On -Partial -Metric Spaces and Related Fixed Point Results with Applications

Abstract

In this paper, we introduce the notion of -partial -metric spaces and prove some related fixed point results in the context of this notion. We also discuss an example to validate our result. Finally, as applications, we evince the importance of our work by discussing some fixed point results on graphical-partial -metric spaces and on partially-ordered-partial -metric spaces.

1. Introduction and Preliminaries

Due to the fact that fixed point theory plays a very crucial role for different mathematical models to obtain their solution existence and has a wide range of applications in different fields related to mathematics, this theory has intrigued many researchers.

By the inception of the Banach fixed point theorem [1], researchers are continuously trying to get the generalizations of this classical result through different methodologies. For instance, Czerwik [2] introduced the notion of -metric spaces, with a triangle inequality weaker than that of metric spaces, in a view to generalize the Banach contraction principle. Moving on the same sequel, Matthews [3] introduced the notion of a partial metric space, which was a part of the study for denotational semantics of dataflow networks and gave a generalized version of the Banach contraction principle. The concept of partial metric spaces was further extended to partial -metric spaces by Shukla in [4]. A number of researchers took keen interest in the generalized version of the metric spaces some work is available in [5–27].

Recently, Gordgi et al. [28] introduced the notion of orthogonal sets and gave a new extension for the classical Banach contraction principle. More details can be found in [29, 30].

After looking into the structure of orthogonal metric spaces, introduced by [29, 30], and the binary relation used with a metric, [31, 32], we introduce the notion of -partial -metric spaces. We are also improving and generalizing the concept of orthogonal contractions in the sense of -partial -metric spaces and establish some fixed point theorems for the proposed contractions.

Throughout this paper, we denote by , and the set of natural numbers, real numbers, integer numbers, and nonnegative real numbers, respectively.

Definition 1 (see [2]. Letbe a nonempty set andSuppose a mappingsatisfies the following conditions for all
if and only if


Then is called a -metric on , and is called a -metric space with coefficient

Definition 2 (see [3]. Letbe a nonempty set. Letsatisfy the following for all :
if and only if



Then is called a partial metric space.

Definition 3 [4]. A partial-metric onis a functionsuch that for all , and for some, we have
if and only if



A partial -metric space is denoted with The number is called the coefficient of

Remark 4 (see [4]. It is clear that every partial metric space is a partial-metric space with coefficientand every-metric space is a partial-metric space with the same coefficient and a zero self-distance. However, the converse of this fact need not hold.

Example 1 [4]. Let be a constant andbe defined byThen, is a partial -metric space with coefficient but it is neither a -metric nor a partial metric space.

Definition 5 [33]. Letbe a nonempty set. A subsetofis called a binary relation on. Then, for any , we say thatis-related to, that is,, orrelates tounderif and only if. means thatis not-related tooris not related tounder

Definition 6 [33]. A binary relationdefined on a nonempty setis calledreflexive if
irreflexive if for some
symmetric if implies
antisymmetric if and imply
transitive if and imply
preorder if is reflexive and transitive;
partial order if is reflexive, antisymmetric, and transitive.

Definition 7 [32]. Letbe a nonempty set and letbe a binary relation on.
(a) A sequence is called an -sequence if (b) A map is -preserving if

Definition 8 [32]. Letbe a metric space andbe a binary relation on. Then,is called an-metric space.

Definition 9 [31]. A mappingis-continuous atif for each-sequenceinwith, we get. Thus,is-continuous onifis-continuous at each.

Definition 10 [31]. A mapis an-contraction, iffor all with , where .

Khalehoghli et al. [31] extended the result of Banach in the following way.

Theorem 11 [31]. Ifis an-preserving and-continuous-contraction on an-complete-metric space withsuch thatfor each. Then,has a unique fixed point.

2. Main Results

Let us begin this section with the definition of -partial -metric spaces.

Definition 12. Letandbe a reflexive binary relation on, denoted as. A mapis called an-partial-metric on the set, if the following conditions are satisfied for allwith either (or), either (or) and either (or):
if and only if ;
;
;
, where .
Then, is called -partial -metric space with the coefficient

Remark 13. In the above definition, a setis endowed with a reflexive binary relationandsatisfies-only for those elements which are comparable under the reflexive binary relation. Hence, the-partial-metric may not be a partial-metric, but the converse is true.
The following simplest example shows that the -partial -metric with need not to be a partial -metric with .

Example 2. Letand let the binary relation be defined byif and only ifor. It is easy to prove thatis an-partial-metric onwith, butis not a partial-metric onwith. Indeed, forand, we have.
In the coming definitions, let be an -partial -metric space with the coefficient .

Definition 14. Letbe an-sequence in, that is,orfor each. Then(i) is a convergent sequence to some if and for each (ii) is Cauchy if exists and is finite

Definition 15. is said to be-complete if for every Cauchy-sequence in, there iswithandfor each.

Definition 16. We say thatis an-property map, if for any iterative-sequenceinwith, for someand, we have thator.

Definition 17. We say thatis-0-continuous atif for each-sequenceinwith, we have. Also,is-0-continuous onifis-0-continuous for each

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

Theorem 18. Letbe an-complete-partial-metric space with the coefficientand letbe such thatfor each. Letbe an-preserving and an-property map satisfying the followingwhere Then, has a fixed point and

Proof. As is such that for each , then by using the -preserving nature of , we construct an -sequence such that and for each . We consider for each . Thus, by (5), we get for all This inequality yields for all To discuss the Cauchy criteria, we will consider an arbitrary integer with and use along (7) in the following way. As and , it follows from the above inequality that Therefore, is a Cauchy -sequence. Since is -complete, there exists such that and for each (for some value of ). Thus, from above, we obtain and for each . As for each , from (5), we get This inequality and the above findings imply As is an -property map, so we get or . Without any loss of generality, we take . Thus, by using with (5), we get the following for each When tends to infinity, the above inequality yields . Hence, we get , and . Therefore, , that is, is a fixed point of .

Remark 19. Note that the fixed point ofis unique if in the above theorem we add (I): for each fixed pointsandof, we haveor.
Since and are fixed points of such that . Then, we have for all By the nature of , we obtain

Since is -preserving, we have for all Therefore, by the triangle inequality and (5), we get

Taking limit as in the above inequality, we obtain and so

Remark 20. Note that the condition “letbe such thatfor eachof Theorem18may be replaced with “letbe such that.”

Example 3. Letand definebyThe relation on is defined by if and only if or . Clearly, is an -complete partial -metric space. Define a map by Then, it is very simple to verify the following: (1)If , then . While if , then . Thus, is an -preserving map(2)Suppose that for any iterative -sequence in with , for some , and , then we get (3)Consider any real number, then . Thus, we have , that is, (4)For each with , we havecase (a) : case (b) :

Hence, by Theorem 18, must has a fixed point.

Example 4. Letand definebyThe relation on is defined by if and only if or .
Clearly, is an -complete partial -metric space. Define a map by Then, one can verify the following: (1)If , then . While if , then . Thus, is an -preserving map(2)Suppose that for any iterative -sequence in with , for some , and , then we get (3)If be some real number, then . Thus, we get , that is, (4)For each with , we haveCase (a) If , then . Thus, Case (b) If , then . Thus, Case (c) If , then . Thus,

Hence, by Theorem 18, must has a fixed point.

Remark 21. Note that the functiondefined in the above example is neither a metric nor a-metric nor a partial-metric on. Indeed,, , , , that is,anddo not exist.

Theorem 22. Letbe an-complete-partial-metric space with the coefficientand letbe such thatfor each. Letbe an-preserving and-0-continuous map satisfying the followingfor all with , , and , where Also, let for each -sequence in with and , we have either or . Then, has a fixed point and

Proof. As is such that for each , then by using the -preserving nature of , we obtain an -sequence such that and for each . We take for each . Then by (27), for each , we get If , then from the above inequality, we obtain that , which is a contradiction. Therefore, we must have . Again, from the above inequality, we have On repeating this process, we obtain For with by , we obtain Using (30) in the above inequality, we obtain As and , it follows from the above inequality that Therefore, is a Cauchy -sequence. Since is -complete, there exists such that and for each . Thus, from above, we obtain and for each . Since is -0-continuous, one gets that , which leads to . Obviously, we have for each . Thus, for each . Since and for each , we have either or . By using , we get the following for each : When tends to infinity, the above inequality yields . Hence, we get , , and . Therefore, we say that , i.e., is a fixed point of .

Remark 23. Note that the fixed point ofis unique if in the above result, we add the condition: for each fixed pointsandof, we haveor.
Since , we have and . From (27), we get

It is a contradiction in the case . Therefore, we must have , that is, .

3. Applications to Graphical Partial -Metric Spaces and Partially-Ordered-Partial -Metric Spaces

In this section, we define a directed graph on , denoted by , with the vertex set and the edge set such that and . Also, has no parallel edge. Note that denotes the path between and , that is, there exists a finite sequence , for some finite , such that , , and for .

Definition 24. Letbe associated the above-defined, denoted as. A mapis called a-partial-metric on the set, if the following conditions are satisfied for allwithand:
if and only if ;
;
;
, where .
Then, is called a -partial -metric space with the coefficient

Remark 25. Ifand, then we getand. Also note ifand, then we have.
Thus, is a preorder relation on . Therefore, is also an -partial -metric space.

Definition 26. Letbe a-sequence in, that is,orfor each. Then, we say that(i) is a convergent sequence to if and for each (ii) is Cauchy if exists and is finite

Definition 27. is said to be-complete if for each Cauchy-sequence inthere iswithandfor each.

Note that for a map , the -0-continuity and -property are defined in the same way as explained in the last section.

Theorem 28. Letbe a-complete-partial-metric space with the coefficientand letbe such thatfor each. Letbe an edge preserving (if, then) and a-property map satisfying the followingwhere Then, has a fixed point and
By Remark 25, we know that is a preorder relation on and is an -partial -metric space. Also, an edge preserving map is path preserving. Thus, all the conditions of Theorem 18 hold. Hence, has a fixed point.
In the following, we obtain partially-ordered-partial -metric spaces from -partial -metric spaces, by considering as a partial order on .

Definition 29. Letbe associated with a partial order, denoted as. Given a map. If the following conditions are satisfied for allwithand:
if and only if ;
;
;
, where ,
then is called a partially-ordered-partial -metric space with the coefficient

As we discussed in the above, we state the following result.

Theorem 30. Letbe an-complete partially-ordered-partial-metric space with the coefficientand letbe such thatfor each. Letbe order preserving (ifthen), and an-property map satisfying the following:where Then, has a fixed point and

Remark 31. completeness is defined in the same way as-completeness.

4. Conclusion

By combining the concepts of orthogonality and the binary relation, we introduced the notion of -partial -metric spaces. We presented some related fixed point results. Some illustrated examples and an application to graphical partial -metric spaces and partially-ordered-partial -metric spaces have been provided. As perspectives, it would be interesting to consider in this setting more generalized contraction mappings involving simulation functions or more control functions.

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 research has been supported by the China Postdoctoral Science Foundation (Grant No. 2019M661047), the Natural Science Foundation of Hebei Province (Grant No. A2019404009), and Postdoctoral Foundation of Hebei Province (Grant No. B2019003016).