Some fixed-point theorems for a general class of mappings in modular G-metric spaces

Godwin Amechi Okeke (Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, Nigeria)
Daniel Francis (Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, Nigeria)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 14 July 2021

Issue publication date: 29 June 2022

1135

Abstract

Purpose

This paper aims to prove some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results of this paper generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. Furthermore, the authors produce an example to demonstrate the applicability of the results.

Design/methodology/approach

The results of this paper are theoretical and analytical in nature.

Findings

The authors established some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. An example was constructed to demonstrate the applicability of the results.

Research limitations/implications

Analytical and theoretical results.

Practical implications

The results of this paper can be applied in science and engineering.

Social implications

The results of this paper is applicable in certain social sciences.

Originality/value

The results of this paper are new and will open up new areas of research in mathematical sciences.

Keywords

Citation

Okeke, G.A. and Francis, D. (2022), "Some fixed-point theorems for a general class of mappings in modular G-metric spaces", Arab Journal of Mathematical Sciences, Vol. 28 No. 2, pp. 203-216. https://doi.org/10.1108/AJMS-02-2021-0037

Publisher

:

Emerald Publishing Limited

Copyright © 2021, Godwin Amechi Okeke and Daniel Francis

License

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

In search for the generalization of classical metric spaces, in 1966, Gahler [2], introduced the concept of 2-metric spaces and proved that its results exists. Dhage [3] extend the work in [2] in which D-metric spaces were introduced. These authors claimed that their results generalized the concept of metric spaces.

In 2003, Mustafa and Sims [4] claimed that the fundamental topological properties of D-metric spaces introduced by Dhage [3] were incorrect. To ameliorate the drawbacks about D-metric spaces, Mustafa and Sims [5] introduced a generalization of metric spaces, which they called G-metric spaces and proved some fixed-point theorems, and in [6], Mustafa et al. proved some fixed-point results on complete G-metric spaces.

Modular theories on linear spaces were given by Nakano in his two monographs [7, 8], where he developed a spectral theory in semiordered linear spaces (vector lattices) and established the integral representation for projections acting in this modular spaces. Nakano [7] established some modulars on real linear spaces, which are convex functionals. Nonconvex modulars and the corresponding modular linear spaces were constructed by Musielak and Orlicz [9]. Orlicz spaces and modular linear spaces have already become classical tools in modern nonlinear functional analysis.

In 2010, a remarkable work of Chistyakov [10] introduced an aspect of metric spaces called modular metric spaces or parameterized metric spaces with the time parameter λ (say), and his purpose was to define the notion of a modular on an arbitrary set and developed the theory of metric spaces generated by modulars, called modular metric spaces and, on the basis of it, defined new metric spaces of (multi-valued) functions of bounded generalized variation of a real variable with values in metric semigroups and abstract convex cones.

In the same year, Chistyakov [11], as an application, presented an exhausting description of Lipschitz continuous and some other classes of superposition (Nemytskii) operators, acting in these modular metric spaces. Chistyakov developed the theory of metric spaces generated by modulars and extended the results given by Nakano [7], Musielak and Orlicz [9] and Musielak [12] to modular metric spaces. Modular spaces are extensions of Lebesgue, Riesz and Orlicz spaces of integrable functions.

The development of theory of metric spaces generated by modulars, called modular metric spaces attracted many research mathematicians still investigating fixed-point results in this area, including Chistyakov himself. Chistyakov [13] also established some fixed-point theorems for contractive maps in modular spaces. It is related to contracting, rather generalized average velocities than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence in [13] and other fixed-point results in modular metric spaces can be found in [1, 14]. Considering applicability, these fixed-point results are applied in finding the fixed-point solution of nonlinear integral equations see [1416] and references therein, while [17] deals with application to partial differential equation in modular metric spaces. Interested readers may see [16, 1821] and the references therein for further studies in modular function spaces.

In 2013, Azadifar et al. [22] introduced the concept of modular G-metric space and obtained some fixed-point theorems of contractive mappings defined on modular G-metric spaces. Our intention in this paper is to extend the fixed-point theorem of Mutlu et al. [1] from the setting of modular metric spaces to modular G-metric spaces. Our results extend and generalize several known results in the literature. For results in non-unique fixed-point theorems in modular metric spaces, readers should also see Hussain [23] and references therein.

Zhao [24] 2019 applied the exponential dichotomy, and Tikhonov and Banach fixed-point theorems are used to study the existence and uniqueness of pseudo almost periodic solutions of a class of iterative functional differential equations of the form x(t)=n=1kl=1Cl,n(t)(x[n](t))l+G(t), where x[n](t) is the nth iterate of x(t).

Recently, Combettes and Glaudin [25] constructed iteratively, a common fixed-point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed-point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. They propose a method that achieves this goal and analyzed its asymptotic behavior. Weak, strong and linear convergence results are established by exploiting a connection with the theory of concentrating arrays. Applications to several nonlinear and nonsmooth analysis problems are presented, ranging from monotone inclusions and inconsistent feasibility problems to variational inequalities and minimization problems arising in data science.

2. Preliminaries

Definition 2.1.

[22] Let X be a nonempty set, and let ωG : (0, ) × X × X × X → [0, ] be a function satisfying;

  • (1)

    ωλG(x,y,z)=0 for all x, y, z ∈ X and λ > 0 if x = y = z,

  • (2)

    ωλG(x,x,y)>0 for all x, y ∈ X and λ > 0 with xy,

  • (3)

    ωλG(x,x,y)ωλG(x,y,z) for all x, y, z ∈ X and λ > 0 with zy,

  • (4)

    ωλG(x,y,z)=ωλG(x,z,y)=ωλG(y,z,x)= for all λ > 0 (symmetry in all three variables),

  • (5)

    ωλ+μG(x,y,z)ωλG(x,a,a)+ωμG(a,y,z), for all x, y, z, aX and λ, ν > 0,

then the function ωλG is called a modular G-metric on X.

Remarks 2.1.

(a)The pair (X, ωG) is called a modular G-metric space, and without any confusion, we will take XωG as a modular G-metric space. From condition (5) above, if ωG is convex, then we have a strong form as,

(b)ωλ+μG(x,y,z)ωλλ+μG(x,a,a)+ωμλ+μG(a,y,z),

(c)If x = a, then (5) above becomes ωλ+μG(a,y,z)ωμG(a,y,z) and

(d)Condition (5) is called rectangle inequality.

Definition 2.2.

[22] Let (X, ωG) be a modular G-metric space. The sequence {xn}nN in X is modular G-convergent to x, if it converges to x in the topology τ(ωλG).

A function T:XωGXωG at xXωG is called modular G-continuous if ωλG(xn,x,x)0 then ωλG(Txn,Tx,Tx)0, for all λ > 0.

Remark 2.1.

The sequence {xn}nN is modular G-converges to x as n, if limnωλG(xn,xm,x)=0. That is for all ϵ > 0 there exists n0N such that ωλG(xn,xm,x)<ϵ for all n, m ≥ n0. Here we say that x is modular G-limit of {xn}nN.

Definition 2.3.

[22] Let (X, ωG) be a modular G-metric space, then {xn}nNXωG is said to be modular G-Cauchy if for every ϵ > 0, there exists nϵN such that ωλG(xn,xm,xl)<ϵ for all n, m, l ≥ nϵ and λ > 0.

A modular G-metric space XωG is said to be modular G-complete if every modular G-Cauchy sequence in XωG is modular G-convergent in XωG.

Proposition 2.1.

[22] Let (X, ωG) be a modular G-metric space, for any x, y, z, aX, it follows that

  • (1)

    If ωλG(x,y,z)=0 for all λ > 0, then x = y = z.

  • (2)

    ωλG(x,y,z)ωλ2G(x,x,y)+ωλ2G(x,x,z) for all λ > 0.

  • (3)

    ωλG(x,y,y)2ωλ2G(x,x,y) for all λ > 0.

  • (4)

    ωλG(x,y,z)ωλ2G(x,a,z)+ωλ2G(a,y,z) for all λ > 0.

  • (5)

    ωλG(x,y,z)23(ωλ2G(x,y,a)+ωλ2G(x,a,z)+ωλ2G(a,y,z)) for all λ > 0.

  • (6)

    ωλG(x,y,z)ωλ2G(x,a,a)+ωλ4G(y,a,a)+ωλ4G(z,a,a) for all λ > 0.

Proposition 2.2.

[22] Let (X, ωG) be a modular G-metric space and {xn}nN be a sequence in XωG. Then the following are equivalent:

  • (1)

    {xn}nN is ωG-convergent to x,

  • (2)

    ωλG(xn,x)0 as n, i.e. {xn}nN converges to x relative to modular metric ωλG(.),

  • (3)

    ωλG(xn,xn,x)0 as n for all λ > 0,

  • (4)

    ωλG(xn,x,x)0 as n for all λ > 0 and

  • (5)

    ωλG(xm,xn,x)0 as m, n for all λ > 0.

The following construction was motivated by conditions (3) and (4) of Proposition 2.2 above and [1].

Let ωG : (0, ) × X × X × X → [0, ] be a modular G-metric on X, XωG be a modular G-metric space, BXωG and κ:BR+{} be a function on B. κ is called lower semicontinuous on B if limnωλG(x,xn,xn)=0κ(x)limninfκ(xn), or limnωλG(x,x,xn)=0κ(x)limninfκ(xn) for all λ > 0 and {xn}n1B. B is closed, if the limit of a modular G-convergent sequence in B always belongs to B. Also B is modular G-bounded, if δωG(B)=sup{ωλG(x,y,y):x,yB,λ>0} is finite.

3. Main results

We begin this section with the following results, which extends the results of Mutlu et al. [1] from the setting of modular metric spaces to modular G-metric spaces.

Theorem 3.1.

Let ωG be a modular G-metric on X, XωG be a complete modular G-metric space, κ:XωGR+{} be a lower semicontinuous function on XωG and T:XωGXωG be a self-map such that

(3.1) κ(Tx)+ωλG(x,Tx,Tx)κ(x)
for all xXωG and λ > 0. Then T has a fixed point in XωG.

Proof.

For any xXωG, let F(x)={yXωG:ωλG(x,y,y)κ(x)κ(y),λ>0} and η(x) =  inf{κ(y) : y ∈ F(x)}. Since x ∈ F(x), therefore, F(x) ≠ Ø and 0 ≤ η(x) ≤ κ(x). Let xXωG be an arbitrary point. Now, we construct a sequence {xn}n1 in XωG as follows. Let x = x1 and when x1, x2, …, xn have been chosen, choose xn+1 ∈ F(x) such that κ(xn+1)η(xn)+12n for all nN. By the process above, we get a sequence {xn}n1 satisfying the conditions.

(3.2) ωλG(xn,xn+1,xn+1)κ(xn)κ(xn+1),η(xn)κ(xn+1)η(xn)+12n
for all nN and λ > 0. Then {κ(xn)}n1 is a nonincreasing sequence in R, and it is bounded blow by zero. So, the sequence {κ(xn)}n1 is convergent to a real number M ≥ 0 (say). By inequality (3.2), we get
(3.3) M=limnκ(xn)=limnη(xn)

Now, let kN be arbitrary, from inequalities (3.2) and (3.3), there exits at least a positive number Nk such that κ(xn)<M+12k for all n ≥ Nk. Since κ(xn) is monotone, we get Mκ(xm)κ(xn)<M+12k for m ≥ n ≥ Nk. It follows that

(3.4) κ(xn)κ(xm)<12kforallmnNk.

Without loss of generality, suppose that m > n and m,nN. From inequality (3.2), we get

(3.5) ωλmnG(xn,xn+1,xn+1)κ(xn)κ(xn+1),forλmnλn>0.

Suppose that m,nN and m>nN. Applying rectangle inequality repeatedly, i.e. condition (5) of Definition (2.1) we have

(3.6) ωλG(xn,xm,xm)ωλmnG(xn,xn+1,xn+1)+ωλmnG(xn+1,xn+2,xn+2)+ωλmnG(xn+2,xn+3,xn+3)+ωλmnG(xn+3,xn+4,xn+4)++ωλmnG(xm1,xm,xm)ωλnG(xn,xn+1,xn+1)+ωλnG(xn+1,xn+2,xn+2)+ωλnG(xn+2,xn+3,xn+3)+ωλnG(xn+3,xn+4,xn+4)++ωλnG(xm1,xm,xm)κ(xn)κ(xn+1)+κ(xn+1)κ(xn+2)++κ(xm1)κ(xm)=κ(xn)κ(xm)
for all m > n ≥ Nk for some NkN. Then by inequality (3.4), we have
(3.7) ωλG(xn,xm,xm)<12k,
for all m, l, n ≥ Nk for some NkN, so that by condition (2) of proposition (2.1), we have
(3.8) ωλG(xn,xm,xl)ωλ2G(xn,xm,xm)+ωλ2G(xl,xm,xm),
so that
(3.9) limn,m,lωλG(xn,xm,xl)limn,mωλ2G(xn,xm,xm)+liml,mωλ2G(xl,xm,xm)limn,mωλG(xn,xm,xm)+liml,mωλG(xl,xm,xm)<12k+12k=22k=21k.

Thus, as k, we have

(3.10) limn,m,lωλG(xn,xm,xl)=0.

Therefore, we can say straightaway that {xn}nN is modular G-Cauchy sequence. The completeness of (Xω, ωG) implies that for any λ > 0, limn,mωλG(xn,xm,u)=0, i.e. for any ϵ > 0, there exists n0N such that ωλG(xn,xm,u)<ϵ for all n,mN and n, m ≥ n0, which implies that limnxnuXωG as n. But κ:XωGR+{} is a lower semicontinuous function on XωG, using inequality (3.6), we get

(3.11) κ(u)limminf(κ(xm))limminf(κ(xn)ωλG(xn,xm,xm))=κ(xn)ωλG(xn,u,u)

Thus, we have that ωλG(xn,u,u)κ(xn)κ(u). So that u ∈ F(xn) for all nN and hence η(xn) ≤ κ(u). Then by inequality (3.3), we get M ≤ κ(u). Moreover, by lower semicontinuity of κ and inequality (3.3), we have κ(u)limninfκ(xn)=M. So κ(u) = M. From inequality (3.1), we know that Tu ∈ F(u), such u ∈ F(u). For nN, we have

(3.12) ωλG(xn,Tu,Tu)ωλ2G(xn,u,u)+ωλ2G(u,Tu,Tu)ωλG(xn,u,u)+ωλG(u,Tu,Tu)κ(xn)κ(u)+κ(u)κ(Tu)=κ(xn)κ(Tu).

Thus, Tu ∈ F(xn), and this implies that η(xn) ≤ κ(Tu). Hence, we obtain M ≤ κ(Tu). From inequality (3.1), we get κ(Tu) ≤ κ(u). As κ(u) = M, we have κ(u) = M ≤ κ(Tu) ≤ κ(u). Therefore, κ(Tu) = κ(u). Then from inequality (3.1), we get ωλG(u,Tu,Tu)κ(u)κ(Tu)=κ(u)κ(u)=0. Thus, Tu = u. Therefore, T has a fixed point in XωG. □

Remark 3.1.

Suppose that ωG is a modular G-metric on X, XωG be a complete modular G-metric space, κ:XωGR+{} be a lower semicontinuous function on XωG and T:XωGXωG be a self-map. To get inequality (3.1) of Theorem 3.1 in Mutlu et al. [1], we invoke the definition of modular G metric space as follows for any λ > 0, define ωλ(x,y,z)=12λxy+yz+xz. Take y = Tx and z = Tx, then inequality (3.1) transform into

(3.13) ωλ(x,Tx)κ(x)κ(Tx)
for all x ∈ Xω and λ > 0. Then T has a fixed point in Xω, which is clearly the result in Mutlu et al. [1].

Theorem 3.2.

Let ωG be a modular G-metric on X, XωG be a complete modular G-metric space, κ:XωGR+{} be a lower semicontinuous function on XωG and T:XωGXωG be a self-map such that for some positive integer, m ≥ 1,

(3.14) ωλG(x,Tmx,Tmx)κ(x)κ(Tmx)
for all xXωG and λ > 0. Then T has a fixed point in XωG for some positive integer m ≥ 1.

Proof.

By Theorem 3.1, Tm has a fixed point say uXωG for some positive integer m ≥ 1, by using inequality (3.14) for some positive integer m ≥ 1. Now Tm(Tu) = Tm+1u = T(Tmu) = Tu, so Tu is a fixed point of Tm. Hence, we have Tu = u. Therefore, u is a fixed point of T because fixed point of T is also fixed point of Tm for some positive integer m ≥ 1. □

Next, we produce the following example to demonstrate the applicability of our results.

Example 3.1.

Let XωG=R and we define the mapping ωG:(0,)×R×R×R[0,] by ωλG(x,y,y)=2λxy for all x,yR and λ > 0. So we can see that (R,ωG) is a complete modular G-metric space and let us define T:(R,ωG)(R,ωG) by Tx=1x for xR+\{0} and κ:(R,ωG)R+{} by κ(x)=32x for which κ(x) defined above is lower semicontinuous. Now we verify the inequality (3.1) of Theorem 3.1 as follows; For xR+\{0} and λ > 0, we have

ωλG(x,Tx,Tx)=ωλGx,1x,1x=1λx1x+1x1x+x1x=2λx1x=2λx21x=2λ(x1)(x+1)x=2λx1x+1x2λx1x+1x+1=2λx1x.
And
κ(x)κ(Tx)=32x321x=32x1x=32x21x=32(x1)(x+1)x32(x1)(x+1)x+1=32x132x.
Therefore, ωλG(x,Tx,Tx)κ(x)κ(Tx) for all λ > 0. Hence, the mapping T has a fixed point. The trivial fixed point of this map, T is 1.

Remark 3.2.

As we can see clearly in this Example 3.1 that the map T has a trivial fixed point at 1.

Proposition 3.3.

Let ωG be a modular G-metric on X, and XωG be a complete modular G-metric space, κ:XωGR+{} be a lower semicontinuous function on XωG, which is bounded from below, then there exists a point uXωG such that κ(u)<κ(z)+ωλG(u,z,z) for each zXωG,zu and for all λ > 0.

Proof.

Following the proof of Theorem 3.1, we get a sequence {zn}n1 such that znuXωG as n. Now for any uXωG, define F(u)={zXωG:ωλG(u,z,z)κ(u)κ(z)λ>0} and η(u) =  inf{κ(z) : z ∈ F(u)}. We will show that uF(u) as zu. Suppose, if possible, otherwise. Let v ∈ F(u) for some vu. Then we have that for all λ > 0, 0<ωλG(u,v,v)κ(u)κ(v) implies κ(v) < κ(u) = M, since

(3.15) ωλG(zn,v,v)ωλ2G(zn,u,u)+ωλ2G(u,v,v)ωλG(zn,u,u)+ωλG(u,v,v)κ(zn)κ(u)+κ(u)κ(v)=κ(zn)κ(v).
for all λ > 0, v ∈ F(zn) for n ≥ 1. So η(zn) ≤ κ(v) for all n ≥ 1. Therefore, M=limnη(zn)κ(v). Hence, M ≤ κ(v), which is a contradiction to the fact that κ(v) < κ(u) = M. Therefore, for each zXωG,zuzF(u), that is zuωλG(u,z,z)>κ(u)κ(z). Hence, κ(u)<κ(z)+ωλG(u,z,z) for each zXωG,zu and for all λ > 0. □

Proposition 3.4.

Let ωG be a modular G-metric on X, and XωG be a complete modular G-metric space, κ:XωGR+{} be a lower semicontinuous function on XωG, which is bounded from below, then for every yXωG and γ > 0, there exists x0XωG such that κ(x0)<κ(x)+γωλG(x,x,x0) on XωG\{x0} and κ(x0)κ(y)γωλG(x0,y,y), for all λ > 0.

Proof.

Define XωGγ={zXωG:κ(z)κ(y)γωλG(z,y,y),λ>0}. Then XωGγ is a nonempty complete modular G-metric space and κ:XωGR+{} be a lower semicontinuous function on XωG, which is bounded from below. Let F(x)={zXωGγ:κ(x)κ(z)+γωλG(z,x,x),λ>0}. Then for every xXωGγ, F(x) ≠ Ø and closed. Also z ∈ F(x) implies F(z) ⊆ F(x). Choose x1XωGγ with κ(x1) <  and when x1, x2, …, xn have been chosen, we can find xn+1 ∈ F(x) such that κ(xn+1)<inf{κ(u):uF(xn)}+12n for n ≥ 1. For any z ∈ F(xn+1) ⊆ F(xn), we get that for all λ > 0,

(3.16) γωλG(z,xn+1,xn+1)γωλ2G(z,xn,xn)+γωλ2G(xn,xn+1,xn+1)γωλG(z,xn,xn)+γωλG(xn,xn+1,xn+1)κ(xn)κ(z)+κ(xn+1)κ(xn)=κ(xn+1)κ(z)inf{κ(u):uF(xn)}κ(z)+12n12n.
So that δωG(F(xn))0 as n, for all λ > 0. Since XωGγ be a complete modular G-metric space, n=1F(xn)={x0}. Since the intersection is a singleton set, we proceed as follows. Now, x0 ∈ F(xn) implies that F(x0) ⊆ F(xn) for n ≥ 1, we get F(x0) = {x0}, so that for all λ > 0, κ(x0)<κ(x)+γωλG(x,x,x0) on XωGγ\{x0}. Again, the inequality κ(x0)<κ(x)+γωλG(x,x,x0) hold on XωG\XωGγ since for zXωGγ, for all λ > 0, we have κ(y)γωλG(z,y,y)<κ(z) and thus, together with the fact that x0XωGγ, we have
(3.17) κ(x0)κ(y)γωλG(x0,y,y)κ(y)γωλ2G(z,y,y)γωλ2G(x0,z,z)κ(y)γωλG(z,y,y)γωλG(x0,z,z)<κ(z)γωλG(x0,z,z).

We are now at home since for all λ > 0, κ(x0)<κ(z)γωλG(x0,z,z). □

Theorem 3.5.

Let ωG be a modular G-metric on X, and XωG,YωG are complete modular G-metric spaces. Let T:XωGXωG be an arbitrary self mapping. Suppose that there exists a closed mapping L:XωGYωG, and κ:L(XωG)R+{} be a lower semicontinuous function on XωG, which is bounded from below, and for every γ > 0, there exists xXωG such that

(3.18) ωλG(x,Tx,Tx)κ(Lx)κ(LTx),
(3.19) γωλG(Lx,LTx,LTx)κ(Lx)κ(LTx),
for all λ > 0. Then, T has a fixed point in XωG.

Proof.

For any xXωG, put Tx = y and let F(x)={yXωG:ωλG(x,y,y)κ(x)κ(Ly)andγωλG(Lx,Ly,Ly)κ(Lx)κ(Ly)λ>0} and η(x) =  inf{κ(Ly) : y ∈ F(x)}. Since x ∈ F(x), therefore, F(x) ≠ Ø and 0 ≤ η(x) ≤ κ(Lx). Let xXωG be an arbitrary point. Now, we construct a sequence {xn}n1 in XωG as follows. Let x = x1 and when x1, x2, …, xn have been chosen, choose xn+1 ∈ F(x) such that κ(Lxn+1)η(xn)+12n for all nN. By the process above, we get a sequence {xn}n1 satisfying the conditions.

(3.20) ωλG(xn,xn+1,xn+1)κ(xn)κ(Lxn+1),
(3.21) γωλG(Lxn,Lxn+1,Lxn+1)κ(Lxn)κ(Lxn+1),
and
(3.22) κ(Lxn+1)12nη(xn)κ(Lxn+1),
for all nN and λ > 0. Then from inequalities, (3.20), (3.21), {κ(Lxn)}n1 is a nonincreasing sequence in R, and it is bounded blow by zero. So, the sequence {κ(Lxn)}n1 is a modular G-convergent and converges to a real number β ≥ 0 (say). By inequality (3.22), we get
(3.23) β=limnκ(Lxn)=limnη(xn)

Now, let kN be arbitrary, from inequalities (3.20),(3.21) and (3.23), there exits at least a positive number Nk such that κ(Lxn)<β+12k for all n ≥ Nk. Since κ(Lxn) is monotone for m ≥ n ≥ Nk, we get βκ(Lxm)κ(Lxn)<β+12k for m ≥ n ≥ Nk. It follows that

(3.24) κ(Lxn)κ(Lxm)<12kforallmnNk.

Without loss of generality, suppose that m > n and m,nN. From inequalities (3.20) and (3.21), we get

(3.25) ωλmnG(xn,xn+1,xn+1)κ(Lxn)κ(Lxn+1),
(3.26) ωλmnG(Lxn,Lxn+1,Lxn+1)κ(Lxn)κ(Lxn+1),forλmn>0,
or since λmnλn, we have
(3.27) ωλnG(xn,xn+1,xn+1)κ(Lxn)κ(Lxn+1),
(3.28) ωλnG(Lxn,Lxn+1,Lxn+1)κ(Lxn)κ(Lxn+1),forλn>0.

Suppose that m,nN and m>nN. Using rectangle inequality repeatedly, i.e. condition 5 of Definition (2.1), we have

(3.29) ωλG(xn,xm,xm)ωλmnG(xn,xn+1,xn+1)+ωλmnG(xn+1,xn+2,xn+2)+ωλmnG(xn+2,xn+3,xn+3)+ωλmnG(xn+3,xn+4,xn+4)++ωλmnG(xm1,xm,xm)ωλnG(xn,xn+1,xn+1)+ωλnG(xn+1,xn+2,xn+2)+ωλnG(xn+2,xn+3,xn+3)+ωλnG(xn+3,xn+4,xn+4)++ωλnG(xm1,xm,xm)κ(Lxn)κ(Lxn+1)+κ(Lxn+1)κ(Lxn+2)++κ(Lxm1)κ(Lxm)=κ(Lxn)κ(Lxm),
for all m > n ≥ Nk for some NkN. Then by inequality (3.24), we have
(3.30) ωλG(xn,xm,xm)<12k,
for all m, l, n ≥ Nk for some NkN, so that by condition (2) of proposition 2.1, we have
(3.31) ωλG(xn,xm,xl)ωλ2G(xn,xm,xm)+ωλ2G(xl,xm,xm),
so that
(3.32) limn,m,lωλG(xn,xm,xl)limn,mωλ2G(xn,xm,xm)+liml,mωλ2G(xl,xm,xm)limn,mωλG(xn,xm,xm)+liml,mωλG(xl,xm,xm)<12k+12k=22k=21k.

Thus, as k, we have

(3.33) limn,m,lωλG(xn,xm,xl)=0.

Therefore, we can say straightaway that {xn}nN is modular G-Cauchy sequence in XωG. Again, using the same procedure, we get

(3.34) γωλG(Lxn,Lxm,Lxm)γωλmnG(Lxn,Lxn+1,Lxn+1)+γωλmnG(Lxn+1,Lxn+2,Lxn+2)+γωλmnG(Lxn+2,Lxn+3,Lxn+3)+γωλmnG(Lxn+3,Lxn+4,Lxn+4)++γωλmnG(Lxm1,Lxm,Lxm)γωλnG(Lxn,Lxn+1,Lxn+1)+γωλnG(Lxn+1,Lxn+2,Lxn+2)+γωλnG(Lxn+2,Lxn+3,Lxn+3)+γωλnG(Lxn+3,Lxn+4,Lxn+4)++γωλnG(Lxm1,Lxm,Lxm)κ(Lxn)κ(Lxn+1)+κ(Lxn+1)κ(Lxn+2)++κ(Lxm1)κ(Lxm)=κ(Lxn)κ(Lxm),
for all m > n ≥ Nk for some NkN. Then by inequality (3.24), we have
(3.35) γωλG(Lxn,Lxm,Lxm)<12k,
for all m, l, n ≥ Nk for some NkN, so that by condition (2) of proposition 2.1, we have
(3.36) γωλG(Lxn,Lxm,Lxl)γωλ2G(Lxn,Lxm,Lxm)+γωλ2G(Lxl,Lxm,Lxm),
so that
(3.37) limn,m,lγωλG(Lxn,Lxm,Lxl)limn,mγωλ2G(Lxn,Lxm,Lxm)+liml,mγωλ2G(Lxl,Lxm,Lxm)limn,mγωλG(Lxn,Lxm,Lxm)+liml,mγωλG(Lxl,Lxm,Lxm)<12k+12k=22k=21k.

Thus, as k, we have

(3.38) limn,m,lγωλG(Lxn,Lxm,Lxl)=0.

Therefore, we can say straightaway that {Lxn}nN is modular G-Cauchy sequence in YωG. The completeness of (Xω, ωG) and (Yω, ωG) implies that for any λ > 0, limn,mωλG(xn,xm,u)=0, i.e. for any ϵ > 0, there exists n0N such that ωλG(xn,xm,u)<ϵ for all n,mN and n, m ≥ n0, which implies that limnxnuXωG as n and for any λ > 0, limn,mωλG(Lxn,Lxm,v)=0, i.e. for any ϵ > 0, there exists n0N such that ωλG(Lxn,Lxm,v)<ϵ for all n,mN and n, m ≥ n0, which implies that limnLxnvXωG as n. The fact that L is closed mapping implies that Lu = v. But κ:XωGR+{} is a lower semicontinuous function on XωG, using inequality (3.29), we get

(3.39) κ(v)=κ(Lu)limminf(κ(Lxm))limminf(κ(Lxn)ωλG(xn,xm,xm))=κ(Lxn)ωλG(xn,u,u)

Thus, we have that ωλG(xn,u,u)κ(Lxn)κ(Lu) for all λ > 0. Again, using inequality, (3.34), we have

(3.40) κ(v)=κ(Lu)limminf(κ(Lxm))limminf(κ(Lxn)γωλG(Lxn,Lxm,Lxm))=κ(Lxn)γωλG(Lxn,u,u)

Thus, we have that γωλG(Lxn,u,u)κ(Lxn)κ(Lu) for all λ > 0. So that u ∈ F(xn) for all nN, and hence, η(xn) ≤ κ(Lu). So by inequality (3.23), we get β ≤ κ(Lu). Meanwhile, by lower semicontinuity of κ and inequality (3.23), we have κ(v)=κ(Lu)limninfκ(xn)=β. Therefore, κ(Lu) = β. By Proposition 3.3, we have that xuxF(u) and Proposition 3.4 for yXωGγ. From inequalities (3.18), (3.19), we know that LTu ∈ F(u), such u ∈ F(u). For nN, we have

(3.41) ωλG(xn,Tu,Tu)ωλ2G(xn,u,u)+ωλ2G(u,Tu,Tu)ωλG(xn,u,u)+ωλG(u,Tu,Tu)κ(Lxn)κ(Lu)+κ(Lu)κ(LTu)=κ(Lxn)κ(LTu).

Thus, LTu ∈ F(xn), and this implies that η(Lxn) ≤ κ(LTu). Hence, we obtain β ≤ κ(LTu). From inequalities (3.18), (3.19), we get κ(LTu) ≤ κ(Lu). As κ(Lu) = β, we have κ(Lu) = β ≤ κ(LTu) ≤ κ(Lu). Therefore, κ(LTu) = κ(Lu). Then from inequality (3.18) and (3.19), we get ωλG(u,Tu,Tu)κ(Lu)κ(LTu)=κ(Lu)κ(Lu)=0. Thus, Tu = u. Therefore, T has a fixed point in XωG. □

Theorem 3.6.

Let ωG be a modular G-metric on X, and XωG,YωG are complete modular G-metric spaces. Let T:XωGXωG be an arbitrary self-mapping for some positive integer m ≥ 1. Suppose that there exists a closed mapping L:XωGYωG for each integer m ≥ 1, and κ:L(XωG)R+{} be a lower semicontinuous function on XωG, which is bounded from below, and for every γ > 0, there exists xXωG such that

(3.42) ωλG(x,Tmx,Tmx)κ(Lmx)κ(LmTmx),
(3.43) γωλG(Lmx,LmTmx,LmTmx)κ(Lmx)κ(LmTmx),
for all λ > 0. Then, T has a fixed point in XωG for some positive integer m ≥ 1.

Proof.

By Theorem 3.5, Tm has a fixed point say uXωG for some positive integer m ≥ 1, by using inequalities (3.42) and (3.43) for some positive integer m ≥ 1. Now Tm(Tu) = Tm+1u = T(Tmu) = Tu, so Tu is a fixed point of Tm. Thus, we have Tu = u. Therefore, u is a fixed point of T. because fixed point of T is also fixed point of Tm for some positive integer m ≥ 1. □

Remark 3.3.

The results of Theorem 3.6 improve and generalize several known results in the literature, including the results of Mutlu et al. [1].

4 . Conclusion and future work

All fixed-point results obtained in this paper do not require the uniqueness of the fixed point of mappings under consideration. As a future direction of study, it will be of interest to prove some new fixed-point results for the nonunique fixed-point theorems established in this paper. More precisely, geometric properties of the set Fix(T) can be investigated as a future problem for a self-mapping T on a modular G-metric space in the case of nonunique fixed point.

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Acknowledgements

The authors wish to thank the editor and the referees for their useful comments and suggestions.

Conflicts of interest: The authors declare no conflict of interests.

Authors’ contributions: All authors contributed equally to the writing of this paper.

Data availability: The data used to support the findings of this study are included within the article.

Corresponding author

Godwin Amechi Okeke can be contacted at: gaokeke1@yahoo.co.uk

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