Abstract

In this paper, we initiate the concept of orthogonal partial -metric spaces. We ensure the existence of a unique fixed point for some orthogonal contractive type mappings. Some useful examples are given, and an application is also provided in support of the obtained results.

1. Introduction

The notion of a metric space plays a vital role in metric fixed point theory. The Banach contraction principle [1] is a very famous result in the literature. Fixed point hypotheses are significant apparatuses for demonstrating the presence and uniqueness of solutions for different numerical models. Given a nonempty set and a map from into itself, the problem of finding a point such that is considered as a fixed point problem, and the point is called a fixed point of .

A natural question is that, under what conditions on and does a fixed point exist? Theorems which establish the existence (and uniqueness) of such points are called fixed point theorems. These results allow us to find the existence of solutions that satisfy certain conditions for operator equations.

There exist many generalizations of the concept of a metric space in the literature. In [2], Matthews introduced the notion of a partial metric space as part of the study of denotational semantics of dataflow networks. In this setting, the self-distance of any point may not be zero. A lot of fixed point theorems have been investigated in partial spaces (see [3–7] and references therein). Another important generalization of a metric space introduced by Czerwik [8] is a -metric space, where the triangular inequality was replaced by with . Since then, many articles dealing with fixed point theory and variation principle in the setting of -metric spaces for single- and multivalued operators have been appeared (see [9–20]).

Recently, Eshaghi Gordji et al. [21] initiated the concept of orthogonal sets and gave an extension of the Banach contraction principle. Furthermore, they presented applications for their results to ensure the existence and uniqueness of solutions for first-order differential equations.

The purpose of this paper is to improve and generalize the concept of an orthogonal contraction in the sense of metric spaces due to Gordji et al. [22]. Namely, we introduce the concept of an orthogonal partial -metric and establish some fixed point theorems for related contractions. We also enrich this paper with a nontrivial example involving an orthogonal partial -metric, which is not a partial -metric. We set up some hypotheses for the proposed construction, and additionally, we present a potential application for the arrangement of Volterra integral equation to guarantee the legitimacy of the outcomes.

2. Preliminaries

In 1993, Czerwik [8] presented the idea of -metric spaces (in short, b-M.Ss).

Definition 1. (see [8]). Let be a nonempty universal set and be a mapping so that for all , , and :Then, is said as a -metric on and is said as a -M.S.

The notion of a partial metric space (in short, -M.S) has been initiated by Matthews [2].

Definition 2. (see [2]). Let be a universal set and be a mapping so that for all , , :Then, is said as a -metric on and is said as a -M.S.

In 2014, Shukla [23] introduced the following concept of partial -metric spaces (in short, -M.Ss) and proved some fixed point results.

Definition 3. (see [23]). Let be a nonempty universal set and be a mapping so that for all , , and :Then, is said as a -metric on and is said as a -M.S with coefficient .

Remark 1. (see [23]). If and are in a -M.S so that , then . However, the converse may not be true.

Example 1. (see [23]). Let , be a constant, and be defined byThen, is a -M.S with coefficient . However, it is neither a -M.S nor a -M.S.

In 2017, the authors of [22] introduced the notion of orthogonal sets and gave a real generalization of Banach fixed point theorem.

Definition 4. (see [22]). Let and be a binary relation defined on . Then, is called an orthogonal set if there is so thatThe element is said to be an O-element.

Definition 5. (see [22]). Let be an OS. A sequence is said to be an orthogonal sequence if

Definition 6. (see [21]). A mapping is orthogonal continuous in if for each O-sequence in such that , then . Also, is said to be OC on if is OC at each .

Definition 7. (see [22]). Let be an orthogonal M.S. Then, is called O-complete if every Cauchy O-seq is convergent.

Definition 8. (see [21]). Let be an orthogonal M.S and . A mapping is called an orthogonal contraction with the Lipschitz constant iffor all , with .

Definition 9. (see [22]). Let be an OS. A mapping is said to be O-preserving if , whenever .

3. Main Results

Throughout the paper, O-comp designs orthogonal complete. First, we introduce the concept of an orthogonal partial -metric space (in short, orthogonal -M.S).

Definition 10. A map is called an orthogonal -M.S on the OS if the following axioms are satisfied:for all , , with .
The pair is said as an orthogonal -M.S with a coefficient .

Remark 2. Every orthogonal partial -metric space is a partial -metric, but the converse is not true in general. Example 2 describes an orthogonal partial -metric, which is not a partial -metric. In [22], Eshaghi Gordji and Habibi considered orthogonal metric spaces to prove some fixed point theorems, while in Theorem 3.2 and Theorem 3.3 of the paper [21], the authors considered generalized orthogonal metric spaces to prove some related fixed point theorems. In this paper, we consider orthogonal partial metric spaces to prove some fixed point results. Note that an orthogonal partial -metric is more generalized than an orthogonal metric, an orthogonal -metric, and an orthogonal partial metric. That is, our results are generalizations and extensions of the results given in [21, 22].

Example 2. Let and let the binary relation be defined by iff or , . It is easy to prove that is an orthogonal -metric on (with a coefficient ). However, is not a -metric on (with a coefficient ). Indeed, for and , we have .

As related topological notions for this new setting, we state the following definitions.

Definition 11. Let be an orthogonal -M.S with . Then, an O-seq is called(i)Convergent iff there exists such that as (ii)Cauchy iff as ,

Definition 12. Let be an orthogonal -M.S. Then, is called O-continuous at if for each O-seq in with , we have . Also, is said to be OC on if is OC at each .

Definition 13. Let be an orthogonal -M.S with . Then, is called O-complete if every Cauchy O-seq is convergent in .

Our first essential main result is as follows.

Theorem 1. Let be an O-comp -M.S with and be an OP and OC mapping so thatwhere . Then, has a unique fixed point and .

Proof. By the definition of orthogonality, there is such that for all , or for all , . It follows that or . Let , , , , , , for all . Since is OP, is an O-seq. Then, by (9), we obtainfor all . For all and , it follows thatTaking limit as , we haveTherefore, is a Cauchy O-seq. Since is O-comp, there is so that as . Since is OC, we obtainHence, is a fixed point of . Next, we demonstrate its uniqueness. Let be a fixed point of . So, we obtain and for all . By the definition of orthogonality, there is so thatSince is OP, one writesorfor all . Therefore, by the triangle inequality, we obtainTaking limit as , we obtainand so .

Example 3. Consider . Given asDefine on by iff or , . Then, is an O-comp -M.S with coefficient . Define the mapping byWe have the following cases:(a)If , then . If , , then , . Thus, is OP.(b)If any O-seq in with , for some and , then we obtain .(c)If any real number, then . Thus, and , that is, .(d)Let , with .If , then the following result holds:If , , then the following result holds:By Theorem 1, has a fixed point.

Our second result as a generalization of Theorem 1 is as follows.

Theorem 2. Let be an O-comp -M.S with and be an OP and OC mapping so thatfor all , , where . Then, has a unique fixed point and .

Proof. By the definition of orthogonality, there is so that for all , or for all , . It follows that or . Let , , , for all . Since is OP, is an O-seq. Then, by (23), we haveIf for some , , we obtain that , which is a contradiction. Thus,Again, we haveRepeating this cycle, we obtainfor all . For , with , we obtainUsing (27) in the above inequality,As and , it follows from the above inequality thatTherefore, is a Cauchy O-seq. Since is O-comp, there is so that as . As is OC, one writesTherefore, is a fixed point. To show that it is unique, consider as a fixed point of . So, we obtain and for all . By the definition of orthogonality, there exists so thatAs is OP, we obtainorfor all . Then, we obtainIt is a contradiction. So, we need to have , that is, . Thus, if fixed point of exists, then it is unique.