Some New Aspects of Metric Fixed Point Theory

Abstract

In this manuscript, we prove fixed point results in -metric spaces endowed with an amorphous binary relation. Moreover, we give an example to highlight the utility of our main results. Finally, we apply our result to examine the existence and uniqueness of the solution for a Fredholm integral equation.

1. Introduction and Preliminaries

In metric fixed point theory, Banach contraction mapping principle [1] is one of the most fundamental tools to investigate the existence and uniqueness of solutions for contraction maps in a complete metric space. Since the appearance of this classical result, researchers have taken keen interest in generalizing and extending this result in different ways, either by improving contraction conditions or by relaxing the axioms of metric space. One may recall the existing notions of, namely, partial metric space [2], partial -metric space [3], partially ordered -metric space [4, 5], partial symmetric space [6], partial JS-metric space [7], metric-like space [8], -metric space [9], rectangular metric space [10], and several others. In 2014, Asadi et al. [11] extended the concept of partial metric space and introduced the notion of -metric spaces to investigate fixed point. This concept was extended in many different ways, such as -metric space [12], -metric space [13, 14], and rectangular -metric space [15]. Also in this sequel, Patle et al. [16] extended the notion of -metric by proving fixed point results for Nadler and Kannan type set valued mappings in -metric spaces.

Recently, Gordji et al. [17] introduced the notion of orthogonal sets and gave a new extension for the classical Banach contraction principle; more details can be found in [18–20]. Utilizing the structure of orthogonal metric spaces, which appeared in [18, 19], and the binary relation used with a metric, in [21], Ali et al. [22] introduced the notion of -partial -metric space and proved fixed point results in this space. Recently, Javed et al. [23] studied fixed point results in fuzzy -metric space using ordered-theoretic relation, which can also be seen in the work of Alam et al. [24].

Inspired by the metric structure used by Ali et al. [22] and using the concept of -metric space, we introduce the notion of -metric space. We improve and generalize some well-known results and establish fixed point theorems in the sense of -metric space. We also impart some illustrative examples and a possible application for Fredholm integral equations to demonstrate the validity of our results.

Definition 1 ([2]). Let be a nonempty set. A function is called a partial metric , if the following conditions are satisfied: (1) iff (2) for all (3) for all (4) for all

The pair is called a partial metric space (pms).

Notation 2 [11]. (1)(2)

In [11], a generalization of the pms was introduced as follows.

Definition 3 ([11]). Let be a nonempty set. A function is called an -metric, if the following conditions are satisfied: (1) iff (2) for all (3) for all (4) for all

The pair is called an -ms.

Example 4. Let Then on is an -metric.

Definition 5 ([25]). Let be a nonempty set. A subset of is called a binary relation on . Then, for any , we say that “ is -related to ,” that is, , or “ relates to under ” iff . means that “ is not -related to ” or “ is not related to under .”

Definition 6 ([25]). A binary relation defined on a nonempty set is called (a)reflexive if (b)irreflexive if for some (c)symmetric if implies (d)antisymmetric if and imply (e)transitive if and imply (f)preorder if is reflexive and transitive.

Definition 7 ([21]). Let be an -set. A sequence is called an -sequence if

Definition 8 ([21]). A map is -continuous in if for each in such that then Also, is said to be on if is for each .

Definition 9 ([18]). Let be a ms and be a binary relation on . Then is called an -ms.

Definition 10 ([18]). Let be an -ms. Then is said to be -complete, if every Cauchy is convergent.

Definition 11 ([18]). Let be an -set. A map is said to be -preserving if whenever Also, is said to be weakly if or whenever

Definition 12 [21]. A map is an -contraction, if for all with , where .

2. Main Results

We will start this section with the definition of an -ms, but first, we introduce the following notations, which would be helpful during the proof.

Notation 13. (1);(2).

Definition 14. Let and be a reflexive binary relation (br) on , denoted as . A map is called an -metric on , if the following conditions are satisfied, for all with either or , either or and either or :
iff

Then, is called an -ms.

Remark 15. In the above definition, a set is endowed with a reflexive br and satisfies only for those elements which are comparable under the reflexive br . Hence, the -ms may not be a -metric, but the converse is true.

The next example shows that the -ms does not need to be an -ms.

Example 16. Let and a br be defined by: iff or . It is easy to prove that is an -metric on , but is not an -metric on ; since for and , we have .

Definition 17. Let be an -ms. Let be an in , that is, or for each . Then, is said to be (i)-convergent to some if(ii)-Cauchy ifexist and are finite.

Definition 18. is said to be -complete -ms, if for every -Cauchy sequence in is convergent.

Definition 19. Let be an -ms. We say that is at if for each in with , we have . Also, is on if is for each

The following results help us to ensure the existence of fixed point for self-maps. Throughout, we assume that is a preorder relation.

Theorem 20. Let be an -complete -ms and be and satisfying the following condition: where Then, has a unique fp and

Proof. Let , for all Since is , is . Then, by (5), we get for all For any and it follows that Hence, As it follows from the above inequality that Similarly, Thus, is a -Cauchy in . Since is -complete, there exists such that Now, we show that is a fp of . Consider By using we get implies Hence, By we get as desired.
Now, we show that if is a fp of then Then, and hence,
Finally, we show that fixed point of is unique. Assume that there exists such that Hence, we have for all By the choice of in the first part of the proof, we obtain Since is , we have or for all Thus, which implies that and so Thus, ☐

Example 21. Let , and be given by Define the binary relation on by if or Then is -complete -ms. Define by

Then, it is very simple to verify the following:

If then If then Thus, is .

Also, one can see that for all with Hence, the inequality (5) is satisfied with . But is not a contraction. Otherwise, for two points and and for all , we have and one can conclude that it is a contradiction.

Let be an arbitrary in such that converges to Since the inequality (5) is satisfied, for each we have

As goes to infinity, is . But it can be easily seen that is not continuous. Therefore, all the conditions of Theorem 20 are satisfied. Hence, we can conclude that has a unique fixed point in which is

Remark 22. Note that the function defined in the above example is neither a metric, nor an -metric on . Since , , , and .

Theorem 23. Let be an -complete -ms and be and satisfying the following condition: for all , where Then has a unique fp and

Proof. Let , , for all Since is , then is . Then by (24), we get for all which implies where as By repeating this process, Then By (24), for any two natural number it follows that and so For every we can find a natural number such that and for all . Therefore, it follows that for all , and as Now, So which implies that Similarly, Thus, is a -Cauchy in . Since is -complete, there exists such that Now, we show that is a fp of . Consider, implies Hence, By we get .
Now, we show that if is a fp of then Consider, that is,
Finally, we show that fixed point of is a unique. Assume that there exists such that Since is , we have or for all Thus, which implies that and so Thus, ☐

Theorem 24. Let be an -complete -ms and be and satisfying the following condition: for all , where Then, has a unique fp and

Proof. Let for all Since is , then is . By (41), we get If . Then, from the above inequality, we obtain that , a contradiction. Hence, . From the above inequality, we have On repeating this process, we obtain for all As it follows from the above inequality that Similarly, Thus, is a -Cauchy in . Since is -complete, there exists such that Now, we show that is a fp of . Consider So, Thus, By we get as desired.
Now, we show that if is a fp of then Then, and hence,
Finally, we show that fixed point of is a unique. Assume that there exists such that Hence, we have for all By the choice of in the first part of the proof, we obtain Since is , we have or for all Thus, which is a contradiction. Hence, and so Thus, ☐

3. Application

Within this section, we are attempting to apply Theorem 20 to investigate the presence and uniqueness of solution for a Fredholm integral equation. The space of all continuous real valued functions defined on is considered to be with the -metric given by for all . It can also be equipped with a relation given by Then, is -complete -ms.

Consider the following Fredholm integral equation: where .

Theorem 25. If there exists , such that for all . Then the integral equation (58) has a unique solution.

Proof. Define by Observe that the presence of fixed point of an operator is identical to that of solution of the integral equation (58). Now, for all with , Thus, all the conditions of Theorem 20 are fulfilled. Therefore, the operator has a unique fp, meaning that the integral equation (58) has a unique solution.☐

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 to have no competing interests.

Authors’ Contributions

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