On Relational Weak $\left({\mathit{F}}_{\Re }^{\mathit{m}},\mathsf{\eta }\right)$-Contractive Mappings and Their Applications

Abstract

In this article, we introduce the concept of weak $\left(F_{\Re }^m,\eta \right)$-contractions on relation-theoretic m-metric spaces and establish related fixed point theorems, where $\eta$ is a control function and ℜ is a relation. Then, we detail some fixed point results for cyclic-type weak $\left(F_{\Re }^m,\eta \right)$-contraction mappings. Finally, we demonstrate some illustrative examples and discuss upper and lower solutions of Volterra-type integral equations of the form $\xi \left(\alpha \right)={\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right),\alpha \in \left[0,1\right].$

1. Introduction and Preliminaries

The classical Banach contraction theorem [1] is an important and fruitful tool in nonlinear analysis. In the past few decades, many authors have extended and generalized the Banach contraction mapping principle in several ways (see [2,3,4,5,6,7,8,9,10,11,12]). On the other hand, several authors, such as Boyd and Wong [13], Browder [14], Wardowski [15], Jleli and Samet [16], and many other researchers have extended the Banach contraction principle by employing different types of control functions (see [17,18,19,20,21] and the references therein). Alam et al. [22] introduced the concept of the relation-theoretic contraction principle and proved some well known fixed-point results in this area. Afterward, many researchers focused on fixed-point theorems in relation-theoretic metric spaces. Here, we will present some basic knowledge of relation-theoretic metric spaces (see more detail in [23,24,25,26]). Furthermore, Sawangsup et al. [27] introduced the concept of the ${\left(F,\gamma \right)}_{\Re }$-contractive of mappings to extend F-contractions in metric spaces endowed with binary relations. One of the latest extensions of metric spaces and partial metric spaces [10] was given in paper [28], which completed the concept of m-metric spaces. Using this concept, several researchers have proven some fixed point results in this area (see [20,29,30,31,32,33]). Subsequently, since every F-contraction mapping is contractive and also continuous, Secelean et al. [34] proved that the continuity of an F-contraction can be obtained from condition $F_2$. After that, Imdad et al. [35] introduced the idea of a new type of F-contraction by dropping the condition of $F_1$ and replacing condition $\left(F_3\right)$ with the continuity of F. They also proved some new fixed point results in relation to theoretic metric spaces.

In this paper, we introduce weak $\left(F_{\Re }^m,\eta \right)$-contractive mappings and cyclic-type weak $\left(F_{\Re }^m,\eta \right)$-contractions and provide some new fixed point theorems for such mappings in relation to theoretic m-metric spaces. Finally, as an application, we discuss the lower and upper solutions of Volterra-type integral equations.

Throughout this article, $\mathbb{N}$ indicates a set of all natural numbers, $\mathbb{R}$ indicates a set of real numbers and ${\mathbb{R}}^{+}$ indicates a set of positive real numbers. We also denote ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}.$ Henceforth, U will denote a non-empty set and the self mapping $\gamma :U\to U$ with a Picard sequence based on an arbitrary ${\xi }_0$ in U is given by ${\xi }_n=\gamma \left({\xi }_{n-1}\right)={\gamma }^n\left({\xi }_0\right)$, where all n are members of $\mathbb{N}$ and ${\gamma }^n$ denotes the $n^{th}$-iteration of $\gamma .$

The notion of m-metric spaces was introduced by Asadi et al. [28] as a real generalization of a partial metric space and they supported their claim by providing some constructive examples. For more detail, see, e.g., [29,31].

Definition 1 

([28]). An m-metric space on a non-empty set U is a mapping $m:U\times U\to {\mathbb{R}}^{+}$ such that for all $\xi ,\Im ,\aleph \in U,$

(i)

$\xi =\Im ⟺m\left(\xi ,\xi \right)=m\left(\Im ,\Im \right)=m\left(\xi ,\Im \right)\left(T_0\mathit{-separation}\mathit{axiom}\right);$

(ii)

$m_{\xi \Im }\le m(\xi ,\Im )$$\left(\mathit{minimum}\mathit{self}\mathit{distance}\mathit{axiom}\right);$

(iii)

$m\left(\xi ,\Im \right)=m\left(\Im ,\xi \right)$$\left(\mathit{symmetry}\right);$

(iv)

$m\left(\xi ,\Im \right)-m_{\xi \Im }\le (m\left(\xi ,\aleph \right)-m_{\xi \aleph })+(m\left(\aleph ,\Im \right)-m_{\aleph \Im })$$\left(\mathit{modified}\mathit{triangle}\mathit{inequality}\right)$

where

$$\begin{array}{ccc}\hfill m_{\xi \Im }& =& min\left\{m(\xi ,\xi ),m(\Im ,\Im )\right\};\hfill \\ \hfill M_{\xi \Im }& =& max\left\{m(\xi ,\xi ),m(\Im ,\Im )\right\}.\hfill \end{array}$$

The pair $\left(U,m\right)$ is called an m-metric space on nonempty $U.$

Lemma 1 

([28]). Each partial metric forms an m-metric space but the converse is not true.

Among the classical examples of an m-metric space is a pair $\left(U,m\right)$, where $U=\left\{\xi ,\Im ,\aleph \right\}$ and m is a self mapping on U given by $m\left(\xi ,\xi \right)=1$, $m\left(\Im ,\Im \right)=9$ and $m\left(\aleph ,\aleph \right)=5.$ It is clear that m is an m-metric space. Note that m does not form a partial metric space.

Every m-metric space m on U generates a $T_0$ topology, e.g., ${\tau }_m$, on U which is based on a collection of m-open balls:

$$\left\{B_m\left(\xi ,ϵ\right):\xi \in U,ϵ>0\right\},$$

where

$$B_m\left(\xi ,ϵ\right)=\{\Im \in U:m\left(\xi ,\Im \right)<m_{\xi \Im }+ϵ\}\mathrm{for}\mathrm{all}\xi \in U,\epsilon >0.$$

If m is an m-metric space on $U,$ then the functions $m^w$ and $m^s:U\times U\to {\mathbb{R}}^{+}$ given by

$$m^w\left(\xi ,\Im \right)=m\left(\xi ,\Im \right)-2m_{\xi \Im }+M_{\xi \Im },$$

$$m^s=\left\{\begin{array}{c}m\left(\xi ,\Im \right)-m_{\xi \Im },\mathrm{if}\xi \ne \Im \hfill \\ 0,\mathrm{if}\xi =\Im .\hfill \end{array}\right.,$$

define ordinary metrics on U. It is easy to see that $m^w$ and $m^s$ are equivalent metrics on $U.$

Definition 2 

([28]). Let $\left\{{\xi }_n\right\}$ be a sequence in an m-metric space $\left(U,m\right),$ then

(i)

$\left\{{\xi }_n\right\}$ is said to be convergent with respect to ${\tau }_m$ to ξ if and only if

$$\underset{\mu \to \infty }{lim}\left(m\left({\xi }_n,\xi \right)-m_{{\xi }_n\xi }\right)=0.\mathit{for}\mathit{all}n\in \mathbb{N}.$$

(ii)

If ${lim}_{n,m\to \infty }\left(m\left({\xi }_n,{\xi }_m\right)-m_{{\xi }_n{\xi }_m}\right)$ and ${lim}_{n,m\to \infty }\left(M_{{\xi }_n,{\xi }_m}-m_{{\xi }_n{\xi }_m}\right)$ for all $n,m\in \mathbb{N}$ exists and is finite, then the sequence $\left\{{\xi }_n\right\}$ in a m-metric space $\left(U,m\right)$ is m-Cauchy.

(iii)

If every m-Cauchy $\left\{{\xi }_n\right\}$ in U is m-convergent with respect to ${\tau }_m$ to ξ in U such that

$$\underset{n\to \infty }{lim}m\left({\xi }_n,\xi \right)-m_{{\xi }_n\xi }=0,\mathit{and}\underset{n\to \infty }{lim}\left(M_{{\xi }_n,\xi }-m_{{\xi }_n\xi }\right)=0.\mathit{for}\mathit{all}n\in \mathbb{N},$$

then $\left(U,m\right)$ is said to be complete.

(iv)

$\left\{{\xi }_n\right\}$ is an m-Cauchy sequence if and only if it is a Cauchy sequence in the metric space $\left(U,m^w\right),$

(v)

$\left(U,m\right)$ is M-complete if and only if $\left(U,m^w\right)$ is complete.

Denote $\nabla \left(F\right)$ by the collection of all mappings $F:\left(0,\infty \right)\to R$ satisfying [15]:

• (F₁) $F\left(\xi \right)<F\left(\Im \right)$ for all $\xi <\Im ;$
• (F₂) For each sequence $\left\{{\xi }_n\right\}$ of positive numbers

  $$\underset{n\to \infty }{lim}{\xi }_n=0\mathrm{if}\underset{n\to \infty }{lim}F\left({\xi }_n\right)=-\infty ;$$
• (F₃) There exists $p\in \left(0,1\right)$ such that ${lim}_{n\to 0^{+}}{\xi }^{p}F\left(\xi \right)=0.$

As in [27], we denote $\nabla \left(\rho \right)$ and $\nabla \left(\pi \right)$ (where $\rho$ and $\pi$ are two new control functions) by the collection of all mappings $F:\left(0,\infty \right)\to R,$$\eta :\left(0,\infty \right)\to R$, respectively, satisfying:

• (F₂) For each sequence $\left\{{\xi }_n\right\}$ of positive numbers, ${lim}_{n\to \infty }{\xi }_n=0$ if ${lim}_{n\to \infty }F\left({\xi }_n\right)=-\infty ;$
• (F₃) F is lower semicontinuous;
• (η₁) For each sequence $\left\{{\xi }_n\right\}$ of positive numbers, ${lim}_{n\to \infty }{\xi }_n=0$ if ${lim}_{n\to \infty }\eta \left({\xi }_n\right)=-\infty ;$
• (η₂) $\eta$ is right upper semicontinuous.

Now, we present some extensive examples of control functions in $\rho$ and $\eta$.

Example 1. 

The following functions belong to $\nabla \left(\rho \right)$ and $\nabla \left(\pi \right)$

$$\begin{array}{cc}\left(1\right)F_1\left(\xi \right)=\left\{\begin{array}{cc}\frac{-1}{\xi },& \mathrm{if}\xi \in \left[3,\infty \right)\\ \frac{-1}{\left(\xi +1\right)},& \mathrm{if}\xi \in \left(3,\infty \right)\end{array}\right.& \left(2\right)F_2\left(\xi \right)=\left\{\begin{array}{cc}\frac{-1}{\xi }+\xi ,& \mathrm{if}\xi \in \left[2.8,\infty \right)\\ 2\xi -3,& \mathrm{if}\xi \in \left(3,\infty \right)\end{array}\right.\\ \left(3\right){\eta }_1\left(\varpi \right)=\left\{\begin{array}{cc}\frac{-1}{\xi },& \mathrm{if}\xi \in \left(0,4.6\right)\\ cos\xi ,& \mathrm{if}\xi \in \left[4.6,\infty \right)\end{array}\right.& \left(4\right){\eta }_2\left(s\right)=\left\{\begin{array}{cc}ln\left(\frac{\xi }{3}+sin\xi \right),& \mathrm{if}\xi \in \left(0,3.2\right)\\ sin\xi ,& \mathrm{if}\xi \in \left[3.2,\infty \right)\end{array}\right..\end{array}$$

Let $\Re =\left\{\left(\xi ,\Im \right)\in U^2:\xi ,\Im \in U\right\}$ be a relation on $U.$ If $\left(\xi ,\Im \right)\in \Re$ then we say that $\xi ⪯\Im$ $\left(\xi \mathrm{precede}\Im \right)$ under ℜ denoted by $\xi \Re \Im ,$ and the inverse of ℜ is denoted by ${\Re }^{-1}=\left\{\left(\xi ,\Im \right)\in U^2:\left(\Im ,\xi \right)\in \Re \right\}.$ The set $S=\Re \cup {\Re }^{-1}\subseteq U^2$ consequently illustrates another relation $S^{*}$ on U given by $\xi S^{*}\Im \iff \Im S\xi$ with $\xi \ne \Im .$

As ${\left(\gamma \right)}_{Fix}$ denotes a set of all fixed points of $\gamma$, $\Theta \left(\left[\Psi ,S\right]\right)=\left\{\xi \in U:\xi S\gamma \left(\xi \right)\right\}$ and $ϝ\left(\xi ,\Im ,\nabla \right)$ denotes the fashion of all paths in ∇ from $\xi$ to $\Im .$

Definition 3 

([22]). Let U $\ne \varphi$ and $\gamma :U\to U$, and ℜ is a binary relation on U. Then, ℜ is γ-closed if for any $\Omega ,\Im \in U$,

$$\xi \Re \Im \Rightarrow \gamma \left(\xi \right)\Re \gamma \left(\Im \right).$$

Definition 4 

([22]). Let $U\ne \varphi$ and ℜ be a binary relation on U. Then, ℜ is transitive if $\xi \Re \aleph \in$ and $\aleph \Re \Im \Rightarrow$$\aleph \Re \Im$ for all $\xi ,\Im ,\aleph \in U$.

Definition 5 

([22]). Let $\xi ,\Im \in U$. A path of length $n\in \mathbb{N}$ in ℜ: ξ$\to \Im$ is a finite sequence $\left\{t_0,t_1,t_2,\dots ,t_n\right\}\subseteq U$ such that

(i)

$t_0=\xi$ and $t_n=\Im$;

(ii)

$\left(t_j,t_{j+1}\right)\in \Re$ for all j in this set $\left\{0,1,2,\dots ,n-1\right\}.$

Consider that a class of all paths from ξ to ℑ in ℜ is written as $\nabla \left(\xi ,\Im ,\Re \right).$ Note that a path of length n involves $n+1$ elements of U, although they are not necessarily distinct.

Definition 6 

([36]). Let $\left(U,m\right)$ be a relation theoratic m-metric space endowed with binary relation ℜ on U, which is regular if for each sequences $\left\{{\xi }_n\right\}$ in U, we have

$$\left.\begin{array}{c}{\xi }_n\Re {\xi }_{n+1}\mathit{for}\mathit{all}n\in \mathbb{N}\\ \underset{n\to \infty }{lim}\left(m\left({\xi }_n,\xi \right)-m_{{\xi }_n\xi }\right)=0\mathit{i.e.},{\xi }_n\stackrel{t_m}{\to }\xi \in \Re \end{array}\right\}\Rightarrow {\xi }_n\Re \xi \mathit{for}\mathit{all}n\in \mathbb{N}.$$

Definition 7 

([36]). Let $\left(U,m\right)$ be a relation theoratic m-metric space endowed with binary relation ℜ on U. A sequence ${\xi }_n\in U$ is called ℜ-preserving if ${\xi }_n\Re {\xi }_{n+1}.$

Definition 8 

([36]). Let $\left(U,m\right)$ be a relation theoratic m-metric space endowed with binary relation ℜ on U, which is said to be ℜ-complete if for each ℜ-preserving m-Cauchy sequence $\left\{{\xi }_n\right\}$ in U, there exists some ξ in U such that

$$\underset{n\to \infty }{lim}m\left({\xi }_n,\xi \right)-m_{{\xi }_n\xi }=0,\mathit{and}\underset{n\to \infty }{lim}\left(M_{{\xi }_n,\xi }-m_{{\xi }_n\xi }\right)=0.$$

Definition 9 

([36]). Let U $\ne \varphi$ and $\gamma :U\to U$. Then, γ is said to be ℜ-continuous at ξ if, for ℜ-preserving sequence $\left\{{\xi }_n\right\}$ with ${\xi }_n$→ξ, we have $\gamma \left({\xi }_n\right)\to \gamma \left(\xi \right)$ as μ$\to \infty .$γ is ℜ-continuous if it is ℜ-continuous at each point of $U.$

2. Weak $\left({\mathit{F}}_{\mathbf{\Re }}^{\mathit{m}},\mathsf{\eta }\right)$-Contractions

In this section, we introduce the concept of weak $\left(F_{\Re }^m,\eta \right)$-contraction relations and establish related fixed point theorems in relation theoretic m-metric space, where $\eta$ is a control function and ℜ is a relation. We begin with the following Lemma.

Lemma 2. 

Assume that $\left(U,m\right)$ is an m-metric space and let $\left\{{\xi }_n\right\}$ be a sequence in U such that ${lim}_{n\to \infty }m\left({\xi }_n,{\xi }_{n+1}\right)=0.$ If $\left\{{\xi }_n\right\}$ is not an m-Cauchy sequence in U, then there exists $\epsilon >0$ and two subsequences $\left\{{\xi }_{\alpha \left(\chi \right)}\right\}$ and $\left\{{\xi }_{\beta \left(\chi \right)}\right\}$ of positive integers such that $\left\{{\alpha }_{\chi }\right\}>\left\{{\beta }_{\chi }\right\}>\chi$ and the following sequences converges to ${\epsilon }^{+}$ as $\chi$ converges to $+\infty .$ With $M^{*}\left(\xi ,\Im \right)=m\left(\xi ,\Im \right)-m_{\xi \Im };$

$$\begin{array}{ccc}\hfill & & M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right),M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)+1}\right),M^{*}\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)}\right),\hfill \\ & & M^{*}\left({\xi }_{\beta \left(\chi \right)+1}{\xi }_{\beta \left(\chi \right)-1}\right),M^{*}\left({\xi }_{\beta \left(\chi \right)+1},{\xi }_{\beta \left(\chi \right)+1}\right).\hfill \end{array}$$

(1)

Proof. 

If $\left\{{\xi }_n\right\}$ is not an m-Cauchy sequence in U, there exists $\epsilon >0$ and two sequences $\left\{{\alpha }_{\chi }\right\}$ and $\left\{{\beta }_{\chi }\right\}$ of positive integers such that $\left\{{\alpha }_{\chi }\right\}>\left\{{\beta }_{\chi }\right\}>\chi$ and

$$M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)-1}\right)<\epsilon ,M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\ge \epsilon ,$$

(2)

for all positive integers $\chi$. Using the triangle inequality of m-metric space, we obtain

$$\begin{array}{ccc}\hfill \epsilon & \le & M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\le M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)+M^{*}\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)}\right)\hfill \\ & <& M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)+\epsilon .\hfill \end{array}$$

Thus,

$$\underset{\chi \to \infty }{lim}{M}^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)=\epsilon ,$$

which implies

$$\underset{\chi \to \infty }{lim}\left(m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)-m_{{\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}}\right)=\epsilon .$$

Furthermore,

$$\underset{\chi \to \infty }{lim}{m}_{{\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}}=0.$$

Hence,

$$\underset{\chi \to \infty }{lim}m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)=\epsilon .$$

(3)

Again, using the triangle inequality,

$$\begin{array}{ccc}\hfill M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)& \le & M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)+1}\right)+M^{*}\left({\xi }_{\alpha \left(\chi \right)+1},{\xi }_{\beta \left(\chi \right)+1}\right)\hfill \\ & & +M^{*}\left({\xi }_{\alpha \left(\chi \right)+1},{\xi }_{\beta \left(\chi \right)}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill M^{*}\left({\xi }_{\alpha \left(\chi \right)+1},{\xi }_{\beta \left(\chi \right)+1}\right)& \le & M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)+1}\right)+M^{*}\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\hfill \\ & & +M^{*}\left({\xi }_{\alpha \left(\chi \right)+1},{\xi }_{\beta \left(\chi \right)}\right).\hfill \end{array}$$

Taking $\chi \to +\infty$ in the above inequality and from (3), we have

$$\underset{\chi \to \infty }{lim}{M}^{*}\left({\xi }_{\alpha \left(\chi \right)+1},{\xi }_{\beta \left(\chi \right)+1}\right)=\epsilon .$$

□

Now, we introduce the concept of weak $\left(F_{\Re }^m,\eta \right)$-contractions.

Definition 10. 

Given a relation theoretic m-metric space $\left(U,m\right)$ endowed with binary relation ℜ on $U.$ Suppose

$$\Xi =\left\{\xi S^{*}\Im :m\left(\xi ,\Im \right)>0\right\}.$$

We can say that a self mapping $\gamma :U\to U$ is a weak $\left(F_{\Re }^m,\eta \right)$-contraction if there exists $F_{\Re }^m\in \nabla \left(\rho \right)$, $\eta \in \nabla \left(\pi \right)$ and

$$\tau +F_{\Re }^m\left(m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\right)\le \eta \left(m\left(\xi ,\Im \right)\right),$$

(4)

for all $\left(\xi ,\Im \right)\in \Xi .$

Our main result is demonstrated in the following.

Theorem 1. 

Let $\left(U,m\right)$ be a complete relation theoretic m-metric space endowed with transitive binary relation ℜ on U, $\gamma :U\to U$, satisfying the following conditions:

(i)

$\Theta \left(\left[\gamma ,\Re \right]\right)$ is non-empty;

(ii)

ℜ is γ-closed;

(iii)

γ is ℜ-continuous;

(iv)

γ is a weak $\left(F_{\Re }^m,\eta \right)$-contraction mapping with $F_{\Re }^m\left(\xi \right)>\eta \left(\xi \right)$ for all $\xi >0$.

Then, $\gamma$ possesses a fixed point in U.

Proof. 

Let ${\xi }_0\in \Theta \left(\left[\gamma ,\Re \right]\right)$. Define a sequence $\left\{{\xi }_{n+1}\right\}$ in U by ${\xi }_{n+1}=\gamma \left({\xi }_n\right)={\gamma }^{n+1}\left({\xi }_0\right)$ for each $n\in \mathbb{N}.$ If there exists a member $n_0$ of $\mathbb{N}$ such that $\gamma \left({\xi }_{n_0}\right)={\xi }_{n_0}$, then $\gamma$ has a fixed point ${\xi }_{n_0}$ and the proof is complete. Let

$${\xi }_{n+1}\ne {\xi }_n,$$

(5)

for all member n of $\mathbb{N}$ such that $m\left({\xi }_{n+1},{\xi }_n\right)>0.$ Since $\gamma \left({\Omega }_0\right)S^{*}{\Omega }_0$, and by the $\gamma$-closedness of ℜ, ${\Omega }_{n+1}{S}^{*}{\Omega }_n$ for all $n\in \mathbb{N}$. Thus, $\left({\xi }_n,{\xi }_{n+1}\right)\in \Xi$ and from ($iv$) we obtain

$$\begin{array}{ccc}\hfill F_{\Re }^m\left(m\left({\xi }_{n+1},{\xi }_n\right)\right)& =& F_{\Re }^m\left(m\left(\gamma \left({\xi }_n\right),\gamma \left({\xi }_{n-1}\right)\right)\right)\hfill \\ & \le & F_{\Re }^m\left(m\left({\xi }_n,{\xi }_{n-1}\right)\right)-\tau \hfill \end{array}$$

Let ${\delta }_n=m\left({\xi }_n,{\xi }_{n+1}\right)$ for all $n\in \mathbb{N}.$ Then, ${\delta }_{\mu }>0$ for all $n\in \mathbb{N},$ and using (5), one obtains

$$F_{\Re }^m\left({\delta }_n\right)\le \left({\delta }_{n-1}\right)-\tau <F_{\Re }^m\left({\delta }_{n-1}\right)-\tau \le \eta \left({\delta }_{n-2}\right)-2\tau \le \dots \le \eta \left({\delta }_{n-2}\right)-n\tau .$$

From the above inequality, we obtain ${lim}_{n\to \infty }{F}_{\Re }^m\left({\delta }_n\right)=-\infty$. Then, by $\left(F_2\right),$ we have

$$\underset{n\to \infty }{lim}{\delta }_n=0.$$

(6)

From (3) and (6), we have ${\xi }_{n+1}\ne {\xi }_n$ for all $n,m\in \mathbb{N}$ with $n\ne m.$ Now, we shall prove that $\left\{{\xi }_n\right\}$ is am m-Cauchy sequence in $\left(U,m\right)$. Assume, in contrast, that $\left\{{\xi }_n\right\}$ is not an m-Cauchy sequence. By Lemmas 2.1 and 2.6, there exists $\epsilon >0$ and two subsequences $\left\{{\xi }_{\alpha \left(\chi \right)}\right\}$ and $\left\{{\xi }_{\beta \left(\chi \right)}\right\}$ of $\left\{{\xi }_n\right\}$ such that $\left\{{\xi }_{\alpha \left(\chi \right)}\right\}>\left\{{\xi }_{\beta \left(\chi \right)}\right\}>\chi$ and

$$\begin{array}{ccc}\hfill \underset{\chi \to \infty }{lim}m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)& =& \epsilon \hfill \\ \hfill \underset{\chi \to \infty }{lim}m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)& =& \epsilon .\hfill \end{array}$$

Since ℜ is a transitive relation, $\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\in \Re .$ From condition $\left(iv\right)$, we have

$$\tau +F_{\Re }^m\left(m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\right)\le \eta \left(m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\right)$$

and so

$$\begin{array}{ccc}\hfill \tau +\underset{\chi \to \infty }{lim}inf{F}_{\Re }^m\left(m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\right)& \le & \underset{\chi \to \infty }{lim}inf\eta \left(m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\right)\hfill \\ & \le & \underset{\chi \to \infty }{lim}sup\eta \left(m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \tau +F_{\Re }^m\left({\epsilon }^{*}\right)& \le & \eta \left({\epsilon }^{*}\right)\hfill \\ & <& F_{\Re }^m\left({\epsilon }^{*}\right)\hfill \end{array}$$

is a contradiction; hence, $\left\{{\xi }_n\right\}$ is an m-Cauchy sequence in $\left(U,m\right)$. Since $\left(U,m\right)$ is ℜ-complete, there exists ${\xi }^{*}\in U$ such that $\left\{{\xi }_{\mu }\right\}$ converges to ${\xi }^{*}$ with respect to $t_m$; that is, $m\left({\xi }_n,{\xi }^{*}\right)-m_{{\xi }_n,{\xi }^{*}}\to 0$ as $n\to \infty .$ Now, the ℜ-continuity of $\gamma$ implies that

$$\xi =\underset{n\to \infty }{lim}{\xi }_{n+1}=\underset{n\to \infty }{lim}\gamma \left({\xi }_n\right)=\gamma \left(\xi \right).$$

Therefore, $\xi$ is a fixed point of $\gamma .$ □

Example 2. 

Let $U=\left[0,\infty \right)$ and m be a relation theoretic m-metric space defined by $m\left(\xi ,\Im \right)=\frac{\xi +\Im }{2}$ for all $\xi ,\Im \in U.$ Then, $\left(U,m\right)$ is a complete m-metric space. Consider a sequence $\left\{{\varpi }_n\right\}\subseteq U$ given by ${\varpi }_n=\frac{n\left(n+1\right)\left(n+2\right)}{3}$ for all $\mu \in \mathbb{N}.$ Set a binary relation ℜ on U by $\Re =\left\{\left(1,1\right)\right\}\cup \left\{\left(1,{\varpi }_{\Gamma }\right):\Gamma \in \mathbb{N}\right\}\cup \left\{\left({\varpi }_{\Gamma },{\varpi }_{\Lambda }\right):\Gamma <\Lambda \mathrm{for}\mathrm{each}\Gamma ,\Lambda \in \mathbb{N}\right\}$. Define a mapping $\gamma :U\to U$ by

$$\gamma \left(\xi \right)=\left\{\begin{array}{cc}\xi ,& \mathit{if}\xi \in \left[0,1\right]\\ ceil\left(ln\xi \right),& \mathit{if}\xi \in \left[1,{\varpi }_1\right]\\ \left(\frac{\xi -{\varpi }_1}{{\varpi }_2-{\varpi }_1}\right)+1,& \mathit{if}\xi \in \left[{\varpi }_1,{\varpi }_2\right]\\ \frac{{\varpi }_{n-1}\left({\varpi }_{n+1}-\xi \right)+{\varpi }_n\left(\xi -{\varpi }_n\right)}{{\varpi }_{n+1}-{\varpi }_n},& \mathit{if}\xi \in \left[{\varpi }_n,{\varpi }_{\mu +1}\right]\mathit{for}\mathit{all}n=2,3,\dots 100.\end{array}\right.$$

Obviously, ℜ is γ-closed and γ is continuous. Define $F_{\Re }^m,\eta :\left(0,\infty \right)\to R$ by

$$\begin{array}{ccc}\hfill F_{\Re }^m\left(\varpi \right)& =& \{\frac{-1}{\varpi }+\frac{4}{5}\varpi \mathit{if}\varpi \in \left(0,1.1\right]\frac{-1}{\varpi }+\varpi \mathit{if}\varpi \in \left(1.1,\infty \right)\mathit{and}\hfill \\ \hfill \eta \left(\varpi \right)& =& \{\frac{-1}{\varpi }+\frac{1}{3}\varpi \mathit{if}\varpi \in \left(0,6.5\right)\frac{-2}{\varpi }+\varpi \mathit{if}\varpi \in \left[6.5,\infty \right)\hfill \end{array}$$

Now, we will show that γ is a $\left(F_{\Re }^m,\eta \right)$-contraction mapping. Assume that $\left(\xi ,\Im \right)\in \Xi =\left\{\xi S^{*}\Im :m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)>0\right\}.$ Therefore, we will discuss four cases.

Case 1 If $\xi =1$ and $\Im ={\varpi }_2$, then $m\left(\xi ,\Im \right)=4.5$ and $m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)=1.5,$

$$\begin{array}{ccc}\hfill 2+F_{\Re }^m\left(m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\right)& =& 2-\frac{1}{m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)}+\frac{4}{5}m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\hfill \\ & \le & -\frac{2}{m\left(\xi ,\Im \right)}+m\left(\xi ,\Im \right)=\eta \left(m\left(\xi ,\Im \right)\right)\hfill \end{array}$$

Case 2 If $\xi =1$ and $\Im ={\varpi }_{\Gamma }$ for all $\Gamma >2,$ then $m\left(\xi ,\Im \right)=\left|\frac{1+{\varpi }_{\Gamma }}{2}\right|\ge 10.5$ and $m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)=\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|\ge 4.5$,

$$\begin{array}{ccc}\hfill 2\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|-\left|\frac{1+{\varpi }_{\Gamma }}{2}\right|& <& 2\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|<\left|\frac{1+{\varpi }_{\Gamma }}{2}\right|\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|\hfill \\ & <& \left|\frac{1+{\varpi }_{\Gamma }}{2}\right|\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|\left(\left|\frac{1+{\varpi }_{\Gamma }}{2}\right|\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|-2\right)\hfill \end{array}$$

which implies

$$2+\frac{2}{\left|\frac{1+{\varpi }_{\Gamma }}{2}\right|}-\frac{1}{\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|}\le \left|\frac{1+{\varpi }_{\Gamma }}{2}\right|-\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|,$$

and thus,

$$2-\frac{1}{\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|}-\left|\frac{1+{\varpi }_{\Gamma -1}}{2}\right|\le -\frac{2}{\left|\frac{1+{\varpi }_p}{2}\right|}-\left|\frac{1+{\varpi }_{\Gamma }}{2}\right|.$$

Then,

$$\begin{array}{ccc}\hfill 2+F_{\Re }^m\left(m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\right)& =& 2-\frac{1}{m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)}+m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\hfill \\ & \le & -\frac{2}{m\left(\xi ,\Im \right)}+m\left(\xi ,\Im \right)=\eta \left(m\left(\xi ,\Im \right)\right).\hfill \end{array}$$

Case 3 If $\xi ={\varpi }_1$ and $\Im ={\varpi }_2$, then $m\left(\xi ,\Im \right)=5$ and $m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)=1$,

$$\begin{array}{ccc}\hfill 2+F_{\Re }^m\left(m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\right)& =& 2-\frac{1}{m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)}+\frac{4}{5}m\left(\gamma \left(\Omega \right),\gamma \left(\Im \right)\right)\hfill \\ & \le & -\frac{2}{m\left(\xi ,\Im \right)}+m\left(\xi ,\Im \right)=\eta \left(m\left(\xi ,\Im \right)\right).\hfill \end{array}$$

Case 4 If $\xi ={\varpi }_{\Gamma }$ and $\Im ={\varpi }_{\Lambda }$ for all Γ and Λ in $\mathbb{N}$ and $\left(\Gamma ,\Lambda \right)$ is not equal to $\left(1,2\right)$ with $\Gamma <\Lambda ,$ then $m\left(\xi ,\Im \right)=\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|\ge 14$ and $m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)=\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|\ge 7$,

$$\begin{array}{ccc}\hfill 2\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Gamma -1}}{2}\right|-\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|& <& 2\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|<\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|\hfill \\ & <& \left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|\left(\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|-2\right),\hfill \end{array}$$

which implies

$$2+\frac{2}{\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|}-\frac{1}{\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|}\le \left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|-\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|.$$

Then,

$$2-\frac{1}{\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|}+\left|\frac{{\varpi }_{\Gamma -1}+{\varpi }_{\Lambda -1}}{2}\right|\le -\frac{2}{\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|}+\frac{2}{\left|\frac{{\varpi }_{\Gamma }+{\varpi }_{\Lambda }}{2}\right|}.$$

Hence,

$$\begin{array}{ccc}\hfill 2+F_{\Re }^m\left(m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\right)& =& 2-\frac{1}{m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)}+m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\hfill \\ & \le & -\frac{2}{m\left(\xi ,\Im \right)}+m\left(\xi ,\Im \right)=\eta \left(m\left(\xi ,\Im \right)\right).\hfill \end{array}$$

Therefore, from all cases, we deduce that

$$\tau +F_{\Re }^m\left(m(\gamma \left(\xi \right),\gamma \left(\Im \right))\right)\le \eta \left(m\left(\xi ,\Im \right)\right),$$

for all $\xi ,\Im \in \Xi$. Then, γ is a weak $\left(F_{\Re }^m,\eta \right)$-contraction mapping with $\tau =2.$ Furthermore, there exists ${\xi }_0=1$ in U such that ${\Omega }_0{S}^{*}\gamma \left({\Omega }_0\right)$ and the class $\Theta \left(\left[\gamma ,\Re \right]\right)$ is non-empty. Thus, all conditions of Theorem 2.3 hold and γ has a fixed point.

Theorem 2. 

Theorem 1 remains true if the condition $\left(ii\right)$ is replaced by the following: (ii)′ $\left(X,\kappa ,\nabla \right)$ is regular.

Proof. 

Similar to the argument of Theorem 1 we will show the sequence $\left\{{\xi }_n\right\}$ is m-cauchy and converges to some $\xi$ in U such that $m\left({\xi }_n,\xi \right)-m_{{\xi }_n,\xi }$ as $n\to \infty$. Now,

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}m\left({\xi }_n,\xi \right)& =& \underset{n\to \infty }{lim}{m}_{{\xi }_n,\xi }=\underset{n\to \infty }{lim}min\left\{m\left({\xi }_n,{\xi }_n\right),m\left(\xi ,\xi \right)\right\}=m\left(\xi ,\xi \right)\hfill \\ & =& \underset{n,m\to \infty }{lim}m\left({\xi }_n,{\xi }_m\right)=0\mathrm{and}\underset{n,m\to \infty }{lim}{m}_{{\xi }_n,{\xi }_m}=0.\hfill \end{array}$$

As ${\xi }_n{S}^{*}{\xi }_{n+1},$ then ${\xi }_n{S}^{*}\xi$ for all $n\in \mathbb{N}$. Set $L=\left\{n\in \mathbb{N}:\gamma \left({\xi }_n\right)=\gamma \left(\xi \right)\right\}$. We have two cases dependent on L.

Case 1: If $\left\{L\mathrm{is}\mathrm{finite}\right\}$, then there exists $n_0$$\in \mathbb{N}$ such that $\gamma \left({\xi }_n\right)\ne \gamma \left(\xi \right)$ for every $n\ge n_0.$ Moreover, ${\xi }_n{S}^{*}\xi$ and $\gamma \left({\xi }_n\right)S^{*}\gamma \left(\xi \right)$ for all $n\ge n_0.$ Since $\gamma$ is a weak $\left(F_{\Re }^m,\eta \right)$-contraction mapping, we have

$$\tau +F_{\Re }^m\left(m\left(\gamma \left({\xi }_{\mu }\right),\gamma \left(\xi \right)\right)\right)\le \eta \left(m\left({\xi }_{\mu },\xi \right)\right).$$

Since, ${lim}_{n\to \infty }m\left({\xi }_n,\xi \right)=0$,

$$\underset{n\to \infty }{lim}{F}_{\Re }^m\left(m\left({\xi }_n,\xi \right)\right)=-\infty .$$

Hence,

$$\underset{n\to \infty }{lim}{F}_{\Re }^m\left(m\left(\gamma \left({\xi }_n\right),\gamma \left(\xi \right)\right)\right)=-\infty .$$

Therefore, ${lim}_{n\to \infty }m\left(\gamma \left({\xi }_n\right),\gamma \left(\xi \right)\right)=0$ and $\gamma \left(\xi \right)=\xi$, where $\xi$ is a fixed point of $\gamma .$

Case 2: If $\left\{L\mathrm{is}\mathrm{infinite}\right\}$, then there exists a subsequence $\left\{{\xi }_{n_k}\right\}$⊂$\left\{{\xi }_n\right\}$ such that ${\xi }_{n_k+1}=\gamma \left({\xi }_{n_k}\right)=\gamma \left(\xi \right)$ for all $k\in \mathbb{N}.$ Thus, $\gamma \left({\xi }_{n_k}\right)\to \gamma \left(\xi \right)$ with respect to $t_m$ as ${\xi }_n\to \xi ,$ then $\gamma \left(\xi \right)=\xi$, i.e., $\gamma$ has a fixed point. Hence, the proof is complete. □

Now, we discuss various results to ensure the uniqueness of the fixed points:

Theorem 3. 

If $ϝ\left(\xi ,\Im ,\nabla \right)\ne \varphi$ for all $\xi ,\Im \in {\left(\gamma \right)}_{Fix}$ in Theorem 1 and Theorem 2, then γ possesses a unique fixed point.

Proof. 

Let $\xi ,\Im \in$ Fix $\left(\gamma \right)$ such that $\xi \ne \Im .$ Since $ϝ\left(\xi ,\Im ,\nabla \right)\ne \varphi ,$ then there exists a path $\left(\left\{a_0,a_1,\dots a_n\right\}\right)$ of some finite length μ in ∇ from ξ to ℑ $\left(\mathrm{with}{a}_s\ne a_{s+1}\mathrm{for}\mathrm{all}s\in \left[0,p-1\right]\right)$. Then, $a_0=\xi$, $a_k=\Im ,$$a_s{S}^{*}{a}_{s+1}$ for every $s\in \left[0,p-1\right].$ As $a_s\in \gamma \left(U\right),$$\gamma \left(a_s\right)=a_s$ for all $s\in \left[0,p-1\right]$ and since $F_{\eta }^m\left(\xi \right)>\eta \left(\xi \right),$ we obtain

$$F_R^m\left(m\left(a_s,a_{s+1}\right)\right)=F_{\Re }^m\left(m\left(\gamma \left(a_s\right),\gamma \left(a_{s+1}\right)\right)\right)\le \eta \left(m\left(a_s,a_{s+1}\right)\right)$$

Since $F_{\Re }^m\left(a\right)>\eta \left(a\right)$ for all $a>0,$

$$F_{\Re }^m\left(m\left(a_s,a_{s+1}\right)\right)<F_{\Re }^m\left(m\left(a_s,a_{s+1}\right)\right).$$

Hence, γ possesses a unique fixed point. □

Theorem 4. 

Let $\left(U,m\right)$ be a complete relation theoretic m-metric space endowed with a transitive binary relation ℜ on U. Let $\gamma :U\to U$ satisfy the following:

(i)

The class $\Theta \left(\left[\gamma ,\Re \right]\right)$ is nonempty;

(ii)

The binary relation ℜ is γ-closed;

(iii)

The mapping γ is ℜ-continuous;

(iv)

There exists $F_{\Re }^m\in \nabla \left(\rho \right)$, $\eta \in \nabla \left(\pi \right)$ and $\xi >0$ such that

$$\tau +F_{\Re }^m\left(\kappa \left(m\left(\xi \right),{\gamma }^2\left(\xi \right)\right)\right)\le \eta \left(m\left(\xi ,\gamma \left(\xi \right)\right)\right)$$

for all $\xi \in U,$ with $\gamma \left(\xi \right)S^{*}{\gamma }^2\left(\xi \right)$ and $F_{\eta }^m\left(\xi \right)>\eta \left(\xi \right)$ for all $\xi >0.$

Then, $\gamma$ has a fixed point.

Furthermore, if the following conditions are satisfied:

(v)

${\left(iv\right)}^{{}^{\prime }}$

(vi)

$\xi \in {\left({\gamma }^n\right)}_{\mathrm{Fix}}$$\left(\mathrm{for}\mathrm{some}n\in \mathbb{N}\right)$ which implies that $\xi S^{*}\gamma \left(\xi \right).$

Then, ${\left({\gamma }^n\right)}_{\mathrm{Fix}}={\left(\gamma \right)}_{\mathrm{Fix}}$ for each n is a member of $\mathbb{N}.$

Proof. 

Let ${\xi }_0\in \Theta \left(\left[\gamma ,\Re \right]\right)$, i.e., ${\xi }_0{S}^{*}\gamma \left({\xi }_0\right)$, then, from $\left(ii\right)$, we obtain ${\xi }_n{S}^{*}{\xi }_{n+1}$ for each $n\in \mathbb{N}$. Denote ${\xi }_{n+1}=\gamma \left({\xi }_n\right)=$${\gamma }^{n+1}\left({\xi }_0\right)$ for all $n\in \mathbb{N}.$ If there exists $n_0\in \mathbb{N}$ such that $\gamma \left({\xi }_{n_0}\right)={\xi }_{n_0},$ then $\gamma$ has a fixed point ${\xi }_{n_0}$. Now, assume that

$${\xi }_{n+1}\ne {\xi }_n,$$

(7)

for every $n\in \mathbb{N}$. Then, ${\xi }_n{S}^{*}{\xi }_{n+1}\left(\mathrm{for}\mathrm{all}n\in \mathbb{N}\right)$. Continuing this process and from $\left(iv\right)$ we have,

$$F_{\Re }^m\left(m\left(\gamma \left({\xi }_{n-1}\right),{\gamma }^2\left({\xi }_{n-1}\right)\right)\right)\le F_{\Re }^m\left(m\left({\xi }_{n-1},\gamma \left({\xi }_{n-1}\right)\right)\right)\le m\left({\xi }_{n-1},{\xi }_n\right)-\tau ,$$

for all $n\in \mathbb{N}$, which implies,

$$\begin{array}{ccc}\hfill F_{\Re }^m\left(m\left({\xi }_n,{\xi }_{n+1}\right)\right)& \le & \eta \left(m\left({\xi }_{n-1},{\xi }_n\right)\right)-\tau \hfill \\ & <& F_{\Re }^m\left(m\left({\xi }_{n-2},{\xi }_{n-1}\right)\right)-\tau \hfill \\ & \le & \eta \left(m\left({\xi }_{n-1},{\xi }_n\right)\right)-2\tau \hfill \\ & & \dots \hfill \\ & \le & \eta \left(m\left({\xi }_0,{\xi }_1\right)\right)-n\tau .\hfill \end{array}$$

Setting $n\to \infty$ in the above inequality, we deduce that ${lim}_{n\to \infty }{F}_{\Re }^m\left(m\left({\xi }_n,{\xi }_{n+1}\right)\right)=-\infty .$ Since $F_{\Re }^m\in \nabla \left(\rho \right),$ then

$$\underset{n\to \infty }{lim}m\left({\xi }_n,{\xi }_{n+1}\right)=0.$$

(8)

From conditions (7) and (8), we have ${\xi }_{n+1}\ne {\xi }_n$ for all $n,m\in \mathbb{N}$ with $n\ne m.$ Now, we will prove that $\left\{{\xi }_n\right\}$ is an m-Cauchy sequence in $\left(U,m\right)$. Assume, in contrast, that $\left\{{\xi }_n\right\}$ is not an m-Cauchy sequence; then, by Lemma 2 and (6), there exists $\epsilon >0$ and two subsequences $\left\{{\xi }_{\alpha \left(\chi \right)}\right\}$ and $\left\{{\xi }_{\beta \left(\chi \right)}\right\}$ of $\left\{{\xi }_n\right\}$ such that $\left\{\alpha \left(\chi \right)\right\}>\left\{\beta \left(\chi \right)\right\}>\chi$ and

$$\begin{array}{ccc}\hfill \underset{\chi \to \infty }{lim}m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)& =& \epsilon \mathrm{and}\hfill \\ \hfill \underset{\chi \to \infty }{lim}m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)& =& \epsilon .\hfill \end{array}$$

Since ℜ is a transitive relation, $\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\in \Re .$ From condition $\left(iv\right)$,

$$\tau +F_{\Re }^m\left(m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\right)\le \eta \left(m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\right)$$

and hence,

$$\begin{array}{ccc}\hfill \tau +\underset{\chi \to \infty }{lim}inf{F}_{\Re }^m\left(m\left({\xi }_{\alpha \left(\chi \right)},{\xi }_{\beta \left(\chi \right)}\right)\right)& \le & \underset{\chi \to \infty }{lim}inf\eta \left(m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\right)\hfill \\ & \le & \underset{\chi \to \infty }{lim}sup\eta \left(m\left({\xi }_{\alpha \left(\chi \right)-1},{\xi }_{\beta \left(\chi \right)-1}\right)\right).\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill \tau +F_{\Re }^m\left({\epsilon }^{*}\right)& \le & \eta \left({\epsilon }^{*}\right)\hfill \\ & <& F_{\Re }^m\left({\epsilon }^{*}\right)\hfill \end{array}$$

it is contradiction. Hence, $\left\{{\xi }_n\right\}$ is an m-Cauchy sequence in $\left(U,m\right).$ Since $\left(U,m\right)$ is ℜ-complete, there exists $\xi \in U$ such that $\left\{{\xi }_n\right\}$ converges to ${\xi }^{*}$ with respect to $t_m$; that is, $m\left({\xi }_n,{\xi }^{*}\right)-m_{{\xi }_n,{\xi }^{*}}\to 0$ as $n\to \infty .$ By using the ℜ-continuity of $\gamma$,

$$\xi =\underset{n\to \infty }{lim}{\xi }_{n+1}=\underset{n\to \infty }{lim}\gamma \left({\xi }_n\right)=\gamma \left(\xi \right).$$

Finally, we will prove that ${\left({\gamma }^n\right)}_{Fix}={\left(\gamma \right)}_{Fix}$ where $n\in \mathbb{N}.$ Assume, in contrast, that $\xi \in {\left({\gamma }^n\right)}_{Fix}$ and $\xi \notin {\left(\gamma \right)}_{Fix}$ for some $n\in \mathbb{N}$. Then, from condition ${\left(iv\right)}^{{}^{\prime }}$, $m\left(\xi ,\gamma \left(\xi \right)\right)>0$ and $\xi S^{*}\gamma \left(\xi \right)$. Using $\left(ii\right)$ and $\left(iv\right)$, we obtain ${\gamma }^n\left(\xi \right)S^{*}{\gamma }^{n+1}\left(\xi \right)$ for all $n\in \mathbb{N}$,

$$\begin{array}{ccc}\hfill F_{\Re }^m\left(m\left(\xi ,\gamma \left(\xi \right)\right)\right)& =& F_{\Re }^m\left(m\left(\gamma \left({\gamma }^{n-1}\left(\xi \right)\right),{\gamma }^2\left({\gamma }^{n-1}\left(\xi \right)\right)\right)\right)\le \eta \left(m\left(\gamma \left({\gamma }^{n-1}\left(\xi \right)\right),{\gamma }^2\left({\gamma }^{n-1}\left(\xi \right)\right)\right)\right)-\tau \hfill \\ & <& F_{\Re }^m\left(m\left({\gamma }^{n-1}\left(\xi \right)\right),{\gamma }^n\left(\xi \right)\right)-\tau \hfill \\ & \le & \eta \left(m\left({\gamma }^{n-2}\left(\xi \right)\right),{\gamma }^{n-1}\left(\xi \right)\right)-2\tau \hfill \\ & <& F_{\Re }^m\left(m\left({\gamma }^{n-2}\left(\xi \right)\right),{\gamma }^{n-1}\left(\xi \right)\right)-2\tau \hfill \\ & \le & \eta \left(m\left({\gamma }^{n-3}\left(\xi \right)\right),{\gamma }^{n-2}\left(\xi \right)\right)-3\tau \hfill \\ & & \dots \hfill \\ & \le & \eta \left(m\left(\xi ,\gamma \left(\xi \right)\right)\right)-n\tau \hfill \end{array}$$

Taking $n\to \infty$ in the above inequality, we obtain

$$F_{\Re }^m\left(m\left(\xi ,\gamma \left(\xi \right)\right)\right)=-\infty$$

as a contradiction. Therefore, ${\left({\gamma }^n\right)}_{Fix}={\left(\gamma \right)}_{Fix}$ for any $n\in \mathbb{N}.$ □

3. Cyclic-Type Weak $\left({\mathit{F}}_{\Re }^{\mathit{m}},\mathit{\eta }\right)$-Contraction Mappings

In 2003, Kirk et al. [37] introduced cyclic contractions in metric spaces and investigated the existence of proximity points and fixed points for cyclic contraction mappings. Inspired by [37] and our Theorems 1 and 5 we obtained the following fixed point results for cyclic-type weak $\left(F_{\Re }^m,\eta \right)$-contraction mappings.

Theorem 5 

([37] ). Assume that $\left(U,m\right)$ is a compete m-metric space and G, H are two non-empty closed subsets of U and $\gamma :U\to U$. Suppose that the following conditions hold:

(i)

$\gamma \left(B\right)\subseteq D$ and $\gamma \left(D\right)\subseteq B;$

(ii)

There exists a constant $k\in \left(0,1\right)$ such that

$$m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\le km\left(\xi ,\Im \right)\mathit{for}\mathit{all}\xi \in B,\Im \in D.$$

(9)

Then, $B\cap D$ is non-empty and $\xi$ in $B\cap D$ is a fixed point of γ.

Theorem 6. 

Let $\left(U,m\right)$ be a complete relation theoretic m-metric space endowed with a transitive binary relation ℜ on U, G and H are two non-empty closed subsets of U and $\gamma :U\to U$. Assume that the following axioms hold:

(i)

$\gamma \left(G\right)\subseteq H$ and $\gamma \left(H\right)\subseteq G;$

(ii)

There exists $F_{\Re }^m\in \nabla \left(\rho \right)$ and $\eta \in \nabla \left(\pi \right)$ and $\xi >0$ such that

$$\tau +F_{\Re }^m\left(m(\gamma \left(\xi \right),\gamma \left(\Im \right))\right)\le \eta \left(m\left(\xi ,\Im \right)\right)$$

(10)

for all ξ in $G,$ℑ in H, with $F_{\eta }^m\left(\xi \right)>\eta \left(\xi \right)$ for all $\xi >0$.

Then, ${\xi }^{*}\in Z=G\cup H$ is a fixed point of $\gamma$. Moreover, $\xi \in B\cap D.$

Proof. 

From $\left(i\right)$, $Z=G\cup H$ is closed, so Z is a closed subspace of U. Therefore, $\left(U,m\right)$ is a complete m-metric space. Set the a binary relation ℜ on Z by

$$\Re =G\times H.$$

This implies that

$$\xi \Re \Im \in \iff \left(\xi ,\Im \right)\in B\times D\mathrm{for}\mathrm{all}\xi ,\Im \in Z.$$

The set $S=\Re \cup {\Re }^{-1}$ is an asymmetric relation. Directly, we set $\left(U,m,S\right)$ as regular. Let $\left\{{\xi }_n\right\}\in Z$ be any sequence and $\xi \in Z$ be a point such that

$${\xi }_{n}S{\xi }_{n+1}\mathrm{for}\mathrm{all}n\in \mathbb{N}$$

and

$$\underset{n\to \infty }{lim}m\left({\xi }_n,\xi \right)=\underset{n\to \infty }{lim}min\left\{m\left({\xi }_n,{\xi }_n\right),m\left(\xi ,\xi \right)\right\}=m\left(\xi ,\xi \right).$$

Using the definition of $S,$ we have

$$\left({\xi }_n,{\xi }_{n+1}\right)\in \left(B\times D\right)\cup \left(D\times B\right)\mathrm{for}\mathrm{all}n\in \mathbb{N}$$

(11)

Immediately, we obtain the product of $Z\times Z$ in the m-metric space m as

$$m\left(\left({\xi }_1,{\Im }_1\right),\left({\xi }_2,{\Im }_2\right)\right)=\frac{m\left({\xi }_1,{\Im }_1\right)+m\left({\xi }_2,{\Im }_2\right)}{2}.$$

Since $\left(U,m\right)$ is a complete m-metric space, $\left(Z\times Z,m\right)$ is complete. Furthermore, $G\times H$ and $H\times G$ are close in $\left(Z\times Z,m\right)$ because G and H are closed in $\left(U,m\right).$ Applying the limit $n\to \infty$ to (11), we have $\left(\xi ,\Im \right)\in \left(B\times D\right)\cup \left(D\times B\right).$ This implies that $\xi \in B\cap D.$ Furthermore, from (11), we have ${\xi }_n\in B\cup D.$ Thus, we obtain ${\xi }_n{S}^{*}\xi$ for all $n\in \mathbb{N}.$ Therefore, our theorem is proven. Furthermore, since $\gamma$ is self mapping, from condition $\left(i\right)$, for all $\xi ,\Im \in U,$ we obtain

$$\begin{array}{ccc}\hfill \left(\xi ,\Im \right)\mathrm{in}G\times H& \Rightarrow & \left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\in H\times G\hfill \\ \hfill \left(\xi ,\Im \right)\mathrm{in}H\times G& \Rightarrow & \left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\in G\times H.\hfill \end{array}$$

The binary relation ℜ is $\gamma$-closed, and as $B\ne \varphi$, there exists ${\xi }_0\in B$ such that $\gamma \left({\xi }_0\right)\in D$, i.e., ${\xi }_0{S}^{*}\gamma \left({\xi }_0\right).$ Therefore, all the hypotheses of Theorem $\left(2.8\right)$ are satisfied. Hence, ${\left(\gamma \right)}_{Fix}$$\ne \varphi$ and also ${\left(\gamma \right)}_{Fix}\subseteq B\cap D$. Finally, as $\xi S^{*}\Im$ for all $\xi ,\Im \in G\cap H.$ Hence, $G\cap H$ is ∇-directed. Hence, all conditions of Theorem 3 are satisfied and $\gamma$ has a unique fixed point. □

4. Application

In this section, we study existence of a solution for a Volterra-type integral equation by using Theorem 2.6. Consider the following Volterra-type integral equation:

$$\xi \left(\alpha \right)={\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right),\alpha \in \left[0,1\right],$$

(12)

where $A:\left[0,1\right]\times \left[0,1\right]\times \left[0,1\right]\to \left[0,1\right]$ and $\Psi :\left[0,1\right]\to \left[0,1\right].$ Consider the Banach contraction $\delta =C\left(\left[0,1\right],\left[0,1\right]\right)$ of all continuous functions $\xi :$$\left[0,1\right]\to \left[0,1\right]$ equipped with norm $∥\xi ∥={max}_{0\le \alpha \le 1}\left|\xi \left(\alpha \right)\right|.$ Define an m-metric space m on $\delta$ by $m\left(\xi ,\Im \right)=∥\frac{\xi +\Im }{2}∥$ for each $\xi ,\Im$ in $\delta .$ Then $\left(\delta ,m\right)$ is a complete m-metric space.

Definition 11. 

Lower and upper solutions of (9) are functions Λ and Θ in Banach space δ, respectively, such that

$$\Lambda \left(\alpha \right)\le {\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)\kappa \sigma +\Psi \left(\alpha \right)\mathrm{and}\Theta \left(\alpha \right)\ge {\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right),\alpha \in \left[0,1\right]$$

In this section, we prove the existence and unique solution to the Volterra-type integral Equation (12).

Theorem 7. 

Consider Volterra-type integral Equation (12). Assume that there is a positive real number τ such that

$$\left|\frac{A\left(\alpha ,\sigma ,\xi \right)+A\left(\alpha ,\sigma ,\Im \right)}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\Omega +\Im }{2}\right|\right]}-\tau },$$

(13)

for all $\alpha ,\sigma$ in $\left[0,1\right]$ and $\xi ,\Im$ in $\delta .$ if (12) has a lower solution, then a solution exists for the integral Equation (12).

Proof. 

We define an operator $\gamma :\delta \to \delta$, $F_{\Re }^m,\eta :R^{+}\to R$ by

$$\gamma \left(\xi \left(\alpha \right)\right)={\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right),\xi \in \delta ,$$

$$\eta \left(\varpi \right)=ln\varpi -\frac{1}{\left[1+\varpi \right]}$$

and

$$F_{\Re }^m\left(\varpi \right)=ln\varpi$$

for all $\varpi \in R^{+},$$F_{\Re }^m\in \nabla \left(\rho \right)$ and $\eta \in \nabla \left(\pi \right)$, respectively. We can verify easily that $\gamma$ is well defined and ⪯ on ℜ is $\gamma$-closed. Note that $\xi$ is a fixed point of $\gamma$ if and only if there is a solution to (12). Now, we want to prove that $\gamma$ is a $F_{\Re }^m$-contraction mapping with $\eta .$ Let

$$\left(\xi ,\Im \right)\in \Xi =\left\{\xi S^{*}\Im :m\left(\xi ,\Im \right)>0,\mathrm{where}m\mathrm{is}\mathrm{Banach}\mathrm{space}\right\},$$

which implies that $\xi ⪯\Im .$ Since ℜ is $\gamma$-closed, then $\gamma \left(\xi \right)⪯\gamma \left(\Im \right),$

$$\begin{array}{ccc}\hfill \left|\frac{\gamma \left(\xi \left(\alpha \right)\right)+\gamma \left(\Im \left(\alpha \right)\right)}{2}\right|& =& \left|\frac{{\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right)+{\int }_0^{\alpha }A\left(\alpha ,\sigma ,\Im \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right)}{2}\right|\hfill \\ & & \left|\frac{{\int }_0^{\alpha }A\left(\alpha ,\sigma ,\xi \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right)+{\int }_0^{\alpha }A\left(\alpha ,\sigma ,\Im \left(\sigma \right)\right)m\sigma +\Psi \left(\alpha \right)}{2}\right|\hfill \\ & \le & {\int }_0^{\alpha }\left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau }\hfill \\ & \le & {\int }_0^{\alpha }\left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau }\hfill \\ & \le & {\int }_0^{\alpha }\underset{\alpha \in \left[0,1\right]}{max}\left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau }\hfill \\ & \le & ∥\frac{\xi +\Im }{2}∥e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau },\hfill \end{array}$$

and so

$$\left|\frac{\gamma \left(\xi \left(\alpha \right)\right)+\gamma \left(\Im \left(\alpha \right)\right)}{2}\right|\le ∥\frac{\xi +\Im }{2}∥e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau }.$$

Taking the supremum norm on both sides, we have

$$∥\frac{\gamma \left(\xi \left(\alpha \right)\right)+\gamma \left(\Im \left(\alpha \right)\right)}{2}∥\le ∥\frac{\xi +\Im }{2}∥e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau }.$$

This implies that

$$ln\left(∥\frac{\gamma \left(\xi \left(\alpha \right)\right)+\gamma \left(\Im \left(\alpha \right)\right)}{2}∥\right)\le ln\left(∥\frac{\xi +\Im }{2}∥e^{-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau }\right),$$

then

$$ln\left(∥\frac{\gamma \left(\xi \left(\alpha \right)\right)+\gamma \left(\Im \left(\alpha \right)\right)}{2}∥\right)=ln\left(∥\frac{\xi +\Im }{2}∥\right)-\frac{1}{\left[1+∥\frac{\xi +\Im }{2}∥\right]}-\tau .$$

Consequently,

$$\tau +F_{\Re }^m\left({∥\frac{\gamma \left(\xi \right)+\gamma \left(\Im \right)}{2}∥}_{tr}\right)\le \eta \left({∥\frac{\xi +\Im }{2}∥}_{tr}\right).$$

Thus,

$$\tau +F_{\Re }^m\left(m\left(\gamma \left(\xi \right),\gamma \left(\Im \right)\right)\right)\le \eta \left(m\left(\xi ,\Im \right)\right).$$

Therefore, $\gamma$ is an $\left(F_R^m,\eta \right)$-contraction and thus, Inequality (4) holds. Since $\left\{{\xi }_{\mu }\right\}$ is an ℜ-preserving sequence $\left\{{\xi }_n\right\}$ in $Z\left(\left[0,1\right]\right)$ such that ${\xi }_n$ converges with respect to $t_m$ to $\xi$ for some $\xi$ in $Z\left(\left[0,1\right]\right),$ we obtain

$${\xi }_0\left(\alpha \right)⪯{\xi }_1\left(\alpha \right)⪯{\xi }_2\left(\alpha \right)⪯\dots ⪯{\xi }_n\left(\alpha \right)⪯{\xi }_{n+1}\left(\alpha \right)⪯\dots ,$$

for all $\alpha \in \left[0,1\right].$ Which implies,

$${\xi }_n\left(\alpha \right)⪯\xi \left(\alpha \right)\mathrm{for}\mathrm{all}\alpha \in \left[0,1\right].$$

Thus, $\xi ,\Im \in {\left(\gamma \right)}_{Fix}.$ Then, $\aleph =max\left\{\xi ,\Im \right\}\in Z\left(\left[0,1\right]\right)$, and thus $\xi ⪯\aleph$, $\Im ⪯\aleph$, $\xi S^{*}\aleph$ and $\Im S^{*}\aleph .$ Hence, all axioms of Theorem 3 hold and the integral Equation (12) has a solution. □

Theorem 8. 

Consider Volterra-type integral Equation (12). Assume that A is non-decreasing in the third variables; then, there is positive real number τ such that

$$\left|\frac{A\left(\alpha ,\sigma ,\xi \right)+A\left(\alpha ,\sigma ,\Im \right)}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-\tau },$$

for all $\alpha ,\sigma$ in $\left[0,1\right]$ and $\xi ,\Im$ in $\delta .$ If (12) has an upper solution, then a solution exists for the integral Equation (12).

Proof. 

Define a binary relation on Banach space as follows

$$\left(\xi ,\Im \right)\in \Xi =\left\{\xi S^{*}\Im \mathrm{with}\alpha \left(\xi \right)⪰\alpha \left(\Im \right):m\left(\xi ,\Im \right)>0,\mathrm{where}m\mathrm{is}\mathrm{a}\mathrm{Banach}\mathrm{space}\right\}.$$

Now, due to the proof of the above Theorem, then all conditions of Theorem 8 and integral Equation (12) have unique solutions. □

Example 3. 

Assume that a function

$$\xi \left(\alpha \right)=\frac{\alpha }{2},\mathit{for}\mathit{all}\alpha \mathit{in}\left[0,1\right]$$

is a solution of Equation (12)

$$\xi \left(\alpha \right)=\frac{3}{2}\left(\alpha \right)-\left(1+\alpha \right)ln\left(1+\alpha \right)+{\int }_0^{\alpha }ln\left(1+\xi \left(\sigma \right)\right)m\sigma ,\mathit{for}\mathit{all}\alpha \mathit{in}\left[0,1\right].$$

(14)

Proof. 

Let γ be a self operator from δ to δ, which is given by

$$\gamma \left(\xi \left(\alpha \right)\right)=\frac{3}{2}\left(\alpha \right)-\left(1+\alpha \right)ln\left(1+\alpha \right)+{\int }_0^{\alpha }ln\left(1+\xi \left(\sigma \right)\right)m\sigma ,\mathrm{for}\mathrm{all}\alpha \mathrm{in}\left[0,1\right].$$

Now, we take $\tau \in \left[0.0091,\infty \right),$

$$A\left(\alpha ,\sigma ,\xi \right)=ln\left(1+\xi \left(\sigma \right)\right)$$

and

$$\Psi \left(\alpha \right)=\frac{3}{2}\left(\alpha \right)-\left(1+\alpha \right)ln\left(1+\alpha \right).$$

Observe that given function $A\left(\alpha ,\sigma ,\xi \right)=ln\left(1+\xi \left(\sigma \right)\right)$ in the third variable is non-decreasing and that $\frac{\alpha }{2}\le$$\frac{3}{2}\left(\alpha \right)-\left(1+\alpha \right)ln\left(1+\alpha \right)+{\int }_0^{\sigma }ln\left(1+\xi \left(\sigma \right)\right)m\sigma$ for all α in $\left[0,1\right]$ such that $\xi \left(\alpha \right)=\frac{\alpha }{2}$ is a lower solution of (16), then the following below inequality holds,

$$\left|\frac{A\left(\alpha ,\sigma ,\xi \right)+A\left(\alpha ,\sigma ,\Im \right)}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-\tau }.$$

(15)

Now, from the non-decreasing function $\alpha \mapsto e^{-\frac{1}{\left[1+\left|\frac{\alpha }{2}\right|\right]}-0.091},$ we have

$$\left|\frac{ln\left(1+\xi \right)+ln\left(1+\Im \right)}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-0.091}.$$

Hence, all conditions of Theorem 7 hold and the integral Equation (12) has a unique solution $\xi \left(\alpha \right)=\frac{\alpha }{2}$ for all α in $\left[0,1\right].$ □

Example 4. 

Assume that a function

$$\xi \left(\alpha \right)=\alpha ,\mathit{for}\mathit{all}\alpha \in \left[0,1\right]$$

is a solution of Equation (12):

$$\xi \left(\sigma \right)=\alpha -\left(1-\alpha \right)ln\left(2-\alpha \right)-ln\left(2\right)+{\int }_0^{\alpha }ln\left(2-\xi \left(\sigma \right)\right)m\sigma ,\mathit{for}\mathit{all}\alpha \mathit{in}\left[0,1\right].$$

(16)

Proof. 

In view of the above example, the following below inequality holds for all $\xi ,\Im$ in $\left[0,1\right]$ and $\tau =0.091$

$$\left|\frac{ln\left(2-\xi \right)+ln\left(2-\Im \right)}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-\tau }.$$

Using the arguments of the above example, we can say that the all conditions of Theorem 8 hold. Hence, the integral Equation (12) has a unique solution $\xi \left(\alpha \right)=\alpha$ for all $\alpha$ in $\left[0,1\right].$ □

Finally, we give an example different to the above example and others given in the literature [38] which satisfies all conditions of Theorem 15.

Example 5. 

Assume that a function

$$\xi \left(\alpha \right)=\frac{1}{3}\alpha ,\mathit{for}\mathit{all}\alpha \mathit{in}\left[0,1\right]$$

is a solution of Equation (12):

$$\xi \left(\alpha \right)=\frac{5}{3}\alpha -\frac{\alpha }{1+\alpha }+{\int }_0^{\alpha }\left(\frac{\xi \left(\sigma \right)}{1+\xi \left(\sigma \right)}\right)m\sigma ,\mathit{for}\mathit{all}\alpha \mathit{in}\left[0,1\right].$$

(17)

Proof. 

Let $\gamma$ be a self operator from $\delta$ to $\delta$, which is given by

$$\gamma \left(\xi \left(\alpha \right)\right)=\frac{5}{3}\alpha -\frac{\alpha }{1+\alpha }+{\int }_0^{\alpha }\left(\frac{\xi \left(\sigma \right)}{1+\xi \left(\sigma \right)}\right)m\sigma ,\mathrm{for}\mathrm{all}\alpha \mathrm{in}\left[0,1\right].$$

Now, we take $\tau \in \left[0.091,\infty \right),$

$$A\left(\alpha ,\sigma ,\xi \right)=\frac{\xi \left(\sigma \right)}{1+\xi \left(\sigma \right)}$$

and

$$\Psi \left(\alpha \right)=\frac{5}{3}\alpha -\frac{\alpha }{1+\alpha }.$$

Observe that given the function $A\left(\alpha ,\sigma ,\xi \right)=\frac{\xi \left(\sigma \right)}{1+\xi \left(\sigma \right)}$ in the third variable is non-decreasing and that $\frac{1}{3}\alpha \le$$\frac{5}{3}\alpha -\frac{\alpha }{1+\alpha }+{\int }_0^{\alpha }\left(\frac{\xi \left(\sigma \right)}{1+\xi \left(\sigma \right)}\right)m\sigma$ for all $\alpha$ in $\left[0,1\right]$ such that $\xi \left(\alpha \right)=\frac{1}{3}\alpha$ is a lower solution of (16), then the following below inequality holds:

$$\left|\frac{A\left(\alpha ,\sigma ,\xi \right)+A\left(\alpha ,\sigma ,\Im \right)}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-\tau }.$$

(18)

Now, from the non-decreasing function $\alpha \mapsto e^{-\frac{1}{\left[1+\left|\frac{\alpha }{2}\right|\right]}-0.9},$ we have

$$\left|\frac{\frac{\xi }{1+\xi }+\frac{\Im }{1+\Im }}{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-0.9}.$$

Hence, all axioms of Theorem 7 hold and the integral Equation (12) has a unique solution $\xi \left(\alpha \right)=\frac{\alpha }{3}$ for all $\alpha$ in $\left[0,1\right].$ □

Example 6. 

Assume that a function

$$\xi \left(\alpha \right)=\frac{3}{5}\alpha +\frac{1}{3},\mathit{for}\mathit{all}\alpha \in \left[0,1\right]$$

is a solution of Equation (12):

$$\xi \left(\sigma \right)=\frac{3}{5}\alpha +\frac{1}{3}-\left(1-\alpha \right)\left(2-\alpha \right)+2+{\int }_0^{\alpha }\left(1+\xi \left(\sigma \right)\right)m\sigma ,\mathit{for}\mathit{all}\alpha \mathit{in}\left[0,1\right].$$

(19)

Proof. 

In view of the above example, the following below inequality holds for all $\xi ,\Im$ in $\left[0,1\right]$ and $\tau =0.9$

$$\left|\frac{1+\xi +1+\Im }{2}\right|\le \left|\frac{\xi +\Im }{2}\right|e^{-\frac{1}{\left[1+\left|\frac{\xi +\Im }{2}\right|\right]}-\tau }.$$

Using the arguments of the above example, we can say that the all conditions of Theorem 7 hold. Hence, the integral Equation (12) has a unique solution $\xi \left(\alpha \right)=\frac{3}{5}\alpha +\frac{1}{3}$ for all $\alpha$ in $\left[0,1\right].$ □

5. Conclusions

In this article, we have introduced the notion of weak $\left(F_{\Re }^m,\eta \right)$-contractions and proved related fixed point theorems in relation theoretic m-metric space endowed with a relation ℜ using a control function $\eta$. Examples and applications to Volterra-type integral equations are given to validate our main results. Analogously, such results can be extended to generalized distance spaces (such as symmetric spaces, $m_{b}m$-spaces, $rmm$-spaces, $r{m}_{b}m$- spaces, $pm$-spaces and $p_{b}m$-spaces) endowed with relations.