

Abstract

In this study, we obtain some coincidence point theorems for weakly --admissible contractive mappings in an orthogonal extended -metric space. An example and an application are provided to illustrate the usability of the obtained results. Our results generalize the results of several studies from metric and -metric frameworks to the setting of orthogonal extended -metric spaces.

1. Introduction

In 1922, Banach [1] proposed a theorem known as the Banach fixed point theorem (or Banach contraction principle or BCP for short) to prove the existence of solutions for nonlinear operator equations and integral equations. Since then, it has become a very popular tool due to its simplicity and practicality in solving a variety of problems such as control theory, economic theory, nonlinear analysis, and global analysis. Later, there is a large body of literature on the applications, generalizations, and expansions of this theorem. In [2], for example, the authors introduced the definition of a cyclic p-contraction pair for single-valued mappings and presented some best proximity point results for such mappings on a proximally complete pair of subsets in a metric space. In [3], Dehici and Redjel obtained some fixed-point results for nonexpansive mappings defined on an intersection of a finite number of closed bounded and convex nonempty subsets in Banach spaces. These research studies have been done by several authors in different directions, for example, by weakening the hypothesis, using different settings, considering different contractions, and so on.

Many mathematical problems require finding distances between two or more objects that are not easy to measure exactly in general. There are different approaches to get the right concept of a metric structure. Due to the need to create an appropriate framework for modeling several salient problems of a practical nature, the study of metric spaces has attracted the attention of many researchers. In the last few decades, a number of generalizations of metric spaces in several articles, such as 2-metric spaces [4], -metric spaces [5], -metric spaces [4], partial metric spaces [6], and cone metric spaces [7], have been illustrated.

Sedghi et al. [8] introduced the concept of an metric space and proved that this concept is a generalization of -metric space and -metric space. Also, they have proven some properties on -metric spaces and some fixed-point theorems for mappings defined on an -metric space.

The concept of -metric space, as one of the useful generalizations of standard metric spaces, was first used by Bakhtin in [9] and Czerwik in [10]. This idea has been extended by Parvaneh and Ghoncheh [11] to the -metric settings.

The concept of weak contraction mapping was introduced by Alber and Guerre-Delabrere in [12] in the arrangement of Hilbert spaces. Rhoades [13] proved that every weak contraction mapping on a complete metric space has a unique fixed point. Zhang and Sang [14] have generalized this concept to weakly contractive pairs.

In this study, we prove some coincidence point results for -admissible -contractive mappings in orthogonal extended -metric spaces where we were motivated by the work in [15]. Using an auxiliary function, commonly referred to as , in the article [16] by Samet et al., which continued in several articles, we want to define some related concepts such as orthogonal -compatibility and orthogonal -regularity.

2. Preliminaries

From now on,

Definition 1 (see [11]). Let be a nonempty set. A function is an extended -metric (-metric, for short) if there exists an , such that(1) iff (2)(3)for all .
In this case, the pair is called an extended -metric space or, briefly, a -metric space.
A -metric is a -metric when for some fixed , and a metric is a -metric when . The following example shows that a -metric need not be a -metric.

Example 1 (See [11]). Let be a -metric space with coefficient and let .
For each , we haveSo, is a -metric with .
Note that is not a metric on , as, e.g.,Similarly, although is a -metric on with , there is no , such that is a -metric with parameter . Indeed, putting and , we should have which cannot hold for any fixed and sufficiently large.

Definition 2 (see [8]). Let be a nonempty set. An -metric on is a function so that(1)(2) if and only if (3)For all .
The pair is called an -metric space.

Example 2. (see [8]).(1)Let and be a norm on . Then, is an -metric on .(2)Let and be a norm on . Then, is an -metric on .(3)Let be a nonempty set and be an ordinary metric on . Then, is an -metric on .

Definition 3 (see [17]). Let be a nonempty set and be a given real number. Suppose that a mapping satisfies(1)(2) if and only if (3)For all .
Then, is called a -metric, and the pair is called a -metric space.

Definition 4 (see [18]). Let be a nonempty set and . Suppose that satisfies(1)(2) if and only if (3)For all .
Then, is called an extended -metric (-metric) and the pair is called an extended -metric space (-metric space).
Each -metric space is an -metric space (-MS) with and every -MS with parameter is an -MS with .

Remark 1. In -MS, we have and for all , i.e, any -metric is not symmetry in general.

Proposition 1 (see [18]). Let be -MS with coefficient and let , where . Then, is an extended -metric with .

Proof. For all ,Therefore, is an extended -metric with .

Example 3 (see [18]). Let be -MS with coefficient . Then, , , , , and , and are some -metrics.

Definition 5 (see [18]). Let be -MS and . The open ball and the closed ball with center and radius are as follows:

Example 4 (see [18]). Let . Denote by for all and let . Thus,

Definition 6 (see [18]). Let be -MS. A sequence in is said to be(1)-Cauchy if, for each , there exists a positive integer , such that for all , (2)-convergent to a point if, for each , there exists a positive integer , such that for all , (3)An -MS is called -complete, if every -Cauchy sequence is -convergent in .Recently, Eshaghi Gordji et al. [15] introduced the notion of orthogonal sets.

Definition 7. (see [15]). Let be a binary relation. If there exists , such that for all or for all , then is called an orthogonal set (briefly, -set). We denote this -set by .

Definition 8 (see [15]). Let be an -set. A sequence is called an orthogonal sequence (briefly, -sequence) if or for all .

Definition 9 (see [15]). Let be an orthogonal set with metric d. Then, is orthogonally complete (briefly, -complete) if every Cauchy -sequence is convergent.
Orthogonal continuous mappings and other concepts can be defined as above.
Motivated by [11], we present the following definitions.

Definition 10. Let be an --MS, be a self-mapping on , and let be a function.(1) is an --admissible mapping if for all .(2)The space is said to be --complete if every Cauchy -sequence in with for all converges in (3) is an --continuous mapping on , if as , , and for all , for given and arbitrary sequence (4) is --regular if , , and for all ; then, for all .Let be a nonempty set and be a given mapping. For every , let .
Motivated by the work in [19], we present the following definition.

Definition 11. Let be a set, be mappings such that , and let be a function. The ordered pair is said to be(i)Weakly ---admissible if for all , and for all , and and for all ,(ii)Partially weakly ---admissible if for all , and for all (iii)The ordered pair is said to be rectangular weakly ---admissible (rectangular partially weakly --admissible) if it is weakly --admissible (partially weakly --admissible) with respect to and , , , and imply and for all .If, in the previous conditions, (the identity mapping), then we can omit the function in the respective notions.

Definition 12. Let be an --MS and . The pair is said to be --compatible if , whenever is an -sequence in , such that for all and for some .
In general, an -metric mapping with nontrivial function need not be jointly continuous in all its variables [18]. Thus, in some proofs, we will need the following simple lemma about the -convergent sequences.

Lemma 1. Let a be --MS.(1)Suppose that and are -convergent to and , respectively. Then, we have In particular, if , then we have .(2)Suppose that is -convergent to and is arbitrary. Then, we have(3)Suppose that is -convergent to and is arbitrary. Then, we have

Proof. (1)Using the rectangle inequality in the -metric space, it is easy to see that Making in the above inequalities, we acquire the desired result.(2)According to the rectangle inequality, we see that(3)According to the rectangle inequality, we see thatRecall that the self-mappings of a set are said to be weakly compatible if , whenever [20].
The function is called an altering distance function (a.d.f) [21], if(1) is continuous and nondecreasing(2) iff

3. Main Results

Let be --MS, and let be four self-mappings. Throughout this study, unless otherwise stated, for all , will denote an arbitrary element of the set

Theorem 1. Let be an --complete --MS, , and let be such that and .
Suppose that for all with and ,where are two a.d.f. Assume that , and are --continuous, the pairs and are --compatible, and the pairs and are rectangular partially weakly ---admissible and ---admissible, respectively. Then, the pairs and have a coincidence point in . Moreover, if and , then is a coincidence point of , and .

Proof. Let . Choose , such that , and , such that . Continuing in this way, we construct a sequence defined byfor all . As and and the pairs and are partially weakly ---admissible and ---admissible, respectively, we haveContinuing this way, we acquire thatfor all .
Three steps will be taken to finish the proof.

Step 1. We will prove that .
Let . Suppose that for some . Then, .
If , then gives . Indeed, whereIf , then (19) becomeswhich implies that , that is, .
If , then (19) becomeswhich implies that , that is, .
Similarly, if , then gives . Continuing this process, we find that is a constant sequence for . Hence, is correct.
Now, suppose thatfor all . We show thatfor each
Let and , for . Then, as and , using (15), we acquire thatwhereIfAs , then from (25), we havewhich implies that . This is possible only if , that is, , a contradiction to (23). Hence, for all .
Therefore, (24) is proved for .
Analogously,for all .
Similarly, for other values of , we can see that is a nondecreasing sequence of nonnegative real numbers. Therefore,for some .
We know thatSubstituting possible values of in (25) and making , we acquire that . For instance, letThen, from (25), we haveLetting in (33), using (30) and the continuity of and , we haveHence, .
Now, taking into account (33) and letting , we find that . Hence, . Similarly, for the other values of , we can show that

Step 2. We will show the -Cauchyness of in . Assume that for some , we have subsequences , such that andin which is the least index, i.e.,From the rectangle inequality and (36) and (37), we haveTaking the limit as in (38), from (35), we acquire thatUsing the rectangle inequality again, we haveLetting in the above inequality, we haveFinally,Letting in the above and using (39), we haveWe know that andfor all . On the other hand, the pairs and are rectangular partially weakly --admissible with respect to and , respectively. So,which imply thatAlso,which imply thatContinuing in this manner, we obtain that and . Now, we can apply (15) to acquire thatwhereIfthen from (35), we get that . Hence, according to (49), we have , which contradicts (41).
Ifthen from (39) and (43), we get thatTaking the limit as in (49), we havewhich implies that . Hence, , a contradiction to (39).
Ifthen from (37), by taking the limit as in (49), we havewhich implies that . Hence, . Therefore, from the rectangular inequality, we can conclude that which contradicts (39). Hence, is a -Cauchy -sequence.

Step 3. We will show that , , and have a point of coincidence.
Since is a -Cauchy -sequence in the --complete --MS and , then there exists , such thatHence,As is --compatible and , soMoreover, from , , and the --continuity of and , we acquire thatBy the rectangle inequality, we haveTaking the limit as in (61), we deduce thatwhich yields that , that is, is a coincidence point of and .
Similarly, it can be proved that . Now, let and . From (15), we havewhereIn all three cases, the above inequality yields that
In what follows, we replace the obligation of --continuity of , and by an hypothesis on the underlying space and replace the --compatibility by weak compatibility.

Theorem 2. Let be an --regular --complete -MS, , and let be such that , , and and are two -closed subsets of . Suppose thatfor all and with and . Then, the pairs and admit a coincidence point in whenever the pairs and are weakly compatible and the pairs and are rectangular partially weakly ---admissible and ---admissible, respectively. Moreover, if and , then is a coincidence point of , and .

Proof. From the proof of Theorem (15), for some , we haveSince is -closed and , therefore, . Hence, for some , one has , andSimilarly, for , we have andWe shall show that is a coincidence point of and .
Since , as , from --regularity of , . Therefore, from (65), we havewhereFrom Lemma 1,Taking the upper limit as in (69), using Lemma 1 and the continuity of and , we acquire thatwhich implies that . Hence, .
As and are weakly compatible, we have . Thus, is a coincidence point of and .
Analogously, it can be shown that is a coincidence point of the pair .

Example 5. Let , and on be given by , for all , with . Let be given by and , iff . Define self-maps , and on byLet be such that , that is, . By the definition of and , we have , and so, . Therefore,Thus, and . Hence, is partially weakly ---admissible.
To prove that is partially weakly ---admissible, let be such that , that is, . Hence, we have , and so, . Therefore,Thus, and .
Furthermore,Define as and for all .
Using the mean value theorem, for all and with , we haveThus, (15) is true for all and . Therefore, all the conditions of Theorem 1 are satisfied. Moreover, is a coincidence point of , and .
We arrange the following example, which supports Theorem 2, based on Example 2.8 of [22].

Example 6. Consider the set and chooseDefine the mapping and the relation on byLet be defined by and .
Obviously,(1) is an --regular --complete -MS. Indeed, let , and for all . Then, for all . So, is --regular.(2), , and and are two -closed subsets of (3) for all and with and , i.e., for all and with .(4)The pairs and are weakly compatible(5)The pair is rectangular partially weakly ---admissible(6)The pair is rectangular partially weakly ---admissibleNote that in parts 5 and 6, it is sufficient to take .
Therefore, , and satisfy all the conditions given in Theorem 2. Thus, 1, 2, and 3 are coincidence points of the pairs and . Moreover, and , so is a coincidence point of , and .

4. Application

In this part of the research, we study the following boundary value problem:where are the continuous functions.

The above system is metamorphosed to the following system of Fredholm integral equations:where the kernel is given by

See [23] for details.

Now, we present an existence result for a common solution of (82) in (the set of continuous real functions defined on ). Note that we consider the space , with the -metric given byfor all which is --complete --MS. Note that , , and iff .

Define byfor all and .

Clearly, a function is a common solution of (82) if and only if it is a common fixed point of .

Consider the following obligations: For all and for all , are continuous for all

Theorem 3. Assume that the above presumptions and hold. Then, the system (82) possesses a common solution in .

Proof. To show that all assumptions of Theorem 2 are satisfied, it remains to prove that satisfy the contractive condition in Theorem 2. Let . For each , we haveVia a careful calculation, we get thatSo, we acquire thatTaking the supremum on , we deduce thatNow, by taking , , and , we infer thatThus, all the hypotheses of Theorem 2 are fulfilled, and we deduce the existence of an element , such that . Note that regularity of the space can be done as in [24].

5. Conclusions

In this study, we prove certain coincidence point theorems in an orthogonal extended -metric space for four weakly --admissible contractive mappings. Many authors consider spaces with an extrastructure, such as partially ordered spaces, graphical spaces, and orthogonal spaces, in addition to working on different spaces and different contractive conditions to avoid exploring the contractive conditions with all elements in the space. In reality, they significantly reduce the calculations in contractive conditions. Only elements that are orthogonal to each other are included in this study. Also, -admissibility is a generalization of monotonicity in the regular case.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this study. All authors read and approved the final manuscript.

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Copyright © 2021 Mohammad Mahdi Rezaei et al. 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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