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##  eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(2): 441-453

Published online June 30, 2021

### Coupled Fixed Point Theorems in Modular Metric Spaces Endowed with a Graph

Yogita Sharma*, Shishir Jain

Department of Computer Science, Shri Vaishnav Institute of Management, Gumashta Nagar, Indore
e-mail : yogitasharma2006@gmail.com

Department of Mathematics, Shri Vaishnav Vidyapeeth Vishwavidyalaya, Gram Baroli, Sanwer Road, Indore
e-mail : jainshishir11@rediffmail.com

Received: March 25, 2020; Revised: September 19, 2020; Accepted: January 6, 2021

### Abstract

In this work, we define the concept of a mixed G-monotone mapping on a modular metric space endowed with a graph, and prove some fixed point theorems for this new class of mappings. Results of this paper extend coupled fixed point theorems from partially ordered metric spaces into the modular metric spaces endowed with a graph. An example is presented to illustrate the new result.

Keywords: coupled fixed point, G-monotone mapping, connected graph, modular metric space, partial order relation

### 1. Introduction

The Banach contraction principle is one of the most useful principle in applied mathematics. Because of its simplicity and usefulness, and its compatibility in modeling various problems, it has been generalized and extended by several researchers in various directions. Many mathematicians have done remarkable work on fixed point results for partially ordered metric spaces. The very first foray in this direction was taken by Ran and Reurings , it was a combination of Banach contraction principle and Knaster-Tarski fixed point theorem. They proved fixed point results for monotone mapping $F:X→X$ on a complete metric space (X,d) endowed with a partial order relation ≼. The results of Ran and Reurings were extended by Neito et al.  to functions which are not necessarily continuous.

In 2009, Harjani and Sadarangani  considered the result of Rhoades  in the setting of partially ordered metric spaces. The concept of coupled fixed point was introduced by Geo and Lakshmikantham  in 1987. After that, Bhaskar and Lakshmikantham  studied applications of coupled fixed point theorems for binary mappings. They introduced the concept of the mixed monotone property, and proved certain coupled fixed point theorems. These theorems are among the most interesting coupled fixed point theorems for mappings in ordered metric spaces having the mixed monotone property. In particular, they manifested the existence of a unique solution for a periodic boundary value problem. Ansari et al.  proved some coupled coincidence point results for mixed g-monotone mappings in partially ordered metric spaces via new functions. Jachymski  and Jachymski and Lukawska  introduced the concept of graph theory in the study of fixed point results. They generalized the above mentioned results and presented applications to the theory of linear operators, They studied the class of generalized Banach contractions on a metric space with a directed graph. This work on the fixed point theory of a metric space endowed with a graph, has since been extended by Alfuraidan  and Alfuraidan and Khamsi . Alfuraidan and Khamsi  also proved coupled fixed point results of monotone mappings in a metric space with a graph.

Many coupled fixed point theorems were extended to modular metric space, which was introduced by Chistyakov via F-modular mappings  in 2008. This theory was developed further in  and . In 2012, Chistyakov  established some fixed point theorems for contractive maps in modular metric spaces. Many authors have since considered this space. Ali Mutlu et al.  extended to partially ordered modular metric spaces certain coupled fixed point theorems for mappings having the mixed monotone property, and proved the existence of a unique solution for a given nonlinear integral equation. In this paper, we extend certain the coupled fixed point results of Ali Mutlu et al.  to a mapping having the mixed monotone property in modular metric spaces endowed with a graph.

### 2. Preliminaries

Let X be a nonempty set, λ be in $(0,∞)$, and the function $ω:(0,∞)×X×X→[0,∞]$ will be written as $ωλ(a,b)=ω(λ,a,b)$ for all $λ>0$ and $a,b∈X$.

Definition 2.1.() Let X be a nonempty set, a function $ω:(0,∞)×X×X→[0,∞]$ is said to be a modular metric on X if it satisfies the following axioms, for all a,b,c ∈ X:

• (i) $ωλ(a,b)=0$ for all $λ>0$ if and only if a = b.

• (ii) $ωλ(a,b)=ωλ(b,a)$ for all $λ>0$.

• (iii) $ωλ+μ(a,b)≤ωλ(a,c)+ωμ(c,b)$ for all $λ,μ>0$.

If instead of (i), we have the condition $(i*)$

then ω is said to be pseudomodular on X. The main property of a (pseudo) modular function ω on a set X is that for given a,b∈ X, the function $0<λ↦ωλ(a,b)∈[0,∞]$ is non-increasing on $(0,∞)$.

In fact, if $0<μ<λ$, then (iii), $(i*)$ and (ii) imply

$ωλ(a,b)≤ωλ−μ(a,a)+ωμ(a,b)=ωμ(a,b).$

Definition 2.2.() Let ω be a psedomodular function on X. Fix a0∈ X, and set

$Xω*=Xω*(a0)={a∈X:ωλ(a,a0)→0 as λ→∞}.$

A modular (pseudomodular, strict modular) function ω on X is said to be convex if, instead of (iii), for all $λ,μ>0$ and a,b,c∈ X it satisfies the inequality

$(iv) ωλ+μ(a,b)≤λλ+μωλ(a,c)+μλ+μωμ(c,b).$

The set

$Xω=Xω(a0)={(a∈X:∃λ=λ(a)>0) such that ωλ(a,a0)<∞}$

is called a modular metric space (around a0).

It is clear that $Xω*⊂Xω$, and it is known that this inclusion is proper in general. Also, if ω is a modular function on X, then the modular space Xω can be equipped with a (nontrivial) metric $dω*$, generated by ω and given by

$dω*(a,b)=inf{λ>0:ωλ(a,b)≤λ}, a,b∈Xω*.$

If ω is a convex modular function on X, then the two modular spaces coincide, $Xω=Xω$, and this common set can be endowed with a metric dω given by

$dω(a,b)=inf{λ>0:ωλ(a,b)≤1}, a,b∈Xω.$

Even if ω is a nonconvex modular on X, then $dω*(a,a)=0$ and $dω(a,b)=dω(b,a)$.

Definition 2.3.() let Xω be a modular metric space, and ${an}n∈ℕ$ be a sequence of $Xω$. Then,

• (i) ${an}n∈ℕ$ in $Xω$ or $Xω*$, is said to be modularly convergent to an element $a∈Xω$ if $ωλ(an,a)→0$ as and any such element a will be called a modular limit of the sequence {an}.

• (ii) ${an}n∈ℕ⊂Xω$ is a modular Cauchy sequence (ω-Cauchy) if there exists a number $λ=λ({an})>0$ such that $ωλ(an,am)→0$ as $n,m→∞$, i.e.,

• (iii) A modular space $Xω$ is called modularly complete if every modular Cauchy sequence {an} in $Xω$ is modularly convergent in the following sense - if ${an}⊂Xω$ and there exists $λ=λ({an})>0$ such that $limn,m→∞ωλ(an,am)=0$, then there exists an $a∈Xω$ such that $limn→∞ωλ(an,a)=0.$

Mongkolkeha et al.  introduced Banach contraction in modular metric spaces.

Definition 2.4.() Let $Xω$ be a modular metric space. A self mapping F on $Xω$ is said to be a contraction if there exists $0≤k<1$ such that

$ωλ(Fa,Fb)≤kωλ(a,b)$

for all $a,b∈Xω$ and $λ>0$.

We use the following terminology for graphs (see,).

Let (X,d) be a metric space and $△$ be the diagonal of $X×X$. Let G be a directed graph such that the set V(G) of vertices coincides with X and the set E(G) of edges contains all loops, i.e. $(a,a)∈E(G)$ for every a∈ V(G). Such a digraph is called reflexive. Assume that G has no parallel edges, so we have G = (V(G),E(G)). Let $G−1$ denote the graph obtained from G by reversing the direction of edges. Thus we have $E(G−1)={(b,a)∣(a,b)∈E(G)}$. Also, $G˜$ denotes the undirected graph defined by G by ignoring the direction of edges and we have,

$E(G˜)=E(G)∪E(G−1).$

If a and b are vertices in a graph G, then a (directed) path in G from a to b of length {N} is a sequence ${ai}i=0N$ of {N}+1 vertices such that a0 = a, $aN=b$ and $(an−1,an)∈E(G)$ for $i=1,2,⋯,N$. A graph G is connected if there is a directed path between any two vertices. G is weakly connected if $G˜$ is connected.

The operator $F:X→X$ is called continuous if for all a,b ∈ X, there exist any sequences ${an},{bn}∈X$, for any $n∈ℕ$ such that,

$limn→∞an=a and limn→∞bn=b,$

implies that

$limn→∞F(an,bn)=F(a,b).$

Definition 2.5.() Let (X,d) be a metric space and G = (V(G),E(G)) be a directed graph such that V(G) = X and E(G) contains all loops, that is $△⊆E(G)$. We say that a mapping $F:X→X$ is a G-contraction if F preserves edges of G, i.e., for every a,b∈ X,

$(a,b)∈E(G)⇒(Fa,Fb)∈E(G)$

and there exists $α∈(0,1)$ such that $a,b∈X$,

$(a,b)∈E(G)⇒d(Fa,Fb)≤αd(a,b).$

Remark 2.6. Elements are said to be comparable if for every $(a1,b1),(a2,b2)∈X×X$ there exists $(c1,c2)∈X×X$ such that

Definition 2.7.() Let $(X,≺_)$ be a partially ordered set and $F:X×X→X$ be a mapping. We say that F has the mixed monotone property if F(a, b) is monotone nondecreasing in a and is monotone nonincreasing in b, that is, for any a, b ∈ X,

$a1,a2∈X,a1≺_a2⇒F(a1,b)≺_F(a2,b)$

and

$b1,b2∈X,b1≺_b2⇒F(a,b1)≽F(a,b2).$

Definition 2.8.() Let X be a nonempty set. An element $(a,b)∈X×X$ is called a coupled fixed point of a mapping $F:X×X→X$ if $F(a,b)=a$ and $F(b,a)=b$.

Note that if G is a directed graph defined on X, we can construct another graph on $X×X$, still denoted by G, by

for any $(a,b),(u,v)∈X×X$.

{\bf Remark 2.9.} It is noted that if $(a0,b0)∈Xω$ such that $a0≺_F(a0,b0)$ and $b0≽F(b0,a0)$ and let $a1=F(a0,b0)$ and $b1=F(b0,a0)$, then $a0≺_a1$ and $b0≽b1$. Again let $a2=F(a1,b1)$ and $b2=F(b1,a1)$, we denote

$F2(a0,b0)=F(F(a0,b0),F(b0,a0))=F(a1,b1)=a2$

and

$F2(b0,a0)=F(F(b0,a0),F(a0,b0))=F(b1,a1)=b2.$

Due to the mixed monotone property of F, we have

$a2=F2(a0,b0)=F(a1,b1)≽F(a0,b0)=a1$

and

$b2=F2(b0,a0)=F(b1,a1)≺_F(b0,a0)=b1.$

Further for $n=1,2,⋯$, we get

$an+1=Fn+1(a0,b0)=F(Fn(a0,b0),Fn(b0,a0))=F(an,bn)$

and

$bn+1=Fn+1(b0,a0)=(Fn(b0,a0),Fn(a0,b0))=F(bn,an).$

### 3. Main Results

In this section, we assume that $(Xω,G)$ is a modular metric space endowed with a graph G such that $V(G)=Xω$, $△⊆E(G)$ and G is transitive, i.e., $(a,b)∈E(G),(b,c)∈E(G)$ implies that $(a,c)∈E(G)$.

Definition 3.1. Let X be a nonempty set endowed with a graph G.

• (i) A mapping $F:Xω×Xω→Xω$ has G-preserving property if

$(a1,a2)∈E(G)⇒(F(a1,b),F(a2,b))∈E(G),$

for all $a1,a2,b∈Xω$ and

$(b1,b2)∈E(G)⇒(F(a,b1),F(a,b2))∈E(G),$

for all $a,b1,b2∈Xω.$

• (ii) The mapping F has G-inverting property if

$(a1,a2)∈E(G)⇒(F(a2,b),F(a1,b))∈E(G),$

for all $a1,a2,b∈Xω$ and

$(b1,b2)∈E(G)⇒(F(a,b2),F(a,b1))∈E(G),$

for all $a,b1,b2∈Xω.$

• (iii) And we say that a mapping F has mixed G-monotone property if

$(a1,a2)∈E(G)⇒(F(a1,b),F(a2,b))∈E(G),$

for all $a1,a2,b∈Xω$, and

$(b1,b2)∈E(G)⇒(F(a,b2),F(a,b1))∈E(G),$

for all $a,b1,b2∈Xω.$

Example 3.2. Let $Xω=[0,∞)$ and G be a graph such that $V(G)=Xω$ and $E(G)={(a,b)∈Xω×Xω:a≤b}$. Define mappings $F1,F2,F3:Xω×Xω→Xω$ by:

Then, F1 has G-preserving property, F2 has G-inverting property, while F3 has mixed G-monotone property. Note that each of these three mappings has exactly one property, therefore we can say that these three properties are independent of each other.

Theorem 3.3. Let (Xω,G) be a complete modular metric space with a graph G. Suppose that $F:Xω×Xω→Xω$ is a continuous mapping which has mixed G-monotone property in Xω and k,l be nonnegative constants such that $k+l<1.$ Suppose that the following condition is satisfied for all $a,b,p,q∈Xω$ and $λ>0$:

$ωλ(F(a,b),F(p,q))≤k ωλ(a,p)+l ωλ(b,q),$

where $((a,p),(q,b))∈E(G).$ If there exist $a0,b0∈Xω$ such that

$((a0,b0),(F(a0,b0),F(b0,a0)))∈E(G),$ then F has a coupled fixed point.

Proof. Let $a0,b0∈Xω$ be such that $((a0,b0),(F(a0,b0),F(b0,a0)))∈E(G)$, i.e.,

$(a0,F(a0,b0))∈E(G) and (F(b0,a0),b0)∈E(G).$

We take $a1,b1∈Xω$ with $a1=F(a0,b0)$ and $b1=F(b0,a0)$, then $(a0,a1)∈E(G)$, $(b1,b0)∈E(G).$ Let $a2,b2∈Xω$, where $a2=F(a1,b1)$ and $b2=F(b1,a1)$. Then, by mixed monotone property of F we have

$(F(a0,b0),F(a1,b0))∈E(G),(F(a1,b0),F(a1,b1))∈E(G)⇒(F(a0,b0),F(a1,b1))∈E(G)⇒(a1,a2)∈E(G).$

Similarly, we can obtain that $(b2,b1)∈E(G)$. By induction, we construct two sequences {an} and {bn} in $Xω$ such that

$(ai−1,ai),(bi,bi−1)∈E(G), i=1,2,….$

Then by (3.1), we get

$ωλ(an,an+1)=ωλ(F(an−1,bn−1),F(an,bn)) ≤kωλ(an−1,an)+lωλ(bn−1,bn), n∈ℕ.$

Similarly

Thus, for any $n∈ℕ$ from (3.2) and (3.3), we get

$ωλ(an,an+1)+ωλ(bn,bn+1)≤(k+l)ωλ(an−1,an)+(k+l)ωλ(bn−1,bn) =(k+l)ωλ(an−1,an)+ωλ(bn−1,bn).$

By successive applications of the above inequality we obtain

$0≤ωλ(an,an+1)+ωλ(bn,bn+1)≤(k+l)ωλ(an−1,an)+ωλ(bn−1,bn)≤⋯≤(k+l)nωλ(a0,a1)+ωλ(b0,b1).$

It follows from (3.5) that

$limn→∞ωλ(an,an+1)+ωλ(bn,bn+1)=0.$

Therefore, if $ε>0$ is given then there exists $n0∈ℕ$ such that

Without loss of generality, suppose $m,n∈ℕ$ and $n, there exist $nλm−n∈ℕ$ satisfying

We get

$ωλ(an,am)≤ωλm−n(an,an+1)+ωλm−n(an+1,an+2)+⋯+ωλm−n(am−1,am)$

and

$ωλ(bn,bm)≤ωλm−n(bn,bn+1)+ωλm−n(bn+1,bn+2)+⋯+ωλm−n(bm−1,bm), n

Thus from inequalities (3.6) and (3.7), we get

$ωλ(an,am)+ωλ(bn,bm)≤ωλm−n(an,an+1)+ωλm−n(bn,bn+1) +⋯+ωλm−n(am−1,am)+ωλm−n(bm−1,bm) <εm−n+εm−n+⋯+εm−n =ε$

for all $n>nλm−n$. The above inequality shows that

$ωλ(an,am)<ε, ωλ(bn,bm)<ε$

for all $n>nλm−n$.

This shows that {an} and {bn} are modular Cauchy sequences in Xω. Using completeness of Xω, for $a,b∈Xω$ we have

$limn→∞an=a and limn→∞bn=b.$

Since F is continuous, we obtain:

$a=limn→∞an+1=limn→∞F(an,bn)=F(a,b)$

and

$b=limn→∞bn+1=limn→∞F(bn,an)=F(b,a).$

Thus,$(a,b)∈Xω×Xω$ is a coupled fixed point of F.

The assumption of continuity of F may be relaxed by applying the condition of ω-regularity of graph G which is inspired by Neito and Rodŕiguez-López .

Definition 3.4. Let $(Xω,G)$ be a complete modular metric space with a graph G. Then, the graph G is called ω-regular if for every sequence {an} in $Xω$ such that $limn→∞an=a$ we have:

• (i) if ;

• (ii) if .

The following theorem uses the ω-regularity of graph G instead the continuity of F.

Theorem 3.5. Let $(Xω,G)$ be a complete modular metric space with a graph G. Suppose that $F:Xω×Xω→Xω$ is a mapping which has mixed monotone property in $Xω$ and k,l be nonnegative constants such that $k+l<1.$ Suppose that the following condition is satisfied for all $a,b,p,q∈Xω$ and $λ>0$:

$ωλ(F(a,b),F(p,q))≤k ωλ(a,p)+l ωλ(b,q),$

where $((a,p),(q,b))∈E(G).$ If there exist $a0,b0∈Xω$ such that $((a0,b0), (F(a0,b0), F(b0,a0))∈E(G)$, and the graph G is ω-regular, then F has a coupled fixed point.

Proof. The construction of the sequence {an} and {bn}, which converge to a and b respectively, following the similar process used in Theorem 3.3. Thus, we have two sequences {an} and {bn} such that $limn→∞an=a, limn→∞bn=b$ and

Since the graph G is ω-regular, therefore $(an,a)∈E(G)$ and $(b,bn)∈E(G)$ for all $n≥0$. Let $ϵ>0$ be given, then there exist $n0∈ℕ$ with $ωλ2(an,a)<ϵ2$ and $ωλ2(b,bn)<ϵ2$, for all $n≥n0$, $λ>0$. So, from (iii) and using (3.1) we get

$ωλ(a,F(a,b))≤ωλ2(a,an+1)+ωλ2(an+1,F(a,b)) =ωλ2(an+1,a)+ωλ2(F(an,bn),F(a,b)) ≤ωλ2(an+1,a)+kωλ2(an,a)+lωλ2(bn,b) <ϵ2+kϵ2+lϵ2 =ϵ2+(k+l)ϵ2 <ϵ as k+l<1,$

for all $λ>0$. Hence, $ωλ(a,F(a,b))=0$. So, $F(a,b)=a$. Similarly, we get $F(b,a)=b$. Thus, $(a,b)∈Xω×Xω$ is a coupled fixed point of F.

Remark 3.6. Since the contractivity assumption is made only on comparable elements in $Xω×Xω$, Theorems 3.3 and 3.5, don't guarantee the uniqueness of the coupled fixed point. However, the uniqueness of the coupled fixed point can be establish with the following condition:

For the uniqueness of coupled fixed point we endow product space $Xω×Xω$ with the graph G such that for every $(a,b),(a*,b*)∈Xω×Xω$ there exists $(u,v)∈Xω×Xω$ such that

$(a,u)∈E(G),(v,b)∈E(G) and (a*,u)∈E(G),(v,b*)∈E(G).$

Here, we discuss the uniqueness of the coupled fixed point.

Theorem 3.7. Suppose that all the conditions of Theorem 3.3 (respectively Theorem 3.5) are satisfied. In addition, suppose that the condition (3.10) is satisfied, then F has a unique coupled fixed point.

Proof. It follows from Theorem 3.3 (respectively Theorem 3.5). Suppose that (a,b) and (a*,b*) are two distinct coupled fixed of F. We consider two cases: Case I: If $((a,a*),(b*,b))∈E(G)$. Then, we have from (3.1)

$ωλ(F(a,b),F(a*,b*))≤kωλ(a,a*)+lωλ(b,b*)$

and

$ωλ(F(b*,a*),F(b,a))≤kωλ(b*,b)+lωλ(a*,a) with k+l<1.$

Since (a,b) and (a*,b*) are coupled fixed points of F, we get

$ωλ(a,a*)=ωλ(F(a,b),F(a*,b*))≤kωλ(a,a*)+lωλ(b,b*)$

and

$ωλ(b,b*)=ωλ(F(b,a),F(b*,a*))≤kωλ(b,b*)+lωλ(a,a*).$

Therefore, we have

$ωλ(a,a*)+ωλ(b,b*)≤(k+l)ωλ(a,a*)+(k+l)ωλ(b*,b) =(k+l)(ωλ(a,a*)+ωλ(b,b*)) <ωλ(a,a*)+ωλ(b,b*).$

This is a contradiction as $k+l<1$ and yields the conclusion that the coupled fixed point is unique.

Case II: If (a,b) is not comparable to (a*,b*) such that $(a,b),(a*,b*)∉E(G)$, then there exists $(u,v)∈Xω×Xω$ such that $(a,u)∈E(G)$, $(v,b)∈E(G)$ and $(a*,u)∈E(G)$, $(v,b*)∈E(G)$. From the monotonic property of F it follows that $Fn(u,v)$ is comparable to $Fn(a,b)=a,Fn(b,a)=b$ and $Fn(a*,b*)=a*,Fn(b*,a*)=b*$. Then, we have

$ωλ((a,b),(a*,b*))=ωλ(Fn(a,b),Fn(b,a),Fn(a*,b*),Fn(b*,a*)) ≤ωλ2(Fn(a,b),(Fn(b,a),Fn(u,v),Fn(v,u)) + ωλ2(Fn(u,v),Fn(v,u),Fn(a*,b*),Fn(b*,a*)) ≤(k+l)n(ωλ(a,u)+ωλ(b,v)+ωλ(u,a*)+ωλ(v,b*)).$

Taking $n→∞$, it follows that $ωλ((a,b),(a*,b*))≤0$ $⇒(a,b)=(a*,b*)$. It follows that coupled fixed point is unique. Therefore for given $(a0,b0)∈Xω×Xω$ such that $((a0,b0),(F(a0,b0),F(b0,a0)))∈E(G)$, there exist a unique coupled fixed point (a,b) of F.

Corollary 3.8. Let $(Xω,G)$ be a complete modular metric space with a graph, A continuous mapping $F:Xω×Xω→Xω$ has mixed monotone property in $Xω$ and $k∈[0,1).$ Suppose that we have the following condition for all $a,b,p,q∈Xω$ and $λ>0.$

$ωλ(F(a,b),F(p,q))≤k2(ωλ(a,p)+ωλ(b,q)).$

Here $(p,a),(b,q)∈E(G).$ If there exist $a0,b0∈Xω$ such that $((a0,b0),(F(a0,b0), F(b0,a0)))∈E(G)$. In addition suppose that the condition (3.10) is satisfied, then F has a unique coupled fixed point.

Corollary 3.9. let $(Xω,G)$ be a complete modular metric space with graph G. Suppose that $Xω$ satisfies the following conditions:

• (i) if a non-decreasing sequence ${an}→a$ then $(an,a)∈E(G)$ for all n,

• (ii) if a non-increasing sequence ${bn}→b$ then $(b,bn)∈E(G)$ for all n,

let a mapping $F:Xω×Xω→Xω$ has mixed monotone property in $Xω$ and $k∈[0,1).$ Suppose that we have the following condition for all $a,b,p,q∈Xω$ and $λ>0$

$ωλ(F(a,b),F(p,q))≤k2(ωλ(a,p)+ωλ(b,q))$

where $(a,p),(q,b)∈E(G).$ if there exist $a0,b0∈Xω$ with

$((a0,b0),(F(a0,b0),F(b0,a0)))∈E(G),$

In addition suppose that the condition (3.10) is satisfied, then F has a unique coupled fixed point.

Remark 3.10. Let for a complete modular metric space $Xω=ℝ,$ we define a metric modular function $ω:(0,∞)×ℝ×ℝ→[0,∞)$ by and $λ>0$. Define a mapping $F:ℝ×ℝ→ℝ$ such that $F(a,b)=a−2b4,(a,b)∈Xω×Xω.$ Then F is continuous. Let G be the reflexive digraph defined on $Xω$ with $((a,p),(q,b))∈E(G)$. Then we easily see that F has the mixed G-monotone property and satisfies condition (3.1) but does not satisfy the condition (3.13). Assume there exists $k∈[0,1),$ such that (3.1) holds. Then, we must have

$|a−2b4−p−2q4|≤k2[|a−p|+|b−q|],a≥p,b≤q,$

by which, for a = p, we get

$∣b−q∣≤k∣b−q∣,b≤q,$

which is a contradiction, since $k∈[0,1).$ Hence F does not satisfy the contractive condition (3.13).

Now, we prove that (3.1) holds. Indeed, for a ≥ p and b ≤ q, we have

$∣a−2b4−p−2q4∣≤14∣a−p∣+12∣b−q∣,∣b−2a4−q−2p4∣≤14∣b−q∣+12∣a−p∣,$

that is, the inequality (3.1) holds for $k=14$ and $l=12$, so by Theorem 3.3 we obtain that F has a coupled fixed point (0,0) but none of the Corollary 3.8 and 3.9 can be applied to F in this example.

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