Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations

Abstract

This article consists of some new fixed point theorems verifying relation-theoretic strict almost nonlinear contractions that are preserved by a locally finitely transitive relation. Our results improve certain well-known results in the literature. To confirm the reliability of our results, some examples are delivered. We carry out our findings to solve a nonlinear Fredholm integral equation.

1. Introduction

In fact, the Banach contraction principle (BCP) is one of the most celebrated fixed point theorems ever proved in Mathematics and still remains an inspiration to the workers of metric fixed point theory. As an instance of recent developments in BCP, we merely refer the work of Younis et al. [1]. Alam and Imdad [2] explored a new variant of BCP in the framework of relation metric space. Because of creativity, the results of Alam and Imdad [2] have been expanded and developed by numerous researchers, e.g., [3,4,5]. The contraction that occurs in these outcomes needs to merely be satisfied for comparative elements. It turns out that the class of relational contractions continues to be larger than the class of usual contractions.

Berinde [6] extended the BCP by initiating the concept of almost contraction, which is further generalized by various authors. Indeed, the almost contraction map does not possess a unique fixed point but a sequence of Picard’s iterations of such a map converges to a fixed point. Babu et al. [7] investigated the corresponding uniqueness theorem by initiating a restricted class of almost contraction as follows:

Definition 1

([7]). A map $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$, where $(\mathrm{{\rm Y}},\varsigma )$ remains a metric space, is named as a strict almost contraction if there exists $\alpha \in (0,1)$ and $\lambda \in [0,\infty )$ such that

$$\varsigma (\varphi y,\varphi z)\le \alpha ·\varsigma (y,z)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\},forally,z\in \mathrm{{\rm Y}}.$$

Theorem 1

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

Through the use of a class of local finitely transitive relations, Alam et al. [5] presented a couple of fixed point results under a relational nonlinear contraction. Their choice of control functions was the following family:

$$\mathrm{\Sigma }=\{\mathsf{ϝ}:[0,\infty )\to [0,\infty ):t\in (0,\infty )\Rightarrow t>\mathsf{ϝ}\left(t\right)\mathrm{and}t>\underset{r\to t}{lim\; sup}\mathsf{ϝ}\left(r\right)\}.$$

The manuscript’s purpose is to encompass the concepts involved in Alam et al. [5] and Babu et al. [7] by proving the fixed point results under relational strict almost nonlinear contraction using a locally finitely $\varphi$-transitive relation. Our results are illustrated via certain examples. Our results allow us to come across the unique solution of a specialized nonlinear Fredholm integral equation.

2. Preliminaries

As usual, $\mathbb{N}$, ${\mathbb{N}}_0$ and $\mathbb{R}$ will symbolize the set of natural numbers, whole numbers and real numbers. A subset of ${\mathrm{{\rm Y}}}^2$ is termed as a relation on the set $\mathrm{{\rm Y}}$. In the sequel, let $\mathrm{{\rm Y}}$ be ambient set, $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ a map, £ a relation on $\mathrm{{\rm Y}}$ and $\varsigma$ a metric on $\mathrm{{\rm Y}}$. We say that:

Definition 2

([8]). The relation ${\pounds }^{-1}:=\{(y,z)\in {\mathrm{{\rm Y}}}^2:(z,y)\in \pounds \}$ is inverse of £.

Definition 3

([8]). The symmetric relation ${\pounds }^s:=\pounds \cup {\pounds }^{-1}$ is symmetric closure of £.

Definition 4

([2]). Elements $y,z\in \mathrm{{\rm Y}}$ are £-comparative if $(y,z)\in \pounds$ or $(z,y)\in \pounds$. Such a pair is usually denoted by $[y,z]\in \pounds$. Obviously, $(y,z)\in {\pounds }^s⟺[y,z]\in \pounds .$

Definition 5

([2]). £ is ϕ-closed if for every $y,z\in \mathrm{{\rm Y}}$ with $(y,z)\in \pounds$, we have

$$(\varphi y,\varphi z)\in \pounds .$$

Definition 6

([2]). $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ is a £-preserving sequence if $(y_{\eta },y_{\eta +1})\in \pounds$ for all $\eta \in \mathbb{N}$.

Definition 7

([3]). $(\mathrm{{\rm Y}},\varsigma )$ is £-complete metric space if all the £-preserving Cauchy sequences in $\mathrm{{\rm Y}}$ converge.

Definition 8

([3]). ϕ is £-continuous at $y\in \mathrm{{\rm Y}}$ if for each £-preserving sequence $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ with $y_{\eta }\stackrel{\varsigma }{⟶}y$,

$$\varphi \left(y_{\eta }\right)\stackrel{\varsigma }{⟶}\varphi \left(y\right).$$

Definition 9

([3]). ϕ is £-continuous if it remains £-continuous function at every point.

Definition 10

([2]). £ is ς-self-closed if every £-preserving convergent sequence $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ verifying $y_{\eta }\stackrel{\varsigma }{⟶}y\in \mathrm{{\rm Y}}$ admits a subsequence $\left\{y_{{\eta }_k}\right\}$ with $[y_{{\eta }_k},y]\in \pounds$.

Definition 11

([9]). Given $\mathcal{Ƶ}\subseteq \mathrm{{\rm Y}}$, the set ${\pounds |}_{Ƶ}:=\pounds \cap {Ƶ}^2$ being a relation on $Ƶ$ is restriction of £ on $Ƶ$.

Definition 12

([4]). £ is locally ϕ-transitive if for each £-preserving sequence $\left\{z_{\eta }\right\}\subset \varphi \left(\mathrm{{\rm Y}}\right)$ with range $Ƶ=\{z_{\eta }:\eta \in \mathbb{N}\}$, ${\pounds |}_{Ƶ}$ is transitive.

Definition 13

([10]). Given $n\in \mathbb{N}-\left\{1\right\}$, £ is n-transitive if for every $y_0,y_1,\dots ,y_n\in \mathrm{{\rm Y}}$,

$$(y_{i-1},y_i)\in \pounds \mathrm{for}\mathrm{each}i(1\le i\le n)\Rightarrow (y_0,y_n)\in \pounds .$$

Thus far, 2-transitivity coincides with transitivity.

Definition 14

([11]). £ is finitely transitive if there exists $n\in \mathbb{N}-\left\{1\right\}$ such that £ is n-transitive.

Definition 15

([5]). £ is locally finitely ϕ-transitive if for each £-preserving sequence $\left\{z_{\eta }\right\}\subset \varphi \left(\mathrm{{\rm Y}}\right)$ with range $Ƶ=\{z_{\eta }:\eta \in \mathbb{N}\}$, ${\pounds |}_{Ƶ}$ is finitely transitive.

Remark 1.

The following implications are straightforward:

finitely transitivity⟹ locally finitely ϕ-transitivity;

locally ϕ-transitivity⟹ locally finitely ϕ-transitivity.

Definition 16

([12]). $Ƶ\subseteq \mathrm{{\rm Y}}$ is termed as £-directed if for every $y,z\in Ƶ$, there exists $w\in \mathrm{{\rm Y}}$ which satisfies $(y,w)\in \pounds$ and $(z,w)\in \pounds$.

Now, we indicate the following result of Alam et al. [5].

Theorem 2

([5]). Consider $(\mathrm{{\rm Y}},\varsigma )$ serves as a metric space, $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ as a map, and £ as a relation on $\mathrm{{\rm Y}}$. Also,

(i)

There exists $y_0\in \mathrm{{\rm Y}}$ satisfying $(y_0,\varphi y_0)\in \pounds$;

(ii)

£ is locally finitely ϕ-transitive and ϕ-closed;

(iii)

$(\mathrm{{\rm Y}},\varsigma )$ remains £-complete;

(iv)

Y remains £-continuous, or £ remains ς-self-closed;

(v)

There exists $\mathsf{ϝ}\in \mathrm{\Sigma }$ verifying.

$$\begin{array}{c}\hfill \varsigma (\varphi y,\varphi z)\le \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right),forally,z\in \mathrm{{\rm Y}}with(y,z)\in \pounds .\end{array}$$

Then, ϕ admits a fixed point. Moreover, if $\varphi \left(\mathrm{{\rm Y}}\right)$ is ${\pounds }^s$-directed, then ϕ admits a unique fixed point.

Proposition 1

([4]). If £ is ϕ-closed, then £ is ${\varphi }^{\eta }$-closed, for each $\eta \in {\mathbb{N}}_0$.

Lemma 1

([11]). Assume that $\mathrm{{\rm Y}}$ is a set endowed with a relation £. If $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ is £-preserving sequence and £ is to be continued an n-transitive on $Ƶ=\{y_{\eta }:\eta \in {\mathbb{N}}_0\}$, then

$$(y_{\eta },y_{\eta +1+\upsilon (n-1)})\in \pounds ,forall\eta ,\upsilon \in {\mathbb{N}}_0.$$

Lemma 2

([10]). In a metric space $(\mathrm{{\rm Y}},\varsigma )$, if a sequence $\left\{y_{\eta }\right\}$ is not Cauchy, then there exist $ϵ>0$ and subsequences $\left\{y_{{\eta }_k}\right\}$ and $\left\{y_{{\mu }_k}\right\}$ of $\left\{y_{\eta }\right\}$ enjoying the properties:

(i)

$k\le {\mu }_k<{\eta }_{k}forallk\in \mathbb{N}$,

(ii)

$\varsigma (y_{{\mu }_k},y_{{\eta }_k})\ge ϵforallk\in \mathbb{N}$,

(iii)

$\varsigma (y_{{\mu }_k},y_{{\nu }_k})<ϵ$ for all ${\nu }_k\in \{{\mu }_k+1,{\mu }_k+2,\dots ,{\eta }_k-2,{\eta }_k-1\}$.

Moreover, if $\underset{\eta \to \infty }{lim}\varsigma (y_{\eta },y_{\eta +1})=0$, then

$$\underset{k\to \infty }{lim}\varsigma (y_{{\mu }_k},y_{{\eta }_k+\upsilon })=ϵ,forall\upsilon \in {\mathbb{N}}_0.$$

Proposition 2.

Given $\mathsf{ϝ}\in \mathrm{\Sigma }$ and $\lambda \ge 0$, (A) and (B) are equivalent:

(A)

$$\begin{gathered} \varsigma (\varphi y,\varphi z)\le \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\}, \\ forally,z\in \mathrm{{\rm Y}}with(y,z)\in \pounds . \end{gathered}$$

(B)

$$\begin{gathered} \varsigma (\varphi y,\varphi z)\le \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\}, \\ forally,z\in \mathrm{{\rm Y}}with[y,z]\in \pounds . \end{gathered}$$

Proof. 

The conclusion (B)⇒(A) holds trivially. Conversely, let (A) holds. Assume that $y,z\in \mathrm{{\rm Y}}$ with $[y,z]\in \pounds$. Then, in case $(y,z)\in \pounds$, (A) implies (B). Otherwise, in case $(z,y)\in \pounds$, by symmetric character of metric $\varsigma$ and (A), we obtain

$$\begin{array}{ccc}\hfill \varsigma (\varphi y,\varphi z)& =& \varsigma (\varphi z,\varphi y)\hfill \\ & \le & \mathsf{ϝ}\left(\varsigma \right(z,y\left)\right)+\lambda ·min\left\{\varsigma \right(z,\varphi z),\varsigma (y,\varphi y),\varsigma (z,\varphi y),\varsigma (y,\varphi z\left)\right\}\hfill \\ & =& \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\}.\hfill \end{array}$$

It follows that (A)⇒(B). □

3. Main Results

This section includes the results on existence and uniqueness of fixed point under relational strict almost nonlinear contraction.

Theorem 3.

Consider $(\mathrm{{\rm Y}},\varsigma )$ serves as a metric space, $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ as a map, and £ as a relation on$\mathrm{{\rm Y}}$. Also,

(i)

There exists $y_0\in \mathrm{{\rm Y}}$ satisfying $(y_0,\varphi y_0)\in \pounds$;

(ii)

£ is locally finitely ϕ-transitive and ϕ-closed;

(iii)

$(\mathrm{{\rm Y}},\varsigma )$ remains £-complete;

(iv)

Y remains £-continuous, or £ remains ς-self-closed;

(v)

There exist $\mathsf{ϝ}\in \mathrm{\Sigma }$ and $\lambda \ge 0$ verifying.

$$\begin{array}{ccc}& & \varsigma (\varphi y,\varphi z)\le \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\},\hfill \\ & & forally,z\in \mathrm{{\rm Y}}with(y,z)\in \pounds .\hfill \end{array}$$

Then, ϕ admits a fixed point.

Proof. 

Initiating with the point $y_0\in \mathrm{{\rm Y}}$, we construct a sequence $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ satisfying

$$y_{\eta }:={\varphi }^n\left(y_0\right)=\varphi \left(y_{\eta -1}\right),\mathrm{for}\mathrm{all}\eta \in \mathbb{N}.$$

(1)

Utilizing (i), (ii) and Proposition 1, we get

$$({\varphi }^n{y}_0,{\varphi }^{n+1}{y}_0)\in \pounds ,$$

which using (1) becomes

$$(y_{\eta },y_{\eta +1})\in \pounds ,\mathrm{for}\mathrm{all}\eta \in {\mathbb{N}}_0.$$

(2)

It yields that $\left\{y_{\eta }\right\}$ is a £-preserving sequence.

Define the following sequence of positive real numbers:

$${\varsigma }_{\eta }:=\varsigma (y_{\eta },y_{\eta +1}).$$

Whenever ${\varsigma }_{{\eta }_0}=0$ for some ${\eta }_0\in {\mathbb{N}}_0$, we have $\varphi \left(y_{{\eta }_0}\right)=y_{{\eta }_0}$ so that $y_{{\eta }_0}$ is a fixed point of $\varphi$ and hence the proof is finished.

Otherwise, we have ${\varsigma }_{\eta }>0$ for all $\eta \in {\mathbb{N}}_0$. In this case, by (1), (2) and assumption (v), we obtain

$$\begin{array}{ccc}\hfill {\varsigma }_{\eta }& =& \varsigma (y_{\eta },y_{\eta +1})=\varsigma (\varphi y_{\eta -1},\varphi y_{\eta })\hfill \\ & \le & \mathsf{ϝ}\left(\varsigma (y_{\eta -1},y_{\eta })\right)+\lambda ·min\{\varsigma (y_{\eta -1},y_{\eta }),\varsigma (y_{\eta },y_{\eta +1}),\varsigma (y_{\eta -1},y_{\eta +1}),0\}\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill {\varsigma }_{\eta }\le \mathsf{ϝ}\left({\varsigma }_{\eta -1}\right),\mathrm{for}\mathrm{all}\eta \in {\mathbb{N}}_0.\end{array}$$

(3)

Using the property of $\mathsf{ϝ}$ in (3), we have

$${\varsigma }_{\eta }\le \mathsf{ϝ}\left({\varsigma }_{\eta -1}\right)<{\varsigma }_{\eta -1},\mathrm{for}\mathrm{all}\eta \in \mathbb{N}.$$

It yields that $\left\{{\varsigma }_{\eta }\right\}$ is a decreasing sequence. Now, $\left\{{\varsigma }_{\eta }\right\}$ being bounded below ensures the existence of $\overline{\varsigma }\ge 0$ such that

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

(4)

Our claim is that $\overline{\varsigma }=0.$ Otherwise, in case $\overline{\varsigma }>0$ using the upper limit in (3) and owing to (4) and the definition of $\mathrm{\Sigma }$, we find

$$\overline{\varsigma }=\underset{\eta \to \infty }{lim\; sup}{\varsigma }_{\eta }\le \underset{\eta \to \infty }{lim\; sup}\mathsf{ϝ}\left({\varsigma }_{\eta -1}\right)=\underset{{\varsigma }_{\eta }\to l^{+}}{lim\; sup}\mathsf{ϝ}\left({\varsigma }_{\eta -1}\right)<\overline{\varsigma },$$

which is not possible. Hence,

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

(5)

On contrary, assume that $\left\{y_{\eta }\right\}$ is not Cauchy. Using Lemma 2, there exist $ϵ>0$ and subsequences $\left\{y_{{\eta }_k}\right\}$ and $\left\{y_{{\mu }_k}\right\}$ of $\left\{y_{\eta }\right\}$ satisfying

$$k\le {\mu }_k<n_k,\varsigma (y_{{\mu }_k},y_{{\eta }_k})\ge ϵ>\varsigma (y_{{\mu }_k},y_{{\nu }_k}),$$

$$\mathrm{for}\mathrm{all}k\in \mathbb{N},{\nu }_k\in \{{\mu }_k+1,{\mu }_k+2,\dots ,{\eta }_k-2,{\eta }_k-1\}.$$

By (5) and Lemma 2, we get

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\varsigma (y_{{\mu }_k},y_{{\eta }_k+\upsilon })=ϵ,\mathrm{for}\mathrm{all}\upsilon \in {\mathbb{N}}_0.\end{array}$$

(6)

By (1), we have $Ƶ:=\{y_{\eta }:\eta \in {\mathbb{N}}_0\}\subset \varphi \left(\mathrm{{\rm Y}}\right)$. Thus, using locally finitely $\varphi$-transitivity of £, there exists $n\in \{2,3,\dots \}$ such that ${\pounds |}_{Ƶ}$ is n-transitive.

Using the fact: ${\mu }_k<{\eta }_k$ and $n-1>0$ and applying division algorithm, we have

$${\eta }_k-{\mu }_k=(n-1)({\alpha }_k-1)+(n-{\beta }_k)$$

$${\alpha }_k-1\ge 0,0\le n-{\beta }_k<n-1$$

$$⟺\left\{\begin{array}{c}{\eta }_k+{\beta }_k={\mu }_k+1+(n-1){\alpha }_k\hfill \\ {\alpha }_k\ge 1,1<{\beta }_k\le n.\hfill \end{array}\right.$$

Clearly, ${\beta }_k\in (1,n]$. Consequently the subsequences $\left\{y_{{\eta }_k}\right\}$ and $\left\{y_{{\mu }_k}\right\}$ of $\left\{y_n\right\}$ (verifying (6)) can be such chosen for which ${\beta }_k=\beta$ (a constant) such that

$$\begin{array}{c}\hfill {\mu }_k^{\prime }={\eta }_k+\beta ={\mu }_k+1+(n-1){\alpha }_k.\end{array}$$

(7)

By (6) and (7), we get

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})=\underset{k\to \infty }{lim}\varsigma (y_{{\mu }_k},y_{{\eta }_k+\beta })=ϵ.\end{array}$$

(8)

Use of the triangular inequality yields that

$$\varsigma (y_{{\mu }_k+1},y_{{\mu }_k^{\prime }+1})\le \varsigma (y_{{\mu }_k+1},y_{{\mu }_k})+\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})+\varsigma (y_{{\mu }_k^{\prime }},y_{{\mu }_k^{\prime }+1})$$

and

$$\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})\le \varsigma (y_{{\mu }_k},y_{{\mu }_k+1})+\varsigma (y_{{\mu }_{k+1}},y_{{\mu }_k^{\prime }+1})+\varsigma (y_{{\mu }_k^{\prime }+1},y_{{\mu }_k^{\prime }}).$$

Therefore, we have

$$\begin{array}{cc}\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})-\varsigma (y_{{\mu }_k},y_{{\mu }_k+1})-\varsigma (y_{{\mu }_k^{\prime }+1},y_{{\mu }_k^{\prime }})\hfill & \le \varsigma (y_{{\mu }_{k+1}},y_{{\mu }_k^{\prime }+1})\hfill \\ & \le \varsigma (y_{{\mu }_k+1},y_{{\mu }_k})+\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})+\varsigma (y_{{\mu }_k^{\prime }},y_{{\mu }_k^{\prime }+1})\hfill \end{array}$$

which one (by letting $k\to \infty$ and with the help of (5) and (8) reduces to

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\varsigma (y_{{\mu }_k+1},y_{{\mu }_k^{\prime }+1})=ϵ.\end{array}$$

(9)

By (7) and Lemma 2, we get $(y_{{\mu }_k},y_{{\mu }_k^{\prime }})\in \pounds .$ Denote ${\delta }_k:=\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})$. Using the contractivity condition (v), we obtain

$$\begin{array}{ccc}\hfill \varsigma (y_{{\mu }_k+1},y_{{\mu }_k^{\prime }+1})& =& \varsigma (\varphi y_{{\mu }_k},\varphi y_{{\mu }_k^{\prime }})\hfill \\ & \le & \mathsf{ϝ}\left(\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }})\right)+\lambda ·min\{\varsigma (y_{{\mu }_k},\varphi y_{{\mu }_k}),\varsigma (y_{{\mu }_k^{\prime }},\varphi y_{{\mu }_k^{\prime }}),\varsigma (y_{{\mu }_k},\varphi y_{{\mu }_k^{\prime }}),\varsigma (y_{{\mu }_k^{\prime }},\varphi y_{{\mu }_k})\}\hfill \end{array}$$

so that

$$\varsigma (y_{{\mu }_k+1},y_{{\mu }_k^{\prime }+1})\le \mathsf{ϝ}\left({\delta }_k\right)+\lambda ·min\{{\varsigma }_{{\mu }_k},{\varsigma }_{{\mu }_k^{\prime }},\varsigma (y_{{\mu }_k},y_{{\mu }_k^{\prime }+1}),\varsigma (y_{{\mu }_k^{\prime }},y_{{\mu }_k+1})\}.$$

(10)

Using upper limit in (10) and by Lemma 2 and the property of $\mathsf{ϝ}$, we get

$${\displaystyle ϵ=\underset{k\to \infty }{lim\; sup}\varsigma (y_{{\mu }_k+1},y_{{\mu }_k^{\prime }+1})\le \underset{k\to \infty }{lim\; sup}\mathsf{ϝ}\left({\delta }_k\right)+\lambda ·min\{0,0,ϵ,ϵ\}=\underset{s\to {ϵ}^{+}}{lim\; sup}\mathsf{ϝ}\left(s\right)<ϵ,}$$

which arises a contradiction.

Therefore, $\left\{y_{\eta }\right\}$ remains Cauchy, which is also £-preserving. It turns out by £-completeness of $\mathrm{{\rm Y}}$ that there exists $\overline{y}\in \mathrm{{\rm Y}}$ such that $y_{\eta }\stackrel{\varsigma }{⟶}\overline{y}$.

Finally, the proof will be concluded using (iv). Suppose firstly that $\varphi$ is £-continuous. Since $\left\{y_{\eta }\right\}$ is a £-preserving sequence such that $y_{\eta }\stackrel{\varsigma }{⟶}\overline{y}$; therefore, by £-continuity of $\varphi$, we get

$$y_{\eta +1}=\varphi \left(y_{\eta }\right)\stackrel{\varsigma }{⟶}\varphi \left(\overline{y}\right).$$

This implies that $\varphi \left(\overline{y}\right)=\overline{y}$.

Secondly, assume that £ is $\varsigma$-self-closed. Then we determine a subsequence $\left\{y_{{\eta }_k}\right\}$ of $\left\{y_{\eta }\right\}$ such that $[y_{{\eta }_k},\overline{y}]\in \pounds ,\mathrm{for}\mathrm{all}k\in \mathbb{N}.$ By condition (v), Proposition 2 and $[y_{{\eta }_k},\overline{y}]\in \pounds$, we get

$$\begin{array}{ccc}\hfill \varsigma (y_{{\eta }_k+1},\varphi \overline{y})& =& \varsigma (\varphi y_{{\eta }_k},\varphi \overline{y})\hfill \\ & \le & \mathsf{ϝ}\left(\varsigma (y_{{\eta }_k},\overline{y})\right)+\lambda ·min\{\varsigma (y_{{\eta }_k},y_{{\eta }_k+1}),0,\varsigma (y_{{\eta }_k},\overline{y}),\varsigma (\overline{y},y_{{\eta }_k+1})\}\hfill \\ & =& \mathsf{ϝ}\left(\varsigma (y_{{\eta }_k},\overline{y})\right).\hfill \end{array}$$

We claim that

$$\varsigma (y_{{\eta }_k+1},\varphi \overline{y})\le \varsigma (y_{{\eta }_k},\overline{y}),\mathrm{for}\mathrm{all}k\in \mathbb{N}.$$

(11)

If for some $k_0\in \mathbb{N}$, $\varsigma (y_{{\eta }_{k_0}},\overline{y})=0$, then we get $\varsigma (\varphi y_{{\eta }_{k_0}},\varphi \overline{y})=0$, i.e., $\varsigma (y_{{\eta }_{k_0}+1},\varphi \overline{y})=0$ and hence (11) holds for this $k_0\in \mathbb{N}$. If $\varsigma (y_{{\eta }_k},\overline{y})>0\mathrm{for}\mathrm{all}k\in \mathbb{N}$, then using property of $\mathsf{ϝ}$, we obtain $\varsigma (y_{{\eta }_k+1},\varphi \overline{y})\le \mathsf{ϝ}\left(\varsigma (y_{{\eta }_k},\overline{y})\right)<\varsigma (y_{{\eta }_k},\overline{y})\mathrm{for}\mathrm{all}k\in \mathbb{N}$. Therefore, (11) remains valid for every $k\in \mathbb{N}.$ Using limit of (11) and by $y_{{\eta }_k}\stackrel{\varsigma }{⟶}\overline{y}$, we obtain $y_{{\eta }_k+1}\stackrel{\varsigma }{⟶}\varphi \left(\overline{y}\right)$. This implies that $\varphi \left(\overline{y}\right)=\overline{y}$. Thus, $\overline{y}$ remains a fixed point of $\varphi$. □

Theorem 4.

In addition to the assertions of Theorem 3, if $\varphi \left(\mathrm{{\rm Y}}\right)$ is ${\pounds }^s$-directed, then ϕ admits a unique fixed point.

Proof. 

Let $\overline{y},\overline{z}\in \mathrm{{\rm Y}}$ be two fixed points of $\varphi$. We have

$$\varphi \left(\overline{y}\right)=\overline{y}\mathrm{and}\varphi \left(\overline{z}\right)=\overline{z}.$$

(12)

As $\overline{y},\overline{z}\in \varphi \left(\mathrm{{\rm Y}}\right)$, by our hypothesis there exists $w\in \mathrm{{\rm Y}}$ verifying

$$[\overline{y},w]\in \pounds \mathrm{and}[\overline{z},w]\in \pounds .$$

(13)

Set ${\varrho }_{\eta }:=\varsigma (\overline{y},{\varphi }^{\eta }w)$. By (12), (13), (v) and Proposition 2, we conclude

$$\begin{array}{ccc}\hfill {\varrho }_{\eta }=\varsigma (\overline{y},{\varphi }^{\eta }w)& =& \varsigma (\varphi \overline{y},\varphi \left({\varphi }^{\eta -1}w\right))\hfill \\ & \le & \mathsf{ϝ}\left(\varsigma (\overline{y},{\varphi }^{\eta -1}w)\right)+\lambda ·min\{0,\varsigma ({\varphi }^{\eta -1}w,{\varphi }^{\eta }w),\varsigma (\overline{y},{\varphi }^{\eta }w),\varsigma ({\varphi }^{\eta -1}w,\overline{y})\}\hfill \\ & =& \mathsf{ϝ}\left({\varrho }_{\eta -1}\right)\hfill \end{array}$$

so that

$${\varrho }_{\eta }\le \mathsf{ϝ}\left({\varrho }_{\eta -1}\right).$$

(14)

Assuming first that for some ${\eta }_0\in \mathbb{N}$, ${\varrho }_{{\eta }_0}=0$. Then ${\varrho }_{{\eta }_0}\le {\varrho }_{{\eta }_0-1}$. Secondly, we have ${\varrho }_{\eta }>0,$ for all $\eta \in \mathbb{N}$. Then, using the property of $\mathsf{ϝ}$, (14) becomes ${\varrho }_{\eta }<{\varrho }_{\eta -1}$. Thus, in both cases, we get

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

As a result, similar to Theorem 3, the above relation becomes

$${\displaystyle \underset{\eta \to \infty }{lim}{\varrho }_{\eta }=\underset{\eta \to \infty }{lim}\varsigma (\overline{y},{\varphi }^{\eta }w)=0.}$$

(15)

In similar manner, we find

$${\displaystyle \underset{\eta \to \infty }{lim}\varsigma (\overline{z},{\varphi }^{\eta }w)=0.}$$

(16)

Combining (15) and (16), we get

$$\varsigma (\overline{y},\overline{z})=\varsigma (\overline{y},{\varphi }^{\eta }w)+\varsigma ({\varphi }^{\eta }w,\overline{z})\to 0\mathrm{as}\eta \to \infty$$

i.e., $\overline{y}=\overline{z}$. Thus, uniqueness of fixed point is concluded. □

Remark 2.

Particularly, for $\lambda =0$, Theorems 3 and 4 reduce to corresponding results of Alam et al. [5] (i.e., Theorem 2).

Remark 3.

Under trivial relation $\pounds ={\mathrm{{\rm Y}}}^2$ and for $\mathsf{ϝ}\left(t\right)=\alpha ·t$, Theorem 4 deduces the main result of Babu et al. [7] (i.e., Theorem 1).

4. Examples

With a view of demonstrating Theorems 3 and 4, we attempt the subsequent examples.

Example 1.

Assuming that $\mathrm{{\rm Y}}=[0,\infty )$ with standard metric ς and a relation $\pounds :=\{(y,z)\in {\mathrm{{\rm Y}}}^2:y-z>0\}$. Define the map $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ by $\varphi \left(y\right)=\frac{y}{y+1}$. Clearly, the relation £ is locally finitely ϕ-transitive, the metric space $(\mathrm{{\rm Y}},\varsigma )$ is £-complete and the map ϕ is £-continuous.

Take $(y,z)\in \pounds$. Then, we have $y-z>0$. Now, we have

$$\varphi \left(y\right)-\varphi \left(z\right)=\frac{y-z}{(y+1)(z+1)}>0$$

so that $(\varphi y,\varphi z)\in \pounds$. It follows that £ ϕ-closed.

Let the function $\mathsf{ϝ}:[0,\infty )\to [0,\infty )$ be defined by $\mathsf{ϝ}\left(t\right)=\frac{t}{t+1}$ and $\lambda \ge 0$ be arbitrarily. Now, for all $(y,z)\in \pounds$, we have

$$\begin{array}{ccc}\hfill \varsigma (\varphi y,\varphi z)& =& |\frac{y}{y+1}-\frac{z}{z+1}|=\left|\frac{y-z}{1+y+z+yz}\right|\hfill \\ & \le & \frac{y-z}{1+(y-z)}=\frac{\varsigma (y,z)}{1+\varsigma (y,z)}\hfill \\ & \le & \varphi \left(\varsigma \right(y,z\left)\right)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\}.\hfill \end{array}$$

It means that the presumption (v) of Theorem 3 is fulfilled. Also, $y_0=1$ verifies the presumption (i) of Theorem 3. Finally, $\varphi \left(\mathrm{{\rm Y}}\right)$ is ${\pounds }^s$-directed as for any arbitrary $y,z\in \varphi \left(\mathrm{{\rm Y}}\right)$, the element $w:=(y+z)/2$ satisfies $[y,w]\in \pounds$ and $[z,w]\in \pounds$. Therefore, all the assumptions of Theorems 3 and 4 hold and hence ϕ has a unique fixed point, $\overline{y}=0$.

Example 2.

Assuming that $\mathrm{{\rm Y}}=[0,1]$ with standard metric ς and a relation $\pounds :=\le$. Define the map $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ by

$$\varphi \left(y\right)=\left\{\begin{array}{c}{y}^2,\mathrm{if}0\le y<1/4\hfill \\ 0,\mathrm{if}1/4\le y\le 1.\hfill \end{array}\right.$$

Then $(0,\varphi 0)\in \pounds$. Clearly, the relation £ is locally finitely ϕ-transitive and ϕ-closed. Also, the metric space $(\mathrm{{\rm Y}},\varsigma )$ is £-complete.

Let $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ be £-preserving convergent sequence verifying $y_{\eta }\stackrel{\varsigma }{⟶}y$. Then $\left\{y_{\eta }\right\}$ being monotone increasing convergent sequence satisfies $y_{\eta }\le y$ so that $(y_{\eta },y)\in \pounds$ for each $\eta \in \mathbb{N}$. It follows that £ is ς-self-closed.

Let the function $\mathsf{ϝ}:[0,\infty )\to [0,\infty )$ be defined by $\mathsf{ϝ}\left(t\right)=t/2$ and chose $\lambda =1$. Then one can easily verify the presumption (v) of Theorem 3. Finally, $\varphi \left(\mathrm{{\rm Y}}\right)$ is ${\pounds }^s$-directed as for any arbitrary $y,z\in \varphi \left(\mathrm{{\rm Y}}\right)$, the element $w:=max\{y,z\}$ satisfies $(y,w)\in \pounds$ and $(z,w)\in \pounds$. Therefore, all the assumptions of Theorems 3 and 4 hold and hence ϕ has a unique fixed point, $\overline{y}=0$.

5. An Application to Nonlinear Integral Equations

This section covers an application of earlier theorems to compute a unique solution of the following nonlinear integral equation:

$$y\left(\mathrm{X}\right)=H\left(\mathrm{X}\right)+{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\hslash (\tau ,y\left(\tau \right))d\tau ,\mathrm{X}\in I:=[a,b]$$

(17)

where $H:I\to \mathbb{R}$, $\yen :I^2\to \mathbb{R}$ and $\hslash :I\times \mathbb{R}\to \mathbb{R}$ are functions.

Definition 17.

$\overline{y}\in \mathcal{C}\left(I\right)$ is viewed as a lower solution of (17) if

$$\overline{y}\left(\mathrm{X}\right)\le H\left(\mathrm{X}\right)+{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\hslash (\tau ,\overline{y}\left(\tau \right))d\tau ,forall\mathrm{X}\in I.$$

Definition 18.

$\underline{y}\in \mathcal{C}\left(I\right)$ is viewed as an upper solution of (17) if

$$\underline{y}\left(\mathrm{X}\right)\ge H\left(\mathrm{X}\right)+{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\hslash (\tau ,\underline{y}\left(\tau \right))d\tau ,forall\mathrm{X}\in I.$$

We now proceed to present our main results concerning this section.

Theorem 5.

Assume, in the course of Problem (17), that

(a)

H, ℏ and ¥ are continuous;

(b)

$\mathrm{\yen }(\mathrm{X},\tau )>0,forall\mathrm{X},\tau \in I$;

(c)

There exist $0<\ell \le 1$ and an increasing function $\mathsf{ϝ}\in \mathrm{\Sigma }$ such that

$$0\le \hslash (\mathrm{X},u)-\hslash (\mathrm{X},v)\le \frac{1}{\ell }\mathsf{ϝ}(u-v),forall\mathrm{X}\in Iand\forall u,v\in \mathbb{R}withu\ge v,$$

(d)

$\underset{\mathrm{X}\in I}{sup}{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )d\tau \le \ell$.

If there exists a lower solution of (17), then the problem possesses a unique solution.

Proof. 

Indicate $\mathrm{{\rm Y}}:=\mathcal{C}\left(I\right)$. Undertake metric $\varsigma$ on $\mathrm{{\rm Y}}$ by

$$\varsigma (y,z)=\underset{\mathrm{X}\in I}{sup}|y\left(\mathrm{X}\right)-z\left(\mathrm{X}\right)|,\mathrm{for}\mathrm{all}y,z\in \mathrm{{\rm Y}}.$$

(18)

Undertake a relation £ on $\mathrm{{\rm Y}}$ by

$$\pounds =\{(y,z)\in {\mathrm{{\rm Y}}}^2:y\left(\mathrm{X}\right)\le z\left(\mathrm{X}\right),\forall \mathrm{X}\in I\}.$$

(19)

Define a map $\varphi :\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ by

$$\left(\varphi y\right)\left(\mathrm{X}\right)=H\left(\mathrm{X}\right)+{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\hslash (\tau ,y\left(\tau \right))d\tau ,\mathrm{for}\mathrm{all}\mathrm{X}\in \mathrm{{\rm Y}}.$$

(20)

Obviously, $y_0\in \mathrm{{\rm Y}}$ will form a solution of Problem (17) if and only if $y_0$ remains a fixed point of $\varphi$.

Now, we are going to validate all premises of Theorem 3.

(i)

Let $\overline{y}\in \mathrm{{\rm Y}}$ be a lower solution of (17). Then

$$\overline{y}\left(\mathrm{X}\right)\le H\left(\mathrm{X}\right)+{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\hslash (\tau ,\overline{y}\left(\tau \right))d\tau =\left(\mathsf{ϝ}\overline{y}\right)\left(\mathrm{X}\right)$$

yielding thereby $(\overline{y},\varphi \overline{y})\in \pounds$.

(ii)

Take $y,z\in \mathrm{{\rm Y}}$ such that $(y,z)\in \pounds$. Employing item (c), we obtain

$$\hslash (\mathrm{X},y(\tau \left)\right)-\hslash (\mathrm{X},z(\tau \left)\right)\le 0,\mathrm{for}\mathrm{all}\mathrm{X},\tau \in I.$$

(21)

Using (20), (21) and item (b), we obtain

$$\begin{array}{c}\hfill \left(\varphi y\right)\left(\mathrm{X}\right)-\left(\varphi z\right)\left(\mathrm{X}\right)={\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )[\hslash (\tau ,y\left(\tau \right))-\hslash (\tau ,z\left(\tau \right))]d\tau \le 0,\end{array}$$

following that $\left(\varphi y\right)\left(\mathrm{X}\right)\le \left(\varphi z\right)\left(\mathrm{X}\right)$. Consequently, we have $(\varphi y,\varphi z)\in \pounds$ so that £ remains $\varphi$-closed.

(iii)

$(\mathrm{{\rm Y}},\varsigma )$ is £-complete as it remains complete.

(iv)

Assuming that $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ is a £-preserving sequence, which converges to $\tilde{y}\in \mathrm{{\rm Y}}$. Then for every $\mathrm{X}\in I$, $\left\{y_{\eta }\left(\mathrm{X}\right)\right\}$ is an increasing sequence (in $\mathbb{R}$) which converges to $\tilde{y}\left(\mathrm{X}\right)$. This implies that $y_{\eta }\left(\mathrm{X}\right)\le \tilde{y}\left(\mathrm{X}\right),\mathrm{for}\mathrm{all}\eta \in \mathbb{N}$ and $\mathrm{X}\in I$. Using (19), we conclude $(y_{\eta },\tilde{y})\in \pounds ,\mathrm{for}\mathrm{all}\eta \in \mathbb{N}$; thereby, £ will be is $\varsigma$-self-closed.

(v)

Take $y,z\in \mathrm{{\rm Y}}$ such that $(y,z)\in \pounds$. Using item (c), (18) and (20), we find

$$\begin{array}{ccc}\hfill \varsigma (\varphi y,\varphi z)& =& \underset{\mathrm{X}\in I}{sup}|\left(\varphi y\right)\left(\mathrm{X}\right)-\left(\varphi z\right)\left(\mathrm{X}\right)|=\underset{\mathrm{X}\in I}{sup}[\left(\varphi z\right)\left(\mathrm{X}\right)-\left(\varphi y\right)\left(\mathrm{X}\right)]\hfill \\ \hfill & =& \underset{\mathrm{X}\in I}{sup}{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )[\hslash (\tau ,z\left(\tau \right))-\hslash (\tau ,y\left(\tau \right))]d\tau \hfill \\ & \le & \underset{\mathrm{X}\in I}{sup}{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\frac{1}{\ell }\mathsf{ϝ}(z\left(\tau \right)-y\left(\tau \right))d\tau .\hfill \end{array}$$

(22)

As $0\le z\left(\tau \right)-y\left(\tau \right)\le \varsigma (y,z)$, using increasing property of $\mathsf{ϝ}$, we get

$$\mathsf{ϝ}\left(z\right(\tau )-y(\tau \left)\right)\le \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right).$$

Therefore, (22) becomes

$$\begin{array}{ccc}\hfill \varsigma (\varphi y,\varphi z)& \le & \frac{1}{\ell }\mathsf{ϝ}\left(\varsigma (y,z)\right)\underset{\mathrm{X}\in I}{sup}{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )d\tau \hfill \\ & \le & \frac{1}{\ell }\mathsf{ϝ}\left(\varsigma (y,z)\right).\ell \hfill \\ & =& \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right)\hfill \end{array}$$

yielding thereby

$$\begin{array}{ccc}\hfill \varsigma (\varphi y,\varphi z)& \le & \mathsf{ϝ}\left(\varsigma \right(y,z\left)\right)+\lambda ·min\left\{\varsigma \right(y,\varphi y),\varsigma (z,\varphi z),\varsigma (y,\varphi z),\varsigma (z,\varphi y\left)\right\},\hfill \\ & \mathrm{for}\mathrm{all}& y,z\in \mathrm{{\rm Y}}\mathrm{such}\mathrm{that}(y,z)\in \pounds ,\hfill \end{array}$$

where $\lambda \ge 0$ is arbitrary.

Finally, we will verify the assumption of Theorem 4. Let $y,z\in$ be chosen arbitrarily. Write $w:=max\{\varphi y,\varphi z\}\in \mathrm{{\rm Y}}$. We conclude that $(\varphi y,w)\in \pounds$ and $(\varphi z,w)\in \pounds$. This yields that $\varphi \left(\mathrm{{\rm Y}}\right)$ is ${\pounds }^s$-directed and hence by Theorem 4, $\varphi$ possesses a unique fixed point, which constitutes a unique solution for (17). □

Theorem 6.

In the collaboration to assertions (a)–(d) of Theorem 5, if there exists an upper solution of (17), then the problem possesses a unique solution.

Proof. 

Consider $\mathrm{{\rm Y}}:=\mathcal{C}\left(I\right)$ with a metric $\varsigma$ and a map $\mathsf{ϝ}:\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ to Theorem 5. Take the following relation ${\pounds }^{\prime }$ on $\mathrm{{\rm Y}}$:

$${\pounds }^{\prime }=\{(y,z)\in {\mathrm{{\rm Y}}}^2:y\left(\mathrm{X}\right)\ge z\left(\mathrm{X}\right),\forall \mathrm{X}\in I\}.$$

(23)

If $\underline{y}\in \mathrm{{\rm Y}}$ is an upper solution of (17), then we conclude

$$\underline{y}\left(\mathrm{X}\right)\ge H\left(\mathrm{X}\right)+{\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )\hslash (\tau ,\underline{y}\left(\tau \right))d\tau =\left(\varphi \underline{y}\right)\left(\mathrm{X}\right)$$

yielding thereby $(\underline{y},\varphi \underline{y})\in {\pounds }^{\prime }$.

Take $y,z\in \mathrm{{\rm Y}}$ such that $(y,z)\in {\pounds }^{\prime }$. Employing item (c), we obtain

$$\hslash (\mathrm{X},y(\tau \left)\right)-\hslash (\mathrm{X},z(\tau \left)\right)\ge 0,\mathrm{for}\mathrm{all}\mathrm{X},\tau \in I.$$

(24)

Using (20), (24) and item (b), we obtain

$$\begin{array}{c}\hfill \left(\varphi y\right)\left(\mathrm{X}\right)-\left(\varphi z\right)\left(\mathrm{X}\right)={\int }_a^b\mathrm{\yen }(\mathrm{X},\tau )[\hslash (\tau ,y\left(\tau \right))-\hslash (\tau ,z\left(\tau \right))]d\tau \ge 0,\end{array}$$

which yields that $\left(\varphi y\right)\left(\mathrm{X}\right)\ge \left(\varphi z\right)\left(\mathrm{X}\right)$ so that $(\varphi y,\varphi z)\in {\pounds }^{\prime }$. Therefore, ${\pounds }^{\prime }$ is $\varphi$-closed.

Let $\left\{y_{\eta }\right\}\subset \mathrm{{\rm Y}}$ be a ${\pounds }^{\prime }$-preserving sequence, which converges to $y\in \mathrm{{\rm Y}}$. Then for every $\mathrm{X}\in I$, $\left\{y_{\eta }\left(\mathrm{X}\right)\right\}$ is a decreasing sequence (in $\mathbb{R}$) which converges to $y\left(\mathrm{X}\right)$. Consequently, we have $y_{\eta }\left(\mathrm{X}\right)\ge y\left(\mathrm{X}\right),\mathrm{for}\mathrm{all}\eta \in \mathbb{N}$ and $\mathrm{for}\mathrm{all}\mathrm{X}\in I$. Using (23), we conclude that $(y_{\eta },y)\in {\pounds }^{\prime },\mathrm{for}\mathrm{all}m\in \mathbb{N}$. Thus ${\pounds }^{\prime }$ is $\varsigma$-self-closed.

Thus in all, we have verified all assertions of Theorems 3 and 4 for the metric space $(\mathrm{{\rm Y}},\varsigma )$, the map $\varphi$ and the relation ${\pounds }^{\prime }$. This completes the proof. □

6. Conclusions

Very recently, Alharbi and Khan [13] have established new fixed point theorems for relational almost strict Boyd–Wong contractions via a locally $\varphi$-transitive relation. In proving our results, although the class of contractions is relatively strong, we employed a locally finitely $\varphi$-transitive relation, which remains more general than what is utilized in [13]. Hence, our results are completely different from those proved by Alharbi and Khan [13]. In the near term, learners may extend our results to a pair of mappings.