#### Abstract

The aim is to present a new relational variant of fixed point result that generalizes various fixed point results of the existing theme for contractive type mappings. As an application, we solve a periodic boundary value problem and validate all assertions with the help of nontrivial examples. We also highlight the close connections of the fixed point results equipped with a binary relation to that of graph related metrical fixed point results. Radically, these investigations unify the theory of metrical fixed points for contractive type mappings.

#### 1. Introduction

Alber and Guerre-Delabriere [1] presented the notion of weak contraction in Hilbert spaces and established the compatible fixed point results. Afterwards, Rhoades [2] stated that these results are still valid in the settings of metric spaces which are complete. Weak contractions are also connected to the mappings of Boyd and Wong [3], Geraghty [4], and that of Reich [5]. Further, generalizations of these fixed point results for weakly contractive mappings on this theme was obtained by Dutta and Choudhury [6]. In this continuation, an ordered analog of results due to Reich [5] and Geraghty [4] were presented by Amini-Harandi and Emami [7]. However, an analog of Banach contraction principle [8] in the same settings was investigated by Turinici [9] which was later explored by several authors (see Ran and Reurings [10], Nieto and Rodríguez-López [11], Sabetghadam and Masiha [12], Sabetghadam et al. [13], Harjani and Sadarangani [14], Samet and Turinici [15], Alam and Imdad [16, 17], and Prasad [18, 19]) and this process is still on. Meanwhile, Jachymski [20] presented an interesting metrical fixed point result by incorporating the notion of graphical contraction mapping, and there exist detailed generalization of this settings too (see for instance [21–23]).

Among all these generalizations, we must recite Alam and Imdad [17] in which the authors utilized relational variants of metrical definitions of continuity, contractions, and completeness to obtain some interesting generalizations of the fixed point results. Noticeably, Alam and Imdad [16] presented a relational variant of fixed point result due to Boyd and Wong [3] to such settings. The objective of this work is to investigate a new fixed point theorem in relational metric spaces and to solve a boundary value problem in the light of obtained results. Moreover, we highlight the connection of such findings to the fixed point results obtained under graphical contraction mappings. In this way, we utilize the contractive assumption enjoying only on those elements which are associated with either a binary relation or some graph related structure instead of the entire space.

#### 2. Preliminaries

We use notations for a nonempty binary relation, for the the set , and for the set of real numbers thoroughly in this paper. Also, the triple denotes an -metric space where is a binary relation on a nonempty set , and is a metric on

*Definition 1 (see [24]). *Let be a nonempty set and . Then,
(a) is a binary relation on and “ relates under iff (b) and are -comparative, if either or , and denoted by (c)The inverse of is defined by (d)The symmetric closure of is defined by (e) iff

*Definition 2 (see [17]). *Consider a binary relation and a self-map on a nonempty set . Then, for ,
(a) is -closed if(b) is -closed iff is -closed

*Definition 3 (see [17]). *Consider a binary relation and a sequence on a nonempty set . Then, is an -preserving sequence (shortly, -sequence) if .

*Definition 4 (see [17]). *Consider an -sequence on an -metric space . Then, is -complete if every -Cauchy sequence converges to a point in

*Remark 5. *Every -complete metric space is a complete metric space, and in respect to the universal relation, these notions are the same.

Proposition 6 ([17]). *Consider a binary relation and a self-map on a nonempty set . If is -closed, then is -closed, where and denotes th iterate of .*

*Definition 7 (see [17]). *Consider a self-map on an -metric space . Then, is -continuous at if for any -sequence with , we have . Moreover, is -continuous if it is -continuous at each point of .

*Remark 8. *Noticeably, continuity of implies -continuity, and in respect to the universal relation, these notions are the same.

*Definition 9 (see [17]). *Consider an -metric space . Then, is -self-closed if for any -sequence with , there exists a subsequence of with .

*Definition 10 (see [17]). *Consider a binary relation on a nonempty set . A subset of is -connected if for each pair ; there exists a path in from to .

*Definition 11 (see [25]). *Consider a binary relation on a nonempty set . Then, a subset of is -directed if for each pair ; there exists so that and .

*Definition 12 ([16]). *Consider a self-map on an -metric space . Then, is -transitive if for any .

Motivated by Turinici [26], Alam and Imdad [16] notified the subsequent weaker form of transitivity.

*Definition 13 ([16]). *A binary relation on a nonempty set is locally transitive if for each (effectively) -sequence (with range ), the binary relation is transitive, where is the restriction of to .

*Definition 14 ([16]). *Consider a self-map on an -metric space . Then, is locally -transitive if for each (effectively) -sequence (with range ), the binary relation is transitive.

*Definition 15 (see [25]). *Consider a binary relation on a nonempty set . For , a path of length in from to is a finite sequence satisfying the following:
(i) and (ii) for each .Noticeably, a path of length has elements of , though they are not necessarily distinct.

Lemma 16 (see [16, 27]). *Consider a sequence on a metric space . If is not a Cauchy, then there exist and two subsequences and of so that for *(i)*Moreover, if then*

Consider a binary relation and a self-map on a nonempty set . We use the following notations in the subsequent sections: (i) (the set of all fixed points of ),(ii)

Also, is the class of functions satisfying the assumption implies .

#### 3. Main Results

In this section, we first consider the existence and uniqueness of fixed points for contractive mappings in relational metric spaces. Secondly, we present results related to graphical structure in the similar metric settings.

Theorem 17. *Consider a self-map on an -metric space . Assume that the subsequent assumptions hold:
*(a)* is -complete*(b)* is -closed and locally -transitive*(c)*either is -continuous or is -self-closed*(d)* is nonempty*(e)*there exists so that**for each with . Then, has a fixed point.*

*Proof. *In the light of assumption (), let Define a sequence of joint iterates with initial point , that is,
Since and is -closed, we have
so that
So, is -sequence.

If there exists so that , then is a fixed point of , so the proof is accomplished.

In the other case, assume that . From (), we have
Put Then, we have
So, is a nonnegative nonincreasing and bounded below which possesses the limit From the inequality (9), taking , we have
implies , and so, .

Now, we shall show that is Cauchy. On contrary, assume that is not Cauchy. So, by Lemma 16, there exist and two subsequences and of so that
Next, in view of Lemma 16, we have
Since is -sequence and , so the local -transitivity of gives rise that . By triangular inequality and (), we obtain
that is,
Using the facts that and , we have
which implies that Since we obtain
which is a contradiction in the light of (12). So, is -Cauchy. As is -complete, there exists so that .

Next, we assert that is a fixed point of . At first, we consider is -continuous. As is -sequence with , -continuity of implies that . From the uniqueness of the limit, we obtain that is, is a fixed point of

Alternately, assume that is -self-closed. So, there exists subsequence of with . By using the fact that in the light of (), we have
Taking limit and , we have , and hence, .☐

*Remark 18. *Theorem 17 remains valid if we consider -transitive, locally transitive or simply transitive assumption in place of the locally -transitivity of besides retaining all other assumptions.

##### 3.1. Uniqueness Result

Theorem 19. *Along with the assumptions of Theorem 17, assume that the subsequent assumption holds:**() is -connected. Then, has a unique fixed point.*

*Proof. *Let and be two distinct fixed points of , that is, and , then for , we have
Noticeably, . By assumption (), there exists a path (say ) of finite length in from to so that
As is -closed, then in the light of Proposition 6, we obtain
Now, applying the contractive assumption () to (20), we obtain
For convenience, we put .

We have two cases: Firstly, assume that for some , that is, , which implies that . In this way, . Thus, by induction, we get for every . Hence, .

Secondly, assume that for , then using (20), in view of () and taking on the inequality (21), we have for each

Finally, utilizing the triangular inequality of metric , in view of above conclusion, we obtain
as . Hence, has a unique fixed point.☐

*Remark 20. *Theorem 19 remains valid if we consider is complete or is -directed in place of the assumption () besides retaining the all other assumptions.

*Example 21. *Let equipped with the usual metric for . Define a binary relation on and a mapping by
Clearly, is an -complete metric space and is -continuous. Let . If , that is, then . Also, for all we have This implies that . Thus, the claim holds. In consequence of the above reasonings, Also, we can easily verify that is -transitive and locally -transitive.

Let with . Define , we have
Thus, all the assumptions of Theorems 17 and 19 are satisfied. Hence, has a unique fixed point.

*Example 22. *Let equipped with the usual metric . Define a binary relation on and the mapping by
Let Then, which implies that is -closed. Observe that is not reflexive, antisymmetric, and neither transitive. So, is not partial order.

Now, we shall show is -self-closed. Let be any -sequence with so that which implies that . As is closed, we can take a subsequence of so that , which implies that . Hence, is -self-closed.

Notice that, for , we have
that is, does not satisfy the contractive assumption () of Theorem 17. However, if then the assumption () is satisfied for all Also, by usual calculations, we can easily verify that is -connected.

So, satisfies all assumptions of Theorems 17 and 19. Thus, has a unique fixed point at .

*Remark 23. *Noticeably, in Example 22 binary relation is nonreflexive, nonsymmetric, nonantisymmetric, and nontransitive. So, it is not a partial order, quasiorder, and near-order which indicate the utility of such generalizations over the corresponding several prominent recent fixed point results on this theme.

##### 3.2. Fixed Point Result under Graphical Structure

Jachymski [11] introduced the graphical variant of Banach contraction principle in metric spaces by transforming the order structure into a graphical structure on such spaces.

Let be a nonempty set and denotes the diagonal points of . Then, is a directed graph with the vertex set which coincides with , and the edge set containing its edges with all loops, that is, . Additionally, assuming that has no parallel edges, so we can symbolize as a pair Also, we assume as a weighted graph by assigning to each edge the distance between its vertices. If and are any vertices of a graph then a path in from to of length is a sequence of vertices so that and for Graph is connected if there is a path between any two of its vertices, and is weakly connected if is connected (see for details [21–23]).

The triple denotes a -metric space where is a graph on a nonempty set and is a metric on

*Definition 24 (see [20]). *Consider a self-map on a -metric space . Then, is said to be -contraction if there exists such that
and is -closed, that is,

*Definition 25 (see [20]). *Consider a sequence on a -metric space . Then, is said to be edge-preserving sequence (shortly, -sequence) if .

Also, is -complete if every -Cauchy sequence converges in .

*Definition 26 (see [20]). *Consider a self-map on a -metric space . Then, is -continuous at if for any -sequence with , we have . Moreover, is -continuous if it is -continuous at each point of .

*Definition 27. *Consider a -metric space . Then, is -self-closed if for any -sequence with there exists a subsequence of with .

*Definition 28 ([23]). *Consider a graph on a nonempty set . Then, is transitive if, for any with

*Definition 29. *Consider a self-map on a -metric space . Then, is -transitive if for any .

*Definition 30. *A graph on a nonempty set is locally transitive if for each (effectively) -sequence (with range ), the graph is transitive.

*Definition 31. *Consider a self-map on a -metric space . Then, is locally -transitive if for each (effectively) -sequence (with range ), the graph is transitive.

Theorem 32. *Consider a self-map on a -metric space . Assume that the subsequent assumptions hold:
*(a)* is -complete*(b)* is -closed and locally -transitive*(c)*either is -continuous or is -self-closed*(d)* is nonempty, that is, there exists in so that ,*(e)*there exists so that**for all with ,
*(f)* is weakly connected**Then, has a unique fixed point.*

*Proof. *Define . Then, clearly, the contractive assumption () is same as in Theorem 17. Similarly, -completeness of metric space implies the -completeness. From (), we have , which implies that is nonempty. For with then in the light of assumption (b), that is, if is -closed and locally -transitive, then is -closed and locally -transitive. Also, one can easily verify that -continuity of implies -continuity and -self-closedness of implies -self-closedness of . Moreover, the assumption () implies that is -connected which validates that has only one fixed point.☐

*Remark 33. *In view of the above discussion, if we define a binary relation so that . Then, under this assumption of , Theorem 32 reduces to Theorems 17 and 19. This implies that edge preserving structure of a graph is considered as a particular case of a binary relation .

*Remark 34. *Noticeably, if we define so that Then, under this assumption of , Theorem 32 reduces to their corresponding partial ordered analogous. This implies that partial-order relation-related metrical notions can be considered as a particular case of an edge-preserving structure related to a graph.

#### 4. An Application

The theory of boundary value problems is a substantial field of mathematics, having various applications in numerous branches of physics, biology, chemistry, engineering, and other fields related to the real life problems. Based on this fact, we present a unique solution for the first order periodic boundary value problem by utilizing the main result. For this, we consider a periodic boundary value problem of first order as follows: where and is a continuous function.

Let denote the space of all continuous functions defined on . We recall the subsequent definitions.

*Definition 35 (see [14]). *A function is a lower solution of (30), if

*Definition 36 ([14]). *A function is a upper solution of (30), if

Now, we prove the existence of solution for the problem (30). Let be a class of functions satisfying the subsequent assumptions: (i) is increasing(ii)for each

Examples of such functions are and

Theorem 37. *In addition to the problem (30), assume that there exists so that with where . Then, the existence of a lower or an upper solution of problem (30) validates the existence and uniqueness of a solution of problem (30).*

*Proof. *Problem (30) can be rewritten as
This can be transformed to the integral equation
where
Define by
and a binary relation
(i)Note that with supmetric, that is, for and is an -complete metric space(ii)For an -preserving sequence so that Then, for , we havewhich converges to . This implies that . So, . Hence, is -self-closed.
(iii)For , that is, , then in the light of inequality (33), we haveand for , we have
so that , that is, is -closed.
(iv)Let be a lower solution of (30), then we must haveMultiplying both sides by , we have
so that
As , we have
so that
Using (43) and (45), we have
that is,
Thus, and so

For Define then By the last inequality, we derive
Thus, all assumptions of Theorem 17 are satisfied, so has a fixed point. Finally, in view of the proof of Theorem 19, has a unique fixed point, which is indeed a unique solution of the problem (30).☐

#### 5. Conclusion

In this work, we have proved new relational and graphical variants of fixed point results and validated all the assertions with the help of nontrivial examples. We have also provided a view to connect the theory of fixed point results equipped with a binary relation with that of graph related metrical fixed point theory. Further, inspired by the fact that boundary value problems appear in various branches of science and engineering, we resolve them to verify the genuineness and utility of the established conclusions.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

No competing interests are associated with the article.

#### Authors’ Contributions

All authors done the equal contributions to the article.