Solutions of Integral Equations via Fixed-Point Results on Orthogonal Gauge Structure

Mehmood, Mazhar; Aydi, Hassen; Ali, Muhammad Usman; Fahimuddin, undefined; Shoaib, Abdullah; De La Sen, Manuel

doi:https://doi.org/10.1155/2021/8387262

Mathematical Problems in Engineering

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

Research Article | Open Access

Volume 2021 | Article ID 8387262 | https://doi.org/10.1155/2021/8387262

Solutions of Integral Equations via Fixed-Point Results on Orthogonal Gauge Structure

Mazhar Mehmood,¹Hassen Aydi,^2,3Muhammad Usman Ali,⁴ Fahimuddin,⁵Abdullah Shoaib,¹and Manuel De La Sen⁶

Academic Editor: Francesca Vipiana

Received07 May 2020

Revised03 Dec 2020

Accepted06 Jan 2021

Published28 Jan 2021

Abstract

The main outcome of this paper is to introduce the notion of orthogonal gauge spaces and to present some related fixed-point results. As an application of our results, we obtain existence theorems for integral equations.

1. Introduction

Fixed-point theory is a very important tool for proving the existence and uniqueness of the solutions to various mathematical models, such as integral and partial differential equations, optimization, variational inequalities, and approximation theory. Fixed-point theory has also gained considerable importance in the recent past after the famous Banach contraction theorem [1]. Since then, there have been many results related to mappings satisfying various types of contractive inequalities [2–5]. Recently, Gordji et al. [6] introduced an exciting notion of the orthogonal sets after which, orthogonal metric spaces were introduced. The concept of a sequence, continuity, and completeness has been redefined in this space. Further, they gave an extension of the Banach fixed-point theorem on this newly described shape and also applied their theorem to show the existence of a solution for a differential equation, which cannot be applied by the Banach fixed-point theorem.

On the other hand, many definitions and theorems in the literature do not require that all of the properties of a metric hold true. In the last decades, many concepts of generalized metrics (as controlled and double controlled metrics) have been introduced (see [7, 8]).

Gauge spaces are characterized by the fact that the distance between two points may be zero even if the two points are distinct. For instance, Frigon [9] and Chis and Precup [10] gave a generalization of the Banach contraction principle on gauge spaces. In the same direction, many interesting results have been raised obtained by different authors in [11–17]. In 2013, Ali et al. [18] ensured the existence of fixed points for an integral operator via a fixed-point theorem on complete gauge spaces. In 2012, Wardowski [19] gave a new type of contractions, named as -contractions, and established new related fixed-point results. This contraction type nicely generalizes the most famous Banach contraction condition. Later on, many researchers worldwide generalized this result (see [20–24]).

To give the ongoing research a new direction, we have combined the above statements in two directions of research. For this, we apply the concept of orthogonality in gauge spaces and investigate the existence of solutions of integral equations through the fixed-point theorem on orthogonal complete gauge spaces.

2. Preliminaries and Basic Definitions

First, we include some basic definitions and theorems which are useful to understand the results presented in this paper.

Wardowski [19] introduced the family of all functions so that (: for all with , we have (): for each positive sequence , iff (): there is so that

The following are elements in :(i) for (ii) for (iii) for

Further, Wardowski [19] introduced F-contractions and the related fixed-point theorem in the following way.

Theorem 1 (see [19]). Let be a complete metric space and let be an F-contraction mapping, that is, there are and so that for all impliesThen admits a unique fixed point in .

Minak et al. [25] generalized this result as follows.

Theorem 2 (see [25]). Let be a complete metric space and be a self-mapping. Suppose that there are and so thatfor all , with . If or is continuous, then admits a unique fixed point.

Now, we explain the notion of a pseudometric.

Definition 1 (see [26]). Let be a nonempty set. A function is said to be a pseudometric on if for all :(i)(ii)(iii)

Example 1. Denote by the set of continuous real-valued functions with . This point then induces a pseudometric on defined as .
In 2017, Gordji et al. [6] initiated the notion of an orthogonal set (or o-set).

Definition 2 (see [6]). Let and be a binary relation defined on . The pair is called an orthogonal set (or an o-set), ifThe element is said to be orthogonal. An orthogonal set may have more than one orthogonal element.

Example 2 (see [6]). Let . We write that if there is so that . Note that for each . Hence, is an o-set.

Definition 3 (see [6]). Let be an o-set. Any two elements are said to be orthogonally related iff .

Definition 4 (see [6]). Let be an o-set. A mapping is said to be -preserving if for all orthogonally related elements , we have .

Definition 5 (see [6]). Let be an orthogonal set. is said to be an orthogonal sequence (briefly, an o-sequence) if

3. Orthogonal Gauge Space

The simplest way of defining an orthogonal gauge space is that the gauge space defined on an o-set is called an orthogonal gauge space. The precise discussion is given below.

Let be an o-set and be a pseudometric on , then is said to be an orthogonal pseudometric space (or an o-pseudometric space).

Example 3. Let be the set of continuous real-valued functions with . This element induces an orthogonal pseudometric on defined by and the orthogonality on is given as iff or .

Definition 6. Let be an orthogonal pseudometric space. Then, the orthogonal d-ball (or -ball) of radius and centered at is the set .
Note that when we say that is an o-pseudometric on , it means that is an o-set and is a pseudometric on .

Definition 7. A family of o-pseudometrics on is said to be separating if for each pair such that , there is with .

Definition 8. Let be the family of o-pseudometrics on . Then, the topology having as a subbasis the set of ballsis said to be the topology induced by the set of o-pseudometrics. The pair is called an orthogonal gauge space.

Definition 9. Let be an orthogonal gauge space with respect to the family of o-pseudometrics on .(i)An orthogonal sequence converges to if(ii)An orthogonal sequence is Cauchy if(iii) is an o-complete gauge space iff each Cauchy o-sequence in converges to an element of .(iv)A subset of is called closed if it contains the limit of each convergent o-sequence.

4. Fixed-Point Results on Orthogonal Gauge Structure

In this section, we study the existence of fixed points for a mapping satisfying certain conditions on an orthogonal gauge structure. Throughout this article, is a directed set and is a nonempty o-set with an orthogonal element (say ) and also endowed with a separating o-complete gauge structure of o-pseudometrics.

Theorem 3. Let be a nonempty o-set endowed with a separating o-complete gauge structure of o-pseudometrics. Let be a mapping with and so thatfor all with and for each , whenever for , where
and for all . Further, assume that(i) is -preserving(ii)There is an element with and (iii)For each with and , we have (iv)For any o-sequence in such that for each and , we have and for each Then, possesses a fixed point.

Proof. Due to (ii), there is with and , and by considering (iii), we get . Moreover, we have , since is -preserving. Repetition of the same arguments implies and for each . Consider for each . Then we say that is an o-sequence with for each . Also, note that if there is some so that , then is a fixed point of . Thus, we assume there does not exist any such a natural number. As with and , then from (8), we haveSince is strictly increasing, from above we getthat is,Since , we haveNow, from (9), we haveAlso, we know that and . From (8), we getSince is strictly increasing, again from above we getthat is,As , thus we haveNow, from (14), we haveBy the obtained inequalities, we getWorking on the same steps, we conclude thatLetting in (20), we get for all . Thus, by property , we have . Let for all and for each . Using , there is so thatFrom (20), we haveLetting in (22), we getThis implies that there is with for each and for all . Thus, we haveTo ensure that is a Cauchy o-sequence, take with . Using (24) and triangular inequality, we haveThe series is convergent, so for all . It yields that is a Cauchy o-sequence. Since is o-complete, there is so that as . By (iv), we have and for each . We now claim that for all . On contrary, suppose that there is with , then there exists such that for each . Now, note that we have with , , and for all . Then, from (8), we getThis implies thatThus, by considering the triangular property and (27), we have for each At the limit , one obtainsIt is a contradiction, so for all . As is separating, we obtain .

Example 4. Let be the collection of all twice differentiable real-valued functions. Take the metricfor each . For , consider . Let be defined byGiven asSuppose that with and . Then and are both constant functions with at least one of them is the zero function. Say and . Then, we have the following two cases: Case 1: for and , we have and . Thus, and for each . Case 2: for and , we have and . Thus, and for each .Take and for all . ThenBy taking and , one can conclude that (8) holds for all with and for each .
Also, for , we have for each , then at least one of them is the zero function, and hence, for each , that is, . Thus, is -preserving. Further, for , , and so and . Furthermore, for all with and , we know that and are constant functions, then and are also constant functions, and so .
Moreover, for each o-sequence in with for each and , we have . Therefore, and for each .
Consequently, all the conditions in Theorem 3 are verified. One can conclude that possesses a fixed point.

Remark 1. For the functions defined in the above example, note that (8) does not hold for every . It suffices to take and , then and . Also note that and .

Theorem 4. Let be a nonempty o-set endowed with a separating o-complete gauge structure of o-pseudometrics. Let be a self-mapping with and so thatfor all with and for each , whenever for , where are positive real numbers with for all . Further, assume that(i) is -preserving(ii)There is with and (iii)For each with and , we have (iv)For any o-sequence in so that for each and , we have and for each Then, admits a fixed point.

Proof. Using (ii), there is with and , and by considering (iii), we get . Moreover, we have , since is -preserving. Repetition of the same arguments implies that and for each . Consider . Then is an o-sequence with for each . Also, note that if there is some such that , then is a fixed point of . Thus, we assume that such a natural number does not exist. As with and , then from (34), we getIf we assume that , then we have a contradiction with respect to (35). Thus, for all . Using (35), we haveAgain, we know that and ; then, from (34), we haveIf we assume that , then we have a contradiction to (37). Thus, . Thus, from (37), we haveFrom (36) and (38), we haveWorking with the same steps, we obtainLetting in (40), we get for all . Thus, by property , we have . Let for all and for each . From , there is so thatFrom (40), we haveLetting in (42), we getThis implies that there is so that for each and for all . Thus, we haveWe claim that is a Cauchy o-sequence. Take the integers with . Using (44) and the triangular inequality, one writesThe series converges, so for all . That is, is a Cauchy o-sequence. Since is o-complete, there is so that as . By (iv), we have and for each . We now claim that for all . On contrary, suppose that there is with . Then, there exists such that for each . Now, note that with , and for all . Then, from (34), we getThis implies thatThus, for each , by considering the triangular property and (47), we haveLetting in the above inequality, we haveIt is a contradiction, so for all . As is separating, hence we obtain .
In the following corollaries, we assume that is a nonempty o-set with an orthogonal element (say ) and endowed with a separating o-complete gauge structure . Further, assume that a directed graph is defined on so that the set of its vertices coincides with (i.e., ) and the set of edges is so that , where . Moreover, is supposed that it has no parallel edges.
The following corollaries can be obtained from our results by defining as

Corollary 1. Let be a mapping with and such thatfor each with and for each , whenever for , where and for all . Further, assume that(i) is -preserving(ii)There is an element with and (iii)For each with and , we have (iv)For any o-sequence in with for each and , we have and for each Then, admits a fixed point.

Corollary 2. Let be a mapping with and such thatfor each with and for each , whenever for , where are positive real numbers with for all . Further, assume that(i) is -preserving(ii)There exists an element with and (iii)For each with and , we have (iv)For any o-sequence in such that for each and , we have and for each Then, admits a fixed point.

5. Application to Integral Equations

Consider the following Volterra-type integral equation:where(i) is continuous(ii) is continuous(iii) is a continuous function

Let be the space of all real-valued continuous functions from into . We can define orthogonality relation on by

Define the family of pseudometrics as , for each , where is a positive real number. Clearly, defines a gauge structure on , which is separating and o-complete.

Theorem 6. Take . Define the operator bywhere , , and are continuous functions. Also, assume that there are and so that for all with and , we haveMoreover,for each . Then, (53) admits at least one solution.

Proof. For , takefor every . Note that for every . Hence, we say that if , then . Also, note that for each , we have for every , that is, .
Now, for all with and for each , we haveThus, we haveThis implies thatOne writesThat is,Therefore, it can be concluded that Theorem 4 is applied for the operator with the choice of for all , and for each and . Hence, possesses a fixed point, i.e., (53) admits at least one solution.
Takewhere(i) is continuous(ii) is continuousThe above equation is a Fredholm-type integral equation.

Theorem 7. Let and let the operator be defined bywhere and are continuous functions (). Assume that(i)If then we have(ii)There are and so that for each with and , we have Moreover,(iii)There is so that .(iv)For any sequence in with or for each and , we have or .Then, (64) admits at least one solution.
Let be the set of all real-valued continuous functions. Again, define orthogonality relation on byand family of pseudometrics given as , for each , where is a positive real number. Note that defines a gauge structure on , which is separating and o-complete, so the conclusion of this theorem can be obtained from Theorem 4 by taking for all , and for each and .

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 competing interests regarding the publication of this paper.

Authors’ Contributions

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

Acknowledgments

The authors are grateful to the Basque Government for its support of this work through grant IT1207/19.

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Copyright © 2021 Mazhar Mehmood 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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