Relational Almost (ϕ,ψ)-Contractions and Applications to Nonlinear Fredholm Integral Equations

Abstract

This paper deals with the fixed-point findings under relational strict almost $(\varphi ,\psi )$-contraction. Our findings complement and strengthen prevailing results. In the course of the procedure, we also derive a related fixed point theorem for strict almost $(\varphi ,\psi )$-contraction. We provide several illustrative examples to support the validity of our outcomes. We also argue about the possibility of a unique solution of a nonlinear Fredholm integral equation via our outcomes.

1. Introduction

The foremost and traditional method in nonlinear functional analysis is the classical Banach contraction principle (abbreviated as BCP). There are numerous generalizations of BCP accessible in the literature. Berinde [1] presented a new generalization of BCP in 2004, which is known as almost contraction.

Definition 1

([1,2]). A self-map $\mathbf{f}$ on a metric space $(\mathbb{V},\omega )$ is known as almost contraction if there exist $0<\alpha <1$ and $\ell \ge 0$, enjoying

$$\omega (\mathbf{f}v,\mathbf{f}u)\le \alpha ·\omega (v,u)+\ell ·\omega (v,\mathbf{f}u),\forall v,u\in \mathbb{V}.$$

Making use of symmetry of $\omega$, the aforementioned condition is identical to the following one:

$$\omega (\mathbf{f}v,\mathbf{f}u)\le \alpha ·\omega (v,u)+\ell ·\omega (u,\mathbf{f}v),\forall v,u\in \mathbb{V}.$$

Theorem 1

([1]). Every almost contraction map on a complete metric space possesses a fixed point.

The almost contraction is a weak Picard operator, which means that it does not need to have a unique fixed point but Picard’s iterative sequence remains convergent to a fixed point of the map. An almost contraction map is not necessarily continuous but it remains continuous at each of fixed points (c.f. [2]). In addition to the usual contraction, almost contraction extends not only the usual contraction but also several well-known generalized contractions, which include Kannan contraction [3], Chatterjea contraction [4], Zamfirescu contraction, and [5] and a special class of Ćirić’s quasi-contraction [6]. In recent years, many fixed point results involving almost contraction conditions have been established, e.g., [7,8,9,10,11,12,13,14].

Babu et al. [15] established a notably restricted category of almost contraction in order to derive a uniqueness theorem.

Definition 2

([15]). A self-map $\mathbf{f}$ on a metric space $(\mathbb{V},\omega )$ is known as strict almost contraction if there exist $0<\alpha <1$ and $\ell \ge 0$, enjoying

$$\omega (\mathbf{f}v,\mathbf{f}u)\le \alpha ·\omega (v,u)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},\forall v,u\in \mathbb{V}.$$

Theorem 2

([15]). Every strict almost contraction map on a complete metric space possesses a unique fixed point.

Metric fixed-point theory continues to be an exquisite field of study for constructing fixed-point outcomes in relational metric space. The contraction condition that leads to such results needs to be satisfied for only comparative elements with regard to the relation. As of right now, relational contractions continue to be weaker than usual contractions.

In 2015, Alam and Imdad [16] launched this progression by establishing an analog of BCP in relational metric space. Numerous outcomes have been proven in this strategy since then. Alam and Imdad [17] investigated certain coincidence and common fixed-point theorems in relational metric space. Alam et al. [18] defined relational analogs of completeness and continuity and utilized the same to improve the relation-theoretic contraction principle. Almarri et al. [19] proved fixed-point theorems for relational Geraghty contractions and provided an application to boundary value problems. Hossain et al. [20] presented relation-theoretic variants of weak contractions and provided applications to nonlinear matrix equations. Hasanuzzaman and Imdad [21] proved Feng-–Liu type results in relational metric spaces and gave an application to nonlinear Bernstein operators. Choudhury and Chakraborty [22] established some fixed-point results under multi-valued relational Kannan—Geraghty type contractions employing the concept of w-distance. On the other hand, Antal et al. [23] utilized the idea of w-distance to prove fixed-point results under $(\phi ,\psi ,p)$-weakly contractive mappings in relational metric space. Relation-theoretic aspects of almost contractions are studied in [24,25,26,27]. One of the principal features of relational contractions is that the contraction inequality constitutes acceptable just for comparable elements. As of right now, relational contractions continue to be weaker than corresponding usual contractions; consequently, they have the potential to resolve boundary value problems, nonlinear matrix equations, and integral equations, whereas outcomes about the fixed point in ordinary metric space are not implemented.

Dutta and Choudhury [28] introduced yet another type of contractivity condition depending on a pair $\varphi$ and $\psi$ of auxiliary functions, a so-called $(\varphi ,\psi )$-contraction, which is defined as follows:

$$\varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right).$$

By implementing a new pairing of test functions, Alam et al. [29] enhanced the concept of $(\varphi ,\psi )$-contraction and utilized it to expand the BCP. The relational analog of results due to Alam et al. [29] was subsequently achieved by Sk et al. [30].

In this article, we subsume the concepts of relational contraction, strict almost contraction, and $(\varphi ,\psi )$-contraction as follows:

$$\varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},$$

where the elements v and u are connected via relation on metric space.

Similar to the relation-theoretic contraction principle [16], in order to prove their relation-theoretic formulations, a few generalized contractions require an arbitrary binary relation for the existence of fixed points of such map. Apart from this, in the context of earlier contraction-condition, transitivity of underlying relation is also required. However, the transitivity requirement is very restrictive. With a view to employing an optimal condition of transitivity, we adopt ‘locally finitely $\mathbf{f}$-transitivity’, which is relatively weaker than usual transitivity, local transitivity, $\mathbf{f}$-transitivity, and finitely transitivity (c.f. [30,31,32,33,34,35,36]). Employing the above contraction-condition and locally finitely $\mathbf{f}$-transitive relation, we prove metrical fixed-point results and present several examples that verify the validity of our findings. Through our results, we characterize the possibility of a unique solution of certain nonlinear Fredholm integral equations when a lower or an upper solution exists.

2. Preliminaries

The sets of natural numbers, whole numbers, real numbers, and nonnegative real numbers will be symbolized by $\mathbb{N}$, ${\mathbb{N}}_0$, $\mathbb{R}$ and ${\mathbb{R}}_{+}$, respectively. Note that a subset of ${\mathbb{V}}^2$ is designated as a relation on the set $\mathbb{V}$ (Ref. [37]).

Definition 3.

Assuming that $\mathbb{V}$ remains a set, and ℘ is a relation on $\mathbb{V}$.

• Ref. [16] The elements $v,u\in \mathbb{V}$ are named as ℘-comparative if $(v,u)\in \wp$ or $(u,v)\in \wp$. Such a pair is symbolized by $[v,u]\in \wp$.
• Ref. [16] A sequence $\left\{v_{ɩ}\right\}\subset \mathbb{V}$, satisfying $(v_{ɩ},v_{ɩ+1})\in \wp$, $forall$  $ɩ\in \mathbb{N}$, is named as ℘-preserving.
• Ref. [38] $\mathbb{U}\subseteq \mathbb{V}$ is named as ℘-directed if for any $v,u\in \mathbb{U}$, ∃$w\in \mathbb{V}$ enjoying $(v,w)\in \wp$ and $(u,w)\in \wp$.
• Ref. [39] For $\mathbb{U}\subseteq \mathbb{V}$, the relation ${\wp |}_{\mathbb{U}}:=\wp \cap {\mathbb{U}}^2$ on $\mathbb{U}$ is named as restriction of ℘ on $\mathbb{U}$.
• Ref. [35] For $l\in \mathbb{N}-\left\{1\right\}$, ℘ is named as l-transitive if for any $v_0,v_1,\dots ,v_{ɩ}\in \mathbb{V}$,

  $$(v_{i-1},v_i)\in \wp \mathrm{for}\mathrm{each}i(1\le i\le l)\Rightarrow (v_0,v_l)\in \wp .$$
  Thus, the ideas of usual transitivity and 2-transitivity are equivalent.
• Ref. [36] ℘ is named as finitely transitive if for some $l\in \mathbb{N}-\left\{1\right\}$, ℘ remains l-transitive.

Definition 4.

Assuming that $\mathbb{V}$ remains a set, ℘ is a relation on $\mathbb{V}$, and $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ constitutes a mapping.

• Ref. [16] ℘ is named as $\mathbf{f}$-closed if for every $v,u\in \mathbb{V}$ enjoying $(v,u)\in \wp$, one has

  $$(\mathbf{f}v,\mathbf{f}u)\in \wp .$$
• Ref. [31] ℘ is named as locally $\mathit{f}$-transitive if for every ℘-preserving sequence $\left\{u_{ɩ}\right\}\subset \mathit{f}\left(\mathbb{V}\right)$ with range $\mathbb{U}=\{u_{ɩ}:ɩ\in \mathbb{N}\}$, ${\wp |}_{\mathbb{U}}$ is transitive.
• Ref. [32] ℘ is named as locally finitely $\mathit{f}$-transitive if for every ℘-preserving sequence $\left\{u_{ɩ}\right\}\subset \mathit{f}\left(\mathbb{V}\right)$ with range $\mathbb{U}=\{u_{ɩ}:ɩ\in \mathbb{N}\}$, ${\wp |}_{\mathbb{U}}$ is finitely transitive.

Proposition 1

([31]). If ℘ is $\mathbf{f}$-closed then for every $ɩ\in {\mathbb{N}}_0$, ℘ is ${\mathbf{f}}^{ɩ}$-closed.

Remark 1.

The class of finitely transitive relation and the class of locally $\mathit{f}$-transitive relation both are contained in the class of locally finitely $\mathit{f}$-transitive relation.

Definition 5.

Assuming that $(\mathbb{V},\omega )$ remains metric space, and ℘ is a relation on $\mathbb{V}$.

• Ref. [16] ℘ is named as ω-self-closed if every ℘-preserving convergent sequence in $\mathbb{V}$ has a subsequence, each of its terms is ℘-comparative to the limit of convergence.
• Ref. [17] $(\mathbb{V},\omega )$ is named as ℘-complete metric space if each ℘-preserving Cauchy sequence in $\mathbb{V}$ converges.
• Ref. [17] A map $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ is named as ℘-continuous at $v\in \mathbb{V}$ if for each ℘-preserving sequence $\left\{v_{ɩ}\right\}\subset \mathbb{V}$ with $v_{ɩ}\stackrel{\omega }{⟶}v$,

  $$\mathbf{f}\left(v_{ɩ}\right)\stackrel{\omega }{⟶}\mathbf{f}\left(v\right).$$

  A map, which is ℘-continuous at each point, is named as ℘-continuous.

Lemma 1

([35]). In a metric space $(\mathbb{V},\omega )$, let a sequence $\left\{v_{ɩ}\right\}$ be not Cauchy. Then, we can find subsequences $\left\{v_{{ɩ}_{\kappa }}\right\}$ and $\left\{v_{{ȷ}_{\kappa }}\right\}$ of $\left\{v_{ɩ}\right\}$ and a constant ${\epsilon }_0>0$ such that

(i)

$i\le {ɩ}_{\kappa }<{ȷ}_{\kappa },\forall i\in \mathbb{N}$,

(ii)

$\omega (v_{{ɩ}_{\kappa }},v_{{ȷ}_{\kappa }})\ge {\epsilon }_0,\forall ɩ\in \mathbb{N}$,

(iii)

$\omega (v_{{ɩ}_{\kappa }},v_{{\rho }_{\kappa }})<{\epsilon }_0,\forall$${\rho }_{\kappa }\in \{{ɩ}_{\kappa }+1,{ɩ}_{\kappa }+2,\dots ,{ȷ}_{\kappa }-2,{ȷ}_{\kappa }-1\}$.

Moreover, if $\underset{ɩ\to \infty }{lim}\omega (v_{ɩ},v_{ɩ+1})=0$, then

$$\underset{ɩ\to \infty }{lim}\omega (v_{{ɩ}_{\kappa }},v_{{ȷ}_{\kappa }+\lambda })={\epsilon }_0,\forall \lambda \in {\mathbb{N}}_0.$$

Lemma 2

([36]). Let $\mathbb{V}$ be a set that is associated with a relation ℘. If $\left\{v_{ɩ}\right\}\subset \mathbb{V}$ remains ℘-preserving sequence and the relation ℘ is l-transitive on $\mathbb{U}=\{v_{ɩ}:ɩ\in {\mathbb{N}}_0\}$, then

$$(v_{ɩ},v_{ɩ+1+\lambda (l-1)})\in \wp ,\forall ɩ,\lambda \in {\mathbb{N}}_0.$$

Let $\Phi$ be the collection of auxiliary functions $\varphi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ meeting the requirements listed below:

• Φ₁: $\varphi$ is right continuous;
• Φ₂: $\varphi$ is increasing.

Also, let $\Psi$ be the collection of auxiliary functions $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ meeting the requirements listed below:

• Ψ₁: $\psi \left(t\right)>0,\forall t>0$;
• Ψ₂: $\underset{t\to r}{lim\; inf}\psi \left(t\right)>0,\forall r>0$.

The aforementioned families $\Phi$ and $\Psi$ have been described by Alam et al. [29].

Proposition 2

([29]). Let $\varphi ,\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a pair of auxiliary functions such that ϕ satisfies axiom ${\Phi }_2$ and ψ satisfy axiom ${\Psi }_1$, verifying

$$\varphi \left(s\right)\le \varphi \left(t\right)-\psi \left(t\right),\forall s\in {\mathbb{R}}_{+}\mathrm{and}t>0.$$

Then

$$s<t.$$

By symmetry of metric $\omega$, the following conclusion holds.

Proposition 3.

Given $\varphi \in \Phi$, $\psi \in \Psi$ and $\ell \ge 0$, the following axioms are identical:

(A)

$\varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},forallv,u\in \mathbb{V}with(v,u)\in \wp .$

(B)

$\varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},forallv,u\in \mathbb{V}with[v,u]\in \wp .$

3. Main Results

We look into the fixed point results for relational strict almost $(\varphi ,\psi )$-contractions.

Theorem 3.

Assuming that $(\mathbb{V},\omega )$ is a metric space, $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ is a map and ℘ continues to be a relation on $\mathbb{V}$. Furthermore,

(i)

$(v_0,\mathbf{f}{v}_0)\in \wp$ for some $v_0\in \mathbb{V}$,

(ii)

$(\mathbb{V},\omega )$ is ℘-complete,

(iii)

℘ remains locally finitely $\mathbf{f}$-transitive and $\mathbf{f}$-closed,

(iv)

$\mathbb{V}$ is ℘-continuous, or ℘ is ω-self-closed,

(v)

$\exists \varphi \in \Phi$, $\psi \in \Psi$ and $\ell \ge 0$ enjoying

$$\begin{array}{ccc}& & \varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},\hfill \\ & & \forall v,u\in \mathbb{V}with(v,u)\in \wp .\hfill \end{array}$$

Then, $\mathbf{f}$ admits a fixed point.

Proof. 

Given $v_0\in \mathbb{V}$. Construct a sequence $\left\{v_{ɩ}\right\}\subset \mathbb{V}$, enjoying

$$v_{ɩ}:={\mathbf{f}}^{ɩ}\left(v_0\right)=\mathbf{f}\left(v_{ɩ-1}\right),\forall ɩ\in \mathbb{N}.$$

(1)

By assumption (i), $\mathbf{f}$-closedness of ℘ and Proposition 1, we obtain

$$({\mathbf{f}}^{ɩ}{v}_0,{\mathbf{f}}^{ɩ+1}{v}_0)\in \wp ,$$

which owing to (1) reduces to

$$(v_{ɩ},v_{ɩ+1})\in \wp ,\forall ɩ\in {\mathbb{N}}_0.$$

(2)

Hence, $\left\{v_{ɩ}\right\}$ is a ℘-preserving sequence.

Let us denote ${\omega }_{ɩ}:=\omega (v_{ɩ},v_{ɩ+1})$. If ${\omega }_{{ɩ}_0}=\omega (v_{{ɩ}_0},v_{{ɩ}_0+1})=0$ for some ${ɩ}_0\in {\mathbb{N}}_0$, then in lieu of (1), one has $\mathbf{f}\left(v_{{ɩ}_0}\right)=v_{{ɩ}_0}$. Hence, $v_{{ɩ}_0}$ serves as a fixed point of $\mathbf{f}$ and thus our task is over.

In case ${\omega }_{ɩ}>0$, for every $ɩ\in {\mathbb{N}}_0$, using item (v), (1) and (2), we obtain

$$\begin{array}{ccc}\hfill \varphi \left({\omega }_{ɩ}\right)& =& \varphi \left(\omega (v_{ɩ},v_{ɩ+1})\right)=\varphi \left(\omega (\mathbf{f}{v}_{ɩ-1},\mathbf{f}{v}_{ɩ})\right)\hfill \\ & \le & \varphi \left(\omega (v_{ɩ-1},v_{ɩ})\right)-\psi \left(\omega (v_{ɩ-1},v_{ɩ})\right)+\ell ·min\{\omega (v_{ɩ-1},v_{ɩ}),\omega (v_{ɩ},v_{ɩ+1}),\omega (v_{ɩ-1},v_{ɩ+1}),0\},\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill \varphi \left({\omega }_{ɩ}\right)\le \varphi \left({\omega }_{ɩ-1}\right)-\psi \left({\omega }_{ɩ-1}\right)\forall ɩ\in {\mathbb{N}}_0.\end{array}$$

(3)

Using Proposition 2 in (3), we obtain

$${\omega }_{ɩ}<{\omega }_{ɩ-1},\forall ɩ\in \mathbb{N}.$$

It follows that $\left\{{\omega }_{ɩ}\right\}\subset (0,\infty )$ is a decreasing sequence. Further, as $\left\{{\omega }_{ɩ}\right\}$ is bounded below by ‘0’, ∃$\overline{\omega }\ge 0$, verifying

$$\begin{array}{c}\hfill \underset{ɩ\to \infty }{lim}{\omega }_{ɩ}=\overline{\omega }.\end{array}$$

(4)

We shall verify that $\overline{\omega }=0.$ Assuming on the contrary that $\overline{\omega }>0$. Making use of limit superior in (3), we conclude

$$\begin{array}{ccc}\hfill \underset{ɩ\to \infty }{lim\; sup}\varphi \left({\omega }_{ɩ+1}\right)& \le & \underset{ɩ\to \infty }{lim\; sup}\varphi \left({\omega }_{ɩ}\right)+\underset{ɩ\to \infty }{lim\; sup}[-\psi \left({\omega }_{ɩ}\right)]\hfill \\ & \le & \underset{ɩ\to \infty }{lim\; sup}\varphi \left({\omega }_{ɩ}\right)-\underset{ɩ\to \infty }{lim\; inf}\psi \left({\omega }_{ɩ}\right).\hfill \end{array}$$

Employing right continuity of $\varphi$, we obtain

$$\begin{array}{ccc}\hfill \varphi \left(\overline{\omega }\right)& \le & \varphi \left(\overline{\omega }\right)-\underset{ɩ\to \infty }{lim\; inf}\psi \left({\omega }_{ɩ}\right)\hfill \end{array}$$

implying thereby

$$\begin{array}{c}\hfill \underset{{\omega }_{ɩ}\to \overline{\omega }>0}{lim\; inf}\psi \left({\omega }_{ɩ}\right)=\underset{ɩ\to \infty }{lim\; inf}\psi \left({\omega }_{ɩ}\right)\le 0\end{array}$$

which contradicts axiom ${\Psi }_2$ so that $\overline{\omega }=0$. Thus, we have

$$\begin{array}{c}\hfill \underset{ɩ\to \infty }{lim}{\omega }_{ɩ}=0.\end{array}$$

(5)

Now, we shall verify that $\left\{v_{ɩ}\right\}$ is Cauchy. Assuming on the contrary that $\left\{v_{ɩ}\right\}$ is not Cauchy. In lieu of Lemma 1, we can find subsequences $\left\{v_{{ɩ}_{\kappa }}\right\}$ and $\left\{v_{{ȷ}_{\kappa }}\right\}$ of $\left\{v_{ɩ}\right\}$ and a constant ${\epsilon }_0>0$, which satisfy

$$\kappa \le {ɩ}_{\kappa }<{ȷ}_{\kappa },\omega (v_{{ɩ}_{\kappa }},v_{{ȷ}_{\kappa }})\ge {\epsilon }_0>\omega (v_{{ɩ}_{\kappa }},v_{{\rho }_{\kappa }}),\mathrm{for}\mathrm{all}\kappa \in \mathbb{N},{\rho }_{\kappa }\in \{{ɩ}_{\kappa }+1,{ɩ}_{\kappa }+2,\dots ,{ȷ}_{\kappa }-2,{ȷ}_{\kappa }-1\}.$$

Owing to (5) and Lemma 1, we obtain

$$\begin{array}{c}\hfill \underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }},v_{{ȷ}_{\kappa }+\lambda })={\epsilon }_0,\forall \lambda \in {\mathbb{N}}_0.\end{array}$$

(6)

By (1), we have $\mathbb{U}:=\{v_{ɩ}:ɩ\in {\mathbb{N}}_0\}\subset \mathbf{f}\left(\mathbb{V}\right)$. By locally finitely $\mathbf{f}$-transitivity of ℘, we can find $l\in \{2,3,\cdots \}$ for which ${\wp |}_{\mathbb{U}}$ is l-transitive. Employing the fact: ${ɩ}_{\kappa }<{ȷ}_{\kappa }$ and $l-1>0$ and by division algorithm, we obtain

$${ȷ}_{\kappa }-{ɩ}_{\kappa }=(l-1)(p_{\kappa }-1)+(l-q_{\kappa })$$

$$p_{\kappa }-1\ge 0,0\le l-q_{\kappa }<l-1$$

$$⟺\left\{\begin{array}{c}{ȷ}_{\kappa }+q_{\kappa }={ɩ}_{\kappa }+1+p_{\kappa }(l-1)\hfill \\ p_{\kappa }\ge 1,1<q_{\kappa }\le l.\hfill \end{array}\right.$$

Clearly, $q_{\kappa }\in (1,l]$. Thus, we can determine the subsequences $\left\{v_{{ȷ}_{\kappa }}\right\}$ and $\left\{v_{{ɩ}_{\kappa }}\right\}$ of $\left\{v_{ɩ}\right\}$ (verifying (6)) for which $q_{\kappa }=q$ (a constant). Thus, we have

$$\begin{array}{c}\hfill {ɩ}_{\kappa }^{\prime }={ȷ}_{\kappa }+q={ɩ}_{\kappa }+1+p_{\kappa }(l-1).\end{array}$$

(7)

By (6) and (7), we obtain

$$\begin{array}{c}\hfill \underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})=\underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }},v_{{ȷ}_{\kappa }+q})={\epsilon }_0.\end{array}$$

(8)

Use of triangular inequality yields that

$$\omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }^{\prime }+1})\le \omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }})+\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})+\omega (v_{{ɩ}_{\kappa }^{\prime }},v_{{ɩ}_{\kappa }^{\prime }+1})$$

and

$$\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})\le \omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }+1})+\omega (v_{{ɩ}_{\kappa +1}},v_{{ɩ}_{\kappa }^{\prime }+1})+\omega (v_{{ɩ}_{\kappa }^{\prime }+1},v_{{ɩ}_{\kappa }^{\prime }}).$$

Therefore, we have

$$\begin{array}{ll}\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})-\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }+1})-\omega (v_{{ɩ}_{\kappa }^{\prime }+1},v_{{ɩ}_{\kappa }^{\prime }})& \le \omega (v_{{ɩ}_{\kappa +1}},v_{{ɩ}_{\kappa }^{\prime }+1})\\ & \le \omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }})+\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})+\omega (v_{{ɩ}_{\kappa }^{\prime }},v_{{ɩ}_{\kappa }^{\prime }+1}).\end{array}$$

Employing $\kappa \to \infty$ and by (5) and (8), above inequality becomes

$$\begin{array}{c}\hfill \underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }^{\prime }+1})={\epsilon }_0.\end{array}$$

(9)

Owing to (7) and Lemma 2, we conclude $(v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})\in \wp .$ Denote ${\delta }_{\kappa }:=\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})$. Employing the assumption (v), we obtain

$$\begin{array}{ccc}\hfill \varphi \left(\omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }^{\prime }+1})\right)& =& \varphi \left(\omega (\mathbf{f}{v}_{{ɩ}_{\kappa }},\mathbf{f}{v}_{{ɩ}_{\kappa }^{\prime }})\right)\le \varphi \left(\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})\right)-\psi \left(\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }})\right)\hfill \\ & & +\ell ·min\{\omega (v_{{ɩ}_{\kappa }},\mathbf{f}{v}_{{ɩ}_{\kappa }}),\omega (v_{{ɩ}_{\kappa }^{\prime }},\mathbf{f}{v}_{{ɩ}_{\kappa }^{\prime }}),\omega (v_{{ɩ}_{\kappa }},\mathbf{f}{v}_{{ɩ}_{\kappa }^{\prime }}),\omega (v_{{ɩ}_{\kappa }^{\prime }},\mathbf{f}{v}_{{ɩ}_{\kappa }})\}\hfill \end{array}$$

so that

$$\varphi \left(\omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }^{\prime }+1})\right)\le \varphi \left({\delta }_{\kappa }\right)-\psi \left({\delta }_{\kappa }\right)+\ell ·min\{{\omega }_{{ɩ}_{\kappa }},{\omega }_{{ɩ}_{\kappa }^{\prime }},\omega (v_{{ɩ}_{\kappa }},v_{{ɩ}_{\kappa }^{\prime }+1}),\omega (v_{{ɩ}_{\kappa }^{\prime }},v_{{ɩ}_{\kappa }+1})\}.$$

(10)

Using upper limit in (10), we obtain

$$\begin{array}{ccc}\hfill \underset{\kappa \to \infty }{lim\; sup}\varphi \left(\omega (v_{{ɩ}_{\kappa }+1},v_{{ɩ}_{\kappa }^{\prime }+1})\right)& \le & \underset{\kappa \to \infty }{lim\; sup}\varphi \left({\delta }_{\kappa }\right)+\underset{\kappa \to \infty }{lim\; sup}[-\psi \left({\delta }_{\kappa }\right)]+\ell ·min\{0,0,{\epsilon }_0,{\epsilon }_0\}.\hfill \end{array}$$

Due to right continuity of $\varphi$ and (8), we conclude

$$\begin{array}{ccc}\hfill \varphi \left({\epsilon }_0\right)& \le & \varphi \left({\epsilon }_0\right)-\underset{\kappa \to \infty }{lim\; inf}\psi \left({\delta }_{\kappa }\right)\hfill \end{array}$$

yielding thereby

$$\begin{array}{ccc}\hfill \underset{\kappa \to \infty }{lim\; inf}\psi \left({\delta }_{\kappa }\right)& \le & 0,\hfill \end{array}$$

which arises a contradiction. Thus, $\left\{v_{ɩ}\right\}$ is ℘-preserving Cauchy. Using ℘-completeness of $\mathbb{V}$, ∃$\overline{v}\in \mathbb{V}$ verifying $v_{ɩ}\stackrel{\omega }{⟶}\overline{v}$.

Ultimately, we utilize assumption (iv) to enclose the evidence. Assume that $\mathbf{f}$ is a ℘-continuous map. As $\left\{v_{ɩ}\right\}$ remains ℘-preserving verifying $v_{ɩ}\stackrel{\omega }{⟶}\overline{v}$, ℘-continuity of $\mathbf{f}$ yields that $v_{ɩ+1}=\mathbf{f}\left(v_{ɩ}\right)\stackrel{\omega }{⟶}\mathbf{f}\left(\overline{v}\right)$ so that $\mathbf{f}\left(\overline{v}\right)=\overline{v}$.

Otherwise, assuming that ℘ is $\omega$-self-closed. Consequently, we can find a subsequence $\left\{v_{{ɩ}_{\kappa }}\right\}$ of $\left\{v_{ɩ}\right\}$ satisfying $[v_{{ɩ}_{\kappa }},\overline{v}]\in \wp ,\forall ɩ\in \mathbb{N}.$ Now, we claim that

$$\begin{array}{ccc}\hfill \underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }+1},\mathbf{f}\overline{v})& =& 0.\hfill \end{array}$$

(11)

Whenever $v_{{ɩ}_{\kappa }}=\overline{v}$ for some $\kappa \in \mathbb{N}$. Then, we have $v_{{ɩ}_{\kappa }+1}=\mathbf{f}\left(v_{{ɩ}_{\kappa }}\right)=\mathbf{f}\left(\overline{v}\right)$ yielding thereby

$$\underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }+1},\mathbf{f}\overline{v})=0.$$

i.e., (11) holds. Now, we consider $v_{{ɩ}_{\kappa }}\ne \overline{v}$ so that $\omega (v_{{ɩ}_{\kappa }},\overline{v})>0$ for all $\kappa \in \mathbb{N}$. On the contrary, assume that

$$\underset{\kappa \to \infty }{lim}\omega (v_{{ɩ}_{\kappa }+1},\mathbf{f}\overline{v})=ϵ>0.$$

Using assumption (v), we obtain

$$\begin{array}{ccc}\hfill \varphi \left(\omega (v_{{ɩ}_{\kappa }+1},\mathbf{f}\overline{v})\right)& =& \varphi \left(\omega (\mathbf{f}{v}_{{ɩ}_{\kappa }},\mathbf{f}\overline{v})\right)\le \varphi \left(\omega (v_{{ɩ}_{\kappa }},\overline{v})\right)-\psi \left(\omega (v_{{ɩ}_{\kappa }},\overline{v})\right)\hfill \\ & & +\ell ·min\{{\omega }_{{ɩ}_{\kappa }},\omega (\overline{v},\mathbf{f}\overline{v}),\omega (v_{{ɩ}_{\kappa }},\mathbf{f}\overline{v}),\omega (\overline{v},v_{{ɩ}_{\kappa }+1})\}.\hfill \end{array}$$

Using upper limit in above and right continuity of $\varphi$, we obtain

$$\begin{array}{ccc}\hfill \varphi \left(ϵ\right)& \le & \varphi \left(0\right)-\underset{\kappa \to \infty }{lim\; inf}\psi \left(\omega (v_{{ɩ}_{\kappa }},\overline{v})\right)+\ell ·min\{0,\omega (\overline{v},\mathbf{f}\overline{v}),ϵ,0\}\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill \underset{\kappa \to \infty }{lim\; inf}\psi \left(\omega (v_{{ɩ}_{\kappa }},\overline{v})\right)\le \varphi \left(0\right)-\varphi \left(ϵ\right).\end{array}$$

Using the fact $ϵ>0$ and monotone property of $\varphi$, above inequality implies that

$$\begin{array}{c}\hfill \underset{\kappa \to \infty }{lim\; inf}\psi \left(\omega (v_{{ɩ}_{\kappa }},\overline{v})\right)\le 0,\end{array}$$

which is a contradiction. Therefore, (11) holds and hence, we have

$$v_{{ɩ}_{\kappa }+1}\stackrel{\omega }{⟶}\mathbf{f}\left(\overline{v}\right).$$

This concludes that $\mathbf{f}\left(\overline{v}\right)=\overline{v}$. Thus, $\overline{v}$ is a fixed point of $\mathbf{f}$. □

Theorem 4.

In keeping with the predictions of Theorem 3, $\mathbf{f}$ exhibits a unique fixed point if $\mathbf{f}\left(\mathbb{V}\right)$ is ℘-directed.

Proof. 

By Theorem 3, ∃$\overline{v},\overline{u}\in \mathbb{V}$ which enjoys

$$\mathbf{f}\left(\overline{v}\right)=\overline{v}\mathrm{and}\mathbf{f}\left(\overline{u}\right)=\overline{u}.$$

(12)

As $\overline{v},\overline{u}\in \mathbf{f}\left(\mathbb{V}\right)$, by our assumption, $\exists w\in \mathbb{V}$ verifying

$$(\overline{v},w)\in \wp \mathrm{and}(\overline{u},w)\in \wp .$$

(13)

Denote ${\varrho }_{ɩ}:=\omega (\overline{v},{\mathbf{f}}^{ɩ}w)$. Using (12), (13) and assumption (v), one obtains

$$\begin{array}{ccc}\hfill \varphi \left({\varrho }_{ɩ}\right)& =& \varphi \left(\omega (\mathbf{f}\overline{v},\mathbf{f}\left({\mathbf{f}}^{ɩ-1}w\right))\right)\hfill \\ & \le & \varphi \left(\omega (\overline{v},{\mathbf{f}}^{ɩ-1}w)\right)-\psi \left(\omega (\overline{v},{\mathbf{f}}^{ɩ-1}w)\right)+\ell ·min\{0,\omega ({\mathbf{f}}^{ɩ-1}w,{\mathbf{f}}^{ɩ}w),\omega (\overline{v},{\mathbf{f}}^{ɩ}w),\omega ({\mathbf{f}}^{ɩ-1}w,\overline{v})\}\hfill \\ & =& \varphi \left({\varrho }_{ɩ-1}\right)-\psi \left({\varrho }_{ɩ-1}\right)\hfill \end{array}$$

so that

$$\varphi \left({\varrho }_{ɩ}\right)\le \varphi \left({\varrho }_{ɩ-1}\right)-\psi \left({\varrho }_{ɩ-1}\right).$$

(14)

If $\exists {ɩ}_0\in \mathbb{N}$ for which ${\varrho }_{{ɩ}_0}=0$, then we have ${\varrho }_{{ɩ}_0}\le {\varrho }_{{ɩ}_0-1}$. Otherwise ${\varrho }_{ɩ}>0,\forall ɩ\in \mathbb{N}$. By Proposition 2, (14) reduces to ${\varrho }_{ɩ}<{\varrho }_{ɩ-1}$. Hence, in both cases, we have

$${\varrho }_{ɩ}\le {\varrho }_{ɩ-1}.$$

By applying reasoning similar to Theorem 3, above inequality becomes

$${\displaystyle \underset{ɩ\to \infty }{lim}{\varrho }_{ɩ}=\underset{ɩ\to \infty }{lim}\omega (\overline{v},{\mathbf{f}}^{ɩ}w)=0.}$$

(15)

Similarly, one can find

$${\displaystyle \underset{ɩ\to \infty }{lim}\omega (\overline{u},{\mathbf{f}}^{ɩ}w)=0.}$$

(16)

By using (15), (16) and triangular inequality, we conclude

$$\omega (\overline{v},\overline{u})=\omega (\overline{v},{\mathbf{f}}^{ɩ}w)+\omega ({\mathbf{f}}^{ɩ}w,\overline{u})\to 0\mathrm{as}ɩ\to \infty$$

implying thereby $\overline{v}=\overline{u}$. Therefore, $\mathbf{f}$ possesses a unique fixed point. □

Under full relation $\wp ={\mathbb{V}}^2$, Theorem 4 reduces to the following result in abstract metric space.

Corollary 1.

Assuming that $(\mathbb{V},\omega )$ remains a complete metric space and $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ is a map. If there exist $\varphi \in \Phi$, $\psi \in \Psi$ and $\ell \ge 0$, enjoying

$$\begin{array}{ccc}& & \varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},\hfill \\ & & forallv,u\in \mathbb{V},\hfill \end{array}$$

then $\mathbf{f}$ possesses a unique fixed point.

In particular, for $\varphi \left(t\right)=t$ and $\psi \left(t\right)=(1-\alpha )t$ where $0<\alpha <1$, Corollary 1 reduces to Theorem 2.

Setting $\ell =0$ in Theorem 3, we deduce the following outcome of Sk et al. [30].

Corollary 2

([30]). Assuming that $(\mathbb{V},\omega )$ is a metric space, $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ is a map and ℘ continues to be a relation on $\mathbb{V}$. Furthermore,

(i)

$(v_0,\mathbf{f}{v}_0)\in \wp$ for some $v_0\in \mathbb{V}$,

(ii)

$(\mathbb{V},\omega )$ is ℘-complete,

(iii)

℘ remains locally finitely $\mathbf{f}$-transitive and $\mathbf{f}$-closed,

(iv)

$\mathbb{V}$ is ℘-continuous, or ℘ is ω-self-closed,

(v)

$\exists \varphi \in \Phi$ and $\psi \in \Psi$ enjoying

$$\begin{array}{c}\hfill \varphi \left(\omega \right(\mathbf{f}v,\mathbf{f}u\left)\right)\le \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right),\forall v,u\in \mathbb{V}with(v,u)\in \wp .\end{array}$$

Then, $\mathbf{f}$ admits a fixed point.

Let $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a function verifying $\phi \left(t\right)<t,\mathrm{for}\mathrm{all}t>0$ and $\underset{s\to t^{+}}{lim\; sup}\phi \left(s\right)<t,\mathrm{for}\mathrm{all}t>0$. Taking $\varphi \left(t\right)=t$ and $\psi \left(t\right)=t-\phi \left(t\right)$ in Theorem 3, we deduce the following outcome of Alharbi and Khan [26].

Corollary 3

([26]). Assuming that $(\mathbb{V},\omega )$ is a metric space, $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ is a map and ℘ continues to be a relation on $\mathbb{V}$. Furthermore,

(i)

$(v_0,\mathbf{f}{v}_0)\in \wp$ for some $v_0\in \mathbb{V}$,

(ii)

$(\mathbb{V},\omega )$ is ℘-complete,

(iii)

℘ remains locally $\mathbf{f}$-transitive and $\mathbf{f}$-closed,

(iv)

$\mathbb{V}$ is ℘-continuous, or ℘ is ω-self-closed,

(v)

∃ a function $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, verifying $\phi \left(t\right)<t,forallt>0$ and $\underset{s\to t^{+}}{lim\; sup}\phi \left(s\right)<t,forallt>0$, and $\ell \ge 0$, enjoying

$$\begin{array}{ccc}& & \omega (\mathbf{f}v,\mathbf{f}u)\le \phi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\},\hfill \\ & & \forall v,u\in \mathbb{V}with(v,u)\in \wp .\hfill \end{array}$$

Then, $\mathbf{f}$ admits a fixed point.

4. Examples

The following examples are provided for evidence of the results established in the preceding section.

Example 1.

Consider $\mathbb{V}:={\mathbb{R}}^2$ with the metric ω defined by

$$\omega ((v,u),(w,z))={\displaystyle \frac{|v-w|+|u-z|}{2}}\forall (v,u),(w,z)\in \mathbb{V}.$$

On $\mathbb{V}$, take a relation ℘ given by

$$\wp =\{((v,u),(w,z))\in {\mathbb{V}}^2:w-v\ge 0,u-z\ge 0\}.$$

Then $(\mathbb{V},\omega )$ is ℘-complete metric space. Define a map $\mathit{f}:\mathbb{V}\to \mathbb{V}$ as

$$\mathit{f}(v,u)=\left({\displaystyle \frac{v-2u}{4}},{\displaystyle \frac{u-2v}{4}}\right),\forall (v,u)\in \mathbb{V}.$$

Then, ℘ is $\mathit{f}$-closed as well as ω-self-closed.

Define

$$\varphi \left(t\right)={\displaystyle \frac{t}{2}},\psi \left(t\right)={\displaystyle \frac{t}{16}}.$$

Then $\varphi \in \Phi$ and $\psi \in \Psi$. Take $(v,u),(w,z)\in \mathbb{V}$ verifying $\left((v,u),(w,z)\right)\in \wp$. One has

$$\begin{array}{ccc}\hfill \varphi \left(\omega \right(\mathit{f}(v,u),\mathit{f}(w,z)\left)\right)& =& {\displaystyle \frac{\omega \left(\mathit{f}\right(v,u),\mathit{f}(w,z\left)\right)}{2}}\hfill \\ & =& {\displaystyle \frac{1}{8}}\left(\right|(v-w)+2(z-u)|+|(z-u)+2(v-w)\left|\right)\hfill \\ & =& {\displaystyle \frac{3}{16}}(v-w+z-u),\hfill \end{array}$$

i.e.,

$$\varphi \left(\omega (\mathit{f}(v,u),\mathit{f}(w,z))\right)={\displaystyle \frac{3}{16}}(v-w+z-u).$$

(17)

Additionally,

$$\begin{array}{ccc}\hfill \varphi \left(\omega \right((v,u),(w,z)\left)\right)-\psi \left(\omega \right((v,u),(w,z)\left)\right)& =& {\displaystyle \frac{\omega \left(\right(v,u),(w,z\left)\right)}{2}}-{\displaystyle \frac{\omega \left(\right(v,u),(w,z\left)\right)}{16}}\hfill \\ & =& {\displaystyle \frac{7}{16}}\omega ((v,u),(w,z))\hfill \\ & =& {\displaystyle \frac{7}{32}}(v-w+z-u),\hfill \end{array}$$

i.e.,

$$\varphi \left(\omega ((v,u),(w,z))\right)-\psi \left(\omega ((v,u),(w,z))\right)={\displaystyle \frac{7}{32}}(v-w+z-u).$$

(18)

From (17) and (18), one obtains

$$\begin{array}{ccc}\hfill \varphi \left(\omega \right(\mathit{f}(v,u),\mathit{f}(w,z)\left)\right)& \le & \varphi \left(\omega \right((v,u),(w,z)\left)\right)-\psi \left(\omega \right((v,u),(w,z)\left)\right)+\ell ·min\left\{\omega \right((v,u),\mathit{f}(v,u)),\hfill \\ & & \omega \left(\right(w,z),\mathit{f}(w,z\left)\right),\omega \left(\right(v,u),\mathit{f}(w,z\left)\right),\omega \left(\right(w,z),\mathit{f}(v,u\left)\right)\},\hfill \end{array}$$

where $\ell \ge 0$ is arbitrary. Thus, the contractivity condition (v) holds. Further, here $\mathit{f}\left(\mathbb{V}\right)$ is also ℘-directed and hence by Theorem 4, $\mathit{f}$ enjoys a unique fixed point $v=(0,0).$

Example 2.

Take $\mathbb{V}:=(-1,1]$ with the usual metric ω. On $\mathbb{V}$, define a binary relation ℘ by

$$\wp =\{(v,u)\in {\mathbb{V}}^2:v>u\ge 0\}.$$

Then $(\mathbb{V},\omega )$ is a ℘-complete metric space.

Define

$$\varphi \left(t\right)=\left\{\begin{array}{cc}t,\hfill & if0\le t\le 1\hfill \\ t^2,\hfill & ift>1\hfill \end{array}\right.\mathit{and}\psi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{t^2}{2}},\hfill & if0\le t\le 1,\hfill \\ 4,\hfill & ift>1.\hfill \end{array}\right.$$

Then, $\varphi \in \Phi$ and $\psi \in \Psi$. Define a map $\mathit{f}:\mathbb{V}\to \mathbb{V}$ as

$$\mathit{f}\left(v\right)=\left\{\begin{array}{cc}v+1,\hfill & \mathit{if}-1<v<0,\hfill \\ v-{\displaystyle \frac{v^2}{2}},\hfill & \mathit{if}0\le v\le 1.\hfill \end{array}\right.$$

Take $v,u\in \mathbb{V}$ with $(v,u)\in \wp$, then $v>u\ge 0$. Thus, we have

$$\begin{array}{ccc}\hfill \varphi \left(\omega \right(\mathit{f}v,\mathit{f}u\left)\right)& =& (v-{\displaystyle \frac{1}{2}}{v}^2)-(u-{\displaystyle \frac{1}{2}}{u}^2)\hfill \\ & =& (v-u)-{\displaystyle \frac{1}{2}}(v-u)(v+u)\le (v-u)-{\displaystyle \frac{1}{2}}{(v-u)}^2\hfill \\ & \le & \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\}.\hfill \end{array}$$

Therefore, $\mathit{f}$ verifies condition (v) of Theorem 3. Here ℘ is locally finitely $\mathit{f}$-transitive and $\mathit{f}$-closed. Left over the predictions of Theorems 3 and 4 are satisfied and $\mathit{f}$ enjoys a unique fixed point $v=0$.

Example 3.

Let $\mathbf{V}=[0,1)$ with Euclidean metric ω. Let $\mathit{f}:\mathbf{V}\to \mathbf{V}$ be a map defined by

$$\mathit{f}\left(v\right)=\left\{\begin{array}{c}{\displaystyle \frac{v}{2}},\mathrm{if}v\in \mathbb{Q}\cap \mathbf{V}\hfill \\ 0,\mathrm{if}v\in {\mathbb{Q}}^c\cap \mathbf{V}.\hfill \end{array}\right.$$

On $\mathbb{V}$, define a binary relation ℘ by

$$\wp =\{(v,u)\in {\mathbb{V}}^2:vu\in \{v,u\}.$$

Then, ℘ is locally finitely $\mathit{f}$-transitive, $\mathit{f}$-closed as well as ω-self-closed.

Define

$$\varphi \left(t\right)=t,\psi \left(t\right)={\displaystyle \frac{t}{2}}.$$

Then, $\varphi \in \Phi$ and $\psi \in \Psi$. Here, $\mathit{f}$ verifies condition (v) of Theorem 3 for $\ell =2$. Left over, the predictions of Theorems 3 and 4 are satisfied and $\mathit{f}$ enjoys a unique fixed point $\overline{v}=0$.

Remark 2.

The involved map in Example 3 is not φ-contraction as in particular for the pair $v={\displaystyle \frac{1}{2}}$ and $u={\displaystyle \frac{1}{\sqrt{2}}}$, we have

$$\varphi \left(\omega (\mathbf{f}v,\mathbf{f}u)\right)={\displaystyle \frac{1}{4}}$$

and

$$\begin{array}{ccc}& & \varphi \left(\omega \right(v,u\left)\right)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right\}\hfill \\ & & =\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\sqrt{2}}}\right)-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\sqrt{2}}}\right)+2·{\displaystyle \frac{1}{4}}={\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{2\sqrt{2}}}={\displaystyle \frac{3-\sqrt{2}}{4}}>{\displaystyle \frac{1}{4}}.\hfill \end{array}$$

Thus, Example 3 cannot work in the context of ordinary metric space, which substantiates the utility of fixed-point outcomes in relational metric space over the corresponding outcomes in ordinary metric space.

5. An Application

Consider the nonlinear Fredholm integral equation of the form:

$$v\left(s\right)=\theta \left(s\right)+{\int }_0^{1}M(s,\tau )ϝ(\tau ,v\left(\tau \right))d\tau ,s\in [0,1].$$

(19)

Here $\theta :I\to \mathbb{R}$, $ϝ:I\times \mathbb{R}\to \mathbb{R}$ and $M:I^2\to \mathbb{R}$ remain functions, where $I:=[0,1]$. As usual, $\mathrm{C}\left(I\right)$ indicates the family of real continuous maps on I.

Definition 6.

$\eta \in \mathcal{C}\left(I\right)$ is known as a lower solution of (19) if

$$\eta \left(s\right)\le \theta \left(s\right)+{\int }_0^{1}M(s,\tau )ϝ(\tau ,\eta \left(\tau \right))d\tau ,\forall s\in I.$$

Definition 7.

$\mu \in \mathcal{C}\left(I\right)$ is known as an upper solution of (19) if

$$\mu \left(s\right)\ge \theta \left(s\right)+{\int }_0^{1}M(s,\tau )ϝ(\tau ,\mu \left(\tau \right))d\tau ,\forall s\in I.$$

$\Omega$ shall stand for the family of functions $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that

(i)

$\phi$ is increasing;

(ii)

∃$\psi \in \Psi$ such that $\phi \left(t\right)=t-\psi \left(t\right)$, for all $t\in {\mathbb{R}}_{+}.$

Theorem 5.

In conjunction with Problem (19), assuming that

(I)

θ, ϝ and M are continuous,

(II)

$M(s,\tau )>0,\forall s,\tau \in I$,

(III)

∃$\phi \in \Omega$, verifying

$$0\le ϝ(s,a)-ϝ(s,b)\le \phi (a-b),\forall s\in Iand\forall a,b\in \mathbb{R}witha\ge b,$$

(IV)

$\underset{s\in I}{sup}{\int }_0^{1}M(s,\tau )d\tau =1$.

Furthermore, the problem has a unique solution if (19) has a lower solution.

Proof. 

On $\mathbb{V}:=\mathcal{C}\left(I\right)$, define a metric $\omega$ by

$$\omega (v,u)=\underset{s\in I}{sup}|v\left(s\right)-u\left(s\right)|,\forall v,u\in \mathbb{V}.$$

(20)

On $\mathbb{V}$, define a relation ℘ by

$$\wp =\{(v,u)\in {\mathbb{V}}^2:v\left(s\right)\le u\left(s\right),\forall s\in I\}.$$

(21)

Take a map $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ defined by

$$\left(\mathbf{f}v\right)\left(s\right)=\theta \left(s\right)+{\int }_0^{1}M(s,\tau )ϝ(\tau ,v\left(\tau \right))d\tau ,\forall s\in \mathbb{V}.$$

(22)

We are going to ensure all of the assertions of Theorems 3 and 4.

(i) If $\eta \in \mathbb{V}$ is a lower solution of (19), then

$$\eta \left(s\right)\le \theta \left(s\right)+{\int }_0^{1}M(s,\tau )ϝ(\tau ,\eta \left(\tau \right))d\tau =\left(\mathbf{f}\eta \right)\left(s\right)$$

so that $(\eta ,\mathbf{f}\eta )\in \wp$.

(ii) $(\mathbb{V},\omega )$ being a complete metric space is ℘-complete.

(iii) Take $v,u\in \mathbb{V}$ verifying $(v,u)\in \wp$. Using assumption (III), we obtain

$$ϝ(s,v(\tau \left)\right)-ϝ(s,u(\tau \left)\right)\le 0,\forall s,\tau \in I.$$

(23)

Making use of (22), (23) and condition (II), we find

$$\begin{array}{c}\hfill \left(\mathbf{f}v\right)\left(s\right)-\left(\mathbf{f}u\right)\left(s\right)={\int }_0^{1}M(s,\tau )[ϝ(\tau ,v\left(\tau \right))-ϝ(\tau ,u\left(\tau \right))]d\tau \le 0,\end{array}$$

so that $\left(\mathbf{f}v\right)\left(s\right)\le \left(\mathbf{f}u\right)\left(s\right)$, which, using (21), yields that $(\mathbf{f}v,\mathbf{f}u)\in \wp$ and hence ℘ remains $\mathbf{f}$-closed. Also, ℘ is locally finitely $\mathbf{f}$-transitive.

(iv) Let $\left\{v_{ɩ}\right\}\subset \mathbb{V}$ be a ℘-preserving sequence such that $v_{ɩ}\stackrel{\omega }{⟶}\varpi \in \mathbb{V}$. Then for every $s\in I$, $\left\{v_{ɩ}\left(s\right)\right\}$ is an increasing real sequence such that $v_{ɩ}\left(s\right)\stackrel{\mathbb{R}}{⟶}\varpi \left(s\right)$. This yields that $v_{ɩ}\left(s\right)\le \varpi \left(s\right),\forall ɩ\in \mathbb{N}$ and $\forall s\in I$ so that $(v_{ɩ},\varpi )\in \wp ,\forall ɩ\in \mathbb{N}$. Thus, ℘ is $\omega$-self-closed.

(v) Take $v,u\in \mathbb{V}$ with $(v,u)\in \wp$. By (III), (20) and (22), we conclude

$$\begin{array}{ccc}\hfill \omega (\mathbf{f}v,\mathbf{f}u)& =& \underset{s\in I}{sup}|\left(\mathbf{f}v\right)\left(s\right)-\left(\mathbf{f}u\right)\left(s\right)|=\underset{s\in I}{sup}[\left(\mathbf{f}u\right)\left(s\right)-\left(\mathbf{f}v\right)\left(s\right)]\hfill \\ & =& \underset{s\in I}{sup}{\int }_0^{1}M(s,\tau )[ϝ(\tau ,u\left(\tau \right))-ϝ(\tau ,v\left(\tau \right))]d\tau \hfill \\ & \le & \underset{s\in I}{sup}{\int }_0^{1}M(s,\tau )\phi (u\left(\tau \right)-v\left(\tau \right))d\tau .\hfill \end{array}$$

(24)

As $\phi$ is increasing and $0\le u\left(\tau \right)-v\left(\tau \right)\le \omega (v,u)$, we obtain $\phi \left(u\right(\tau )-v(\tau \left)\right)\le \phi \left(\omega \right(v,u\left)\right)$ and hence (24) reduces to

$$\begin{array}{ccc}\hfill \omega (\mathbf{f}v,\mathbf{f}u)& \le & \phi \left(\omega (v,u)\right)\underset{s\in I}{sup}{\int }_0^{1}M(s,\tau )d\tau =\phi \left(\omega (v,u)\right)\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill \omega (\mathbf{f}v,\mathbf{f}u)& \le & \omega (v,u)-\psi \left(\omega \right(v,u\left)\right)+\ell ·min\left\{\omega \right(v,\mathbf{f}v),\omega (u,\mathbf{f}u),\omega (v,\mathbf{f}u),\omega (u,\mathbf{f}v\left)\right)\},\hfill \\ & \forall & v,u\in \mathbb{V}\mathrm{such}\mathrm{that}(v,u)\in \wp ,\hfill \end{array}$$

where $\ell \ge 0$ is arbitrary. Now take $v,u\in \mathbb{V}$ arbitrary. Set $w:=max\{\mathbf{f}v,\mathbf{f}u\}\in \mathbb{V}$, then we have $(\mathbf{f}v,w)\in \wp$ and $(\mathbf{f}u,w)\in \wp$. Hence, $\mathbf{f}\left(\mathbb{V}\right)$ is ℘-directed. Consequently, by Theorem 4, $\mathbf{f}$ admits a unique fixed point, which in lieu of (22) remains a unique solution of (19). □

Theorem 6.

In conjunction with Problem (I)–(IV) of Theorem 5, the problem (19) admits a unique solution if the problem has an upper solution.

Proof. 

Consider a metric $\omega$ on $\mathbb{V}:=\mathcal{C}\left(I\right)$ and a map $\mathbf{f}:\mathbb{V}\to \mathbb{V}$ like as the proof of Theorem 5. Take a relation ${\wp }^{\prime }$ on $\mathbb{V}$ as:

$${\wp }^{\prime }=\{(v,u)\in {\mathbb{V}}^2:v\left(s\right)\ge u\left(s\right),\forall s\in I\}.$$

(25)

If $\mu \in \mathbb{V}$ is an upper solution of (19), then we have

$$\mu \left(s\right)\ge \theta \left(s\right)+{\int }_0^{1}M(s,\tau )ϝ(\tau ,\mu \left(\tau \right))d\tau =\left(\mathbf{f}\mu \right)\left(s\right)$$

implying thereby $(\mu ,\mathbf{f}\mu )\in {\wp }^{\prime }$.

Take $v,u\in \mathbb{V}$, verifying $(v,u)\in {\wp }^{\prime }$. Using assumption (III), we obtain

$$ϝ(s,v(\tau \left)\right)-ϝ(s,u(\tau \left)\right)\ge 0,\forall s,\tau \in I.$$

(26)

By (22), (26) and assumption (II), we obtain

$$\begin{array}{c}\hfill \left(\mathbf{f}v\right)\left(s\right)-\left(\mathbf{f}u\right)\left(s\right)={\int }_0^{1}M(s,\tau )[ϝ(\tau ,v\left(\tau \right))-ϝ(\tau ,u\left(\tau \right))]d\tau \ge 0,\end{array}$$

so that $\left(\mathbf{f}v\right)\left(s\right)\ge \left(\mathbf{f}u\right)\left(s\right)$, which, using (25), yields that $(\mathbf{f}v,\mathbf{f}u)\in {\wp }^{\prime }$ and hence ${\wp }^{\prime }$ is $\mathbf{f}$-closed.

Let $\left\{v_{ɩ}\right\}\subset \mathbb{V}$ be a ${\wp }^{\prime }$-preserving sequence such that $v_{ɩ}\stackrel{\omega }{⟶}\varpi \in \mathbb{V}$. Then for every $s\in I$, $\left\{v_{ɩ}\left(s\right)\right\}$ is a decreasing real sequence such that $v_{ɩ}\left(s\right)\stackrel{\mathbb{R}}{⟶}\varpi \left(s\right)$. This implies that $v_{ɩ}\left(s\right)\ge \varpi \left(s\right),\forall ɩ\in \mathbb{N}$ and $\forall s\in I$ so that $(v_{ɩ},\varpi )\in {\wp }^{\prime },\forall ɩ\in \mathbb{N}$. Therefore, ${\wp }^{\prime }$ remains $\omega$-self-closed.

Therefore, all the assumptions of Theorems 3 and 4 are verified for the metric space $(\mathbb{V},\omega )$, the map $\mathbf{f}$ and the relation ${\wp }^{\prime }$. This concludes the proof. □

Intending to illustrate Theorem 5, one considers the following example.

Example 4.

Consider the integral equation of the form (19), whereas $\theta \left(s\right)=2(1-2s^2)$, $ϝ(\tau ,\xi )={\displaystyle \frac{1}{3}}\xi$, and $M(s,\tau )=2s\tau$. Define a function $\phi :[0,\infty )\to [0,\infty )$ by $\phi \left(t\right)={\displaystyle \frac{2}{3}}t$. Obviously, assumptions $\left(\mathrm{I}\right)$–$\left(\mathrm{IV}\right)$ of Theorem 5 is satisfied. Moreover, $\theta =0$ forms a lower solution for the present problem. Therefore, Theorem 5 can be applied to the given problem, and hence, $v\left(s\right)=2(1-2s^2)$ forms the unique solution of the integral equation.

6. Conclusions

In this work, we investigated fixed-point outcomes for a strict almost $(\varphi ,\psi )$-contraction map in the relational metric space. The underlying relation in our findings being locally finitely $\mathbf{f}$-transitive is restrictive, but the class of functional contraction is weakened. The findings investigated herewith enrich, improve, and unify several known findings, especially due to Babu et al. [15], Alharbi and Khan [26], Sk et al. [30], and similar others. Several examples are also attempted to convey our outcomes. Our outcomes are applied to compute a unique positive solution of a specific nonlinear Fredholm integral equation where the existence of a unique solution is ensured by the presence of an upper or a lower solution. In the foreseeable future, researchers might extend our findings to a pair of maps or more general distance spaces.