Relation-Theoretic Fixed Point Theorems Involving Certain Auxiliary Functions with Applications

Abstract

This article includes some fixed point results for $(\phi ,\psi ,\theta )$-contractions in the context of metric space endowed with a locally $\mathcal{H}$-transitive relation. We constructed an example for attesting to the credibility of our results. We also discussed the existence and uniqueness of the solution of a Fredholm integral equation using our results.

1. Introduction

The Banach contraction principle (abbreviated as BCP) has been extended to ordered metric spaces by Ran and Reurings [1] and Nieto and Rodríguez-López [2]. The contraction condition involved in such results is weakened as it is desirable to hold for comparable (with respect to a given partial order) elements only but at the expense of the order-preserving property of the underlying mapping. Here, it can be highlighted that these results are indeed natural variants of Turinici’s results [3,4]. In 2015, Alam and Imdad [5] investigated the relation-theoretic analog of BCP and observed that the partial order utilized in results of Ran and Reurings [1] and Nieto and Rodríguez-López [2] is not optimal and can further be weakened to the extent of an arbitrary relation. In the process, the authors of [5,6] initiated the relation-theoretic variants of involved metrical concepts (completeness, contraction, continuity, etc.). Particularly for universal relations, these concepts reduce to their corresponding usual notions. Consequently, under universal relation, relation-theoretic fixed point results deduce to their corresponding classical fixed point results. In a short span of the last seven years, the relation-theoretic contraction principle has attracted the attention of many researchers and by now this paper already earned more than a hundred citations, e.g., [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41].

On the other hand, several authors extended and generalized BCP by improving the class of contraction mappings using certain auxiliary functions. This trend was initiated by Browder [42] and refined by Boyd and Wong [43] and Matkowski [44] by introducing the notion of $\phi$-contractions. Indeed, the class of $\phi$-contractions depends on certain auxiliary function $\phi :[0,\infty )\to [0,\infty )$, which is utilized instead of the Lipschitz constant $0\le k<1$. Dutta and Choudhury [45] introduced the concept of $(\phi ,\psi )$-contractions involving two auxiliary functions $\phi$ and $\psi$. Recently, Alam et al. [46] generalized the notion of $(\phi ,\psi )$-contractions and extended the BCP under $(\phi ,\psi )$-contractions. In this continuation, Sk et al. [47] obtained relation-theoretic variants of the fixed point results of Alam et al. [46]. In 2012, Jleli et al. [48] proved some fixed point results in the context of ordered metric space using the three auxiliary functions $\phi ,\psi$, and $\theta$.

The intent of this manuscript is to prove the fixed point theorems under $(\phi ,\psi ,\theta )$-contractions using a relation-theoretic approach. To illustrate our results, we provide an illustrative example and an application to Fredholm integral equations. Our newly proved results indeed improve and extend Theorems 3.1–3.3 of Jleli et al. [48] in the following respects:

(i)

the partial order is replaced by locally $\mathcal{H}$-transitive relation, which remains an optimal condition of transitivity;

(ii)

completeness and continuity are replaced by their relational analogs, i.e., $\mathfrak{S}$-completeness and $\mathfrak{S}$-continuity;

(iii)

the regularity of the ambient metric space is replaced by $\varrho$-self-closedness, which remains relatively weaker;

(iv)

the directedness property of the whole ambient space utilized in the uniqueness theorem is replaced by the directedness property of the $\mathcal{H}$-image of the ambient space.

2. Preliminaries

The present section contains relevant concepts and auxiliary results needed to prove our main results. The set of natural numbers will be denoted by $\mathbb{N}$, while the set of whole numbers will be denoted by ${\mathbb{N}}_0$. By a relation (or, a binary relation) $\mathfrak{S}$ on a set $\mathbb{A}$, we mean a subset of ${\mathbb{A}}^2$.

In what follows, $\mathbb{A}$ is a set, $\mathfrak{S}$ is a relation on $\mathbb{A}$, and $\mathcal{H}$ is a map from $\mathbb{A}$ into itself.

Definition 1

([5]). A pair of elements $r,t\in \mathbb{A}$ satisfying either $(r,t)\in \mathfrak{S}$ or $(t,r)\in \mathfrak{S}$ is said to be $\mathfrak{S}$-comparative. We shall denote such a pair by $[r,t]\in \mathfrak{S}$.

Definition 2

([49]). For each pair $r,t\in \mathbb{A}$, if $[r,t]\in \mathfrak{S}$, then we say that $\mathfrak{S}$ is a complete relation.

Definition 3

([49]). The inverse of $\mathfrak{S}$ is a relation ${\mathfrak{S}}^{-1}$ defined by

$${\mathfrak{S}}^{-1}:=\{(r,t)\in {\mathbb{A}}^2:(t,r)\in \mathfrak{S}\}.$$

Definition 4

([49]). The symmetric closure of $\mathfrak{S}$ is a relation ${\mathfrak{S}}^s$ defined by

$${\mathfrak{S}}^s:=\mathfrak{S}\cup {\mathfrak{S}}^{-1}.$$

Proposition 1

([5]). $(r,t)\in {\mathfrak{S}}^s⟺[r,t]\in \mathfrak{S}.$

Definition 5

([49]). For any subset $\mathbb{B}\subseteq \mathbb{A}$, the relation on $\mathbb{B}$ defined by

$${\mathfrak{S}|}_{\mathbb{B}}:=\mathfrak{S}\cap {\mathbb{B}}^2.$$

is referred as the restriction of $\mathfrak{S}$ on $\mathbb{B}$.

Definition 6

([5]). For each pair of elements $r,t\in \mathbb{A}$ with $(r,t)\in \mathfrak{S}$, if

$$(\mathcal{H}r,\mathcal{H}t)\in \mathfrak{S},$$

then $\mathfrak{S}$ is termed as $\mathcal{H}$-closed.

Proposition 2

([7]). If $\mathfrak{S}$ is $\mathcal{H}$-closed, then $\mathfrak{S}$ is ${\mathcal{H}}^n$-closed, $\forall n\in \mathbb{N}$.

Definition 7

([5]). If a sequence $\left\{r_n\right\}\subset \mathbb{A}$ verifies $(r_n,r_{n+1})\in \mathfrak{S}$$\forall n\in {\mathbb{N}}_0$, then we say that $\left\{r_n\right\}$ is $\mathfrak{S}$-preserving.

Definition 8

([6]). A metric space $(\mathbb{A},\varrho )$ is referred as $\mathfrak{S}$-complete if each $\mathfrak{S}$-preserving Cauchy sequence in $\mathbb{A}$ converges.

A complete metric space is $\mathfrak{S}$-complete for any arbitrary relation $\mathfrak{S}$. Further, these two concepts coincide under a universal relation (i.e., $\mathfrak{S}={\mathbb{A}}^2$).

Definition 9

([6]). $\mathcal{H}$ is termed as $\mathfrak{S}$-continuous at $r\in \mathbb{A}$ if for every $\mathfrak{S}$-preserving sequence $\left\{r_n\right\}\subset \mathbb{A}$ verifying $r_n\stackrel{\varrho }{⟶}r$,

$$\mathcal{H}\left(r_n\right)\stackrel{\varrho }{⟶}\mathcal{H}\left(r\right).$$

A $\mathfrak{S}$-continuous map at each point of $\mathbb{A}$ is referred as $\mathfrak{S}$-continuous.

A continuous map is $\mathfrak{S}$-continuous for any arbitrary relation $\mathfrak{S}$. Further, these two concepts coincide under a universal relation.

Definition 10

([5]). $\mathfrak{S}$ is called ϱ-self-closed if each $\mathfrak{S}$-preserving convergent sequence in $\mathbb{A}$ has a subsequence whose terms are $\mathfrak{S}$-comparative with the limit.

Definition 11

([50]). A subset $\mathbb{B}\subset \mathbb{A}$ is referred as $\mathfrak{S}$-directed if for each pair $r,t\in \mathbb{B}$, $\exists s\in \mathbb{A}$ verifying $(r,s)\in \mathfrak{S}$ and $(t,s)\in \mathfrak{S}$.

Definition 12

([7]). A subset $\mathbb{B}\subset \mathbb{A}$, in which every pair of elements has a path, is referred to as an $\mathfrak{S}$-connected set.

Inspired by [51,52,53], the following concept was introduced.

Definition 13

([7]). $\mathfrak{S}$ is termed as locally $\mathcal{H}$-transitive if for any $\mathfrak{S}$-preserving sequence $\left\{r_n\right\}\subset \mathcal{H}\left(\mathbb{A}\right)$, the relation ${\mathfrak{S}|}_{\mathbb{B}}$ (whereas $\mathbb{B}:=\{r_n:n\in {\mathbb{N}}_0\}$) is transitive.

The following notations are adopted in the upcoming text.

•

$F\left(\mathcal{H}\right)$:=the set of all fixed points of $\mathcal{H}$;

•

$\mathbb{A}(\mathcal{H},\mathfrak{S}):=\{r\in \mathbb{A}:(r,\mathcal{H}r)\in \mathfrak{S}\}$.

The relation-theoretic version of BCP investigated by Alam and Imdad [5] is stated as:

Theorem 1

([5,6,41]). Assume that $(\mathbb{A},\varrho )$ is a metric space, $\mathfrak{S}$ remains a relation on $\mathbb{A}$ while $\mathcal{H}$ remains a map from $\mathbb{A}$ to itself. Additionally,

(i)

$(\mathbb{A},\varrho )$ remains $\mathfrak{S}$-complete;

(ii)

$\mathbb{A}(\mathcal{H},\mathfrak{S})$ remains nonempty;

(iii)

$\mathfrak{S}$ remains $\mathcal{H}$-closed;

(iv)

$\mathcal{H}$ remains $\mathfrak{S}$-continuous or $\mathfrak{S}$ is ϱ-self-closed;

(v)

$\exists k\in [0,1)$ verifying

$\varrho (\mathcal{H}r,\mathcal{H}t)\le k\varrho (r,t),\forall r,t\in \mathbb{A}with(r,t)\in \mathfrak{S}.$

Then, $\mathcal{H}$ has a fixed point. Additionally, if $\mathcal{H}\left(\mathbb{A}\right)$ remains ${\mathfrak{S}}^s$-connected, then $\mathcal{H}$ has a unique fixed point.

Lemma 1

([48]). If a sequence $\left\{r_n\right\}\subset \mathbb{A}$ (wherein ϱ remains a metric on $\mathbb{A}$), is not Cauchy, then one can find an $ϵ>0$ and two subsequences $\left\{r_{n_{\lambda }}\right\}$ and $\left\{r_{p_{\lambda }}\right\}$ of $\left\{r_n\right\}$ verifying

(i)

$\lambda \le p_{\lambda }<n_{\lambda },\forall \lambda \in \mathbb{N}$,

(ii)

$\varrho (r_{p_{\lambda }},r_{n_{\lambda }})>ϵ,\forall \lambda \in \mathbb{N}$,

(iii)

$\varrho (r_{p_{\lambda }},r_{n_{\lambda }-1})\le ϵ,\forall \lambda \in \mathbb{N}$.

Moreover, if $\underset{n\to \infty }{lim}\varrho (r_n,r_{n+1})=0$, then

(iv)

$\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }},r_{n_{\lambda }})=ϵ,$

(v)

$\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }},r_{n_{\lambda }+1})=ϵ,$

(vi)

$\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }+1},r_{n_{\lambda }})=ϵ,$

(vii)

$\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }+1},r_{n_{\lambda }+1})=ϵ.$

3. Main Results

In what follows, $\mathsf{\Phi }$ will denote the family of functions $\phi :[0,\infty )\to [0,\infty )$ verifying

• ${\mathsf{\Phi }}_1$: $\phi$ remains monotonic increasing;
• ${\mathsf{\Phi }}_2$: $\phi$ remains continuous;
• ${\mathsf{\Phi }}_3$: $\phi \left(c\right)=0\iff c=0$

while $\Psi$ will denote the family of functions $\psi :[0,\infty )\to [0,\infty )$ verifying

• ${\Psi }_1$: $\underset{c\to l^{+}}{lim}\psi \left(c\right)>0,\forall l>0$;
• ${\Psi }_2$: $\underset{c\to 0^{+}}{lim}\psi \left(c\right)=0$

and $\Theta$ will denote the family of functions $\theta :{[0,\infty )}^4\to [0,\infty )$ verifying

• ${\Theta }_1$: $\theta$ remains continuous;
• ${\Theta }_2$: $\theta (a,b,c,d)=0\iff abcd=0.$

These families of auxiliary functions are introduced by Jleli et al. [48]. Using the symmetry of $\varrho$, one proposes the following result:

Proposition 3.

Assume that $(\mathbb{A},\varrho )$ is a metric space, $\mathfrak{S}$ remains a relation on $\mathbb{A}$ while $\mathcal{H}$ remains a map from $\mathbb{A}$ to itself. If $\phi \in \mathsf{\Phi },\psi \in \Psi$ and $\theta \in \Theta$, then the following conditions are equivalent:

$$\begin{array}{ccc}\hfill \left(\mathrm{I}\right)\phi \left(\varrho \right(\mathcal{H}r,\mathcal{H}t\left)\right)& \le & \phi \left(\varrho \right(r,t\left)\right)-\psi \left(\varrho \right(r,t\left)\right)+\theta \left(\varrho \right(r,\mathcal{H}r),\varrho (t,\mathcal{H}t),\varrho (r,\mathcal{H}t),\varrho (t,\mathcal{H}r\left)\right),\hfill \\ & & \forall r,t\in \mathbb{A}with(r,t)\in \mathfrak{S},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(\mathrm{II}\right)\phi \left(\varrho \right(\mathcal{H}r,\mathcal{H}t\left)\right)& \le & \phi \left(\varrho \right(r,t\left)\right)-\psi \left(\varrho \right(r,t\left)\right)+\theta \left(\varrho \right(r,\mathcal{H}r),\varrho (t,\mathcal{H}t),\varrho (r,\mathcal{H}t),\varrho (t,\mathcal{H}r\left)\right),\hfill \\ & & \forall r,t\in \mathbb{A}with[r,t]\in \mathfrak{S}.\hfill \end{array}$$

Theorem 2.

Assume that $(\mathbb{A},\varrho )$ is a metric space, $\mathfrak{S}$ remains a relation on $\mathbb{A}$ while $\mathcal{H}$ remains a map from $\mathbb{A}$ to itself. Additionally,

(i)

$(\mathbb{A},\varrho )$ remains $\mathfrak{S}$-complete;

(ii)

$\mathbb{A}(\mathcal{H},\mathfrak{S})$ remains nonempty;

(iii)

$\mathfrak{S}$ remains $\mathcal{H}$-closed and locally $\mathcal{H}$-transitive;

(iv)

$\mathcal{H}$ remains $\mathfrak{S}$-continuous or $\mathfrak{S}$ is ϱ-self-closed;

(v)

∃$\phi \in \mathsf{\Phi },\psi \in \Psi$ and $\theta \in \Theta$, verifying

$$\begin{array}{c}\hfill \phi \left(\varrho \right(\mathcal{H}r,\mathcal{H}t\left)\right)\le \phi \left(\varrho \right(r,t\left)\right)-\psi \left(\varrho \right(r,t\left)\right)+\theta \left(\varrho \right(r,\mathcal{H}r),\varrho (t,\mathcal{H}t),\varrho (r,\mathcal{H}t),\varrho (t,\mathcal{H}r\left)\right),\\ \forall r,t\in \mathbb{A}with(r,t)\in \mathfrak{S}.\end{array}$$

Then $\mathcal{H}$ has a fixed point.

Proof. 

Using hypothesis (ii), choose $r_0\in \mathbb{A}(\mathcal{H},\mathfrak{S})$, then $(r_0,\mathcal{H}{r}_0)\in \mathfrak{S}$. Define a sequence $\left\{r_n\right\}\subset \mathbb{A}$ as follows:

$$r_n={\mathcal{H}}^n\left(r_0\right)=\mathcal{H}\left(r_{n-1}\right),\forall n\in \mathbb{N}.$$

(1)

As $(r_0,\mathcal{H}{r}_0)\in \mathfrak{S}$, using assumption (ii) and Proposition 2, we have

$$({\mathcal{H}}^n{r}_0,{\mathcal{H}}^{n+1}{r}_0)\in \mathfrak{S}$$

which in lieu of (1) becomes

$$(r_n,r_{n+1})\in \mathfrak{S},\forall n\in {\mathbb{N}}_0$$

(2)

showing that $\left\{r_n\right\}$ remains $\mathfrak{S}$-preserving. If ∃$n_0\in {\mathbb{N}}_0$ such that $\varrho (r_{n_0},r_{n_0+1})=0$, then applying (1), we conclude that $r_{n_0}$ is a fixed point of $\mathcal{H}$. Otherwise, suppose that ${\varrho }_n:=\varrho (r_n,r_{n+1})>0$, $\forall n\in {\mathbb{N}}_0.$ Using (1) and assumption (v), one gets

$$\begin{array}{ccc}\hfill \phi \left(\varrho (r_n,r_{n+1})\right)& =& \phi \left(\varrho (\mathcal{H}{r}_{n-1},\mathcal{H}{r}_n)\right)\hfill \\ & \le & \phi \left(\varrho (r_{n-1},r_n)\right)-\psi \left(\varrho (r_{n-1},r_n)\right)\hfill \\ & & +\theta (\varrho (\mathcal{H}{r}_n,r_n),\varrho (\mathcal{H}{r}_{n-1},r_{n-1}),\varrho (\mathcal{H}{r}_{n-1},r_n),\varrho (\mathcal{H}{r}_n,r_{n-1}))\hfill \end{array}$$

so that

$$\phi \left({\varrho }_n\right)\le \phi \left({\varrho }_{n-1}\right)-\psi \left({\varrho }_{n-1}\right)+\theta ({\varrho }_n,{\varrho }_{n-1},0,\varrho (r_{n-1},r_{n+1})).$$

Using axiom ${\Theta }_2$, we get

$$\phi \left({\varrho }_n\right)\le \phi \left({\varrho }_{n-1}\right)-\psi \left({\varrho }_{n-1}\right),\forall n\in \mathbb{N}$$

(3)

yielding thereby

$$\phi \left({\varrho }_n\right)\le \phi \left({\varrho }_{n-1}\right),\forall n\in \mathbb{N}.$$

Using axiom ${\mathsf{\Phi }}_1$, above inequality gives rise to

$$\begin{array}{ccc}\hfill {\varrho }_n& \le & {\varrho }_{n-1},\forall n\in \mathbb{N},\hfill \end{array}$$

which yields that the sequence $\left\{{\varrho }_n\right\}$ remains decreasing in $[0,\infty )$. As $\left\{{\varrho }_n\right\}$ is also bounded below, ∃$r\ge 0$ verifying

$$\underset{n\to \infty }{lim}{\varrho }_n=r^{+}.$$

(4)

Assume that $r>0$. Taking $n\to \infty$ in (3) and using continuity of $\phi ,$ one finds

$$\phi \left(r\right)\le \phi \left(r\right)-\underset{n\to \infty }{lim}\psi \left({\varrho }_{n-1}\right).$$

(5)

Using axiom ${\Psi }_1$ and (4), one has

$$\underset{n\to \infty }{lim}\psi \left({\varrho }_{n-1}\right)=\underset{t\to r^{+}}{lim}\psi \left(t\right)>0.$$

Hence, (5) reduces to

$$\phi \left(r\right)\le \phi \left(r\right)-\underset{t\to r^{+}}{lim}\psi \left(t\right)<\phi \left(r\right)$$

which is a contradiction. Hence, we have

$$\underset{n\to \infty }{lim}{\varrho }_n=\underset{n\to \infty }{lim}\varrho (r_n,r_{n+1})=0.$$

(6)

If $\left\{r_n\right\}$ is not Cauchy, then by Lemma 1, one can find $ϵ>0$ and two subsequences $\left\{r_{n_{\lambda }}\right\}$ and $\left\{r_{p_{\lambda }}\right\}$ of $\left\{r_n\right\}$ of $\left\{r_n\right\}$ verifying

$$\lambda \le p_{\lambda }<n_{\lambda },\varrho (r_{p_{\lambda }},r_{n_{\lambda }})>ϵ\ge \varrho (r_{p_{\lambda }},r_{n_{\lambda }-1}),\forall \lambda \in \mathbb{N}.$$

Using (6) and Lemma 1, one has

$$\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }},r_{n_{\lambda }})=\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }},r_{n_{\lambda }+1})=\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }+1},r_{n_{\lambda }})=\underset{\lambda \to \infty }{lim}\varrho (r_{p_{\lambda }+1},r_{n_{\lambda }+1})=ϵ.$$

(7)

As $\left\{r_n\right\}$ remains $\mathfrak{S}$-preserving (owing to (2)) and $\left\{r_n\right\}\subset \mathcal{H}\left(\mathbb{A}\right)$ (owing to (1)), by locally $\mathcal{H}$-transitivity of $\mathfrak{S}$, one has $(r_{m_k},r_{n_k})\in \mathfrak{S}$. Hence, applying contractivity condition (v), one obtains

$$\begin{array}{ccc}\hfill \phi (r_{p_{\lambda }+1},r_{n_{\lambda }+1}))& =& \phi \left(\varrho (\mathcal{H}{r}_{p_{\lambda }},\mathcal{H}{r}_{n_{\lambda }})\right)\hfill \\ & \le & \phi \left(\varrho (r_{p_{\lambda }},r_{n_{\lambda }})\right)-\psi \left(\varrho (r_{p_{\lambda }},r_{n_{\lambda }})\right)\hfill \\ & & +\theta (\varrho (r_{p_{\lambda }},\mathcal{H}{r}_{p_{\lambda }}),\varrho (r_{n_{\lambda }},\mathcal{H}{r}_{n_{\lambda }}),\varrho (r_{p_{\lambda }},\mathcal{H}{r}_{n_{\lambda }}),\varrho (r_{n_{\lambda }},\mathcal{H}{r}_{p_{\lambda }}))\hfill \\ & =& \phi \left(\varrho (r_{p_{\lambda }},r_{n_{\lambda }})\right)-\psi \left(\varrho (r_{p_{\lambda }},r_{n_{\lambda }})\right)\hfill \\ & & +\theta (\varrho (r_{p_{\lambda }},r_{p_{\lambda }+1}),\varrho (r_{n_{\lambda }},r_{n_{\lambda }+1}),\varrho (r_{p_{\lambda }},r_{n_{\lambda }+1}),\varrho (r_{n_{\lambda }},r_{p_{\lambda }+1})).\hfill \end{array}$$

Thus, for each $\lambda \in {\mathbb{N}}_0$, we have

$$\begin{array}{ccc}\hfill \phi \left(\varrho (r_{p_{\lambda }+1},r_{n_{\lambda }+1})\right)& & \le \phi \left(\varrho (r_{p_{\lambda }},r_{n_{\lambda }})\right)-\psi \left(\varrho (r_{p_{\lambda }},r_{n_{\lambda }})\right)\hfill \\ & & +\theta ({\varrho }_{p_{\lambda }},{\varrho }_{n_{\lambda }},\varrho (r_{p_{\lambda }},r_{n_{\lambda }+1}),\varrho (r_{n_{\lambda }},r_{p_{\lambda }+1})).\hfill \end{array}$$

Taking $\lambda \to \infty$ in above and using (6), (7) and properties of $\phi ,\psi ,\theta ,$ one gets

$$\begin{array}{ccc}\hfill \phi \left(ϵ\right)& \le & \phi \left(ϵ\right)-\underset{t\to {ϵ}^{+}}{lim}\psi \left(t\right)+\theta (0,0,ϵ,ϵ)\hfill \\ & =& \phi \left(ϵ\right)-\underset{t\to {ϵ}^{+}}{lim}\psi \left(t\right)<\phi \left(ϵ\right),\hfill \end{array}$$

which remains a contradiction. This concludes that $\left\{r_n\right\}$ is an $\mathfrak{S}$-preserving Cauchy sequence. Since $(\mathbb{A},\varrho )$ is $\mathfrak{S}$-complete, ∃$r\in \mathbb{A}$ verifying $r_n\stackrel{\varrho }{⟶}r$.

Finally, by (iv) one has to prove that $r\in \mathcal{H}\left(\mathbb{A}\right)$. If $\mathcal{H}$ is $\mathfrak{S}$-continuous, then we have

$$\mathcal{H}\left(r_n\right)\stackrel{\varrho }{\to }\mathcal{H}\left(r\right).$$

Using (1), above becomes

$$r_{n+1}\stackrel{\varrho }{\to }\mathcal{H}\left(r\right).$$

By uniqueness of limit, one gets $\mathcal{H}\left(r\right)=r$. Otherwise, assume that $\mathfrak{S}$ remains $\varrho$-self closed. As $\left\{r_n\right\}$ is a $\mathfrak{S}$-preserving sequence and $r_n\stackrel{\varrho }{⟶}r$, ∃ a subsequence $\left\{r_{n_k}\right\}\mathrm{of}\left\{r_n\right\}\mathrm{with}[r_{n_k},r]\in \mathfrak{S}$∀$k\in {\mathbb{N}}_0.$ On using triangular inequality, (v), Proposition 3, $[r_{n_k},r]\in \mathfrak{S}$ and $r_{n_k}\stackrel{\varrho }{⟶}r$, we obtain

$$\begin{array}{ccc}\hfill \phi \left(\varrho (\mathcal{H}r,\mathcal{H}{r}_{n_k})\right)& \le & \phi \left(\varrho (r,r_{n_k})\right)-\psi \left(\varrho (r,r_{n_k})\right)\hfill \\ & & +\theta (\varrho (r,\mathcal{H}r),\varrho (r_{n_k},\mathcal{H}{r}_{n_k}),\varrho (r,\mathcal{H}{r}_{n_k}),\varrho (r_{n_k},\mathcal{H}r))\hfill \\ & \le & \phi \left(\varrho (r,r_{n_k})\right)+\theta (\varrho (r,\mathcal{H}r),\varrho (r_{n_k},r_{n_k+1}),\varrho (r,r_{n_k+1}),\varrho (r_{n_k},\mathcal{H}r))\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill & & \phi \left(\varrho (\mathcal{H}r,\mathcal{H}{r}_{n_k})\right)\le \phi \left(\varrho (r,r_{n_k})\right)+\theta (\varrho (r,\mathcal{H}r),\varrho (r_{n_k},\mathcal{H}{r}_{n_k}),\hfill \\ & & \varrho (r,\mathcal{H}{r}_{n_k}),\varrho (r_{n_k},\mathcal{H}r)).\hfill \end{array}$$

(8)

Taking the limit $k\to \infty$ in (8), one gets

$$0\le \underset{k\to \infty }{lim}\phi \left(\varrho (\mathcal{H}r,\mathcal{H}{r}_{n_k})\right)\le \phi \left(0\right)+\theta (\varrho (r,\mathcal{H}r),0,0,\varrho (r,\mathcal{H}r))=0$$

implying thereby

$$\underset{k\to \infty }{lim}\phi \left(\varrho (\mathcal{H}r,r_{n_k+1})\right)=0.$$

(9)

By continuity of $\phi ,$ one gets

$$\underset{k\to \infty }{lim}\phi \left(\varrho (\mathcal{H}r,r_{n_k+1})\right)=\phi \left(\varrho (\mathcal{H}r,r)\right).$$

(10)

On combining (9) and (10), we obtain

$$\phi \left(\varrho \right(\mathcal{H}r,r\left)\right)=0,$$

which by using axiom ${\mathsf{\Phi }}_3$ implies that

$$\varrho (\mathcal{H}r,r)=0$$

so that $\mathcal{H}\left(r\right)=r$. Therefore in each case, r remains a fixed point of $\mathcal{H}$. □

The corresponding uniqueness result runs as:

Theorem 3.

In addition of Theorem 2, if $\mathcal{H}\left(\mathbb{A}\right)$ is ${\mathfrak{S}}^s$-directed, then $\mathcal{H}$ has a unique fixed point.

Proof. 

In lieu of Theorem 2, $F\left(\mathcal{H}\right)\ne Ø$. Take $r,t\in F\left(\mathcal{H}\right)$. We have

$${\mathcal{H}}^n\left(r\right)=r\mathrm{and}{\mathcal{H}}^n\left(t\right)=t,\forall n\in {\mathbb{N}}_0.$$

(11)

Since $r,t\in \mathcal{H}\left(\mathbb{A}\right)$ and $\mathcal{H}\left(\mathbb{A}\right)$ is ${\mathfrak{S}}^s$-directed, ∃$z_0\in \mathbb{A}$ verifying $[r,z_0]\in \mathfrak{S}$ and $[t,z_0]\in \mathfrak{S}$. We define the sequence $\left\{z_n\right\}$ as follows:

$$z_n={\mathcal{H}}^n\left(z_0\right)=\mathcal{H}\left(z_{n-1}\right),\forall n\in {\mathbb{N}}_0.$$

(12)

Using (11) and (12), $\mathcal{H}$-closedness of $\mathfrak{S}$ and Proposition 2, one gets

$$[r,z_n]\in \mathfrak{S}\mathrm{and}[t,z_n]\in \mathfrak{S},\forall n\in {\mathbb{N}}_0.$$

(13)

Using (12) and (13) and applying contractivity condition (v), we obtain

$$\begin{array}{ccc}\hfill \phi \left(\varrho (z_{n+},r)\right)& =& \phi \left(\varrho (\mathcal{H}{z}_n,\mathcal{H}r)\right)\hfill \\ & \le & \phi \left(\varrho (z_n,r)\right)-\psi \left(\varrho (z_n,r)\right)\hfill \\ & & +\theta (\varrho (z_n,T{z}_n),\varrho (r,\mathcal{H}r),\varrho (z_n,\mathcal{H}r),\varrho (r,T{z}_n))\hfill \\ & =& \phi \left(\varrho (z_n,r)\right)-\psi \left(\varrho (z_n,r)\right)+\theta (\varrho (z_n,z_{n+1}),0,\varrho (z_n,\mathcal{H}r),\varrho (r,z_{n+1}))\hfill \\ & =& \phi \left(\varrho (z_n,r)\right)-\psi \left(\varrho (z_n,r)\right)\hfill \end{array}$$

(14)

implying thereby

$$\phi \left(\varrho (z_{n+1},r)\right)\le \phi \left(\varrho (z_n,r)\right),\forall n\in {\mathbb{N}}_0.$$

Using axiom ${\mathsf{\Phi }}_1$, we get

$$\varrho (z_{n+1},r)\le \varrho (z_n,r),\forall n\in {\mathbb{N}}_0.$$

Thus, the sequence $\left\{{\delta }_n\right\}\subset [0,\infty )$, whereas ${\delta }_n:=\varrho (z_n,r)$, remains decreasing and hence ∃$\delta \ge 0$, verifying

$$\underset{n\to \infty }{lim}{\delta }_n=\underset{n\to \infty }{lim}\varrho (z_n,r)={\delta }^{+}.$$

(15)

Now, we show that $\delta =0$. On the contrary, assume that $\delta >0.$ Taking $n\to \infty$ in (14) and by (15), one obtains

$$\phi \left(\delta \right)\le \phi \left(\delta \right)-\underset{s\to {\delta }^{+}}{lim}\psi \left(s\right)<\phi \left(\delta \right)$$

which is a contradiction. Therefore, one has $\delta =0$, i.e.,

$$\underset{n\to \infty }{lim}\varrho (z_n,r)=0.$$

(16)

Similarly, we have

$$\underset{n\to \infty }{lim}\varrho (z_n,t)=0.$$

(17)

Using triangular inequality, (16) and (17), we get

$$\varrho (r,t)\le \varrho (r,z_n)+\varrho (z_n,t)\to 0\mathrm{as}n\to \infty$$

yielding thereby $r=t$. Thus, $\mathcal{H}$ has a unique fixed point. □

Remark 1.

Under $\mathfrak{S}=⪯,$ the partial order, our results reduce to Theorems 3.1–3.3 of Jleli et al. [48].

Example 1.

Consider ${\mathbb{R}}^2$ with the metric ϱ defined by

$$\varrho ((r,t),(u,v))=\frac{|r-u|+|t-v|}{2}\forall (r,t),(u,v)\in {\mathbb{R}}^2.$$

On ${\mathbb{R}}^2$, take a relation $\mathfrak{S}$ given by

$$\mathfrak{S}=\{((r,t),(u,v))\in {\mathbb{A}}^2\times {\mathbb{A}}^2:u-r\ge 0,t-v\ge 0\}.$$

Then $({\mathbb{R}}^2,\varrho )$ is an $\mathfrak{S}$-complete metric space. Define a map $\mathcal{H}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by

$$\mathcal{H}(r,t)=\left(\frac{r-2t}{4},\frac{t-2r}{4}\right),\forall (r,t)\in {\mathbb{R}}^2.$$

Then $\mathfrak{S}$ is $\mathcal{H}$-closed as well as ϱ-self-closed. Additionally, $(-2,3)\in \mathbb{A}(\mathcal{H},\mathfrak{S})$.

Take $(r,t),(u,v)\in {\mathbb{R}}^2$ verifying $\left((r,t),(u,v)\right)\in \mathfrak{S}$. One has

$$\begin{array}{ccc}\hfill \phi \left(\varrho \right(\mathcal{H}(r,t),\mathcal{H}(u,v)\left)\right)& =& \frac{\varrho \left(\mathcal{H}\right(r,t),\mathcal{H}(u,v\left)\right)}{2}\hfill \\ & =& \frac{1}{8}\left(\right|(r-u)+2(v-t)|+|(v-t)+2(r-u)\left|\right)\hfill \\ & =& \frac{3}{16}(r-u+v-t),\hfill \end{array}$$

i.e.,

$$\phi \left(\varrho (\mathcal{H}(r,t),\mathcal{H}(u,v))\right)=\frac{3}{16}(r-u+v-t).$$

(18)

Additionally,

$$\begin{array}{ccc}\hfill \phi \left(\varrho \right((r,t),(u,v)\left)\right)-\psi \left(\varrho \right((r,t),(u,v)\left)\right)& =& \frac{\varrho \left(\right(r,t),(u,v\left)\right)}{2}-\frac{\varrho \left(\right(r,t),(u,v\left)\right)}{16}\hfill \\ & =& \frac{7}{16}\varrho ((r,t),(u,v))\hfill \\ & =& \frac{7}{32}(r-u+v-t),\hfill \end{array}$$

i.e.,

$$\phi \left(\varrho ((r,t),(u,v))\right)-\psi \left(\varrho ((r,t),(u,v))\right)=\frac{7}{32}(r-u+v-t).$$

(19)

From (18) and (19), one gets

$$\phi \left(\varrho \right(\mathcal{H}(r,t),\mathcal{H}(u,v)\left)\right)\le \phi \left(\varrho \right((r,t),(u,v)\left)\right)-\psi \left(\varrho \right((r,t),(u,v)\left)\right).$$

This yields that the contractivity condition (v) holds for $\phi \left(c\right)=\frac{c}{2},\psi \left(c\right)=\frac{c}{16}$ and $\theta (a,b,c,d)=0.$ Further, here $\mathcal{H}\left(\mathbb{A}\right)$ is also ${\mathfrak{S}}^s$-directed. Consequently, by Theorem 3, $\mathcal{H}$ admits a unique fixed point $r=(0,0).$

4. An Application to Fredholm Integral Equations

This section is devoted to studying the existence and uniqueness of solutions of the following Fredholm integral equation:

$$\nu \left(t\right)={\int }_0^1\kappa (t,\xi )\mathfrak{f}(\xi ,\vartheta \left(\xi \right))d\xi ,\forall t\in I,$$

(20)

where kernel $\kappa (t,\xi )$ is given by

$$\kappa (t,\xi )=\frac{1}{6}\left\{\begin{array}{cc}{\xi }^2(3t-\xi ),\hfill & 0\le \xi \le t\le 1,\hfill \\ t^2(3\xi -t),\hfill & 0\le t\le \xi \le 1.\hfill \end{array}\right.$$

(21)

In what follows, $\mathsf{\Omega }$ will denote the class of functions $\omega :[0,\infty )\to [0,\infty )$ satisfying the following:

(i)

$\omega$ remains increasing,

(ii)

∃$\psi \in \Psi$ verifying $\omega \left(c\right)=c-\psi \left(c\right)$, ∀$c\in [0,\infty ).$

Let $\mathrm{C}\left(I\right)$ denotes the class of real continuous maps on $I=[0,1]$.

Theorem 4.

In addition to (20), assume that $\exists \omega \in \mathsf{\Omega }$, satisfying

$$0\le \mathfrak{f}(t,a)-\mathfrak{f}(t,b)\le \omega (a-b),\forall a,b\in \mathbb{R}witha\ge band\forall t\in I.$$

(22)

If $\exists \alpha \in \mathrm{C}\left(I\right)$, satisfying

$$\alpha \left(t\right)\ge {\int }_0^1\kappa (t,\xi )\mathfrak{f}(\xi ,\alpha \left(\xi \right))d\xi ,\forall t\in I,$$

(23)

where κ is kernel given by (21), then the equation (20) admits a unique solution.

Proof. 

One can easily check that

$$0\le \kappa (t,\xi )\le \frac{1}{2}{t}^2\xi \forall t,\xi \in I.$$

(24)

On $\mathbb{A}:=\mathrm{C}\left(I\right)$, define a metric $\varrho$ and a relation $\mathfrak{S}$ as follows:

$$\varrho (\mu ,\vartheta )=\underset{t\in I}{max}|\mu \left(t\right)-\vartheta \left(t\right)|:={\parallel \mu -\vartheta \parallel }_{\infty },\forall \mu ,\vartheta \in \mathbb{A},$$

and

$$(\mu ,\vartheta )\in \mathfrak{S}\iff \mu \left(t\right)\ge \vartheta \left(t\right),\forall \mu ,\vartheta \in \mathbb{A},\forall t\in I.$$

One can verify that $(\mathbb{A},\varrho )$ remains an $\mathfrak{S}$-complete metric space and $\mathfrak{S}$ remains $\varrho$-self-closed binary relation.

Define the mapping $\mathcal{H}:\mathbb{A}\to \mathbb{A}$ by

$$\mathcal{H}\left(\mu \right)\left(t\right)={\int }_0^1\kappa (t,\xi )\mathfrak{f}(\xi ,\mu \left(\xi \right))d\xi ,\forall t\in I,\forall \mu \in \mathbb{A}.$$

Making use of (22), one can show that $\mathcal{H}$ is $\mathfrak{S}$-closed. Moreover, $\mathcal{H}\left(\mathbb{A}\right)$ is ${\mathfrak{S}}^s$-directed. In view of (23), one has $\alpha \left(t\right)\ge \mathcal{H}\left(\alpha \right)\left(t\right)$ so that $\alpha \in \mathbb{A}(\mathcal{H},\mathfrak{S})$. Again, from (22), for all $t\in I$ and for all $\mu ,\vartheta \in \mathbb{A}$ with $(\mu ,\vartheta )\in \mathfrak{S}$, one has

$$\begin{array}{ccc}\hfill \left|\mathcal{H}\right(\mu \left)\right(t)-\mathcal{H}(\vartheta \left)\right(t\left)\right|& =& {\int }_0^1\kappa (t,\xi )(\mathfrak{f}(\xi ,\mu \left(\xi \right))-\mathfrak{f}(\xi ,\vartheta \left(\xi \right)))d\xi \hfill \\ & \le & {\int }_0^1\kappa (t,\xi )\omega (\mu \left(\xi \right)-\vartheta \left(\xi \right))d\xi \hfill \\ & \le & \left({\int }_0^1\kappa (t,\xi )d\xi \right){\left(\omega \right(\parallel \mu -\vartheta \parallel }_{\infty })(\mathrm{from} (\mathrm{i}\left)\right)\hfill \\ & \le & \frac{\omega (\parallel \mu -\vartheta {\parallel }_{\infty })}{4}(\mathrm{using} (24\left)\right)\hfill \\ & \le & \omega (\parallel \mu -\vartheta {\parallel }_{\infty })\hfill \\ & =& \varrho (\mu ,\vartheta )-\psi \left(\varrho \right(\mu ,\vartheta \left)\right)(\mathrm{from} (\mathrm{ii}\left)\right),\hfill \end{array}$$

implying thereby

$$\varrho (\mathcal{H}\mu ,\mathcal{H}\vartheta )\le \varrho (\mu ,\vartheta )-\psi \left(\varrho \right(\mu ,\vartheta \left)\right).$$

Thus, all conditions of Theorems 2 and 3 are verified. Consequently, ∃ a unique $\overline{\nu }\in \mathrm{C}\left(I\right)$ satisfying $\mathcal{H}\left(\overline{\nu }\right)=\overline{\nu }$, which is indeed the unique solution of (20). □

5. Conclusions

In this article, we have proved the fixed point results in the setting of metric space (abbreviated as MS) equipped with a locally $\mathcal{H}$-transitive relation using a triplet of auxiliary functions. Analogously, such results can be extended to generalized metrical structures (such as symmetric space, metric-like space, multiplicative MS, rectangular MS, cone MS, D-MS, G-MS, complex-valued MS, partial MS, b-MS, fuzzy MS) equipped with locally $\mathcal{H}$-transitive relations.