# On Solution of Boundary Value Problems via Weak Contractions

Authors: Gopi Prasad; Hüseyin Işik
Journal: Journal of Function Spaces
 (2022)

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## 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.


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## Body

## 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 notationsR for a nonempty binary relation, ℕ0 for the the set ℕ0=ℕ∪0, and ℝ for the set of real numbers thoroughly in this paper. Also, the triple ϝ,ς,R denotes an R-metric space where R is a binary relation on a nonempty set ϝ, and ς is a metric on ϝ.Definition 1 (see [24]).
Letϝ be a nonempty set and R⊆ϝ×ϝ. Then,
(a)
R is a binary relation on ϝ and “u˘ relates v˘ under R” iff u˘,v˘∈R(b)
u˘ and v˘ are R-comparative, if either u˘,v˘∈R or v˘,u˘∈R, and denoted by u˘,v˘∈R(c)
The inverse ofR is defined by R−1≔u˘,v˘∈ϝ2:v˘,u˘∈R(d)
The symmetric closure ofR is defined by Rs≔R∪R−1(e)
u˘,v˘∈Rs iff u˘,v˘∈RDefinition 2 (see [17]).
Consider a binary relationR and a self-map θ on a nonempty set ϝ. Then, for u˘,v˘∈ϝ,
(a)
R is θ-closed if(1)u˘,v˘∈R⇒θu˘,θv˘∈R,u˘,v˘∈ϝ.(b)
R is θ-closed iff Rs is θ-closedDefinition 3 (see [17]).
Consider a binary relationR and a sequence u˘n on a nonempty set ϝ. Then, u˘n is an R-preserving sequence (shortly, R-sequence) if u˘n,u˘n+1∈R,n∈ℕ0.Definition 4 (see [17]).
Consider anR-sequence u˘n on an R-metric space ϝ,ς,R. Then, ϝ,ς,R is R-complete if every R-Cauchy sequence converges to a point in ϝ.Remark 5.
EveryR-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 relationR and a self-map θ on a nonempty set ϝ. If R is θ-closed, then R is θn-closed, where n∈ℕ0 and θn denotes nth iterate of θ.Definition 7 (see [17]).
Consider a self-mapθ on an R-metric space ϝ,ς,R. Then, θ is R-continuous at u˘ if for any R-sequence u˘n with u˘n⟶ςu˘, we have θu˘n⟶ςθu˘. Moreover, θ is R-continuous if it is R-continuous at each point of ϝ.Remark 8.
Noticeably, continuity ofθ implies R-continuity, and in respect to the universal relation, these notions are the same.Definition 9 (see [17]).
Consider anR-metric space ϝ,ς,R. Then, R is ς-self-closed if for any R-sequence u˘n⊂ϝ with u˘n⟶ςu˘, there exists a subsequence u˘nk of u˘n with u˘nk,u˘∈R,k∈ℕ0.Definition 10 (see [17]).
Consider a binary relationR on a nonempty set ϝ. A subset E of ϝ is R-connected if for each pair u˘,v˘∈E; there exists a path in R from u˘ to v˘.Definition 11 (see [25]).
Consider a binary relationR on a nonempty set ϝ. Then, a subset E of ϝ is R-directed if for each pair u˘,v˘∈E; there exists w˘∈ϝ so that u˘,w˘∈R and v˘,w˘∈R.Definition 12 ([16]).
Consider a self-mapθ on an R-metric space ϝ,ς,R. Then, R is θ-transitive if for any u˘,v˘,w˘∈ϝ,θu˘,θv˘,θv˘,θw˘∈R⇒θu˘,θw˘∈R.Motivated by Turinici [26], Alam and Imdad [16] notified the subsequent weaker form of transitivity.Definition 13 ([16]).
A binary relationR on a nonempty set ϝ is locally transitive if for each (effectively) R-sequence u˘n⊂ϝ (with range U≔u˘n:n∈N0), the binary relation RU is transitive, where RU is the restriction of R to U.Definition 14 ([16]).
Consider a self-mapθ on an R-metric space ϝ,ς,R. Then, R is locally θ-transitive if for each (effectively) R-sequence u˘n⊂θϝ (with range U≔u˘n:n∈N0), the binary relation RU is transitive.Definition 15 (see [25]).
Consider a binary relationR on a nonempty set ϝ. For u˘,v˘∈ϝ, a path of length kk∈ℕ, in R from u˘ to v˘ is a finite sequence w˘0,w˘1,w˘2,⋯,w˘k⊂ϝ satisfying the following:
(i)
w˘0=u˘ and w˘k=v˘(ii)
w˘i,w˘i+1∈R for each i0≤i≤k−1.
Noticeably, a path of lengthk has k+1 elements of ϝ, though they are not necessarily distinct.Lemma 16 (see [16, 27]).
Consider a sequenceu˘n on a metric space ϝ,ς. If u˘n is not a Cauchy, then there exist ε>0 and two subsequences u˘nk and u˘mk of u˘n so that for k∈ℕ,(2)k≤mk<nk,ςu˘mk,u˘nk>ε,ςu˘mk,u˘nk−1≤ε.(i)
Moreover, iflimn⟶∞ςu˘n,u˘n+1=0, then(3)limk⟶∞ςu˘mk,u˘nk=ε,limk⟶∞ςu˘mk+1,u˘nk+1=ε.Consider a binary relationR and a self-map θ on a nonempty set ϝ. We use the following notations in the subsequent sections:
(i)
Fθ≔u˘∈ϝ:u˘=θu˘ (the set of all fixed points of θ),(ii)
Mθ,R≔u˘∈ϝ:u˘,θu˘∈R.Also,C is the class of functions φ:0,+∞⟶0,1 satisfying the assumption φsn⟶1 implies sn⟶0.


## 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-mapθ on an R-metric space ϝ,ς,R. Assume that the subsequent assumptions hold:
(a)
ϝ,ς,R is R-complete(b)
R is θ-closed and locally θ-transitive(c)
eitherθ is R-continuous or R is ς-self-closed(d)
Mθ,R is nonempty(e)
there existsφ∈C so that(4)ςθu˘,θv˘≤φςu˘,v˘ςu˘,v˘,for each u˘,v˘∈ϝ with u˘,v˘∈R. Then, θ has a fixed point.Proof.
In the light of assumption (d), let u˘0∈Mθ,R. Define a sequence u˘n of joint iterates with initial point u˘0, that is,
(5)u˘n=θnu˘0,n∈ℕ0.
Sinceu˘0,θu˘0∈R and R is θ-closed, we have
(6)θu˘0,θ2u˘0,θ2u˘0,θ3u˘0,⋯,θnu˘0,θn+1u˘0,⋯∈R,so that
(7)u˘n,u˘n+1∈R,n∈ℕ0.
So,u˘n is R-sequence.
If there existsn0∈ℕ so that ςu˘n0,u˘n0−1=0, then u˘n0=θu˘n0−1=u˘n0−1 is a fixed point of θ, so the proof is accomplished.
In the other case, assume thatςu˘n,u˘n−1≠0,n∈ℕ. From (e), we have
(8)ςu˘n+1,u˘n=ςθu˘n,θu˘n−1≤φςu˘n,u˘n−1ςu˘n,u˘n−1<ςu˘n,u˘n−1.
Putsn≔ςu˘n+1,u˘n. Then, we have
(9)sn≤φsn−1sn−1<sn−1.
So,sn is a nonnegative nonincreasing and bounded below which possesses the limit s. From the inequality (9), taking n⟶∞, we have
(10)s≤φss<s,
impliesφs=1, and so, s=0.
Now, we shall show thatu˘n is Cauchy. On contrary, assume that u˘n is not Cauchy. So, by Lemma 16, there exist ε>0 and two subsequences u˘nk and u˘mk of u˘n so that
(11)k≤mk≤nk,ςu˘mk,u˘nk>ε≥ςu˘mk,u˘nk−1,k∈ℕ.
Next, in view of Lemma16, we have
(12)limk⟶∞ςu˘mk,u˘nk=limk⟶∞ςu˘mk+1,u˘nk+1=ε.
Sinceu˘n is R-sequence and u˘n⊂θϝ, so the local θ-transitivity of R gives rise that u˘mk,u˘nk∈R. By triangular inequality and (e), we obtain
(13)ςu˘mk,u˘nk≤ςu˘mk,u˘mk+1+ςu˘mk+1,u˘nk+1+ςu˘nk,u˘nk+1≤ςu˘mk,u˘mk+1+φςu˘mk,u˘nkςu˘mk,u˘nk+ςu˘nk,u˘nk+1,that is,
(14)ςu˘mk,u˘nk≤1−φςu˘mk,u˘nk−1ςu˘mk,u˘mk+1+ςu˘nk,u˘nk+1.
Using the facts thatlimsupk⟶+∞ςu˘mk,u˘nk>0 and limn⟶+∞ςu˘nk,u˘nk+1=0, we have
(15)limsupk⟶+∞1−φςu˘mk,u˘nk−1=+∞,which implies that limsupk⟶+∞φςu˘mk,u˘nk=1. Since φ∈C, we obtain
(16)limsupk⟶+∞ςu˘mk,u˘nk=0,which is a contradiction in the light of (12). So, u˘n is R-Cauchy. As ϝ,ς,R is R-complete, there exists u˘∈ϝ so that u˘n⟶ςu˘.
Next, we assert thatu˘ is a fixed point of θ. At first, we consider θ is R-continuous. As u˘n is R-sequence with u˘n⟶ςu˘, R-continuity of θ implies that u˘n+1=θu˘n⟶ςθu˘. From the uniqueness of the limit, we obtain θu˘=u˘, that is, u˘ is a fixed point of θ.
Alternately, assume thatR is ς-self-closed. So, there exists subsequence u˘nk of u˘n with u˘nk,u˘∈R,k∈ℕ0. By using the fact that u˘nk,u˘∈R in the light of (e), we have
(17)ςu˘nk+1,θu˘=ςθu˘nk,θu˘≤φςu˘nk,u˘ςu˘nk,u˘<ςu˘nk,u˘.
Taking limitk⟶∞ and u˘nk⟶ςu˘, we have u˘nk+1⟶ςθu˘, and hence, θu˘=u˘.☐Remark 18.
Theorem17 remains valid if we consider θ-transitive, locally transitive or simply transitive assumption in place of the locally θ-transitivity of R besides retaining all other assumptions.


### 3.1. Uniqueness Result

Theorem 19.
Along with the assumptions of Theorem17, assume that the subsequent assumption holds:
(u) θϝ is Rs-connected. Then, θ has a unique fixed point.Proof.
Letu˘ and v˘ be two distinct fixed points of θ, that is, Fθ≠∅ and u˘,v˘∈Fθ, then for n∈ℕ0, we have
(18)θnu˘=u˘,θnv˘=v˘.
Noticeably,u˘,v˘∈θϝ. By assumption (u), there exists a path (say w˘0,w˘1,w˘2,⋯,w˘k) of finite length k in Rs from u˘ to v˘ so that
(19)w˘0=u˘,w˘k=v˘andw˘i,w˘i+1∈Rforeachi0≤i≤k−1.
AsR is θ-closed, then in the light of Proposition 6, we obtain
(20)θnw˘i,θnw˘i+1∈Rforeachi0≤i≤k−1,n∈ℕ0.
Now, applying the contractive assumption (e) to (20), we obtain
(21)ςθnw˘i,θnw˘i+1≤φςθn−1w˘i,θn−1w˘i+1ςθn−1w˘i,θn−1w˘i+1.
For convenience, we puttni=ςθnw˘i,θnw˘i+1.
We have two cases: Firstly, assume thattn0i=ςθn0w˘i,θn0w˘i+1=0 for some n0∈ℕ0, that is, θn0w˘i=θn0w˘i+1, which implies that θn0+1w˘i=θn0+1w˘i+1. In this way, tn0+1i=ςθn0+1w˘i,θn0+1w˘i+1=0. Thus, by induction, we get tn0i=0 for every n≥n0. Hence, limn⟶∞tni=0.
Secondly, assume thattni>0 for n∈ℕ0, then using (20), in view of (e) and taking n⟶∞ on the inequality (21), we have limn⟶∞tni=0 for each i0≤i≤k−1.
Finally, utilizing the triangular inequality of metricς, in view of above conclusion, we obtain
(22)ςu˘,v˘=ςθnw˘0,θnw˘k≤tn0+tn1+⋯+tnk−1⟶0,as n⟶∞. Hence, θ has a unique fixed point.☐Remark 20.
Theorem19 remains valid if we consider Rθϝ is complete or θϝ is Rs-directed in place of the assumption (u) besides retaining the all other assumptions.Example 21.
Letϝ=0,+∞ equipped with the usual metric ςu˘,v˘=u˘−v˘ for u˘,v˘∈ϝ. Define a binary relation R=u˘,v˘:u˘≥v˘andu˘,v˘∈0,1 on ϝ and a mapping θ:ϝ⟶ϝ by
(23)θu˘=u˘1+u˘,ifu˘∈0,1,3u˘,ifu˘∈1,+∞.
Clearly,ϝ,ς,R is an R-complete metric space and θ is R-continuous. Let u˘,v˘∈ϝ. If u˘,v˘∈R, that is, u˘,v˘∈0,1, then θu˘,θv˘∈R. Also, for all u˘∈0,1, we have θu˘≤1. This implies that u˘,θu˘∈R. Thus, the claim holds. In consequence of the above reasonings, 0,θ0∈R. Also, we can easily verify that R is θ-transitive and locally θ-transitive.
Letu˘,v˘∈ϝ with u˘,v˘∈R. Define φt=1/1+t,t∈0,+∞, we have
(24)ςθu˘,θv˘=θu˘−θv˘=u˘1+u˘−v˘1+v˘=u˘−v˘1+u˘+v˘+u˘v˘≤u˘−v˘1+u˘−v˘=φςu˘,v˘ςu˘,v˘.
Thus, all the assumptions of Theorems17 and 19 are satisfied. Hence, θ has a unique fixed point.Example 22.
Letϝ=1,4 equipped with the usual metric ςu˘,v˘=u˘−v˘. Define a binary relation R=1,1,3/2,2,2,1,2,2,5/2,3,3,3,7/2,4 on ϝ and the mapping θ:ϝ⟶ϝ by
(25)θu˘=1,ifu˘∈1,2,2,ifu˘∈2,3,3,ifu˘∈3,4.
Letu˘,v˘∈R. Then, θu˘,θv˘∈1,1,2,2,3,3 which implies that R is θ-closed. Observe that R is not reflexive, antisymmetric, and neither transitive. So, R is not partial order.
Now, we shall showR is ς-self-closed. Let u˘n be any R-sequence with u˘n⟶ςu˘, so that u˘n,u˘n+1∈R,n∈ℕ0 which implies that u˘n⊂1,2,3. As 1,2,3 is closed, we can take a subsequence u˘nk of u˘n so that u˘nk=u˘,k∈ℕ0, which implies that u˘nk,u˘∈R,k∈ℕ0. Hence, R is ς-self-closed.
Notice that, forφt=1/1+t,t∈0,+∞, we have
(26)ςθ1,θ3=θ1−θ3=1>φς1,3ς1,3=23,
that is,θ does not satisfy the contractive assumption (e) of Theorem 17. However, if u˘,v˘∈R, then the assumption (e) is satisfied for all u˘,v˘∈R. Also, by usual calculations, we can easily verify that θϝ is Rs-connected.
So,θ satisfies all assumptions of Theorems 17 and 19. Thus, θ has a unique fixed point at u˘=1.Remark 23.
Noticeably, in Example22 binary relation R 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.Letϝ be a nonempty set and Δ denotes the diagonal points of ϝ×ϝ. Then, G is a directed graph with the vertex set VG which coincides with ϝ, and the edge set EG containing its edges with all loops, that is, EG⊇Δ. Additionally, assuming that G has no parallel edges, so we can symbolize G as a pair VG,EG. Also, we assume G as a weighted graph by assigning to each edge the distance between its vertices. If u˘ and v˘ are any vertices of a graph G, then a path in G from u˘ to v˘ of length kk∈ℕ is a sequence u˘ii=0k of k+1 vertices so that u˘0=u˘,u˘k=v˘ and u˘n−1,u˘n∈EG for i=1,2,3,⋯,k. Graph G is connected if there is a path between any two of its vertices, and G is weakly connected if G~ is connected (see for details [21–23]).The tripleϝ,ς,G denotes a G-metric space where G is a graph on a nonempty set ϝ and ς is a metric on ϝ.Definition 24 (see [20]).
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, θ is said to be G-contraction if there exists k∈0,1 such that
(27)∀u˘,v˘∈ϝ,u˘,v˘∈EG⇒ςθu˘,θv˘≤kςu˘,v˘,and G is θ-closed, that is,
(28)∀u˘,v˘∈ϝ,u˘,v˘∈EG⇒θu˘,θv˘∈EG.Definition 25 (see [20]).
Consider a sequenceu˘n on a G-metric space ϝ,ς,G. Then, u˘n is said to be edge-preserving sequence (shortly, E-sequence) if u˘n,u˘n+1∈EG,n∈ℕ0.
Also,ϝ,ς,G is G-complete if every E-Cauchy sequence converges in ϝ.Definition 26 (see [20]).
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, θ is G-continuous at u˘ if for any E-sequence u˘n with u˘n⟶ςu˘, we have θu˘n⟶ςθu˘. Moreover, θ is G-continuous if it is G-continuous at each point of ϝ.Definition 27.
Consider aG-metric space ϝ,ς,G. Then, G is ς-self-closed if for any E-sequence u˘n⊂ϝ with u˘n⟶ςu˘, there exists a subsequence u˘nk of u˘n with u˘nk,u˘∈EG,k∈ℕ0.Definition 28 ([23]).
Consider a graphG on a nonempty set ϝ. Then, G is transitive if, for any u˘,v˘,w˘∈VG with u˘,v˘,v˘,w˘∈EG⇒u˘,w˘∈EG.Definition 29.
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, G is θ-transitive if for any u˘,v˘,w˘∈ϝ,θu˘,θv˘,θv˘,θw˘∈EG⇒θu˘,θw˘∈EG.Definition 30.
A graphG on a nonempty set ϝ is locally transitive if for each (effectively) E-sequence u˘n⊂ϝ (with range U≔u˘n:n∈ℕ0), the graph GU is transitive.Definition 31.
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, G is locally θ-transitive if for each (effectively) E-sequence u˘n⊂θϝ (with range U≔u˘n:n∈ℕ0), the graph GU is transitive.Theorem 32.
Consider a self-mapθ on a G-metric space ϝ,ς,G. Assume that the subsequent assumptions hold:
(a)
ϝ,ς,G is G-complete(b)
G is θ-closed and locally θ-transitive(c)
eitherθ is G-continuous or G is ς-self-closed(d)
Mθ,G is nonempty, that is, there exists u˘0 in ϝ so that u˘0,θu˘0∈EG,(e)
there existsφ∈C so that(29)ςθu˘,θv˘≤φςu˘,v˘ςu˘,v˘,for all u˘,v˘∈ϝ with u˘,v˘∈EG,
(f)
G is weakly connected
Then,θ has a unique fixed point.Proof.
DefineR=u˘,v˘:u˘,v˘∈EG,u˘,v˘∈ϝ. Then, clearly, the contractive assumption (e) is same as in Theorem 17. Similarly, G-completeness of metric space implies the R-completeness. From (d), we have u˘0,θu˘0∈EG, which implies that Mθ,R is nonempty. For u˘,v˘∈ϝ with u˘,v˘∈EG⇒u˘,v˘∈R, then in the light of assumption (b), θu˘,θv˘∈EG⇒θu˘,θv˘∈R, that is, if G is θ-closed and locally θ-transitive, then R is θ-closed and locally θ-transitive. Also, one can easily verify that G-continuity of θ implies R-continuity and ς-self-closedness of G implies ς-self-closedness of R. Moreover, the assumption (f) implies that θϝ is Rs-connected which validates that θ has only one fixed point.☐Remark 33.
In view of the above discussion, if we define a binary relationR so that R=u˘,v˘:u˘,v˘∈EG,u˘,v˘∈ϝ. Then, under this assumption of R, 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 R.Remark 34.
Noticeably, if we defineEG so that EG=u˘,v˘:u˘≤v˘,u˘,v˘∈ϝ. Then, under this assumption of EG, 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.


## 3.1. Uniqueness Result

Theorem 19.
Along with the assumptions of Theorem17, assume that the subsequent assumption holds:
(u) θϝ is Rs-connected. Then, θ has a unique fixed point.Proof.
Letu˘ and v˘ be two distinct fixed points of θ, that is, Fθ≠∅ and u˘,v˘∈Fθ, then for n∈ℕ0, we have
(18)θnu˘=u˘,θnv˘=v˘.
Noticeably,u˘,v˘∈θϝ. By assumption (u), there exists a path (say w˘0,w˘1,w˘2,⋯,w˘k) of finite length k in Rs from u˘ to v˘ so that
(19)w˘0=u˘,w˘k=v˘andw˘i,w˘i+1∈Rforeachi0≤i≤k−1.
AsR is θ-closed, then in the light of Proposition 6, we obtain
(20)θnw˘i,θnw˘i+1∈Rforeachi0≤i≤k−1,n∈ℕ0.
Now, applying the contractive assumption (e) to (20), we obtain
(21)ςθnw˘i,θnw˘i+1≤φςθn−1w˘i,θn−1w˘i+1ςθn−1w˘i,θn−1w˘i+1.
For convenience, we puttni=ςθnw˘i,θnw˘i+1.
We have two cases: Firstly, assume thattn0i=ςθn0w˘i,θn0w˘i+1=0 for some n0∈ℕ0, that is, θn0w˘i=θn0w˘i+1, which implies that θn0+1w˘i=θn0+1w˘i+1. In this way, tn0+1i=ςθn0+1w˘i,θn0+1w˘i+1=0. Thus, by induction, we get tn0i=0 for every n≥n0. Hence, limn⟶∞tni=0.
Secondly, assume thattni>0 for n∈ℕ0, then using (20), in view of (e) and taking n⟶∞ on the inequality (21), we have limn⟶∞tni=0 for each i0≤i≤k−1.
Finally, utilizing the triangular inequality of metricς, in view of above conclusion, we obtain
(22)ςu˘,v˘=ςθnw˘0,θnw˘k≤tn0+tn1+⋯+tnk−1⟶0,as n⟶∞. Hence, θ has a unique fixed point.☐Remark 20.
Theorem19 remains valid if we consider Rθϝ is complete or θϝ is Rs-directed in place of the assumption (u) besides retaining the all other assumptions.Example 21.
Letϝ=0,+∞ equipped with the usual metric ςu˘,v˘=u˘−v˘ for u˘,v˘∈ϝ. Define a binary relation R=u˘,v˘:u˘≥v˘andu˘,v˘∈0,1 on ϝ and a mapping θ:ϝ⟶ϝ by
(23)θu˘=u˘1+u˘,ifu˘∈0,1,3u˘,ifu˘∈1,+∞.
Clearly,ϝ,ς,R is an R-complete metric space and θ is R-continuous. Let u˘,v˘∈ϝ. If u˘,v˘∈R, that is, u˘,v˘∈0,1, then θu˘,θv˘∈R. Also, for all u˘∈0,1, we have θu˘≤1. This implies that u˘,θu˘∈R. Thus, the claim holds. In consequence of the above reasonings, 0,θ0∈R. Also, we can easily verify that R is θ-transitive and locally θ-transitive.
Letu˘,v˘∈ϝ with u˘,v˘∈R. Define φt=1/1+t,t∈0,+∞, we have
(24)ςθu˘,θv˘=θu˘−θv˘=u˘1+u˘−v˘1+v˘=u˘−v˘1+u˘+v˘+u˘v˘≤u˘−v˘1+u˘−v˘=φςu˘,v˘ςu˘,v˘.
Thus, all the assumptions of Theorems17 and 19 are satisfied. Hence, θ has a unique fixed point.Example 22.
Letϝ=1,4 equipped with the usual metric ςu˘,v˘=u˘−v˘. Define a binary relation R=1,1,3/2,2,2,1,2,2,5/2,3,3,3,7/2,4 on ϝ and the mapping θ:ϝ⟶ϝ by
(25)θu˘=1,ifu˘∈1,2,2,ifu˘∈2,3,3,ifu˘∈3,4.
Letu˘,v˘∈R. Then, θu˘,θv˘∈1,1,2,2,3,3 which implies that R is θ-closed. Observe that R is not reflexive, antisymmetric, and neither transitive. So, R is not partial order.
Now, we shall showR is ς-self-closed. Let u˘n be any R-sequence with u˘n⟶ςu˘, so that u˘n,u˘n+1∈R,n∈ℕ0 which implies that u˘n⊂1,2,3. As 1,2,3 is closed, we can take a subsequence u˘nk of u˘n so that u˘nk=u˘,k∈ℕ0, which implies that u˘nk,u˘∈R,k∈ℕ0. Hence, R is ς-self-closed.
Notice that, forφt=1/1+t,t∈0,+∞, we have
(26)ςθ1,θ3=θ1−θ3=1>φς1,3ς1,3=23,
that is,θ does not satisfy the contractive assumption (e) of Theorem 17. However, if u˘,v˘∈R, then the assumption (e) is satisfied for all u˘,v˘∈R. Also, by usual calculations, we can easily verify that θϝ is Rs-connected.
So,θ satisfies all assumptions of Theorems 17 and 19. Thus, θ has a unique fixed point at u˘=1.Remark 23.
Noticeably, in Example22 binary relation R 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.Letϝ be a nonempty set and Δ denotes the diagonal points of ϝ×ϝ. Then, G is a directed graph with the vertex set VG which coincides with ϝ, and the edge set EG containing its edges with all loops, that is, EG⊇Δ. Additionally, assuming that G has no parallel edges, so we can symbolize G as a pair VG,EG. Also, we assume G as a weighted graph by assigning to each edge the distance between its vertices. If u˘ and v˘ are any vertices of a graph G, then a path in G from u˘ to v˘ of length kk∈ℕ is a sequence u˘ii=0k of k+1 vertices so that u˘0=u˘,u˘k=v˘ and u˘n−1,u˘n∈EG for i=1,2,3,⋯,k. Graph G is connected if there is a path between any two of its vertices, and G is weakly connected if G~ is connected (see for details [21–23]).The tripleϝ,ς,G denotes a G-metric space where G is a graph on a nonempty set ϝ and ς is a metric on ϝ.Definition 24 (see [20]).
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, θ is said to be G-contraction if there exists k∈0,1 such that
(27)∀u˘,v˘∈ϝ,u˘,v˘∈EG⇒ςθu˘,θv˘≤kςu˘,v˘,and G is θ-closed, that is,
(28)∀u˘,v˘∈ϝ,u˘,v˘∈EG⇒θu˘,θv˘∈EG.Definition 25 (see [20]).
Consider a sequenceu˘n on a G-metric space ϝ,ς,G. Then, u˘n is said to be edge-preserving sequence (shortly, E-sequence) if u˘n,u˘n+1∈EG,n∈ℕ0.
Also,ϝ,ς,G is G-complete if every E-Cauchy sequence converges in ϝ.Definition 26 (see [20]).
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, θ is G-continuous at u˘ if for any E-sequence u˘n with u˘n⟶ςu˘, we have θu˘n⟶ςθu˘. Moreover, θ is G-continuous if it is G-continuous at each point of ϝ.Definition 27.
Consider aG-metric space ϝ,ς,G. Then, G is ς-self-closed if for any E-sequence u˘n⊂ϝ with u˘n⟶ςu˘, there exists a subsequence u˘nk of u˘n with u˘nk,u˘∈EG,k∈ℕ0.Definition 28 ([23]).
Consider a graphG on a nonempty set ϝ. Then, G is transitive if, for any u˘,v˘,w˘∈VG with u˘,v˘,v˘,w˘∈EG⇒u˘,w˘∈EG.Definition 29.
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, G is θ-transitive if for any u˘,v˘,w˘∈ϝ,θu˘,θv˘,θv˘,θw˘∈EG⇒θu˘,θw˘∈EG.Definition 30.
A graphG on a nonempty set ϝ is locally transitive if for each (effectively) E-sequence u˘n⊂ϝ (with range U≔u˘n:n∈ℕ0), the graph GU is transitive.Definition 31.
Consider a self-mapθ on a G-metric space ϝ,ς,G. Then, G is locally θ-transitive if for each (effectively) E-sequence u˘n⊂θϝ (with range U≔u˘n:n∈ℕ0), the graph GU is transitive.Theorem 32.
Consider a self-mapθ on a G-metric space ϝ,ς,G. Assume that the subsequent assumptions hold:
(a)
ϝ,ς,G is G-complete(b)
G is θ-closed and locally θ-transitive(c)
eitherθ is G-continuous or G is ς-self-closed(d)
Mθ,G is nonempty, that is, there exists u˘0 in ϝ so that u˘0,θu˘0∈EG,(e)
there existsφ∈C so that(29)ςθu˘,θv˘≤φςu˘,v˘ςu˘,v˘,for all u˘,v˘∈ϝ with u˘,v˘∈EG,
(f)
G is weakly connected
Then,θ has a unique fixed point.Proof.
DefineR=u˘,v˘:u˘,v˘∈EG,u˘,v˘∈ϝ. Then, clearly, the contractive assumption (e) is same as in Theorem 17. Similarly, G-completeness of metric space implies the R-completeness. From (d), we have u˘0,θu˘0∈EG, which implies that Mθ,R is nonempty. For u˘,v˘∈ϝ with u˘,v˘∈EG⇒u˘,v˘∈R, then in the light of assumption (b), θu˘,θv˘∈EG⇒θu˘,θv˘∈R, that is, if G is θ-closed and locally θ-transitive, then R is θ-closed and locally θ-transitive. Also, one can easily verify that G-continuity of θ implies R-continuity and ς-self-closedness of G implies ς-self-closedness of R. Moreover, the assumption (f) implies that θϝ is Rs-connected which validates that θ has only one fixed point.☐Remark 33.
In view of the above discussion, if we define a binary relationR so that R=u˘,v˘:u˘,v˘∈EG,u˘,v˘∈ϝ. Then, under this assumption of R, 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 R.Remark 34.
Noticeably, if we defineEG so that EG=u˘,v˘:u˘≤v˘,u˘,v˘∈ϝ. Then, under this assumption of EG, 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:(30)u˘′s=qs,u˘s,s∈J=0,S,u˘0=u˘S,where S>0 and q:J×ℝ⟶ℝ is a continuous function.LetCJ denote the space of all continuous functions defined on J. We recall the subsequent definitions.Definition 35 (see [14]).
A functionβ∈C1J is a lower solution of (30), if
(31)β′s≤qs,βs,s∈J,β0≤βS.Definition 36 ([14]).
A functionβ∈C1J is a upper solution of (30), if
(32)β′s≥qs,βs,s∈J,β0≥βS.Now, we prove the existence of solution for the problem (30). Let A be a class of functions ϕ:0,+∞⟶0,+∞ satisfying the subsequent assumptions:
(i)
ϕ is increasing(ii)
for eachu˘>0,ϕu˘<u˘Examples of such functions areϕs=μs,s∈0,1,ϕs=s/1+s and ϕs=ln1+s.Theorem 37.
In addition to the problem (30), assume that there exists δ>0 so that u˘,v˘∈ℝ with u˘≤v˘,(33)0≤qs,v˘+δv˘−qs,u˘+δu˘≤δϕv˘−u˘,where ϕ∈A. 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
(34)u˘′s+δu˘s=qs,u˘s+δu˘s,s∈J=0,S,u˘0=u˘S.
This can be transformed to the integral equation(35)u˘s=∫0SQs,rqr,u˘r+δu˘rdr,where
(36)Qs,r=eδS+r−seδS−1,0≤r<s≤S,eδr−seδS−1,0≤s<r≤S.
Defineθ:CJ⟶CJ by
(37)θu˘s=∫0SQs,rqr,u˘r+δu˘rdr,and a binary relation
(38)R=u˘,v˘∈CJ×CJ:u˘s≤v˘s,s∈J.(i)
Note thatCJ with supmetric, that is, ςu˘,v˘=supu˘s−v˘s for s∈J and u˘,v˘∈CJ, is an R-complete metric space(ii)
For anR-preserving sequence u˘n so that u˘n⟶ςw˘. Then, for s∈J, we have(39)u˘0s≤u˘1s≤u˘2s≤⋯≤u˘ns≤u˘n+1s≤⋯which converges to w˘s. This implies that u˘ns≤w˘s,n∈ℕ0. So, u˘n,w˘∈R,n∈ℕ0. Hence, R is ς-self-closed.
(iii)
Foru˘,v˘∈R, that is, u˘s≤v˘s, then in the light of inequality (33), we have(40)qs,u˘s+δu˘s≤qs,v˘s+δv˘s,s∈J,and Qs,r>0 for s,r∈J×J, we have
(41)θu˘s=∫0SQs,rqr,u˘r+δu˘rdr≤∫0SQs,rqr,v˘r+δv˘rdr=θv˘s,s∈J,so that θu˘,θv˘∈R, that is, R is θ-closed.
(iv)
Letβ∈C1J be a lower solution of (30), then we must have(42)β′s+δβs≤qs,βs+δβs,s∈J.
Multiplying both sides byeδs, we have
(43)βseδs′≤qs,βs+δβseδs,s∈J,so that
(44)βseδs≤β0+∫0sqr,βr+δβreδrdr,s∈J.
Asβ0≤βS, we have
(45)β0eδS≤βSeδS≤β0+∫0Sqr,βr+δβreδrdr,so that
(46)β0≤∫0SeδreδS−1qr,βr+δβrdr.
Using (43) and (45), we have
(47)βseδs≤∫0seδS+reδS−1qr,βr+δβrdr+∫sSeδreδS−1qr,βr+δβrdr,that is,
(48)βs≤∫0seδS+r−seδS−1qr,βr+δβrdr+∫sSeδr−seδS−1qr,βr+δβrdr≤∫0SQs,rqr,βr+δβrdr=θβs.
Thus,βs,θβs∈R,s∈J and so Mθ,R≠∅.
Foru˘,v˘∈R,(49)ςθu˘,θv˘=sups∈Jθu˘s−θv˘s=sups∈Jθv˘s−θu˘s≤sups∈J∫0SQs,rqr,v˘r+δv˘r−qr,u˘r−δu˘rdr≤sups∈J∫0SQs,rδϕv˘r−u˘rdr≤δϕςu˘,v˘sups∈J∫0SQs,rdr=δϕςu˘,v˘sups∈J1eδS−11δeδS+r−s0s+1δeδr−ssS=δϕςu˘,v˘1δeδS−1eδS−1=ϕςu˘,v˘=ϕςu˘,v˘ςu˘,v˘ςu˘,v˘.
Defineφu˘=ϕu˘/u˘, then φ∈C. By the last inequality, we derive
(50)ςθu˘,θv˘≤φςu˘,v˘ςu˘,v˘,u˘,v˘∈CJwithu˘,v˘∈R.
Thus, all assumptions of Theorem17 are satisfied, so θ has a fixed point. Finally, in view of the proof of Theorem 19, θ 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.



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