Relation-Theoretic Weak Contractions and Applications

Abstract

In this article, we discuss the relation-theoretic aspects of weakly contractive mappings to prove fixed point results in the setting of metric spaces endowed with a certain binary relation. We also provide an example and an application to validate of our results. The results proved herewith unify, generalize, improve, extend, sharpen, subsume and enrich some well-known fixed point theorems of the existing literature.

1. Introduction

The branch of metric fixed point theory has been initiated with the appearance of classical Banach contraction principle, which was acclaimed by Banach [1] in 1922. This classical result remains a cornerstone result of nonlinear functional analysis. Later various researchers generalized and extended the Banach contraction principle in different directions, viz., by enlarging the ambient space (e.g., [2,3]), weakening underlying contraction conditions (e.g., [4,5,6]) and enhancing the number of involved mappings (e.g., [7,8,9]) etc.

In the 1970s, an extensive literature was developed in order to enlarge the class of contraction mappings. In 1977, Rhoades [10] came up with the comparison of various types of contractive mappings in which 149 different conditions are analyzed and compared. In 1979, Browder [11] subsumed the major part of the work of Rhoades [10] under an intuitive and simple mode of arguments.

Recall that a self-mapping T on a metric space $(M,d)$ is called weak $\psi$-contraction (or simply, weak contraction) if there exists an auxiliary function $\psi :[0,\infty )\to [0,\infty )$ such that

$$d(T\mathrm{x},T\mathrm{y})\le d(\mathrm{x},\mathrm{y})-\psi \left(d\right(\mathrm{x},\mathrm{y}\left)\right),\forall \mathrm{x},\mathrm{y}\in M.$$

The notion of weak contraction was initiated by Krasnosel’skii [12], wherein he imposed that $\psi$ is a continuous function such that ${\psi }^{-1}\left(\left\{0\right\}\right)=\left\{0\right\}$. Thereafter, Alber and Guerre-Delabriere [13] established a fixed point theorem employing weak contractivity conditions in the setting of Hilbert space. In this continuation, Rhoades [14] extended the Banach contraction principle for weak contraction mappings and observed that the fixed point result of Alber et al. [13] was also valid in complete metric spaces.

Turinici [15] initiated the order-theoretic aspects of metric fixed point theory. Ran and Reurings [16] and Nieto and Rodríguez-López [17] extended Banach contraction principle in the context of ordered metric spaces. Both results [16,17] are indeed natural versions of the results of Turinici [15]. In the same continuation, Harjani and Sadarangani [18] extended the results of Nieto and Rodríguez-López [17] to weak contractions. In 2015, Alam and Imdad [19] investigated a novel generalization of the Banach contraction principle employing an amorphous binary relation instead of partial order. Soon after, various relation-theoretic results were proposed by several researchers (e.g., [20,21,22,23]). Such results involve weak contraction conditions which hold for the pairs of comparative elements only.

Due to such restrictive nature, these results are applied for solving certain matrix equations, boundary value problems in ordinary differential equations and integral equations, fractional differential equations, elastic beam equations, nonlinear elliptic problems and delayed hematopoiesis models satisfying specific prescribed auxiliary conditions, e.g., [24,25,26,27,28,29,30,31,32,33].

The aim of this article is to prove certain results on the existence and uniqueness of fixed points via weak contractions in a metric space endowed with a certain binary relation. An example is also furnished to demonstrate our newly proved results. Finally, as an application of our main results, we investigated a result regarding the unique solution of certain system of matrix equations.

2. Preliminaries

In this section, we recollect some basic notions and auxiliary results, which are required in proving our main results. Throughout this paper, $\mathbb{N}$ stands for the set of all natural numbers and ${\mathbb{N}}_0$ stands for the set of all whole numbers, i.e., ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

Definition 1

([34]). Let M be a nonempty set. A subset ℜ of $M^2$ is called a binary relation on M. If $(\mathrm{x},\mathrm{y})\in \Re$, then we say that “$\mathrm{x}$ is related to $\mathrm{y}$” or “$\mathrm{x}$ relates to $\mathrm{y}$” under ℜ.

The subsets, $M^2$ and ∅ of $M^2$ are called the universal relation and empty relation, respectively.

Definition 2

([35]). M be a nonempty set endowed with a binary relation ℜ. Given a subset $E\subset M$, the restriction of ℜ to E, denoted by ${\Re |}_E$, is defined as ${\Re |}_E=\Re \cap E^2$.

Indeed, ${\Re |}_E$ is a relation on E induced by ℜ.

Definition 3

([34]). Let M be a nonempty set equipped with a binary relation ℜ. Then

(i) 

The dual relation, inverse or transpose of ℜ, denoted by ${\Re }^{-1}$, and is defined by ${\Re }^{-1}=\{(\mathrm{x},\mathrm{y})\in M^2:(\mathrm{y},\mathrm{x})\in \Re \}$.

(ii) 

The symmetric closure of ℜ, often denoted by ${\Re }^s$, is defined to be the set $\Re \cup {\Re }^{-1}$ (i.e., ${\Re }^s:=\Re \cup {\Re }^{-1}$). Indeed, ${\Re }^s$ is the smallest symmetric relation on M containing ℜ.

Definition 4

([19]). Let ℜ be a binary relation on a nonempty set M and $\mathrm{x},\mathrm{y}\in M$. We say that $\mathrm{x}$ and $\mathrm{y}$ are ℜ-comparative if either $(\mathrm{x},\mathrm{y})\in \Re$ or $(\mathrm{y},\mathrm{x})\in \Re$. It is denoted by $[\mathrm{x},\mathrm{y}]\in \Re .$

Definition 5

([34]). Let ℜ be a binary relation on a set M; then, ℜ is said to be complete if each pair of elements of M is ℜ-comparative, i.e.,

$$\left[\mathrm{x},\mathrm{y}\right]\in \Re ,\forall \mathrm{x},\mathrm{y}\in M.$$

Definition 6

([19]). Let M be a nonempty set and T a self-mapping on M. A binary relation ℜ defined on M is called T-closed if for any $\mathrm{x},\mathrm{y}\in M$

$$(\mathrm{x},\mathrm{y})\in \Re \Rightarrow (T\mathrm{x},T\mathrm{y})\in \Re .$$

Proposition 1

([19]). ${\Re }^s$ is T-closed, if ℜ is T-closed.

Proposition 2

([36]). For each $n\in \mathbb{N}$, ℜ is $T^n$-closed, if ℜ is T-closed.

Definition 7

([37]). Let M be a nonempty set endowed with a binary relation ℜ. A subset D of M is called ℜ-directed if for each $\mathrm{x},\mathrm{y}\in D$, there exists $\omega \in M$ such that $(\mathrm{x},\omega )\in \Re$ and $(\mathrm{y},\omega )\in \Re$.

Definition 8

([35]). Let M be a nonempty set and ℜ a binary relation on M. For a pair $\mathrm{x},\mathrm{y}\in M$, a path in ℜ from $\mathrm{x}$ to $\mathrm{y}$ is a finite sequence $\{{\omega }_0,{\omega }_1,\dots ,{\omega }_k\}\subseteq M$ satisfying the following conditions:

(i) 

${\omega }_0=\mathrm{x}$ and ${\omega }_k=\mathrm{y}$;

(ii) 

$({\omega }_i,{\omega }_{i+1})\in \Re$ for all $i\in \{0,1,2,3,\dots ,k-1\}$.

Apparently, a path of length k comprises $k+1$ points of M; it is not obligatory that those elements need to be distinct.

Definition 9

([36]). Given a nonempty set M equipped with a binary relation ℜ, a subset D of M is called ℜ-connected if each pair of elements of D has a path in ℜ.

By using the symmetric property of metric d and Definition 4, we propose the following result.

Proposition 3.

Given a metric space $(M,d)$ equipped with a binary relation ℜ, a mapping T: $M\to M$ and an auxiliary function ψ: $[0,\infty )\to [0,\infty )$, the following contractivity conditions are equivalent:

(a) 

$d(T\mathrm{x},T\mathrm{y})\le d(\mathrm{x},\mathrm{y})-\psi \left(d\right(\mathrm{x},\mathrm{y}\left)\right)\forall \mathrm{x},\mathrm{y}\in M\mathit{with}(\mathrm{x},\mathrm{y})\in \Re$.

(b) 

$d(T\mathrm{x},T\mathrm{y})\le d(\mathrm{x},\mathrm{y})-\psi \left(d\right(\mathrm{x},\mathrm{y}\left)\right)\forall \mathrm{x},\mathrm{y}\in M\mathit{with}[\mathrm{x},\mathrm{y}]\in \Re$.

Definition 10

([19]). Given a nonempty set M and a binary relation ℜ on M, a sequence $\left\{{\mathrm{x}}_n\right\}\subset M$ is called ℜ-preserving if

$$({\mathrm{x}}_n,{\mathrm{x}}_{n+1})\in \Re ,\forall n\in {\mathbb{N}}_0.$$

Definition 11

([38]). Let ℜ be a binary relation defined on M. We say that $(M,d)$ is ℜ-complete if every ℜ-preserving Cauchy sequence in M converges.

Remark 1

([38]). Every complete metric space is ℜ-complete, for any binary relation ℜ. Specifically, under the universal relation, the notion of ℜ-completeness coincides with usual completeness.

Definition 12

([38]). Let $(M,d)$ be a metric space endowed with a binary relation ℜ. A mapping $T:M\to M$ is said to be ℜ-continuous at an element $\mathrm{x}\in M$ if for any ℜ-preserving sequence $\left\{{\mathrm{x}}_n\right\}\subset M$ such that $\left\{{\mathrm{x}}_n\right\}$ is convergent to $\mathrm{x}$, we have $T\left({\mathrm{x}}_n\right)$ is convergent to $T\left(\mathrm{x}\right)$. Moreover, T is called ℜ-continuous if it is ℜ-continuous at each point of M.

Remark 2

([38]). Every continuous mapping is ℜ-continuous, for any binary relation ℜ. Specifically, under the universal relation, the notion of ℜ-continuity coincides with usual continuity.

Definition 13

([19]). Given a metric space $(M,d)$, a binary relation ℜ defined on M is called d-self-closed if for an ℜ-preserving sequence $\left\{{\mathrm{x}}_n\right\}\subset M$ converging to $\mathrm{x}\in M$, there exists a subsequence $\left\{{\mathrm{x}}_{n_k}\right\}$ of $\left\{{\mathrm{x}}_n\right\}$ such that $[{\mathrm{x}}_{n_k},\mathrm{x}]\in \Re$ for all $k\in \mathbb{N}$.

Given ${\mathrm{x}}_0\in M$ and a mapping $T:M\to M$, the set ${\mathcal{O}}_T\left({\mathrm{x}}_0\right):=\left\{T^n{\mathrm{x}}_0:n\in \mathbb{N}\right\}$ is called the orbit of ${\mathrm{x}}_0$. We write $\mathcal{O}\left(x_0\right)$ instead of ${\mathcal{O}}_T\left({\mathrm{x}}_0\right)$, whenever T is understood. A sequence whose range remains $\mathcal{O}\left(\mathrm{x}\right)$ for some $\mathrm{x}\in M$ is called a T-orbital sequence (c.f. [39]).

Definition 14

([39]). Let $(M,d)$ be a metric space endowed with a binary relation ℜ and T:$M\to M$ a mapping. We say that $(M,d)$ is ($\mathcal{O},\Re$)-complete if each T-orbital ℜ-preserving Cauchy sequence in M converges.

Definition 15

([39]). Let $(M,d)$ be a metric space endowed with a binary relation ℜ and T:$M\to M$ a mapping. We say that T is ($\mathcal{O},\Re$)-continuous at a point $z\in M$ if for any T-orbital ℜ-preserving sequence $\left\{x_n\right\}\subset M$ such that ${\mathrm{x}}_n\stackrel{d}{\to }\mathrm{z}$, we have $T\left({\mathrm{x}}_n\right)\stackrel{d}{\to }T\left(\mathrm{z}\right)$.

Definition 16

([39]). Given a metric space $(M,d)$ and a mapping T:$M\to M$, a binary relation ℜ defined on M is called ($\mathcal{O},d$)-self-closed if for every T-orbital ℜ-preserving sequence $\left\{{\mathrm{x}}_n\right\}\subset M$ converging to $\mathrm{x}\in M$, there exists a subsequence $\left\{{\mathrm{x}}_{n_k}\right\}$ such that $[{\mathrm{x}}_{n_k},\mathrm{x}]\in \Re$ for all $k\in \mathbb{N}$.

In the last three definitions, if the orbital properties are ignored, these conventions give us relational notions, i.e., Definitions 11–13, respectively. While under the universal relation, Definition 14 reduces to the notion of $\mathcal{O}$-completeness (initiated by [40]) and Definition 15 reduces to the notion of $\mathcal{O}$-continuity (initiated by [41]).

Definition 17

([42]). Given $N(\ge 2)\in {\mathbb{N}}_0$, a binary relation ℜ defined on a nonempty set M is called N-transitive if for any ${\mathrm{x}}_0,{\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_N\in M$,

$$({\mathrm{x}}_{i-1},{\mathrm{x}}_i)\in \Re \mathit{for}\mathit{each}i(1\le i\le N)⟹({\mathrm{x}}_0,{\mathrm{x}}_N)\in \Re .$$

Thus, the notion of 2-transitivity coincides with transitivity.

Definition 18

([43]). A binary relation ℜ defined on a nonempty set M is called finitely transitive if it is N-transitive for some $N\ge 2$.

Definition 19

([43]). A binary relation ℜ defined on a nonempty set M is called locally finitely transitive if for each denumerable subset E of M, there exists $N=N\left(E\right)\ge 2$, such that ${\Re |}_E$ is N-transitive.

Definition 20

([44]). Let M be a nonempty set and T a self-mapping on M. A binary relation ℜ on M is called locally finitely T-transitive if for each denumerable subset E of $T\left(M\right)$, there exists $N=N\left(E\right)\ge 2$, such that ${\Re |}_E$ is N-transitive.

The following result establishes the superiority of the idea of ‘locally finitely T-transitivity’over other variants of ‘transitivity’:

Proposition 4

([44]). Let M be a nonempty set, ℜ a binary relation on M and T a self-mapping on M. Then

(i)

ℜ is T-transitive ⇔ ${\Re |}_{T\left(M\right)}$ is transitive,

(ii)

ℜ is locally finitely T-transitive ⇔ ${\Re |}_{T\left(M\right)}$ is locally finitely transitive,

(iii)

ℜ is transitive ⟹ℜ is finitely transitive ⟹ℜ is locally transitive ⟹ℜ is locally finitely T-transitive,

(iv)

ℜ is transitive ⟹ℜ is T-transitive ⟹ℜ is locally finitely T-transitive.

Lemma 1

([45]). Let $(M,d)$ be a metric space and $\left\{{\mathrm{x}}_n\right\}$ a sequence in M. If $\left\{{\mathrm{x}}_n\right\}$ is not a Cauchy sequence, then there exist $ϵ>0$ and two subsequences $\left\{{\mathrm{x}}_{n_k}\right\}$ and $\left\{{\mathrm{x}}_{m_k}\right\}$ of $\left\{{\mathrm{x}}_n\right\}$ such that

(i)

$k\le m_k<n_k,\forall$$k\in \mathbb{N}$,

(ii)

$d({\mathrm{x}}_{m_k},{\mathrm{x}}_{n_k})\ge ϵ$,

(iii)

$d({\mathrm{x}}_{m_k},{\mathrm{x}}_{p_k})\le ϵ,\forall p_k\in \{m_k+1,m_k+2,...,n_k-2,n_k-1\}$.

In addition to this, if $\left\{{\mathrm{x}}_n\right\}$ also verifies $\underset{n\to \infty }{lim}d({\mathrm{x}}_n,{\mathrm{x}}_{n+1})=0$, then

(iv)

$\underset{k\to \infty }{lim}d({\mathrm{x}}_{m_k},{\mathrm{x}}_{n_k+p})=ϵ,\forall p\in {\mathbb{N}}_0$.

Lemma 2

([43]). Let M be a nonempty set, ℜ a binary relation on M and $\left\{{\mathrm{x}}_n\right\}\subset M$ an ℜ-preserving sequence. If the relation ℜ is N-transitive on $X:=\{{\mathrm{x}}_n:n\in {\mathbb{N}}_0\}$ for some natural number $N\ge 2$, then

$$({\mathrm{x}}_n,{\mathrm{x}}_{n+1+p(N-1)})\in \Re ,\forall n,p\in {\mathbb{N}}_0.$$

3. A New Class of Weak Contraction

Let $\mathrm{\Psi }$ be a family of functions $\psi$:$[0,\infty )\to [0,\infty )$ satisfying the following axioms:

${\mathrm{\Psi }}_1$:

$\psi \left(a\right)>0,\forall a>0,$

${\mathrm{\Psi }}_2$:

$\underset{r\to a}{lim\; inf}\psi \left(r\right)>0,\forall a>0.$

Notice that this family of auxiliary functions is suggested by Alam et al. [46]. We shall utilize the family $\mathrm{\Psi }$ to define a new class of weak contractions.

Remark 3

([46]). Axiom ${\mathrm{\Psi }}_1$ is equivalent to the following:

${\mathrm{\Psi }}_1^{\prime }$:

If there exists $a\in [0,\infty )$ such that $\psi \left(a\right)=0$, then $a=0$.

Proposition 5

([46]). If $\psi :[0,\infty )\to [0,\infty )$ is a continuous function, which satisfies axiom ${\mathrm{\Psi }}_1$, then

(i)

$\underset{n\to \infty }{lim}\psi \left(a_n\right)>0\mathrm{whenever}\underset{n\to \infty }{lim}{a}_n=a>0$,

(ii)

$\underset{r\to a}{lim}\psi \left(r\right)>0,\forall a>0$.

Proposition 6

([46]). If $\psi :[0,\infty )\to [0,\infty )$ is a lower semicontinuous function, which satisfies axiom ${\mathrm{\Psi }}_1$, then

(i)

$\underset{n\to \infty }{lim\; inf}\psi \left(a_n\right)>0,\mathrm{whenever}\underset{n\to \infty }{lim}{a}_n=a>0$,

(ii)

$\underset{r\to a}{lim\; inf}\psi \left(r\right)>0,\forall a>0$.

Proposition 7.

If there exists an auxiliary function $\psi :[0,\infty )\to [0,\infty )$, which satisfies axiom ${\mathrm{\Psi }}_1$, such that for all $b\in [0,\infty )$ and $a\in (0,\infty )$,

$$b\le a-\psi \left(a\right),$$

then

$$b<a.$$

Proof. 

By ${\mathrm{\Psi }}_1$, we have $\psi \left(a\right)>0$, which is equivalent to $-\psi \left(a\right)<0$. Now,

$$\begin{array}{ccc}\hfill b& \le & a+(-\psi (a\left)\right)\hfill \\ & <& a+0=a\hfill \\ \hfill \Rightarrow b& <& a.\hfill \end{array}$$

□

Proposition 8.

Let $(M,d)$ be a metric space and T a self-mapping on M. If there exists an auxiliary function $\psi :[0,\infty )\to [0,\infty )$, which satisfies axiom ${\mathrm{\Psi }}_1$, such that T is a weak ψ-contraction, then T is contractive and hence is continuous.

Proof. 

Take two distinct elements $\mathrm{x},\mathrm{y}\in M$ so that $d(\mathrm{x},\mathrm{y})>0$. Applying contractivity condition on this pair, we get

$$d(T\mathrm{x},T\mathrm{y})\le d(\mathrm{x},\mathrm{y})-\psi \left(d\right(\mathrm{x},\mathrm{y}\left)\right).$$

By Proposition 7, we have

$$d(T\mathrm{x},T\mathrm{y})<d(\mathrm{x},\mathrm{y}).$$

It follows that T is contractive. Consequently, T is continuous. □

4. Main Results

Given a nonempty set M endowed with a binary relation ℜ and a mapping $T:M\to M$, let us define the following subset of M

$$M(T,\Re ):=\{\mathrm{x}\in M:(\mathrm{x},T\mathrm{x})\in \Re \}.$$

Now, we are equipped to prove the following result regarding the existence of a fixed point in relational metric space satisfying weak contractivity conditions.

Theorem 1.

Let $(M,d)$ be a metric space, ℜ a binary relation on M and T a self-mapping on M. Suppose that the following conditions hold:

(a) 

$(M,d)$ is $(\mathcal{O},\Re )$-complete,

(b) 

$M(T,\Re )$ is nonempty,

(c) 

ℜ is T-closed and locally finitely T-transitive,

(d) 

either T is $(\mathcal{O},\Re )$-continuous or ℜ is ($\mathcal{O},d$)–self closed,

(e) 

there exists $\psi \in \mathrm{\Psi }$ such that

$$d(T\mathrm{x},T\mathrm{y})\le d(\mathrm{x},\mathrm{y})-\psi \left(d\right(\mathrm{x},\mathrm{y}\left)\right),\forall \mathrm{x},\mathrm{y}\in M\mathit{with}(\mathrm{x},\mathrm{y})\in \Re .$$

Then T has a fixed point.

Proof. 

In view of condition $\left(b\right)$, choose ${\mathrm{x}}_0\in M$ such that $({\mathrm{x}}_0,T\left({\mathrm{x}}_0\right)\in \Re$. Then, we can construct a Picard sequence $\left\{{\mathrm{x}}_n\right\}$ with initial point ${\mathrm{x}}_0$ such that

$${\mathrm{x}}_{n+1}=T^n\left({\mathrm{x}}_0\right)=T\left({\mathrm{x}}_n\right),\forall n\in {\mathbb{N}}_0.$$

(1)

Since $({\mathrm{x}}_0,T{\mathrm{x}}_0)\in \Re$ and ℜ is T-closed, using Proposition 2, we have

$$(T^n{\mathrm{x}}_0,T^{n+1}{\mathrm{x}}_0)\in \Re$$

which, by using (1), reduces to

$$({\mathrm{x}}_n,{\mathrm{x}}_{n+1})\in \Re \forall n\in {\mathbb{N}}_0.$$

Hence, $\left\{{\mathrm{x}}_n\right\}$ is a ℜ-preserving sequence. In view of (1), $\left\{{\mathrm{x}}_n\right\}\subset \mathcal{O}\left({\mathrm{x}}_0\right)$. Therefore $\left\{{\mathrm{x}}_n\right\}$ is also T-orbital.

Denote $d_n:=d({\mathrm{x}}_n,{\mathrm{x}}_{n+1})$. If there exists $n_0\in {\mathbb{N}}_0$ such that $d_{n_0}=0$, then by using (1), we conclude that ${\mathrm{x}}_{n_0}={\mathrm{x}}_{n_0+1}=T\left({\mathrm{x}}_{n_0}\right)$ so that ${\mathrm{x}}_{n_0}$ is a fixed point of T. Otherwise, we have $d_n>0$ for all $n\in {\mathbb{N}}_0.$ Applying contractivity condition (e), we get

$$d({\mathrm{x}}_n,{\mathrm{x}}_{n+1})=d(T{\mathrm{x}}_{n-1},T{\mathrm{x}}_n)\le d({\mathrm{x}}_{n-1},{\mathrm{x}}_n)-\psi \left(d({\mathrm{x}}_{n-1},{\mathrm{x}}_n)\right)$$

so that

$$\begin{array}{ccc}\hfill d_{n+1}& \le & d_n-\psi \left(d_n\right).\hfill \end{array}$$

(2)

In view of Proposition 7, Equation (2) gives rise to

$$d_n<d_{n-1},\forall n\in {\mathbb{N}}_0$$

which yields that the sequence $\left\{d_n\right\}$ is a decreasing sequence of positive real numbers. Since it is bounded below by 0 (as a lower bound), there exists an element $\delta \ge 0$ such that

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

(3)

Now, we claim that $\delta =0$. On the contrary, suppose that $\delta >0$. Taking the upper limit in (2), we get

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim\; sup}{d}_{n+1}& \le & \underset{n\to \infty }{lim\; sup}{d}_n+\underset{n\to \infty }{lim\; sup}[-\psi \left(d_n\right)]\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}{d}_n-\underset{n\to \infty }{lim\; inf}\psi \left(d_n\right)\hfill \end{array}$$

Using (3), above inequality reduces to

$$\delta \le \delta -\underset{n\to \infty }{lim\; inf}\psi \left(d_n\right)$$

implying thereby

$$\underset{d_n\to \delta }{lim\; inf}\psi \left(d_n\right)=\underset{n\to \infty }{lim\; inf}\psi \left(d_n\right)\le 0$$

which contradicts the property of ${\mathrm{\Psi }}_2$. Therefore we have

$$\underset{n\to \infty }{lim}{d}_n=\underset{n\to \infty }{lim}d({\mathrm{x}}_n,{\mathrm{x}}_{n+1})=0.$$

(4)

Now, we show that $\left\{{\mathrm{x}}_n\right\}$ is a Cauchy sequence.

Let on contrary that $\left\{{\mathrm{x}}_n\right\}$ is not a Cauchy sequence. Therefore, by Lemma 1, there exists $ϵ>0$ and two subsequences $\left\{{\mathrm{x}}_{n_k}\right\}$ and $\left\{{\mathrm{x}}_{m_k}\right\}$ of $\left\{{\mathrm{x}}_n\right\}$ such that $k\le m_k<n_k,d({\mathrm{x}}_{m_k},{\mathrm{x}}_{n_k})\ge ϵ$ and $d({\mathrm{x}}_{m_k},{\mathrm{x}}_{p_k})<ϵ$ where $p_k\in \{m_k+1,m_k+2,\dots ,n_k-2,n_k-1\}.$ Furthermore, using (4) and Lemma 1, we have

$$\underset{k\to \infty }{lim}d({\mathrm{x}}_{m_k},{\mathrm{x}}_{n_k+p})=ϵ,\forall p\in {\mathbb{N}}_0.$$

(5)

Since $\left\{{\mathrm{x}}_n\right\}$ is ℜ-preserving and $\left\{{\mathrm{x}}_n\right\}\subset T\left(M\right)$ (owing to (1)), therefore, the range $E:=\{{\mathrm{x}}_n:n\in {\mathbb{N}}_0\}$(of the sequence$\left\{{\mathrm{x}}_n\right\}$) is a denumerable subset of $T\left(M\right)$. By locally finitely T-transitivity of ℜ, there exists a natural number $N=N\left(E\right)\ge 2$, such that ${\Re |}_E$ is N-transitive.

As $m_k<n_k$ and $N-1>0$, by Division Algorithm, we have

$$n_k-m_k=(N-1)({\alpha }_k-1)+(N-{\beta }_k)$$

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

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

Here, ${\alpha }_k$ and ${\beta }_k$ are natural numbers such that ${\beta }_k$ can assume a positive integer in interval $(1,N]$. Hence, without loss of generality, we can choose subsequences $\left\{{\mathrm{x}}_{n_k}\right\}$ and $\left\{{\mathrm{x}}_{m_k}\right\}$ of $\left\{{\mathrm{x}}_n\right\}$ satisfying Equation (5) such that ${\beta }_k$ remains constant say $\beta$, which is independent of k. Write

$$m_k^{\prime }=n_k+\beta =m_k+1+(N-1){\alpha }_k$$

(6)

where $\beta (1<\beta \le N)$ is constant. Owing to (5) and (6), we obtain

$$\underset{n\to \infty }{lim}d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})=\underset{n\to \infty }{lim}d({\mathrm{x}}_{m_k},{\mathrm{x}}_{n_k+\beta })=ϵ.$$

(7)

Using triangular inequality, we have

$$\begin{array}{ccc}\hfill d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})& \le & d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k})+d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})+d({\mathrm{x}}_{m_k^{\prime }},{\mathrm{x}}_{m_k^{\prime }+1})\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})& \le & d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k+1})+d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})+d({\mathrm{x}}_{m_k^{\prime }+1},{\mathrm{x}}_{m_k^{\prime }}).\hfill \end{array}$$

The last two inequalities give rise to

$$\begin{gathered} d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})-d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k+1})-d({\mathrm{x}}_{m_k^{\prime }+1},{\mathrm{x}}_{m_k^{\prime }})\le d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})\le \\ d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k})+d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})+d({\mathrm{x}}_{m_k^{\prime }},{\mathrm{x}}_{m_k^{\prime }+1}) \end{gathered}$$

which, on letting $k\to \infty$ and using (4) and (7), yields that

$$\underset{n\to \infty }{lim}d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})=ϵ.$$

(8)

In view of (6) and Lemma 2, we have $({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})\in \Re$. Denote ${\delta }_k=d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})$. Using (1) and the contractivity condition (e), we obtain

$$\begin{array}{ccc}\hfill d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})& =& d(T{\mathrm{x}}_{m_k},T{\mathrm{x}}_{m_k^{\prime }})\hfill \\ & \le & d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})-\psi \left(d({\mathrm{x}}_{m_k},{\mathrm{x}}_{m_k^{\prime }})\right)\hfill \end{array}$$

so that

$$d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})\le {\delta }_k-\psi \left({\delta }_k\right).$$

(9)

Taking upper limit in inequality (9), we get

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}d({\mathrm{x}}_{m_k+1},{\mathrm{x}}_{m_k^{\prime }+1})\le \underset{k\to \infty }{lim\; sup}{\delta }_k+\underset{k\to \infty }{lim\; sup}[-\psi \left({\delta }_k\right)],\end{array}$$

which, on using (7) and (8), becomes

$$ϵ\le ϵ-\underset{k\to \infty }{lim\; inf}\psi \left({\delta }_k\right)$$

so that

$$\underset{{\delta }_k\to ϵ}{lim\; inf}\psi \left({\delta }_k\right)=\underset{k\to \infty }{lim\; inf}\psi \left({\delta }_k\right)\le 0,$$

which contradicts ${\mathrm{\Psi }}_2$. It follows that $\left\{{\mathrm{x}}_n\right\}$ is a Cauchy sequence. As $\left\{{\mathrm{x}}_n\right\}$ is also T-orbital as well as ℜ-preserving, by $(\mathcal{O},\Re )$-completeness of $(M,d)$, there exists $\mathrm{x}\in M$ such that ${\mathrm{x}}_n\stackrel{d}{\to }\mathrm{x}$.

Now, we apply condition $\left(d\right)$. First, we suppose that T is $(\mathcal{O},\Re )$-continuous. Then, we have

$${\mathrm{x}}_{n+1}=T\left({\mathrm{x}}_n\right)\stackrel{d}{\to }T\left(\mathrm{x}\right).$$

But owing to uniqueness of convergence limit, we get $T\left(\mathrm{x}\right)=\mathrm{x}$. Hence, $\mathrm{x}$ is a fixed point of T. Otherwise, suppose that ℜ is $(\mathcal{O},d)$-self closed. As $\left\{{\mathrm{x}}_n\right\}$ is a T-orbital ℜ-preserving sequence in M with ${\mathrm{x}}_n\stackrel{d}{\to }\mathrm{x}$, by $(\mathcal{O},d)$-self closedness property of ℜ, there exists subsequence $\left\{{\mathrm{x}}_{n_k}\right\}$ of $\left\{{\mathrm{x}}_n\right\}$ such that $[{\mathrm{x}}_{n_k},\mathrm{x}]\in \Re$ for all $k\in {\mathbb{N}}_0$. By assumption $\left(e\right)$, and Proposition 3, we have

$$\begin{array}{ccc}\hfill d({\mathrm{x}}_{n_k+1},T\mathrm{x})& =& d(T{\mathrm{x}}_{n_k},T\mathrm{x})\hfill \\ & \le & d({\mathrm{x}}_{n_k},\mathrm{x})-\psi \left(d({\mathrm{x}}_{n_k},\mathrm{x})\right)\hfill \\ & \le & d({\mathrm{x}}_{n_k},\mathrm{x}),\forall k\in {\mathbb{N}}_0.\hfill \end{array}$$

Letting $k\to \infty$ in above inequality and using ${\mathrm{x}}_{n_k}\stackrel{d}{\to }\mathrm{x}$, we obtain

$${\mathrm{x}}_{n_k+1}\stackrel{d}{\to }T\left(\mathrm{x}\right).$$

Again, by uniqueness of convergence limit, we get $T\left(\mathrm{x}\right)=\mathrm{x}$. Thus, $\mathrm{x}$ is a fixed point of T. □

Theorem 2.

In addition to the hypotheses of Theorem 1, if $T\left(M\right)$ is ${\Re }^s$-connected, then T has a unique fixed point.

Proof. 

In view of Theorem 1, the set of fixed point is nonempty. If $\mathrm{x},\mathrm{y}$ are two fixed points of T, then for all $n\in {\mathbb{N}}_0$, we have

$$T^n\left(\mathrm{x}\right)=\mathrm{x},T^n\left(\mathrm{y}\right)=\mathrm{y}.$$

By ${\Re }^s$-connectedness of $T\left(M\right)$, there exists a path $\{{\omega }_1,{\omega }_2,\dots ,{\omega }_l\}$ of finite length l in ${\Re }^s$ from $\mathrm{x}$ to $\mathrm{y}$, so that

$${\omega }_0=\mathrm{x},{\omega }_l=\mathrm{y},[{\omega }_i,{\omega }_{i+1}]\in \Re \mathrm{for}\mathrm{all}i\in [0\le i\le l-1].$$

(10)

Using T-closedness of ℜ, Propositions 1 and 2, we get

$$[T^n{\omega }_i,T^n{\omega }_{i+1}]\in \Re ,\mathrm{for}\mathrm{each}i(0\le i\le l-1),\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0.$$

(11)

By triangular inequality, we have

$$\begin{array}{c}\hfill d(\mathrm{x},\mathrm{y})=d(T^n{\omega }_0,T^n{\omega }_l)\le \sum _{i=0}^{l-1}d(T^n{\omega }_i,T^n{\omega }_{i+1}).\end{array}$$

(12)

For each $n\in {\mathbb{N}}_0$ and for each $i(0\le i\le i-1)$, define $t_n^i:=d(T^n{\omega }_i,T^n{\omega }_{i+1})$. We claim that

$$\underset{n\to \infty }{lim}{t}_n^i=0.$$

To prove the claim, first fix i and consider two cases. First, suppose that $t_{n_0}^i=0$ for some $n_0\in {\mathbb{N}}_0$, i.e., $T^{n_0}\left({\omega }_i\right)=T^{n_0}\left({\omega }_{i+1}\right)$, which implies that $T^{n_0+1}\left({\omega }_i\right)=T^{n_0+1}\left({\omega }_{i+1}\right)$. Consequently, we get $t_{n_0+1}^i=d(T^{n_0+1}{\omega }_i,T^{n_0+1}{\omega }_{i+1})=0$. By induction, we get $t_n^i=0$ for all $n\ge n_0$, yielding thereby $\underset{n\to \infty }{lim}{t}_n^i=0$. On the other hand, suppose that $t_n>0$ for all $n\in {\mathbb{N}}_0$, then by (11) and assumption (e), we get

$$\begin{array}{ccc}\hfill t_{n+1}^i& =& d(T^{n+1}{\omega }_i,T^{n+1}{\omega }_{i+1})\hfill \\ & \le & d(T^n{\omega }_i,T^n{\omega }_{i+1})-\psi \left(d(T^n{\omega }_i,T^n{\omega }_{i+1})\right)\hfill \\ & =& t_n^i-\psi \left(t_n^i\right)\hfill \\ & \le & t_n^i.\hfill \end{array}$$

(13)

Therefore, $\left\{t_n^i\right\}$ is a decreasing sequence of positive real numbers, and hence, it converges to some $t\ge 0$. We conclude that $t=0$. Otherwise if $t\ne 0$, then by taking upper limit in (13), we obtain

$$\underset{n\to \infty }{lim\; sup}{t}_{n+1}^i\le \underset{n\to \infty }{lim\; sup}{t}_n^i+\underset{n\to \infty }{lim\; sup}(-\psi \left(t_n^i\right))$$

$$t\le t-\underset{n\to \infty }{lim\; inf}\left(\psi \left(t_n^i\right)\right)$$

yielding thereby $\underset{n\to \infty }{lim\; inf}\left(\psi \left(t_n^i\right)\right)\le 0$, which contradicts the condition ${\mathrm{\Psi }}_2$. Hence $\underset{n\to \infty }{lim}{t}_n^i=0$. Now (12) can be re-written as

$$\begin{array}{ccc}\hfill d(\mathrm{x},\mathrm{y})=d(T^n{\omega }_0,T^n{\omega }_l)& \le & \sum _{i=0}^{l-1}d(T^n{\omega }_i,T^n{\omega }_{i+1})\hfill \\ & \le & t_n^0+t_n^1+\cdots +t_n^{l-1}\hfill \\ & \to & 0\mathrm{as}n\to \infty \hfill \end{array}$$

which implies that $\mathrm{x}=\mathrm{y}$. Hence T has a unique fixed point. □

Making use of Proposition (4), we have the following consequences of Theorem 1.

Corollary 1.

Theorem 1 remains true if locally finitely T-transitivity condition is replaced by any one of the following conditions:

1. 

ℜ is transitive,

2. 

ℜ is finitely transitive,

3. 

ℜ is T-transitive,

4. 

ℜ is locally finitely transitive.

Corollary 2.

In addition to the hypotheses of Theorem 1, if anyone of the following assumptions holds,

(i)

${\Re |}_{T\left(M\right)}$ is a complete binary relation,

(ii)

$T\left(M\right)$ is ${\Re }^s$-directed,

then T has a unique fixed point.

Proof. 

If (i) is used, then for any $(\mathrm{x},\mathrm{y})\in T\left(M\right)$, $[\mathrm{x},\mathrm{y}]\in \Re$, that is to say that $\{\mathrm{x},\mathrm{y}\}$ is a path of length 1 in ${\Re }^s$ from $\mathrm{x}$ to $\mathrm{y}$. Hence, $T\left(M\right)$ is ${\Re }^s$-connected; consequently, Theorem 2 gives the conclusion.

Else if (ii) holds then for each $(\mathrm{x},\mathrm{y})\in T\left(M\right)$, there exists $\omega \in M$ such that $[\mathrm{x},\omega ]\in \Re$ and $[\omega ,\mathrm{y}]\in \Re$, that amounts to say that $\{\mathrm{x},\omega ,\mathrm{y}\}$ is a path of length 2 in ${\Re }^s$ from $\mathrm{x}$ to $\mathrm{y}$. Thus, $T\left(M\right)$ is ${\Re }^s$-connected and hence by Theorem 2, the conclusion is immediate. □

Example 1.

Suppose $M=[0,1]$ is a metric space with usual metric d. Consider a binary relation $\Re =\{\mathrm{x},\mathrm{y}\in M\mathit{such}\mathit{that}0\le \mathrm{x}<\mathrm{y}\le \frac{1}{2}\}$ and a mapping $T:M\to M$ defined by

$$T\left(\mathrm{x}\right)=\left\{\begin{array}{cc}\frac{1}{5}+\frac{\mathrm{x}}{3},\hfill & \mathit{if}\mathrm{x}\in [0,\frac{1}{2}]\hfill \\ \frac{1}{4},\hfill & \mathit{if}\mathrm{x}\in (\frac{1}{2},1].\hfill \end{array}\right.$$

It is clear that $(M,d)$ is $(\mathcal{O},\Re )$-complete, ℜ is T-closed and T is $(\mathcal{O},\Re )$-continuous. Now, we have

$$\begin{array}{ccc}\hfill d(T\mathrm{x},T\mathrm{y})& =& |\frac{1}{5}+\frac{\mathrm{x}}{3}-\frac{1}{5}-\frac{\mathrm{y}}{3}|\hfill \\ & =& |\frac{\mathrm{x}}{3}-\frac{\mathrm{y}}{3}|\hfill \\ & =& \frac{1}{3}|\mathrm{x}-\mathrm{y}|\hfill \\ & \le & \frac{1}{6}\mathrm{as}(\mathrm{x},\mathrm{y})\in \Re \mathrm{then}|\mathrm{x}-\mathrm{y}|\le \frac{1}{2}\hfill \\ & <& |\mathrm{x}-\mathrm{y}|(1-|\mathrm{x}-\mathrm{y}\left|\right)\hfill \\ & =& {|\mathrm{x}-\mathrm{y}|-|\mathrm{x}-\mathrm{y}|}^2\hfill \\ & =& d(\mathrm{x},\mathrm{y})-\psi \left(d\right(\mathrm{x},\mathrm{y}\left)\right),\hfill \end{array}$$

which implies that T satisfies the assumption $\left(e\right)$ for $\psi \left(a\right)=a^2$. Therefore, all the conditions of Theorem 1 are satisfied and hence T admits a fixed point. Furthermore, the fixed point is also unique as all assumptions of Theorem 2 are also satisfied. Notice that here $\mathrm{x}=\frac{3}{10}$ is the fixed point of T.

Here, it can be pointed out that the weak $\psi$-contractivity condition holds for merely ℜ-comparative elements, not for all elements of M, e.g., if we take $\mathrm{x}=1$ and $\mathrm{y}=0$, then $\left(\mathrm{x},\mathrm{y}\right)\notin \Re$. Also, this pair does not satisfy the weak $\psi$-contractivity condition. The above example can be viewed by Figure 1.

5. An Application

The following notions, considered by Ran and Reurings [16], are utilized in our subsequent discussion.

1.

$\mathcal{M}\left(n\right)$:=the set of all $n\times n$ complex matrices,

2.

$\mathcal{H}\left(n\right)\subset \mathcal{M}\left(n\right)$:=the set of all $n\times n$ Hermitian matrices,

3.

$\mathcal{P}\left(n\right)\subset \mathcal{H}\left(n\right)$:=the set of all $n\times n$ positive definite matrices,

4.

${\mathcal{H}}^{+}\left(n\right)$:=the set of all $n\times n$ positive semidefinite matrices.

On $\mathcal{H}\left(n\right)$, define an ordered relation ≺ by

$$W\prec V\iff W-V\in \mathcal{P}\left(n\right),\forall W,V\in \mathcal{H}\left(n\right).$$

Proposition 9

([16]). For every pair $W,V\in \mathcal{H}\left(n\right)$, there always exist a greatest lower bound and a least upper bound (w.r.t. ≺).

The spectral norm of a matrix B is denoted by $\parallel B\parallel$ = $\sqrt{{\beta }^{+}\left(B^{*}B\right)}$ such that ${\beta }^{+}\left(B^{*}B\right)$ is the largest eigenvalue of $B^{*}B$, where $B^{*}$ is the conjugate transpose of B. We use the metric induced by the trace norm ${\parallel ·\parallel }_{tr}$ defined by ${\parallel B\parallel }_{tr}={\displaystyle \sum _{i=1}^n}{t}_i\left(B\right)$, where $t_i\left(B\right)$, $i=1,2,\dots ,n$ are the singular values of $B\in \mathcal{M}\left(n\right)$. The set $\mathcal{H}\left(n\right)$ with the metric induced by this norm is ≺-complete metric space. Now, consider the matrix equation

$$W=U+\sum _{i=1}^m{B}_i^{*}\xi \left(W\right)B_i,$$

(14)

where $B_i$ is an arbitrary $n\times n$ matrix, U is a Hermitian positive definite matrix and $\xi :\mathcal{H}\left(n\right)\to \mathcal{P}\left(n\right)$ is a continuous ≺-preserving operator such that $\xi \left(0\right)=0$.

Lemma 3

([16]). Let A and B be two positive semidefinite $n\times n$ matrices. Then

$$0\le tr\left(AB\right)\le \parallel A\parallel tr\left(B\right).$$

Lemma 4

([47]). If $B\in \mathcal{H}\left(n\right)$ satisfies $(B,I_n)\in \Re$, then $\parallel B\parallel \le 1.$

Theorem 3.

Suppose that there is a positive number N such that

1. 

there exists $\psi \in \mathrm{\Psi }$ such that for all $W,V\in \mathcal{H}\left(n\right)$ verifying $W\prec V$, we have

$$\left|tr(\xi \left(V\right)-\xi \left(W\right))\right|\le \frac{1}{N}\left\{\left|tr\right(V-W\left)\right|-\psi \left(\right|tr(V-W)\left|\right)\right\}.$$

2. 

$N{I}_n-{\displaystyle \sum _{i=1}^m}{B}_i{B}^{*}\in \mathcal{P}\left(n\right)$ and ${\displaystyle \sum _{i=1}^m}{B}_i^{*}\xi \left(U\right)B_i\in \mathcal{P}\left(n\right).$

Then, the matrix Equation (14) has a unique solution. Moreover, the iteration

$$W_n=U+\sum _{i=1}^m{B}_i^{*}\xi \left(W_{n-1}\right)B_i,$$

(15)

where $W_0\in \mathcal{H}\left(n\right)$ such that $W_0\prec U+{\displaystyle \sum _{i=1}^m}{B}_i^{*}\xi \left(W_0\right)B_i$ converges in the sense of the trace norm ${\parallel ·\parallel }_{tr}$ to the solution of the matrix Equation (14).

Proof. 

Define the mapping $T:\mathcal{H}\left(n\right)\to \mathcal{H}\left(n\right)$ by

$$T\left(W\right)=U+\sum _{i=1}^m{B}_i^{*}\xi \left(W\right)B_i, \mathrm{for} \mathrm{all} W\in \mathcal{H}(n).$$

Then, ≺ is T-closed and the fixed point of T is the solution of (14). Let $W,V\in \mathcal{H}\left(n\right)$ such that $W\prec V$. Then, we have

$$\begin{array}{ccc}\hfill {\parallel T\left(V\right)-T\left(W\right)\parallel }_{tr}& =& tr\left(T\left(V\right)-T\left(W\right)\right)\hfill \\ & =& tr\left(\sum _{i=1}^m{B}_i^{*}\left(\xi \left(V\right)-\xi \left(W\right)\right)B_i\right)\hfill \\ & =& tr\left(\left(\sum _{i=1}^m\left(B_i{B}_i^{*}\right)\left(\xi \left(V\right)-\xi \left(W\right)\right)\right)\right)\hfill \\ & \le & \parallel {\displaystyle \sum _{i=1}^m}(B_i{B}_i^{*}{)\parallel \parallel \xi \left(V\right)-\xi \left(W\right)\parallel }_{tr}\hfill \\ & \le & \frac{\parallel {\displaystyle \sum _{i=1}^m}(B_i{B}_i^{*})\parallel }{N}\left({\parallel V-W\parallel }_{tr}-\psi \left({\parallel V-W\parallel }_{tr}\right)\right)\hfill \\ \hfill {\parallel T\left(V\right)-T\left(W\right)\parallel }_{tr}& <& {\parallel V-W\parallel }_{tr}-\psi \left({\parallel V-W\parallel }_{tr}\right).\hfill \end{array}$$

This shows that T satisfies weak $\psi$-contractivity condition for each ≺-comparative pair.

In particular, if we take $W=T\left(U\right)$ in (14), then we get

$$U-T\left(U\right)=-\sum _{i=1}^m{B}_i^{*}T\left(U\right)B_i\in \mathcal{P}\left(n\right).$$

This yields that $U\prec T\left(U\right)$ and hence $\mathcal{H}\left(n\right)(T;\prec )$ is nonempty. Now from Theorem 1, there exists $W\in \mathcal{H}\left(n\right)$ such that $T\left(W\right)=W$, i.e., the matrix Equation (14) has a solution. Furthermore, in view of Proposition 9, the set $T\left(\mathcal{H}\right(n\left)\right)$ is ${\prec }^s$-connected. Consequently, by Theorem 2, we get a unique solution of the matrix Equation (14). □

Intending to illustrate Theorem 3, we consider the following example:

Example 2.

Let $U=\left(\begin{array}{ccc}1/3& 1/4& 0\\ 1/4& 1/3& 0\\ 0& 0& 1/3\end{array}\right)$, $B_1=\left(\begin{array}{ccc}0& 1/12& 1/4\\ 1/6& 0& 1/6\\ 0& 1/8& 1/10\end{array}\right)$,$B_2=\left(\begin{array}{ccc}-1/20& 1/25& 0\\ 1/30& 1/10& -1/20\\ 1/15& -1/15& 0\end{array}\right)$, $B_3=\left(\begin{array}{ccc}1/6& 0& 1/12\\ 1/12& 1/6& 1/9\\ 1/9& 1/15& 1/6\end{array}\right).$

Define $\psi :[0,\infty )\to [0,\infty )$ by $\psi \left(a\right)=\frac{a}{3}$. Consider the matrix Equation (14) with $\xi \left(W\right)=W$, i.e.,

$$W=U+B_1^{*}\left(W\right)B_1+B_2^{*}\left(W\right)B_2+B_3^{*}\left(W\right)B_3.$$

(16)

It can be easily seen that all the conditions of Theorem 3 are satisfied with $N=\frac{2}{3}$. Now, we consider the iteration

$$W_n=U+B_1^{*}\left(W_{n-1}\right)B_1+B_2^{*}\left(W_{n-1}\right)B_2+B_3^{*}\left(W_{n-1}\right)B_3$$

(17)

where $W_0=U$, and the error $E{r}_n:={\parallel W_n-W_{n-1}\parallel }_{tr}$(See below Figure 2). From the graph(see below) of the error and the number of iteration we have that after five iteration, we can approximate a solution of (16) by

$$\widehat{W}\approx W_5=\left(\begin{array}{ccc}0.3585& 0.2754& 0.0408\\ 0.2839& 0.3726& 0.0548\\ 0.0312& 0.0231& 0.3798\end{array}\right)$$

with the error of $E{r}_5=9.6099\times {10}^{-5}$ (Calculate on MatLab 2021(b)).

To understand the error value generated for different values of n(where $W_n=X_n$) better, see Figure 2, given as below:

6. Conclusions

We provided the existence and uniqueness of the fixed points for a relation-theoretic weak contraction in the $(\mathcal{O},\Re )$–complete metric spaces. Additionally, an application is provided to find the unique solution of certain matrix equations. The similar results of the recent literature utilized the relational contraction conditions, which are weaker than the corresponding usual contractions. This restrictive nature provides the applicability of such results in areas of certain nonlinear matrix equations and boundary value problems.

As some future works, we can demonstrate these results in various spaces such as b-metric space, quasimetric space, cone metric space etc; or we can generalize these results for a pair of self-mappings; or we can apply the same results to obtain a unique solution of certain boundary value problems.

Several existing outcomes can be deduced from Theorems 1 and 2 in the following respects:

• Under $\Re =⪯$, partial order on M, Theorems 1 and 2 reduce to the fixed point theorems of Harjani and Sadarangani [18].
• Corollary 1 improves the result of Prasad and Dimri [5]. In fact, the authors [5] used the locally transitive relation, while we utilized the locally finitely T-transitive relation.
• Under the universal relation $\Re =M^2$, Theorem 2 reduces to a sharpened version of classical fixed point theorem of Rhoades [10], which again by setting $\psi \left(a\right)=a(1-k(a\left)\right)$ reduces to an enriched version of fixed point theorem of Geraghty [48].
• Particularly for $\psi \left(a\right)=(1-\alpha )a(0\le \alpha \le 1)$, Theorems 1 and 2 reduce to the fixed point theorems of Alam and Imdad [19] (after removing the locally finitely T-transitivity requirement on ℜ).
• Putting $\psi \left(a\right)=a-\varphi \left(a\right)$, where $\psi \in \mathrm{\Psi }$ is a lower semicontinuous function from the right, in Theorems 1 and 2, we get the fixed point results of Alam and Imdad [36].