A Class of Relational Functional Contractions with Applications to Nonlinear Integral Equations

Abstract

In this article, we investigate some fixed-point results under certain functional contractive mappings in a relation metric space. In the process, we utilize more general contraction condition which must be verified for comparative elements only. Our results enrich, modify, refine, unify and sharpen several existing fixed-point results. We construct some examples in support of our results. To attest to the applicability of our results, we establish the existence and uniqueness of theorems regarding the solutions of certain nonlinear integral equations.

1. Introduction

The Banach Contraction Principle (BCP) continues to inspire results on fixed points under various types of contractivity conditions, which have enormous applications within and beyond mathematics. There exists a vast literature on the generalizations of BCP; we merely refer to some recent works contained in [1,2,3]. Alam and Imdad [4] extended the BCP to the context of relational metric space and observed that the order-theoretic contraction principles due to Ran and Reurings [5] and Nieto and Rodríguez-López [6] can be extended up to an arbitrary binary relation in place of the partial order. After the appearance of this pioneering result, fixed-point theorems in relational metric spaces have attracted much attention in the last eight years. Consequently, numerous papers have been released in this direction, e.g., [7,8,9,10,11,12,13,14]. Employing the suitable relation-theoretic contractions, several researchers have also discussed the existence and uniqueness of solutions of typical nonlinear integral equations, e.g., [15,16,17,18,19,20,21,22,23].

In recent years, several authors have proved fixed-point results in the context of metric space with a directed graph. This trend was initiated by Jachymski [24] in 2008. In such results, the set of the edges of the underlying graph contains all loops. It follows that the set of all edges remains a reflexive relation on given metric space. Consequently, the results of Jachymski [24] can be obtained from the results of Alam and Imdad [4]. Therefore, the relational approach generalizes the graphic approach. But converse is not necessarily true. As a counter example, one can consider the strict order (i.e., an irreflexive and transitive binary relation) “<” in the relation-theoretic contraction principle; but the same result cannot be from the graphic fixed-point theorem.

A number of generalizations of BCP consist of more general contractivity conditions on the underlying metric space $(\yen ,\varsigma )$ involving the distance $\varsigma (y,v)$ on the R.H.S. (abbreviation of ‘right hand side’), where $y,v\in \yen$. However, some existing contraction conditions contain $\varsigma (y,v)$ together with the displacements of $y,v\in \yen$ under the mapping $\mathsf{ϝ}$: $\varsigma (y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y)$. A general form of such types of contractions, often called ‘functional contraction’, can be expressed as

$$\varsigma (\mathsf{ϝ}y,\mathsf{ϝ}v)\le \mathcal{G}\left(\varsigma \right(y,v),\varsigma (y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y\left)\right),\forall y,v\in ,\yen$$

where the function $\mathcal{G}:{[0,\infty )}^5\to [0,\infty )$ is chosen appropriately. For some possible choices of F, we advise the readers to study the papers of Kannan [25], Reich [26], Chatterjea [27], Zamfirescu [28], Bianchini [29], Hardy and Rogers [30], Ćirić [31], Turinici [32], Husain and Sehgal [33], Rhoades [34], Park [35], Khan et al. [36], Kincses and Totik [37], Collaco and Silva [38], Berinde [39], Turinici [40] and others.

In this manuscript, we undertake a metric space $(\yen ,\varsigma )$ with a relation $\varrho$ and in this setting, we prove the results concerning the existence and uniqueness of fixed points under a new class of relational functional contraction of the form

$$\varsigma (\mathsf{ϝ}y,\mathsf{ϝ}v)\le \psi \left(\varsigma \right(y,v\left)\right)+\theta \left(\varsigma \right(y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y\left)\right),\forall (y,v)\in \varrho ,$$

where $\psi :[0,\infty )\to [0,\infty )$ and $\theta :{[0,\infty )}^4\to [0,\infty )$ are certain auxiliary functions. The binary relation utilized in the existence theorem does not admit any properties at all (i.e. the relation is amorphous). However, the uniqueness part is to be imposed by the ${\varrho }^s$-directedness property of given metric space. As a corollary, we deduce a corresponding fixed point theorem in abstract metric space. As evidence of our results, some illustrative examples are furnished which unmistakably demonstrate the significant enhancements of the existing results in the literature. Finally, by means of our results, we describe the existence and uniqueness of solutions of certain integral equations in the presence of a lower or an upper solution.

2. Preliminaries

In what follows, $\mathbb{R}$ indicates to the set of real numbers, and $\mathbb{N}$ symbolizes the set of natural numbers. Given a set $\yen \ne \varnothing$, the subset $\varrho \subseteq {\yen }^2$ is named as a binary relation (or, a relation) on ¥.

Definition 1. 

Consider ¥ to be a set with a relation ϱ. We say that

• Two points $y,v\in \yen$ are ϱ-comparative, denoted by $[y,v]\in \varrho$, if $(y,v)\in \varrho$ or $(v,y)\in \varrho$ (cf. [4]).
• ${\varrho }^{-1}:=\{(y,v)\in {\yen }^2:(v,y)\in \varrho \}$ is the inverse of ϱ (cf. [41]).
• ${\varrho }^s:=\varrho \cup {\varrho }^{-1}$ is the symmetric closure of ϱ (cf. [41])
• A sequence $\left\{y_m\right\}\subset \yen$ is ϱ-preserving if it verifies $(y_m,y_{m+1})\in \varrho$, ∀$m\in \mathbb{N}$ (cf. [4]).
• A subset $\mathcal{Ƶ}\subseteq \yen$ is ϱ-directed if for any pair $y,v\in \mathcal{Ƶ}$, ∃$w\in \yen$ satisfying $(y,w)\in \varrho$ and $(v,w)\in \varrho$ (cf. [42]).

Definition 2. 

Given a map $\mathsf{ϝ}:\yen \to \yen$, a relation ϱ on ¥ is known as ϝ-closed (cf. [4]) if

$$\forall y,v\in \yen ;(y,v)\in \varrho ⟹(\mathsf{ϝ}y,\mathsf{ϝ}v)\in \varrho .$$

Definition 3. 

Assume that $(\yen ,\varsigma )$ remains a metric space that endows a relation ϱ. We call that

• $(\yen ,\varsigma )$ is ϱ-complete if any ϱ-preserving Cauchy sequence in ¥ converges. (cf. [7]).
• ϱ is ς-self-closed if any ϱ-preserving sequence $\left\{y_m\right\}\subset \yen$ converging to $p\in \yen$ admits a subsequence $\left\{y_{m_k}\right\}$ verifying $[y_{m_k},p]\in \varrho$, $\forall k\in \mathbb{N}$ (cf. [4]).

Definition 4. 

Assume that $(\yen ,\varsigma )$ remains a metric space with a relation ϱ. A map $\mathsf{ϝ}:\yen \to \yen$ is called

• ϱ-continuous at $p\in \yen$ if for any ϱ-preserving sequence $\left\{y_m\right\}\subset \yen$ (cf. [7]).

  $$y_m\stackrel{\varsigma }{⟶}p⟹\mathsf{ϝ}\left(y_m\right)\stackrel{\varsigma }{⟶}\mathsf{ϝ}\left(p\right)$$
• ϱ-continuous if it remains ϱ-continuous at each $p\in \yen$ (cf. [7]).

Proposition 1 

([4]). $(y,v)\in {\varrho }^s⟺[y,v]\in \varrho .$

Proposition 2 

([8]). ϱ remains ${\mathsf{ϝ}}^n$-closed relation if ϱ is ϝ-closed.

We shall adopt the following notions:

•

$Fix(\mathsf{ϝ})$: = the collection of all fixed points of $\mathsf{ϝ}$.

•

$\yen (\mathsf{ϝ},\varrho ):=\{y\in \yen :(y,\mathsf{ϝ}y)\in \varrho \}$.

Following Muresan [43], a monotonic increasing function $\psi :[0,\infty )\to [0,\infty )$ is referred to as a (c)-comparison function if it verifies ${\displaystyle \sum _{m=1}^{\infty }}{\psi }^m\left(t\right)<\infty ,\forall t>0$. We shall denote the family of all (c)-comparison functions by $\Phi$.

Remark 1 

([43]). If $\psi \in \Phi$, then

(i) 

$\psi \left(t\right)<t,\forall t>0$,

(ii) 

$\psi \left(0\right)=0$.

Let us denote $\Omega$ by the class of all continuous functions $\theta :{[0,\infty )}^4\to [0,\infty )$ verifying $\theta (t_1,t_2,t_3,t_4)=0$ if and only if $t_i$, for at least one $i=1,2,3,4$. This class of functions is proposed by Jleli et al. [44].

We can conclude the following fact by making use of the symmetric property of $\varsigma$.

Proposition 3. 

Let $\psi \in \Phi$ and $\theta \in \Omega$. The following assumptions remain equivalent:

(i) 

$\varsigma (\mathsf{ϝ}y,\mathsf{ϝ}v)\le \psi \left(\varsigma \right(y,v\left)\right)+\theta \left(\varsigma \right(y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y\left)\right),\forall y,v\in \yen with$$(y,v)\in \varrho ,$

(ii) 

$\varsigma (\mathsf{ϝ}y,\mathsf{ϝ}v)\le \psi \left(\varsigma \right(y,v\left)\right)+\theta \left(\varsigma \right(y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y\left)\right),\forall y,v\in \yen with$$[y,v]\in \varrho .$

3. Main Results

We present the following results concerning the existence and uniqueness of fixed points for relational functional contractions.

Theorem 1. 

Assume that $(\yen ,\varsigma )$ remains a metric space with a relation ϱ and that $\mathsf{ϝ}:\yen \to \yen$ is a map such that

(I) 

$\yen (\mathsf{ϝ},\varrho )\ne \varnothing$.

(II) 

ϱ is ϝ-closed.

(III) 

$\yen (,\varsigma )$ is ϱ-complete.

(IV) 

ϝ remains ϱ-continuous or ϱ remains ς-self-closed.

(V)

∃$\psi \in \Phi$ and $\exists \theta \in \Omega$ satisfying

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}y,\mathsf{ϝ}v)& \le & \psi \left(\varsigma \right(y,v\left)\right)+\theta \left(\varsigma \right(y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y\left)\right),\hfill \\ \hfill & \forall & y,v\in \yen with(y,v)\in \varrho .\hfill \end{array}$$

Then, ϝ admits a fixed point.

Proof. 

By (I), we choose $y_0\in \yen (\mathsf{ϝ},\varrho )$. We define the sequence $\left\{y_m\right\}\subset \yen$ verifying

$$y_m={\mathsf{ϝ}}^m\left(y_0\right)=\mathsf{ϝ}\left(y_{m-1}\right),\forall m\in \mathbb{N}.$$

(1)

By assumption (I), we have $(y_0,\mathsf{ϝ}{y}_0)\in \varrho$. Employing (II) and Proposition 2, we obtain

$$({\mathsf{ϝ}}^m{u}_0,{\mathsf{ϝ}}^{m+1}{y}_0)\in \varrho$$

such that

$$(y_m,y_{m+1})\in \varrho ,\forall m\in \mathbb{N}.$$

(2)

Therefore, $\left\{y_m\right\}$ is an $\varrho$-preserving sequence. Employing condition (V) for (2), we obtain

$$\begin{array}{ccc}\hfill \varsigma (y_m,y_{m+1})& =& \varsigma (\mathsf{ϝ}{y}_{m-1},\mathsf{ϝ}{y}_m)\le \psi \left(\varsigma (y_{m-1},y_m)\right)\hfill \\ & +& \theta (\varsigma (\mathsf{ϝ}{y}_m,y_m),\varsigma (\mathsf{ϝ}{y}_{m-1},y_{m-1}),\varsigma (\mathsf{ϝ}{y}_{m-1},y_m),\varsigma (\mathsf{ϝ}{y}_m,y_{m-1})),\forall m\in \mathbb{N}\hfill \end{array}$$

such that

$$\begin{array}{c}\hfill \varsigma (y_m,y_{m+1})\le \psi \left(\varsigma (y_{m-1},y_m)\right)+\theta (\varsigma (y_{m+1},y_m),\varsigma (y_m,y_{m-1}),0,\varsigma (y_{m+1},y_{m-1})),\forall m\in \mathbb{N}\end{array}$$

which, owing to the definition of $\Omega$ and the monotonicity of $\psi$, reduces to

$$\varsigma (y_m,y_{m+1})\le {\psi }^n\varsigma (y_0,\mathsf{ϝ}{y}_0),\forall m\in \mathbb{N}.$$

(3)

Taking arbitrary $m,n\in \mathbb{N}$ with $n<m$. Employing (3), we obtain

$$\begin{array}{ccc}\hfill \varsigma (y_n,y_m)& \le & \varsigma (y_n,y_{n+1})+\varsigma (y_{n+1},y_{n+2})+\cdots +\varsigma (y_{m-1},y_m)\hfill \\ & \le & {\psi }^m\left(\varsigma (y_0,\mathsf{ϝ}{y}_0)\right)+{\psi }^{m+1}\left(\varsigma (y_0,\mathsf{ϝ}{y}_0)\right)+\cdots +{\psi }^{n-1}\left(\varsigma (y_0,\mathsf{ϝ}{y}_0)\right)\hfill \\ & =& \sum _{k=m}^{n-1}{\psi }^k\left(\varsigma (y_0,\mathsf{ϝ}{y}_0)\right)\hfill \\ & \le & \sum _{k\ge m}{\psi }^k\left(\varsigma (y_0,\mathsf{ϝ}{y}_0)\right)\hfill \\ & \to & 0\mathrm{as}m\left(\mathrm{and}\mathrm{hence}n\right)\to \infty ,\hfill \end{array}$$

which shows that $\left\{y_m\right\}$ is Cauchy. Since $\left\{y_m\right\}$ is also $\varrho$-preserving, therefore by $\varrho$-completeness of $\yen (,\varsigma )$, ∃$p\in \yen$ verifying $y_m\stackrel{\varsigma }{⟶}p$.

We shall utilize (IV) to conclude that $p\in Fix(\mathsf{ϝ})$. Let $\mathsf{ϝ}$ be $\varrho$-continuous. Then owing to the fact that $\left\{y_m\right\}$ is $\varrho$-preserving verifying $y_m\stackrel{\varsigma }{⟶}p$, and by $\varrho$-continuity of $\mathsf{ϝ}$, we have $y_{m+1}=\mathsf{ϝ}\left(y_m\right)\stackrel{\varsigma }{⟶}\mathsf{ϝ}\left(p\right)$. Consequently, by uniqueness of limit, we obtain $\mathsf{ϝ}\left(p\right)=p$. In either case, $\varrho$ is $\varsigma$-self closed. Then, $\left\{y_m\right\}$, being $\varrho$-preserving along with $y_m\stackrel{\varsigma }{⟶}p$, induces a subsequence $\left\{y_{m_k}\right\}$ verifying $[y_{m_k},p]\in \varrho$, $\forall k\in \mathbb{N}.$ Using contractivity condition (V), Proposition 3 and $[y_{m_k},p]\in \varrho$, we obtain

$$\begin{array}{ccc}\hfill \varsigma (y_{m_k+1},\mathsf{ϝ}p)& =& \varsigma (\mathsf{ϝ}{y}_{m_k},\mathsf{ϝ}p)\hfill \\ & \le & \psi \left(\varsigma (y_{m_k},p)\right)+\theta (\varsigma (p,\mathsf{ϝ}p),\varsigma (y_{m_k},y_{m_k+1}),\varsigma (p,y_{m_k+1}),\varsigma (y_{m_k},\mathsf{ϝ}p)).\hfill \end{array}$$

By Remark 1 (whether $\varsigma (y_{m_k},p)$ is non-zero or zero), the above inequality gives rise to

$$\begin{array}{c}\hfill \varsigma (y_{m_k+1},\mathsf{ϝ}p)\le \varsigma (y_{m_k},p)+\theta (\varsigma (p,\mathsf{ϝ}p),\varsigma (y_{m_k},y_{m_k+1}),\varsigma (p,y_{m_k+1}),\varsigma (y_{m_k},\mathsf{ϝ}p)).\end{array}$$

(4)

Letting $k\to \infty$ in (4) and using $y_{m_k}\stackrel{\varsigma }{⟶}p$ and the definition of $\Omega$, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\varsigma (y_{m_k+1},\mathsf{ϝ}p)=0\end{array}$$

implying $y_{m_k+1}\stackrel{\varsigma }{⟶}\mathsf{ϝ}\left(p\right)$ so that $\mathsf{ϝ}\left(p\right)=p$. □

Theorem 2. 

Under the conditions (I)–(V) of Theorem 1, if $\mathsf{ϝ}(\yen )$ remains ${\varrho }^s$-directed, then ϝ possesses a unique fixed point.

Proof. 

Taking $p_1,p_2\in Fix(\mathsf{ϝ})$, we obtain

$${\mathsf{ϝ}}^m\left(p_1\right)=p_1\mathrm{and}{\mathsf{ϝ}}^m\left(p_2\right)=p_2.$$

(5)

As $\mathsf{ϝ}(\yen )$ is ${\varrho }^s$-directed and contains $p_1,p_2\in \mathsf{ϝ}(\yen )$, $\exists \omega \in \yen$ verifying

$$[p_1,\omega ]\in \varrho \mathrm{and}[p_2,\omega ]\in \varrho .$$

(6)

By assumption (II) and Proposition 2, we obtain

$$[p_1,{\mathsf{ϝ}}^m\omega ]\in \varrho \mathrm{and}[p_2,{\mathsf{ϝ}}^m\omega ]\in \varrho .$$

(7)

Using (7), assumption (V) and Proposition 3, we obtain

$$\begin{array}{ccc}& & \varsigma (p_1,{\mathsf{ϝ}}^{m+1}\omega )=\varsigma (\mathsf{ϝ}{p}_1,\mathsf{ϝ}\left({\mathsf{ϝ}}^m\omega \right))\le \psi (\varsigma (p_1,{\mathsf{ϝ}}^m\omega )\hfill \\ & & +\theta (\varsigma (p_1,p_1),\varsigma ({\mathsf{ϝ}}^m\omega ,{\mathsf{ϝ}}^{m+1}\omega ),\varsigma (p_1,{\mathsf{ϝ}}^{m+1}\omega ),\varsigma ({\mathsf{ϝ}}^m\omega ,p_1))\hfill \end{array}$$

such that

$$\varsigma (p_1,{\mathsf{ϝ}}^{m+1}\omega )\le \psi \left(\varsigma (p_1,{\mathsf{ϝ}}^m\omega )\right).$$

(8)

We set ${\gamma }_m:=\varsigma (p_1,{\mathsf{ϝ}}^m\omega )$. Then, we shall show that

$${\displaystyle \underset{m\to \infty }{lim}{\gamma }_m=0.}$$

(9)

We distinguish between two cases:

Case-I: If ${\gamma }_{m_0}=0$ for some $m_0\in \mathbb{N}$, then $p_1={\mathsf{ϝ}}^{m_0}\left(\omega \right)$, yielding $p_1=\mathsf{ϝ}\left(p_1\right)={\mathsf{ϝ}}^{m_0+1}\left(\omega \right)$. Consequently, we obtain ${\gamma }_{m_0+1}=0$ and hence, by induction, we have ${\gamma }_m=0,\forall m\ge m_0,$ such that $\underset{m\to \infty }{lim}{\gamma }_m=0$.

Case-II: If ${\gamma }_m>0,\forall m\in \mathbb{N}$, then by induction on m in (8) and by monotonic axiom of $\psi$, we obtain

$$\begin{array}{c}\hfill {\gamma }_{m+1}\le \psi \left({\gamma }_m\right)\le {\psi }^2\left({\gamma }_{m-1}\right)\le \cdots \le {\psi }^m\left({\gamma }_1\right)\end{array}$$

such that

$$\begin{array}{c}\hfill {\gamma }_{m+1}\le {\psi }^m\left({\gamma }_1\right).\end{array}$$

(10)

Taking $m\to \infty$ in (10) and by the definition of (c)-comparison function, we have

$${\displaystyle \underset{m\to \infty }{lim}{\gamma }_{m+1}\le \underset{m\to \infty }{lim}{\psi }^m\left({\gamma }_1\right)=0.}$$

Thus, in both cases, (9) is verified. In view of (5), (6), (9) and triangular inequality, we obtain

$$\varsigma (p_1,p_2)=\varsigma (p_1,{\mathsf{ϝ}}^m\omega )+\varsigma ({\mathsf{ϝ}}^m\omega ,p_2)\to 0,asn\to \infty$$

implying $p_1=p_2$. Therefore, $\mathsf{ϝ}$ admits a unique fixed point. □

Remark 2. 

In particular, for $\theta =0$, Theorems 1 and 2 reduce to the main results of Algehynep et al. [14].

Theorem 2 deduces the following results in abstract metric space.

Corollary 1. 

If $(\yen ,\varsigma )$ remains a complete metric space, $\mathsf{ϝ}:\yen \to \yen$ is a map and ∃$\psi \in \Phi$ and $\theta \in \Omega$ verifying

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}y,\mathsf{ϝ}v)& \le & \psi \left(\varsigma \right(y,v\left)\right)+\theta \left(\varsigma \right(y,\mathsf{ϝ}y),\varsigma (v,\mathsf{ϝ}v),\varsigma (y,\mathsf{ϝ}v),\varsigma (v,\mathsf{ϝ}y\left)\right),\hfill \\ \hfill & \forall & y,v\in \yen ,\hfill \end{array}$$

then ϝ possesses a unique fixed point.

Proof. 

Taking $\varrho ={\yen }^2$, the universal relation in Theorems 1 and 2, we notice that the assumptions (I), (II) and (IV) of Theorem 1 are automatically satisfied. Also, the $\varrho$-completeness of metric space coincides with the completeness, and the contractivity conditions of Theorem 1 coincides with the given contractivity conditions. Moreover, $\mathsf{ϝ}(\yen )$ is also ${\varrho }^s$-directed. Consequently, our result follows from Theorem 2. □

4. Illustrative Examples

To illustrate our results proved in the previous section, we furnish them with the following examples.

Example 1. 

Take $\yen =[-1,4]$ with the metric $\varsigma (y,v)=|y-v|$ and a relation $\varrho =\{(y,v)\in {\yen }^2:y-v\ge 0\}$. Let $\mathsf{ϝ}:\yen \to \yen$ be a mapping defined by

$$\mathsf{ϝ}\left(y\right)=\left\{\begin{array}{c}y/2\mathrm{if}-1\le y\le 2\hfill \\ 1\mathrm{if}2<y\le 4.\hfill \end{array}\right.$$

Then, $(\yen ,\varsigma )$ is ϱ-complete and ϱ is ϝ-closed. Consider the auxiliary functions $\psi \left(t\right)=t/2$ and $\theta (t_1,t_2,t_3,t_4)=ln(t_1{t}_2{t}_3{t}_4+1)$. Then, $\psi \in \Phi$ and $\theta \in \Omega$. It can be easily checked that the condition (V) of Theorem 1 holds for these auxiliary functions. Left over assumptions of Theorems 1 and 2 also hold and hence ϝ admits a unique fixed point $p=0$.

Example 2. 

Take $\yen =[1,3]$ with the metric $\varsigma (y,v)=|y-v|$ and a relation $\varrho =\left\{\right(1,1),(1,2),$$(2,1),(2,2),$$(1,3)\}$. Let $\mathsf{ϝ}:\yen \to \yen$ be a mapping defined by

$$\mathsf{ϝ}\left(y\right)=\left\{\begin{array}{c}1\mathrm{if}1\le y\le 2\hfill \\ 2\mathrm{if}2<y\le 3.\hfill \end{array}\right.$$

Then, $(\yen ,\varsigma )$ is ϱ-complete and ϱ is ϝ-closed. Consider the auxiliary functions $\psi \left(t\right)=t/3$ and $\theta (t_1,t_2,t_3,t_4)=t_1{t}_2{t}_3{t}_4$. Then, $\psi \in \Phi$ and $\theta \in \Omega$. It can be easily checked that the condition (V) of Theorem 1 holds for these auxiliary functions.

Let $\left\{y_m\right\}\subset \yen$ be a ϱ-preserving sequence satisfying $y_m\stackrel{\varsigma }{⟶}q$ such that $(y_m,y_{m+1})\in \varrho ,\forall m\in \mathbb{N}$. Here, $(y_m,y_{m+1})\notin \left\{(1,3)\right\}$ implying $(y_m,y_{m+1})\in \{(1,1),(1,2),(2,1),(2,2)\}$, $\forall m\in \mathbb{N}$ such that $\left\{y_m\right\}\subset \{1,2\}$. As $\{1,2\}$ is closed, we have $[y_m,q]\in \varrho$. Therefore, ϱ is ς-self-closed. Leftover assumptions of Theorems 1 and 2 also hold and hence ϝ admits a unique fixed point $p=1$.

Finally, we construct the following example for which the conditions of the existence result (i.e., Theorem 1) hold only when there is no uniqueness.

Example 3. 

Take $\yen =(0,4]$ with the metric $\varsigma (y,v)=|y-v|$ and a relation $\varrho =\left\{\right(y,v):1\le y\le v\le 2\mathrm{or}3\le y\le v\le 4\}$. Let $\mathsf{ϝ}:\yen \to \yen$ be a map defined by

$$\mathsf{ϝ}\left(y\right)=\left\{\begin{array}{c}1,\mathrm{if}0\le y<3\hfill \\ 4,\mathrm{if}3\le y\le 4.\hfill \end{array}\right.$$

Clearly, $(\yen ,\varsigma )$ remains ϱ-complete, ϝ is ϱ-continuous and ϱ is ϝ-closed. We define the auxiliary functions

$$\psi \left(t\right)=\left\{\begin{array}{cc}1/2\hfill & if t\in [0,1]\hfill \\ t-1/2\hfill & if t>1\hfill \end{array}\right.$$

and

$$\theta (t_1,t_2,t_3,t_4)=min\{t_1,t_2,t_3,t_4\}.$$

Then, $\psi \in \Phi$ and $\theta \in \Omega$. It can be easily checked that the condition (V) of Theorem 1 holds for these auxiliary functions. Thus, the conditions (I)–(V) of Theorem 1 are verified and hence ϝ admits a fixed point.

Furthermore, $\mathsf{ϝ}(\yen )$ is not ${\varrho }^s$-directed as ∄ element in ¥ which is simultaneously ϱ-comparative with 1 as well as 4 such that we cannot apply Theorem 2 here. Note that ϝ admits two fixed points, namely $p=1$ and $q=4$.

5. An Application to Nonlinear Integral Equations

This section contains an application of our results proved in Section 3 to determine a unique solution of the nonlinear integral equation:

$$\vartheta \left(s\right)=f\left(s\right)+{\int }_a^{b}K(s,\xi )w(\xi ,\vartheta \left(\xi \right))d\xi ,s\in [a,b].$$

(11)

where $f:I\to \mathbb{R}$, $K:I\times I\to \mathbb{R}$ and $w:I\times \mathbb{R}\to \mathbb{R}$ are functions, where $I:=[a,b]$. The class of the continuous functions from I to $\mathbb{R}$ will be denotes by $\mathcal{C}\left(I\right)$.

Definition 5. 

We say that $\alpha \in \mathcal{C}\left(I\right)$ forms a lower solution of (11) if $\forall s\in I$

$$\alpha \left(s\right)\le f\left(s\right)+{\int }_a^{b}K(s,\xi )w(\xi ,\alpha \left(\xi \right))d\xi .$$

Definition 6. 

We say that $\beta \in \mathcal{C}\left(I\right)$ forms an upper solution of (11) if $\forall s\in I$

$$\beta \left(s\right)\ge f\left(s\right)+{\int }_a^{b}K(s,\xi )w(\xi ,\beta \left(\xi \right))d\xi .$$

We now prove the main results of this section.

Theorem 3. 

Along with the Problem (11), assume that

(i) 

f, w and K are continuous.

(ii) 

$K(s,\xi )>0,\forall s,\xi \in I$.

(iii) 

∃$0<\lambda \le 1$ and ∃ a (c)-comparison function ψ verifying

$$0\le w(s,x)-w(s,y)\le \frac{1}{\lambda }\psi (x-y),\forall s\in Iand\forall x,y\in \mathbb{R}withx\ge y,$$

(iv) 

$\underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )d\xi \le \lambda$.

Moreover, if a lower solution of (11) exists, then the problem enjoys a unique solution.

Proof. 

We endow $\yen :=\mathcal{C}\left(I\right)$ with a metric $\varsigma$ defined by

$$\varsigma (\vartheta ,\mu )=\underset{s\in I}{sup}|\vartheta \left(s\right)-\mu \left(s\right)|,\forall \vartheta ,\mu \in \yen .$$

(12)

On ¥, we define a relation $\varrho$ by

$$\varrho =\{(\vartheta ,\mu )\in {\yen }^2:\vartheta \left(s\right)\le \mu \left(s\right),\forall s\in I\}.$$

(13)

We consider the mapping $\mathsf{ϝ}:\yen \to \yen$ defined by

$$(\mathsf{ϝ}\vartheta )\left(s\right)=f\left(s\right)+{\int }_a^{b}K(s,\xi )w(\xi ,\vartheta \left(\xi \right))d\xi ,\forall s\in \yen .$$

(14)

Thus, $\vartheta \in \yen$ solves the Problem (11) if and only if $\vartheta \in Fix(\mathsf{ϝ})$.

Now, we shall verify all assumptions of Theorems 1 and 2.

(I) If $\alpha \in \yen$ remains a lower solution of (11), then

$$\alpha \left(s\right)\le f\left(s\right)+{\int }_a^{b}K(s,\xi )w(\xi ,\alpha \left(\xi \right))d\xi =(\mathsf{ϝ}\alpha )\left(s\right)$$

implying thereby $(\alpha ,\mathsf{ϝ}\alpha )\in \varrho$ such that $\yen (\mathsf{ϝ},\varrho )$ is nonempty.

(II) Take $\vartheta ,\mu \in \yen$ verifying $(\vartheta ,\mu )\in \varrho$. Using assumption (iii), we obtain

$$w(s,\vartheta (\xi \left)\right)-w(s,\mu (\xi \left)\right)\le 0,\forall s,\xi \in I.$$

(15)

By (14), (15) and assumption (ii), we obtain

$$\begin{array}{c}\hfill (\mathsf{ϝ}\vartheta )\left(s\right)-(\mathsf{ϝ}\mu )\left(s\right)={\int }_a^{b}K(s,\xi )[w(\xi ,\vartheta \left(\xi \right))-w(\xi ,\mu \left(\xi \right))]d\xi \le 0,\end{array}$$

such that $(\mathsf{ϝ}\vartheta )(s)\le (\mathsf{ϝ}\mu )(s)$, which using (13) yields that $(\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )\in \varrho$ and hence $\varrho$ remains $\mathsf{ϝ}$-closed.

(III) $(\yen ,\varsigma )$ being a complete metric space is $\varrho$-complete.

(IV) If $\left\{{\vartheta }_m\right\}\subset \yen$ is a $\varrho$-preserving sequence converging to $\omega \in \yen$, and then for each $s\in I$, $\left\{{\vartheta }_m\left(s\right)\right\}$ remains an increasing sequence in $\mathbb{R}$ converging to $\omega \left(s\right)$. It follows that ${\vartheta }_m\left(s\right)\le \omega \left(s\right),\forall m\in \mathbb{N}$ and $\forall s\in I$. By (13), we have $({\vartheta }_m,\omega )\in \varrho ,\forall m\in \mathbb{N}$. Therefore, $\varrho$ is $\varsigma$-self-closed.

(V) Let $\vartheta ,\mu \in \yen$ such that $(\vartheta ,\mu )\in \varrho$. By assumption (iii), (12) and (14), one has

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )& =& \underset{s\in I}{sup}|(\mathsf{ϝ}\vartheta )\left(s\right)-(\mathsf{ϝ}\mu )\left(s\right)|=\underset{s\in I}{sup}[(\mathsf{ϝ}\mu )\left(s\right)-(\mathsf{ϝ}\vartheta )\left(s\right)]\hfill \\ \hfill & =& \underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )[w(\xi ,\mu \left(\xi \right))-w(\xi ,\vartheta \left(\xi \right))]d\xi \hfill \\ & \le & \underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )\frac{1}{\lambda }\psi (\mu \left(\xi \right)-\vartheta \left(\xi \right))d\xi .\hfill \end{array}$$

(16)

As $\psi$ is increasing and $0\le \mu \left(\xi \right)-\vartheta \left(\xi \right)\le \varsigma (\vartheta ,\mu )$, we obtain $\psi \left(\mu \right(\xi )-\vartheta (\xi \left)\right)\le \psi \left(\varsigma \right(\vartheta ,\mu \left)\right)$ and hence (16) reduces to

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )& \le & \frac{1}{\lambda }\psi \left(\varsigma (\vartheta ,\mu )\right)\underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )d\xi \hfill \\ & \le & \frac{1}{\lambda }\psi \left(\varsigma (\vartheta ,\mu )\right).\lambda \hfill \\ & =& \psi \left(\varsigma \right(\vartheta ,\mu \left)\right)\hfill \end{array}$$

such that

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )& \le & \psi \left(\varsigma \right(\vartheta ,\mu \left)\right)+\theta \left(\varsigma \right(\vartheta ,\mathsf{ϝ}\vartheta ),\varsigma (\mu ,\mathsf{ϝ}\mu ),\varsigma (\vartheta ,\mathsf{ϝ}\mu ),\varsigma (\mu ,\mathsf{ϝ}\vartheta \left)\right),\hfill \\ & \forall & \vartheta ,\mu \in \yen \mathrm{such}\mathrm{that}(\vartheta ,\mu )\in \varrho ,\hfill \end{array}$$

where $\theta \in \Omega$ is arbitrary.

Now, we take $\vartheta ,\mu \in \yen$ as arbitrary. We set $\varpi :=max\{\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu \}\in \yen$, and then we have $(\mathsf{ϝ}\vartheta ,\varpi )\in \varrho$ and $(\mathsf{ϝ}\mu ,\varpi )\in \varrho$. Therefore, $\mathsf{ϝ}(\yen )$ is ${\varrho }^s$-directed. Thus, by applying Theorem 2, $\mathsf{ϝ}$ possesses a unique fixed point, which which retains to be a unique solution of (11). □

Theorem 4. 

Under the assumptions (i)–(iv) of Theorem 3, if an upper solution of (11) exists, then the problem enjoys a unique solution.

Proof. 

Define a metric $\varsigma$ on $\yen :=\mathcal{C}\left(I\right)$ and a mapping $\mathsf{ϝ}:\yen \to \yen$, which is the same as in the proof of Theorem 3. Consider a relation ${\varrho }^{\prime }$ on ¥, defined by

$${\varrho }^{\prime }=\{(\vartheta ,\mu )\in {\yen }^2:\vartheta \left(s\right)\ge \mu \left(s\right),\forall s\in I\}.$$

(17)

Now, we shall verify all assumptions of Theorems 1 and 2.

(I) If $\beta \in \yen$ is an upper solution of (11), then we have

$$\beta \left(s\right)\ge f\left(s\right)+{\int }_a^{b}K(s,\xi )w(\xi ,\beta \left(\xi \right))d\xi =(\mathsf{ϝ}\beta )\left(s\right)$$

implying $(\beta ,\mathsf{ϝ}\beta )\in {\varrho }^{\prime }$ such that $(\mathsf{ϝ},{\varrho }^{\prime })$ is nonempty.

(II) Taking $\vartheta ,\mu \in \yen$, we verify $(\vartheta ,\mu )\in {\varrho }^{\prime }$. Using assumption (iii), we obtain

$$w(s,\vartheta (\xi \left)\right)-w(s,\mu (\xi \left)\right)\ge 0,\forall s,\xi \in I.$$

(18)

By (14), (18) and assumption (ii), we obtain

$$\begin{array}{c}\hfill (\mathsf{ϝ}\vartheta )\left(s\right)-(\mathsf{ϝ}\mu )\left(s\right)={\int }_a^{b}K(s,\xi )[w(\xi ,\vartheta \left(\xi \right))-w(\xi ,\mu \left(\xi \right))]d\xi \ge 0,\end{array}$$

such that $(\mathsf{ϝ}\vartheta )(s)\ge (\mathsf{ϝ}\mu )(s)$, which using (17) yields that $(\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )\in {\varrho }^{\prime }$ and hence ${\varrho }^{\prime }$ is $\mathsf{ϝ}$-closed.

(III) $(\yen ,\varsigma )$ being a complete metric space is ${\varrho }^{\prime }$-complete.

(IV) If $\left\{{\vartheta }_m\right\}\subset \yen$ is a ${\varrho }^{\prime }$-preserving sequence converging to $\omega \in \yen$, then for each $s\in I$, $\left\{{\vartheta }_m\left(s\right)\right\}$ remains a decreasing sequence in $\mathbb{R}$ converging to $\omega \left(s\right)$. It follows that ${\vartheta }_m\left(s\right)\ge \omega \left(s\right),\forall m\in \mathbb{N}$ and $\forall s\in I$. By (17), we have $({\vartheta }_m,\omega )\in {\varrho }^{\prime },\forall m\in \mathbb{N}$. Hence, ${\varrho }^{\prime }$ remains $\varsigma$-self-closed.

(V) Let $\vartheta ,\mu \in \yen$ such that $(\vartheta ,\mu )\in {\varrho }^{\prime }$. By assumption (iii), (12) and (14), one has

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )& =& \underset{s\in I}{sup}|(\mathsf{ϝ}\vartheta )\left(s\right)-(\mathsf{ϝ}\mu )\left(s\right)|=\underset{s\in I}{sup}[(\mathsf{ϝ}\vartheta )\left(s\right)-(\mathsf{ϝ}\mu )\left(s\right)]\hfill \\ \hfill & =& \underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )[w(\xi ,\vartheta \left(\xi \right))-w(\xi ,\mu \left(\xi \right))]d\xi \hfill \\ & \le & \underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )\frac{1}{\lambda }\psi (\vartheta \left(\xi \right)-\mu \left(\xi \right))d\xi .\hfill \end{array}$$

(19)

As $\psi$ is increasing and $0\le \vartheta \left(\xi \right)-\mu \left(\xi \right)\le \varsigma (\vartheta ,\mu )$, we obtain $\psi \left(\vartheta \right(\xi )-\mu (\xi \left)\right)\le \psi \left(\varsigma \right(\vartheta ,\mu \left)\right)$ and hence (19) reduces to

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )& \le & \frac{1}{\lambda }\psi \left(\varsigma (\vartheta ,\mu )\right)\underset{s\in I}{sup}{\int }_a^{b}K(s,\xi )d\xi \hfill \\ & \le & \frac{1}{\lambda }\psi \left(\varsigma (\vartheta ,\mu )\right).\lambda \hfill \\ & =& \psi \left(\varsigma \right(\vartheta ,\mu \left)\right)\hfill \end{array}$$

such that

$$\begin{array}{ccc}\hfill \varsigma (\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu )& \le & \psi \left(\varsigma \right(\vartheta ,\mu \left)\right)+\theta \left(\varsigma \right(\vartheta ,\mathsf{ϝ}\vartheta ),\varsigma (\mu ,\mathsf{ϝ}\mu ),\varsigma (\vartheta ,\mathsf{ϝ}\mu ),\varsigma (\mu ,\mathsf{ϝ}\vartheta \left)\right),\hfill \\ & \forall & \vartheta ,\mu \in \yen \mathrm{such}\mathrm{that}(\vartheta ,\mu )\in {\varrho }^{\prime },\hfill \end{array}$$

where $\theta \in \Omega$ is arbitrary.

Now, we take $\vartheta ,\mu \in \yen$ as arbitrary. We set $\wp :=max\{\mathsf{ϝ}\vartheta ,\mathsf{ϝ}\mu \}\in \yen$, and then we have $(\wp ,\mathsf{ϝ}\vartheta )\in {\varrho }^{\prime }$ and $(\wp ,\mathsf{ϝ}\mu )\in {\varrho }^{\prime }$. Therefore, $\mathsf{ϝ}(\yen )$ is ${\varrho }^{\prime s}$-directed. Thus, by applying Theorem 2, $\mathsf{ϝ}$ possesses a unique fixed point, which which retains to be a unique solution of (11). □

6. Conclusions

We investigated the results concerning the existence and uniqueness of fixed points under a class of functional contractions involving two auxiliary functions in the relational metric space. By means of illustrating the results, several examples are also undertaken. The obtained results have been applied to a nonlinear integral equation wherein the presence of a lower solution or an upper solution ensures the existence of a unique solution.

A significant aspect of the relational-metric fixed-point theorems is that the involved contraction condition in such results must be satisfied only for comparative elements via underlying relation, rather than any pair of elements. Thus, in particular, relational functional contractions remain relatively weaker than the usual functional contractions.

In future works, the results proved herewith can be further extended to more general contractivity conditions (replacing the class of (c)-comparison functions with the class of Matkowski’s comparison functions, or the class of Boyd–Wong’s control functions); to generalized metrical structures (such as quasi-metric space, symmetric space, dislocated space, partial metric space, rectangular metric space, 2-metric space, etc.); and to pair of self-maps by proving coincident point theorems.