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

Published online June 30, 2021

Copyright © Kyungpook Mathematical Journal.

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

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

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 [17], it was a combination of Banach contraction principle and Knaster-Tarski fixed point theorem. They proved fixed point results for monotone mapping F:XX 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. [16] to functions which are not necessarily continuous.

In 2009, Harjani and Sadarangani [11] considered the result of Rhoades [18] in the setting of partially ordered metric spaces. The concept of coupled fixed point was introduced by Geo and Lakshmikantham [10] in 1987. After that, Bhaskar and Lakshmikantham [5] 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. [4] proved some coupled coincidence point results for mixed g-monotone mappings in partially ordered metric spaces via new functions. Jachymski [13] and Jachymski and Lukawska [12] 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 [1] and Alfuraidan and Khamsi [2]. Alfuraidan and Khamsi [3] 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 [6] in 2008. This theory was developed further in [7] and [8]. In 2012, Chistyakov [9] established some fixed point theorems for contractive maps in modular metric spaces. Many authors have since considered this space. Ali Mutlu et al. [15] 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. [15] to a mapping having the mixed monotone property in modular metric spaces endowed with a graph.

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,bX.

Definition 2.1.([8]) 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*)

ωλ(a,a)=0 for all λ>0,aX

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<λωλ(a,b)[0,] is non-increasing on (0,).

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

ωλ(a,b)ωλμ(a,a)+ωμ(a,b)=ωμ(a,b).

Definition 2.2.([8]) Let ω be a psedomodular function on X. Fix a0∈ X, and set

Xω*=Xω*(a0)={aX:ωλ(a,a0)0asλ}.

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)={(aX:λ=λ(a)>0)suchthatωλ(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,bXω*.

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,bXω.

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

Definition 2.3.([8]) let Xω be a modular metric space, and {an}n be a sequence of Xω. Then,

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

  • (ii) {an}nXω 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.,

    for allϵ>0n0(ϵ)such thatfor alln,mn0(ϵ),ωλ(an,am)ϵ.

  • (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,mωλ(an,am)=0, then there exists an aXω such that limnωλ(an,a)=0.

Mongkolkeha et al. [14] introduced Banach contraction in modular metric spaces.

Definition 2.4.([14]) Let Xω be a modular metric space. A self mapping F on Xω is said to be a contraction if there exists 0k<1 such that

ωλ(Fa,Fb)kωλ(a,b)

for all a,bXω and λ>0.

We use the following terminology for graphs (see,[13]).

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 G1 denote the graph obtained from G by reversing the direction of edges. Thus we have E(G1)={(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(G1).

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 (an1,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:XX is called continuous if for all a,b ∈ X, there exist any sequences {an},{bn}X, for any n such that,

limnan=aandlimnbn=b,

implies that

limnF(an,bn)=F(a,b).

Definition 2.5.([13]) 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:XX 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,bX,

(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

(a1,c1)E(G),(c2,b1)E(G), and(a2,c1)E(G),(c2,b2)E(G).

Definition 2.7.([5]) Let (X,_) be a partially ordered set and F:X×XX 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,a2X,a1_a2F(a1,b)_F(a2,b)

and

b1,b2X,b1_b2F(a,b1)F(a,b2).

Definition 2.8.([5]) Let X be a nonempty set. An element (a,b)X×X is called a coupled fixed point of a mapping F:X×XX 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

((a,b),(u,v))E(G)(a,u)E(G) and (v,b)E(G),

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 b0F(b0,a0) and let a1=F(a0,b0) and b1=F(b0,a0), then a0_a1 and b0b1. 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).

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×XωXω has G-preserving property if

    (a1,a2)E(G)(F(a1,b),F(a2,b))E(G),

    for all a1,a2,bXω and

    (b1,b2)E(G)(F(a,b1),F(a,b2))E(G),

    for all a,b1,b2Xω.

  • (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,bXω and

    (b1,b2)E(G)(F(a,b2),F(a,b1))E(G),

    for all a,b1,b2Xω.

  • (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,bXω, and

    (b1,b2)E(G)(F(a,b2),F(a,b1))E(G),

    for all a,b1,b2Xω.

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

F1(a,b)=a+b for all a,bXωF2(a,b)=ea+eb for all a,bXωF3(a,b)=a+11+b for all a,bXω.

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×XωXω 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,qXω and λ>0:

ωλ(F(a,b),F(p,q))kωλ(a,p)+lωλ(b,q),

where ((a,p),(q,b))E(G). If there exist a0,b0Xω such that

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

Proof. Let a0,b0Xω 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,b1Xω with a1=F(a0,b0) and b1=F(b0,a0), then (a0,a1)E(G), (b1,b0)E(G). Let a2,b2Xω, 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

an+1=F(an,bn) and bn+1=F(bn,an),n=0,1,2,, (ai1,ai),(bi,bi1)E(G),i=1,2,.

Then by (3.1), we get

ωλ(an,an+1)=ωλ(F(an1,bn1),F(an,bn))    kωλ(an1,an)+lωλ(bn1,bn),n.

Similarly

ωλ(bn,bn+1)=ωλ(F(bn1,an1),F(bn,an)),    kωλ(bn1,bn)+lωλ(an1,an),for alln.

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

ωλ(an,an+1)+ωλ(bn,bn+1)(k+l)ωλ(an1,an)+(k+l)ωλ(bn1,bn)          =(k+l)ωλ(an1,an)+ωλ(bn1,bn).

By successive applications of the above inequality we obtain

0ωλ(an,an+1)+ωλ(bn,bn+1)(k+l)ωλ(an1,an)+ωλ(bn1,bn)(k+l)nωλ(a0,a1)+ωλ(b0,b1).

It follows from (3.5) that

limnωλ(an,an+1)+ωλ(bn,bn+1)=0.

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

ωλ(an,an+1)+ωλ(bn,bn+1)<ε foralln>n0,λ>0.

Without loss of generality, suppose m,n and n<m, there exist nλmn satisfying

ωλmn(an,an+1)+ωλmn(bn,bn+1)<εmn for all nnλmn.

We get

ωλ(an,am)ωλmn(an,an+1)+ωλmn(an+1,an+2)++ωλmn(am1,am)

and

ωλ(bn,bm)ωλmn(bn,bn+1)+ωλmn(bn+1,bn+2)++ωλmn(bm1,bm),n<m.

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

ωλ(an,am)+ωλ(bn,bm)ωλmn(an,an+1)+ωλmn(bn,bn+1)          ++ωλmn(am1,am)+ωλmn(bm1,bm)          <εmn+εmn++εmn          =ε

for all n>nλmn. The above inequality shows that

ωλ(an,am)<ε,ωλ(bn,bm)<ε

for all n>nλmn.

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

limnan=aandlimnbn=b.

Since F is continuous, we obtain:

a=limnan+1=limnF(an,bn)=F(a,b)

and

b=limnbn+1=limnF(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 [16].

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 limnan=a we have:

  • (i) if (an,an+1)E(G)for allnimplies(an,a)E(G);

  • (ii) if (an+1,an)E(G)for allnimplies(a,an)E(G).

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×XωXω 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,qXω and λ>0:

ωλ(F(a,b),F(p,q))kωλ(a,p)+lωλ(b,q),

where ((a,p),(q,b))E(G). If there exist a0,b0Xω 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 limnan=a,limnbn=b and

an+1=F(an,bn),bn+1=F(bn,an),(an,an+1),(bn+1,bn)E(G),for alln0.

Since the graph G is ω-regular, therefore (an,a)E(G) and (b,bn)E(G) for all n0. Let ϵ>0 be given, then there exist n0 with ωλ2(an,a)<ϵ2 and ωλ2(b,bn)<ϵ2, for all nn0, 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      <ϵask+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)withk+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×XωXω has mixed monotone property in Xω and k[0,1). Suppose that we have the following condition for all a,b,p,qXω 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,b0Xω 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×XωXω has mixed monotone property in Xω and k[0,1). Suppose that we have the following condition for all a,b,p,qXω 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,b0Xω 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=, we define a metric modular function ω:(0,)××[0,) by ωλ(a,b)=|ab|λ for all a,b and λ>0. Define a mapping F:× such that F(a,b)=a2b4,(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

|a2b4p2q4|k2[|ap|+|bq|],ap,bq,

by which, for a = p, we get

bqkbq,bq,

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

a2b4p2q414ap+12bq,b2a4q2p414bq+12ap,

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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