Fixed point theorems for Geraghty-type mappings applied to solving nonlinear Volterra-Fredholm integral equations in modular G-metric spaces

Abstract

Purpose

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.

Design/methodology/approach

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

Findings

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.

Research limitations/implications

The results are theoretical and analytical.

Practical implications

The results were applied to solving nonlinear integral equations.

Social implications

The results has several social applications.

Originality/value

The results of this paper are new.

Citation

Okeke, G.A. and Francis, D. (2021), "Fixed point theorems for Geraghty-type mappings applied to solving nonlinear Volterra-Fredholm integral equations in modular G-metric spaces", Arab Journal of Mathematical Sciences, Vol. 27 No. 2, pp. 214-234. https://doi.org/10.1108/AJMS-10-2020-0098

1. Introduction

In 1973, Geraghty [1] introduced an interesting generalization of Banach contraction mapping principle using the concept of class  of functions, that is α:ℝ+→[0, 1) with the condition that α(tn)→1⇒tn→0 where ℝ+ is the set of all nonnegative real numbers and t∈ℝ+ for all n∈N. In 2012, Gordji et al. [2] proved some fixed point theorems for generalized Geraghty contraction in partially ordered complete metric spaces. Bhaskar and Lakshmikantham [3] proved a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order, using a weak contractivity type of assumption. Yolacan [4] established some new fixed point theorems in 0-complete ordered partial metric spaces. He also remarked on coupled generalized Banach contraction mapping. Faraji et al. [5] extended some fixed point theorems for Geraghty contractive mappings in b-complete b-metric spaces.

Furthermore, Gupta et al. [6], established some fixed point theorems in an ordered complete metric space using distance function. Chaipunya et al. [7] proved some fixed point theorems of Geraghty-type contractions concerning the existence and uniqueness of fixed points under the setting of modular metric spaces which also generalized the results in Gordji et al. [2] under the influence of a modular metric space.

Geraghty-type contractive mappings in metric spaces was generalized to the concept of preordered G-metric spaces in [8] and the authors in [8] obtained unique fixed point results. Furthermore, other interesting fixed point results in G-metric spaces can be found in [9] and the references therein.

In 2010, an essential study by Chistyakov [10] introduced an aspect of metric called modular metric spaces or parameterized metric space with the time parameter λ (say) and his purpose was to define the notion of a modular on an arbitrary set, develop 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 (or Nemytskii) operators, acting in these modular metric spaces. He developed the theory of metric spaces generated by modulars and extended the results given by Nakano [12], Musielak and Orlicz [13], Musielak [14] to modular metric spaces. Modular spaces are extensions of Lebesgue, Riesz and Orlicz spaces of integrable functions.

Modular theories on linear spaces can be found in Nakano [12, 15], where he developed a spectral theory in semi-ordered linear spaces (vector lattices) and established the integral representation for projections acting in this modular space.

Nakano [12] established some modulars on real linear spaces which are convex functionals. Non-convex modulars and the corresponding modular linear spaces were constructed by Musielak and Orlicz [13]. Orlicz spaces and modular linear spaces have already become classical tools in modern nonlinear functional analysis.

Furthermore, the development of theory of metric spaces generated by modulars, called modular metric spaces attracted the attention of several mathematicians (see, e.g. [16–19]).

Okeke et al. [20] established some convergence results for three multi-valued ρ-quasi-nonexpansive mappings using a three step iterative scheme. Moreover, these fixed point results are applicable to nonlinear integral and differential equations see [19, 21–26] and the references therein, while [7] deals with application to partial differential equation in modular metric spaces.

In 2013, Azadifar et al. [27] introduced the notion of modular G-metric space and proved some fixed point theorems for contractive mappings defined on modular G-metric spaces. Based on definitions given in [27], we intend to extend the fixed point theorems obtained in [7] to preordered modular G-metric spaces in this paper. Furthermore, we prove some fixed point theorems for Geraghty-type contraction mappings in the setting of preordered modular G-metric spaces. We apply our results in proving the existence of a unique solution for a system of nonlinear Volterra-Fredholm integral equations in modular G-metric spaces, XωG.

2. Preliminaries

We begin this section with the following results and definitions which will be useful in this paper.

Theorem 2.1.

[28] If {an}n∈N,{bn}n∈N,{cn}n∈N are three sequences in ℝ such that

1. limn→∞an=limn→∞bn=ℓ,

2. for some positive integer N, an≤cn≤bn for all n≥N.

   Then limn→∞cn=ℓ.

Definition 2.1.

[29] A preorder set X is a relation ⪯ that is both,

1. transitive i.e; x ⪯ y and y ⪯ z implies x ⪯ z and,

2. reflexive i.e; x ⪯ x.

A preordered set is a pair (X,⪯) consisting of a set X and a preorder ⪯ on X.

Remark 2.1.

If a preorder ⪯ is antisymmetric i.e; x ⪯ y and y ⪯ x implies x=y, then ⪯ is called a partial order.

Definition 2.2.

[1] Let  be the family of all Geraghty functions, that is functions α:[0, ∞)→[0, 1) satisfying the condition {α(tn)}→1⇒{tn}→0.

For the rest of this paper, we denote the the class of all Geraghty functions by Ger. Such Geraghty class was discussed in [7].

Definition 2.3.

[7] Let  be the family of all Geraghty functions, that is functions βi:ℝ+∪{∞}→[0, 1) satisfying the condition βi(tk)→1n⇒{tk}→0 for all i.

Definition 2.4.

[7] Let Ψ be the class of functions ψ:ℝ+→ℝ+ such that the following conditions hold;

1. ψ is decreasing,

2. ψ is continuous,

3. ψ(t)=0 if and only if t=0.

Extension of Definition 2.2 above is as follows:

Definition 2.5.

[7] Let Ψ¯ be the class of functions ψ:ℝ+∪{∞}→ℝ+∪{∞} such that the following conditions hold;

1. ψ is subadditive,

2. ψ(t) is finite for 0<t<∞,

3. ψ|ℝ+∈Ψ¯.

Definition 2.6.

[27] 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∈X and λ>0 if x=y=z,

2. ωλG(x, x, y)>0 for all x, y∈X and λ>0 with x≠y,

3. ωλG(x, x, y)≤ωλG(x, y, z) for all x,y,z∈X and λ>0 with z≠y,

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, a∈X and λ, ν>0,

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

Remarks 2.1.

1. 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), if ωG is convex, then we have a strong form as,

2. ωλ+μG(x, y, z)≤ωλλ+μG(x, a, a)+ωμλ+μG(a, y, z),

3. If x=a, then (5) above becomes ωλ+μG(a, y, z)≤ωμG(a, y, z),

4. Condition (5) is called rectangle inequality.

Definition 2.7.

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

A function T:XωG→XωG at x∈XωG is called ωG-continuous if ωλG(xn, x, x)→0 then ωλG(Txn, Tx, Tx)→0, for all λ>0.

Remark 2.2.

{xn}n∈N modular G-converges to x as n→∞, if limn→∞ωλG(xn, xm, x)=0. That is for all ϵ>0 there exists n0∈N such that ωλG(xn, xm, x)<ϵ for all n, m≥n0. Here we say that x is modular G-limit of {xn}n∈N.

Definition 2.8.

[27] Let (X, ωG) be a modular G-metric space, then {xn}n∈N⊆XωG is said to be ωG-Cauchy if for every ϵ>0, there exists nϵ∈N 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 ωG-complete if every ωG-Cauchy sequence in XωG is ωG-convergent in XωG.

Proposition 2.2.

[27] Let (X, ωG) be a modular G-metric space, for any x, y, x, a∈X, it follows that:

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

2. ωλG(x, y, z)≤ωλ/2G(x, x, y)+ωλ/2G(x, x, z) for all λ>0.

3. ωλG(x, y, y)≤2ωλ/2G(x, x, y) for all λ>0.

4. ωλG(x, y, z)≤ωλ/2G(x, a, z)+ωλ/2G(a, y, z) for all λ>0.

5. ωλG(x, y, z)≤23(ωλ/2G(x, y, a)+ωλ/2G(x, a, z)+ωλ/2G(a, y, z)) for all λ>0.

6. ωλG(x, y, z)≤ωλ/2G(x, a, a)+ωλ/2G(y, a, a)+ωλ/2G(z, a, a) for all λ>0.

Proposition 2.3.

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

1. {xn}n∈N is ωG-convergent to x,

2. ωλG(xn, x)→0 as n→∞, i.e; {xn}n∈N 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,

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

We give the following definition which will be useful in our results.

Definition 2.9.

An ordered modular G-metric space is a triple (X, ωG, ⪯) where (X,ω) is a modular metric space and ⪯ is a partial order on XωG. If ⪯ is a preorder on XωG, then (X, ωG, ⪯) is a preordered modular G-metric space.

3. Main results

Theorem 3.1.

Let (X,ωG) be a complete modular G-metric space with a preorder, ⪯ and a nondecreasing self-mapping T:XωG→XωG on XωG such that for each λ>0, there is ν(λ)∈[0, λ) such that the following conditions hold:

(1)

(3.1)ψ(ωλG(Tx, Ty, Ty))≤α(ψ(ωλG(x, y, y)))ψ(ωλ+ν(λ)G(x, y, y))+β(ψ(ωλG(x, y, y)))ψ(ωλG(x, Tx, Tx))+γ(ψ(ωλG(x, y, y)))ψ(ωλG(y, Ty, Ty)),

where ψ∈Ψ¯ and {α, β, γ}∈Ger with α(t)+2 max{supt≥0β(t), supt≥0γ(t)}<1, and distinct x, y∈XωG. Assuming that if a nondecreasing sequence {xn}n∈N converges to x*, then xn ⪯ x∗ for each n∈N,

• (2) if ψ is subadditive and for any x, y∈XωG, there exists z∈XωG with z ⪯ Tz and ωλG(z, Tz, Tz) is finite for all λ>0 such that z is comparable to both x and y. Then T has a fixed point u∈XωG and the sequence define by {Tnx0}n≥1 converges to u. Moreover, the fixed point of T is unique.

Proof. Let x0∈XωG be such that x0 ⪯ Tx0 and let xn=Txn−1=Tnx0 for all n∈N. Regarding that T is nondecreasing mapping, we have that x0 ⪯ Tx0=x1, which implies that x1=Tx0 ⪯ Tx1=x2. Inductively, we have

(3.2)x0 ⪯ x1 ⪯ x2 ⪯…⪯ xn−1 ⪯ xn ⪯ xn+1 ⪯….

Assume that there exists n0∈N such that xn0=xn0+1. Since xn0=xn0+1=Txn0, then xn0 is the fixed point of T. Now suppose that xn⪵xn+1 for all n∈N, thus by inequality (3.2), we have that

(3.3)x0≺x1≺x2≺…≺xn−1≺xn≺xn+1≺….

Now for each λ>0, and x0 ≺ Tx0 for all n∈N implies that ωλG(x0, Tx0, Tx0)>0. Again, let x0∈XωG such that ωλG(x0, Tx0, Tx0)<∞ ∀ λ>0.

First, we show that for all n∈N, the sequence ωλG(Tnx0, Tn+1x0, Tn+1x0)=0 for all λ>0, as n→∞. Assume that, for each n∈N, there exists λn>0 such that ωλnG(Tnx0, Tn+1x0, Tn+1x0)≠0. Otherwise the proof is complete. Suppose not, for each n≥1, if 0<λ<λn, then we have that ωλG(Tnx0, Tn+1x0, Tn+1x0)≠0. Since Tnx0⪯Tn+1x0, from inequality (3.1) we can see that ψ(ωλnG(Tnx0, Tn+1x0, Tn+1x0))≤ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))=ψ(ωλG(TTn−1x0, TTnx0, TTnx0)). Take x=Tn−1x0 and y=Tnx0, then inequality (3.1) becomes;

(3.4)ψ(ωλnG(Tnx0, Tn+1x0, Tn+1x0))≤ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλ+ν(λ)G(Tn−1x0, Tnx0, Tnx0))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, TTn−1x0, TTn−1x0))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, TTnx0, TTnx0))=α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλ+ν(λ)G(Tn−1x0, Tnx0, Tnx0))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0,Tnx0,Tnx0))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, Tnx0, Tnx0))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, Tnx0, Tnx0))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0)),

for which we have that

(3.5)ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤δψ(ωλG(Tn−1x0, Tnx0, Tnx0))≤ψ(ωλG(Tn−1x0, Tnx0, Tnx0))⋮≤ψ(ωλG(x0, Tx0, Tx0))<∞,

where

(3.6)δ:=α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))1−γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0))).

Therefore, {ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))}n≥1 is nonincreasing and bounded below, so the sequence {ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))}n≥1 converges to some real number k≥0. Assume k>0, we can see clearly that by using inequality 3, inequality 3 becomes

(3.7)ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤(α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0))))

as n→∞, we get

(3.8)1≤limn→∞inf(α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0))))

So, we have that

(3.9)ψ(ωλG(Tn−1x0, Tnx0, Tnx0))=0,

hence

(3.10)limn→∞ωλG(Tn−1x0, Tnx0, Tnx0)=0

for all λ>0, which is a contradiction to our assumption. Therefore,

(3.11)limn→∞ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))=0,

so,

(3.12)limn→∞ωλG(Tnx0,Tn+1x0,Tn+1x0)=0

for all λ>0. This shows that ωλG(Tnx0, Tn+1x0, Tn+1x0)=0 for all λ>0, n≥1.

Next, we show that {Tnx0}n≥1 is a modular G-Cauchy sequence. Suppose, if possible that {Tnx0}n≥1 not a modular G-Cauchy sequence , then there exists real numbers, λ0>0, ϵ>0 and also there exists two subsequences {Tnkx0}k≥1 and {Tmkx0}k≥1 of the sequence {Tnx0}n≥1 such that, for nk>mk>k, we have that ωλ0G(Tmkx0, Tnkx0, Tnkx0)≥ϵ, but ωλ0G(Tmkx0, Tnk−1x0, Tnk−1x0)<ϵ. Now, since Tmkx0⪯Tnkx0, we have that ϵ≤ωλ0G(Tmkx0, Tnkx0, Tnkx0) which implies that ψ(ϵ)≤ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0))=ψ(ωλ0G(TTmk−1x0, TTnk−1x0, TTnk−1x0)). Set x=Tmk−1x0 and y=Tnk−1x0 into inequality (3.1), then we have

(3.13)ψ(ϵ)≤ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0))≤α(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0+ν(λ0)G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+β(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tmk−1x0, TTmk−1x0, TTmk−1x0))+γ(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tnk−1x0, TTnk−1x0, TTnk−1x0))=α(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0+ν(λ0)G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+β(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+γ(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0))≤α(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))+ψ(ωλ0G(Tmkx0, Tnk−1x0, Tnk−1x0)))+β(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+γ(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0))≤ψ(ων(λ0)G(Tmk−1x0,Tmkx0,Tmkx0))+ψ(ωλ0G(Tmkx0, Tnk−1x0, Tnk−1x0))+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0))≤ψ(ων(λ0)G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ϵ)+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0)),

as k→∞, we obtain

(3.14)ψ(ϵ)≤limk→∞ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0))≤ψ(ϵ),

so that

(3.15)limk→∞ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0))=ψ(ϵ).

Hence

(3.16)limk→∞ωλ0G(Tmkx0, Tnkx0, Tnkx0)=ϵ.

Again, using condition 5 of Definition 2.6, we get

(3.17)ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0))≤α(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0+ν(λ0)G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+β(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+γ(ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)))×ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0))≤ψ(ωλ0+ν(λ0)G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0))≤ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tnk−1x0, Tnkx0, Tnkx0))≤ψ(ωλ0G(Tmk−1x0,Tnk−1x0,Tnk−1x0))+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ02G(Tnk−1x0, Tmkx0, Tmkx0))+ψ(ωλ02G(Tmkx0, Tnkx0, Tnkx0))≤ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tnk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0))≤ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0))+ψ(ωλ0G(Tmk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tnk−1x0, Tmkx0, Tmkx0))+ψ(ωλ0G(Tmkx0, Tnkx0, Tnkx0)),

as k→∞, we have

(3.18)ψ(ϵ)≤limk→∞ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0))≤ψ(ϵ),

so that

(3.19)limk→∞ψ(ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0))=ψ(ϵ).

Hence

(3.20)limk→∞ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)=ϵ.

Thus, it follows that

(3.21)1≤limk→∞inf(α(ψ(ωλ0G(Tmk−1x0,Tnk−1x0,Tnk−1x0)))).

Therefore, we conclude that

(3.22)limk→∞ωλ0G(Tmk−1x0, Tnk−1x0, Tnk−1x0)=0  ∀  λ>0.

This is a contradiction. Therefore, it follows that {Tnx0}n≥1 is a modular G-Cauchy sequence in XωG. Since XωG is complete modular G-metric space, there exists u∈XωG such that Tnx0→u∈XωG. Now we show that u is a fixed point of T for any arbitrary λ>0, using condition 5 of Definition 2.6 and inequality (3.1), we have that

(3.23)ψ(ωλG(Tnx0, Tu, Tu))≤ψ(ωλ2G(Tn+1x0, Tu, Tu))+ψ(ωλ2G(Tnx0, Tn+1x0, Tn+1x0))=ψ(ωλ/2G(Tn+1x0, Tu, Tu))+ψ(ωλ/2G(Tnx0, Tn+1x0, Tn+1x0))≤ψ(ωλG(Tn+1x0, Tu, Tu))+ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤α(ψ(ωλG(Tnx0, u, u)))ψ(ωλ+ν(λ)G(Tnx0, u, u))+β(ψ(ωλG(Tnx0, u, u)))ψ(ωλG(Tnx0, TTnx0, TTnx0))+γ(ψ(ωλG(Tnx0, u, u)))ψ(ωλG(u, Tu, Tu))+ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))=α(ψ(ωλG(Tnx0, u, u)))ψ(ωλ+ν(λ)G(Tnx0, u, u))+β(ψ(ωλG(Tnx0, u, u)))ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))+γ(ψ(ωλG(Tnx0, u, u)))ψ(ωλG(u, Tu, Tu))+ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0)),

as n  →∞, we have that

(3.24)ψ(ωλG(u, Tu, Tu))≤γ(0)ψ(ωλG(u, Tu, Tu))

for all λ>0, which implies that

(3.25)(1−γ(0))ωλG(u,Tu,Tu)≤0  ∀  λ>0.

Therefore,

(3.26)ωλG(u,Tu,Tu)=0  ∀  λ>0,

where 1−γ(0)<1. Hence, u is a fixed point of T for all λ>0, i.e; Tu=u.

Finally, for the uniqueness, we can see from above that T has a fixed point u∈XωG. Suppose that there is another fixed point of T i.e; Tv=v, for v∈XωG, thus condition (2) of Theorem 3.1 tells us that if z∈XωG with z ≤ Tz and it is comparable to both u and v and Tnz is also comparable to u and v for each n∈N. Now for λ>0, then ψ(ωλG(Tn+1z, u, u)) and ψ(ωλG(Tn+1z, v, v)) are finite. Claim : u=v. Indeed, using inequality (3.1), we have by taking x=Tnz and y=u. First consider ψ(ωλG(Tn+1z, u, u))< ∞, so that we have the following inequality by using condition 6 of Proposition 2.2

(3.27)ψ(ωλG(Tn+1z, u, u))=ψ(ωλ(Tn+1z, Tu, Tu))=ψ(ωλG(TTnz, Tu, Tu))≤α(ψ(ωλG(Tnz, u, u)))ψ(ωλ+ν(λ)G(Tnz, u, u))+β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, TTnz, TTnz))+γ(ψ(ωλG(Tnz, u, u)))ψ(ωλG(u, Tu, Tu))=α(ψ(ωλG(Tnz, u, u)))ψ(ωλ+ν(λ)G(Tnz, u, u))+β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, Tn+1z, Tn+1z))≤α(ψ(ωλG(Tnz, u, u)))ψ(ωλ+ν(λ)G(Tnz, u, u))+ψ(ωλ/4G(Tn+1z, u, u))+ψ(ωλ4G(Tn+1z, u, u)))=α(ψ(ωλG(Tnz, u, u)))ψ(ωλ+ν(λ)G(Tnz, u, u))+β(ψ(ωλG(Tnz, u, u)))(ψ(ωλ/2G(Tnz, u, u))+2ψ(ωλ/4G(Tn+1z, u, u))≤α(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, u, u)))+β(ψ(ωλG(Tnz, u, u)))(ψ(ωλG(Tnz, u, u))+2ψ(ωλG(Tn+1z, u, u)))=α(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, u, u)))+β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, u, u))+2β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tn+1z, u, u)),

so, we have that

(3.28)ψ(ωλG(Tn+1z, u, u))≤α(ψ(ωλG(Tnz, u, u)))+β(ψ(ωλG(Tnz, u, u)))1−2β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, u, u))≤ψ(ωλG(Tnz, u, u))≤ψ(ωλG(Tn−1z, u, u))⋮≤ψ(ωλG(z, u, u))<∞.

Therefore, {ψ(ωλG(Tn+1z, u, u))}n≥1 is nonincreasing sequence which is bounded below and converges to some real number ℓ∈[0, ∞) Assume that ℓ>0, using the fact that limn→∞ωλG(Tnx0, Tn+1x0, Tn+1x0)=0 for all λ>0, from inequality 3, we have that

(3.29)ψ(ωλG(Tn+1z, u, u))≤α(ψ(ωλG(Tnz, u, u)))ψ(ωλ+ν(λ)G(Tnz, u, u))+β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, Tn+1z, Tn+1z))≤α(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, u, u))+β(ψ(ωλG(Tnz, u, u)))ψ(ωλG(Tnz, Tn+1z, Tn+1z)).

Using inequality 3 and letting n→∞, inequality 3 becomes

(3.30)1≤limn→∞infα(ψ(ωλG(Tnz, u, u))).

Thus, by condition 4 of Proposition 2.3 we have that

(3.31)limn→∞ωλG(Tnz, u, u)=0

for all λ>0. Therefore, Tnz→u as n→∞.

Secondly consider ψ(ωλG(Tn+1z, v, v))<∞, from inequality (3.1), we have by taking x=Tnz and y=v, so that we have the following inequality by using condition 6 of Proposition 2.2

(3.32)ψ(ωλG(Tn+1z, v, v))=ψ(ωλ(Tn+1z, Tv, Tv))=ψ(ωλG(TTnz, Tv, Tv))≤α(ψ(ωλG(Tnz, v, v)))ψ(ωλ+ν(λ)G(Tnz, v, v))+β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, TTnz, TTnz))+γ(ψ(ωλG(Tnz, v, v)))ψ(ωλG(v, Tv, Tv))=α(ψ(ωλG(Tnz, v, v)))ψ(ωλ+ν(λ)G(Tnz, v, v))+β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, Tn+1z, Tn+1z))≤α(ψ(ωλG(Tnz, v, v)))ψ(ωλ+ν(λ)G(Tnz, v, v))+β(ψ(ωλG(Tnz, v, v)))(ψ(ωλ/2G(Tnz, v, v)))+ψ(ωλ/4G(Tn+1z, v, v))+ψ(ωλ/4G(Tn+1z, v, v)))=α(ψ(ωλG(Tnz, v, v)))ψ(ωλ+ν(λ)G(Tnz, v, v))+β(ψ(ωλG(Tnz, v, v)))(ψ(ωλ/2G(Tnz, v, v))+2ψ(ωλ/4G(Tn+1z, v, v))≤α(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, v, v)))+β(ψ(ωλG(Tnz, v, v)))(ψ(ωλG(Tnz, v, v))+2ψ(ωλG(Tn+1z, v, v)))=α(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, v, v)))+β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, v, v))+2β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tn+1z, v, v)).

Therefore, we have

(3.33)ψ(ωλG(Tn+1z, v, v))≤α(ψ(ωλG(Tnz, v, v)))+β(ψ(ωλG(Tnz, v, v)))1−2β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, v, v))≤ψ(ωλG(Tnz, v, v))≤ψ(ωλG(Tn−1z, v, v))⋮≤ψ(ωλG(z, v, v))<∞.

Hence, {ψ(ωλG(Tn+1z, v, v))}n≥1 is nonincreasing sequence which is bounded below and converges to some real number ℓ0∈[0, ∞) Suppose that ℓ0>0, using the fact that limn→∞ωλG(Tnx0, Tn+1x0, Tn+1x0)=0 for all λ>0. From inequality 3, we have that

(3.34)ψ(ωλG(Tn+1z, v, v))≤α(ψ(ωλG(Tnz, v, v)))ψ(ωλ+ν(λ)G(Tnz, v, v))+β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, Tn+1z, Tn+1z))≤α(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, v, v))+β(ψ(ωλG(Tnz, v, v)))ψ(ωλG(Tnz, Tn+1z, Tn+1z)).

Using inequality 3 and letting n→∞, inequality 3 becomes

(3.35)1≤limn→∞infα(ψ(ωλG(Tnz, v, v))).

Thus, by condition 4 of Proposition 2.3 we have that

(3.36)limn→∞ωλG(Tnz, v, v)=0

for all λ>0. Therefore, Tnz→v as n→∞.

Suppose, if possible, that limn→∞Tnz exists and not unique. Let limn→∞Tnz=u and limn→∞Tnz=v as we have seen above, where u≠v. For each λ>0, u≠v⇒ωλG(u, v, v)>0. If we take ψ(ϵ1)=13ψ(ωλ(u, v, v))>0, then for λ>0, limn→∞Tnz=u⇒ given ϵ1>0,∃ m1∈N such that ψ(ωλ/2G(u, Tnz, Tnz))<ψ(ϵ1) for n>m1. Again, limn→∞Tnz=v⇒ given ϵ1>0, ∃ m2∈N such that ψ(ωλ/4G(v, Tnz, Tnz))<ψ(ϵ1) for n>m2. Set m=max{m1, m2}, then for n≥m, by condition 6 of Proposition 2.2, we have

ψ(ωλG(u, v, v))≤ψ(ωλ/2G(u, Tnz, Tnz)+2ωλ/4G(v, Tnz, Tnz))≤ψ(ωλ/2G(u, Tnz, Tnz))+2ψ(ωλ/4G(v, Tnz, Tnz))<ψ(ϵ1)+2ψ(ϵ1)=3ψ(ϵ1),

which shows that ψ(ωλG(u, v, v))<ψ(ωλG(u, v, v)) for all λ>0. This is a contradiction. Hence, u=v. Therefore the fixed point of T is unique and the proof is complete.□

We shall give an example to support Theorem 3.1 above.

Example 3.1.

Let X=ℝ, define modular G-metric by ωλG(x, y, y)=∞ if λ≤2|x−y| and ωλG(x, y, y)=0 if λ>2|x−y|, and ωλG(x, y, y)=G(x, y, y)λ, where G(x, y, y)=2|x−y| or G(x, y, z)=|x−y|+|y−z|+|x−z| for x,y,z∈ℝ. We can see that XωG=ℝ. So it follows from Theorem 3.1 that ℝ is a complete preordered modular G-metric space. Now define a map T:ℝ→ℝ by Tx=x31+x2. For x, y∈ℝ, then ωλG(x, y, y)=∞ if λ≤2|x−y|, so inequality (3.1) is satisfied. Again, if λ>2|x−y| and x,y∈ℝ, then G(Tx, Ty, Ty)=2|x31+x2−y31+y2|≤4|x−y|<2λ, therefore, ωλG(Tx, Ty, Ty)=2ωλG(x, y, y)≤0. We can take ψ(t)=t, α(t)=β(t)=γ(t)=12. But T has a fixed point at x=0.

Corollary 3.2.

Let (X, ωG) be a complete modular G-metric space with a preorder, ⪯ and a nondecreasing self-mapping T:XωG→XωG on XωG such that for each λ>0, there is ν(λ)∈[0, λ) such that the following conditions hold:

(1)

(3.37)ψ(ωλG(Tx, Ty, Ty))≤α(ψ(ωλG(x, y, y)))ψ(ωλ+ν(λ)G(x, y, y)),

where ψ∈Ψ¯ and α∈Ger and distinct x, y∈XωG. Assuming that if a nondecreasing sequence {xn}n∈N converges to x*, then xn⪯x* for each n∈N,

(2) if ψ is subadditive and for any x, y∈XωG, there exists z∈XωG with z⪯Tz and ωλG(z, Tz, Tz) is finite for all λ>0 such that z is comparable to both x and y. Then T has a fixed point u∈XωG and the sequence define by {Tnx0}n≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: Let x0∈XωG be such that x0⪯Tx0 and let xn=Txn−1=Tnx0 for all n∈N. Regarding that T is nondecreasing mapping, we have that x0⪯Tx0=x1, which implies that x1=Tx0⪯Tx1=x2. Inductively, we have

(3.38)x0 ⪯ x1 ⪯ x2 ⪯⋯⪯ xn−1 ⪯ xn ⪯ xn+1 ⪯⋯.

Assume that there exists n0∈N such that xn0=xn0+1. Since xn0=xn0+1=Txn0, then xn0 is the fixed point of T. Now suppose that xn⪵xn+1 for all n∈N, thus by inequality (3.38), we have that

(3.39)x0≺x1≺x2≺⋯≺xn−1≺xn≺xn+1≺⋯.

Now for each λ>0, and x0≺Tx0 for all n∈N implies that ωλG(x0, Tx0, Tx0)>0. Again, let x0∈XωG such that ωλG(x0, Tx0, Tx0)<∞ ∀ λ>0.

First, we show that for all n∈N, the sequence ωλG(Tnx0, Tn+1x0, Tn+1x0)=0 for all λ>0, as n→∞.

Assume that for each n∈N, there exists λn>0 such that ωλnG(Tn+1x0, Tnx0, Tnx0)≠0. Otherwise we are done. Suppose that for each n≥1, if 0<λ<λn, then we have ωλG(Tn+1x0, Tnx0, Tnx0)≠0. Since Tnx0⪯Tn+1x0, we have from inequality (3.37) that ψ(ωλnG(Tn+1x0, Tnx0, Tnx0))≤ψ(ωλG(Tn+1x0, Tnx0, Tnx0))=ψ(ωλG(TTnx0, TTn−1x0, TTn−1x0)). Take x=Tnx0 and y=Tn−1x0, then inequality (3.37) becomes;

(3.40)ψ(ωλG(Tn+1x0, Tnx0, Tnx0))≤α(ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)))×ψ(ωλ+ν(λ)G(Tnx0, Tn−1x0, Tn−1x0))≤α(ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)))×ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0))<ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)).

Therefore, {ψ(ωλG(Tn+1x0, Tnx0, Tnx0))}n≥1 is nonincreasing and bounded below and converges to some real number τ≥0. Assume that τ>0. In such a case,

(3.41)τ<ψ(ωλG(Tn+1x0, Tnx0, Tnx0))≤α(ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)))ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0))<ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)),

which implies that

(3.42)1<ψ(ωλG(Tn+1x0, Tnx0, Tnx0))τ≤α(ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)))ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0))τ<ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0))τ.

Letting, n→∞, Theorem 2.1 ensure that {α(ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0)))}n≥1→1 or from inequality 3,

(3.43)1≤limn→∞infα(ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0))).

As α∈Ger, then

(3.44)limn→∞ψ(ωλG(Tnx0, Tn−1x0, Tn−1x0))→0

for all λ>0, which contradicts the fact that τ>0. Thus τ=0, so that

(3.45)limn→∞ωλG(Tn+1x0, Tnx0, Tnx0)=0.

Hence ωλG(Tn+1x0, Tnx0, Tnx0)=0 for all λ>0, n≥1. Following the proof of Theorem 3.1 above, we see that T has a unique fixed point in XωG. □

Theorem 3.3.

Let (X, ωG) be a complete modular G-metric space with a preorder, ⪯, and a nondecreasing self-mapping T:XωG→XωG on XωG such that for each λ>0, there is ν(λ)∈[0,λ) such that the following conditions hold:

(1)

(3.46)ψ(ωλG(Tmx, Tmy, Tmy))≤α(ψ(ωλG(x, y,y)))ψ(ωλ+ν(λ)G(x, y, y)),

where ψ∈Ψ¯ and α∈Ger and distinct x, y∈XωG. Assuming that if a nondecreasing sequence {xn}n∈N converges to x*, then xn⪯x* for each n∈N,

(2) if ψ is subadditive and for any x, y∈XωG, there exists z∈XωG with z⪯Tz and ωλG(z, Tz, Tz) is finite for all λ>0 such that z is comparable to both x and y.Then T has a fixed point u∈XωG for some positive integer m and the sequence define by {Tnx0}n≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: By Corollary 3.2, Tm has a fixed point say u∈XωG for some positive integer m≥1. Now Tm(Tu)=Tm+1u=T(Tmu)=Tu, so Tu is a fixed point of Tm. By the uniqueness of fixed point of Tm, we have Tu=u. Therefore, u is a fixed point of T. Since fixed point of T is also fixed point of Tm, hence T has a unique fixed point in XωG. □

Theorem 3.4.

Let (X, ωG) be a complete modular G-metric space with a preorder, ⪯ and a nondecreasing self-mapping T:XωG→XωG on XωG such that for each λ>0, there is ν(λ)∈[0, λ) such that the following conditions hold:

(1)

(3.47)ψ(ωλG(Tmx, Tmy, Tmy))≤α(ψ(ωλG(x, y, y)))ψ(ωλ+ν(λ)G(x, y, y))+β(ψ(ωλG(x,y,y)))ψ(ωλG(x, Tmx, Tmx))+γ(ψ(ωλG(x,y,y)))ψ(ωλG(y, Tmy, Tmy)),

where ψ∈Ψ¯ and {α, β, γ}∈Ger with α(t)+2max{supt≥0β(t), supt≥0γ(t)}<1, and distinct x, y∈XωG. Assuming that if a nondecreasing sequence {xn}n∈N converges to x*, then xn⪯x* for each n∈N,

(2) if ψ is subadditive and for any distinct x, y∈XωG, there exists z∈XωG with z⪯Tz and ωλG(z, Tz, Tz) is finite for all λ>0 such that z is comparable to both x  and y. Then T has a fixed point u∈XωG for some positive integer m≥1 and the sequence define by {Tnx0}n≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: By Theorem 3.1, Tm has a fixed point say u*∈XωG for some positive integer m≥1. Now Tm(Tu*)=Tm+1u*=T(Tmu*)=Tu*, so Tu* is a fixed point of Tm. By the uniqueness of fixed point of Tm, we have Tu*=u*. Therefore, u* is a fixed point of T. Since fixed point of T is also fixed point of Tm, hence T has a unique fixed point in XωG. □

Theorem 3.5.

Let (X, ωG) be a complete modular G-metric space with a preorder, ⪯ and a nondecreasing self-mapping T: XωG→XωG on XωG such that for each λ>0, there is ν(λ)∈[0, λ) such that the following conditions hold:

(1)

(3.48)ψ(ωλG(Tx, Ty, Tz))≤α(ψ(ωλG(x, y, z)))ψ(ωλ+ν(λ)G(x, y, z))+β(ψ(ωλG(x, y, z)))ψ(ωλG(x, Tx, Tx))+γ(ψ(ωλG(x, y, z)))ψ(ωλG(y, Ty, Ty))+δ(ψ(ωλG(x, y, z)))ψ(ωλG(z, Tz, Tz)),

where ψ∈Ψ¯ and {α, β, γ, δ}∈Ger with α(t)+2max{supt≥0β(t), supt≥0γ(t), supt≥0δ(t)}<1, and x, y, z∈XωG. Assuming that if a nondecreasing sequence {xn}n∈N converges to x, then xn⪯x for each n∈N,

(2) if ψ is subadditive and for any x, y, z∈XωG, there exists w∈XωG with w⪯Tw and ωλG(w, Tw, Tw) is finite for all λ>0 such that w is comparable to both x, y and z. Then T has a fixed point u∈XωG and the sequence define by {Tnx0}n≥1 converges to u. Moreover, the fixed point of T is unique.

Proof. Let x0∈XωG be such that x0⪯Tx0 and let xn=Txn−1=Tnx0 for all n∈N. Regarding that T is nondecreasing mapping, we have that x0⪯Tx0=x1, which implies that x1=Tx0⪯Tx1=x2. Inductively, we have

(3.49)x0 ⪯ x1 ⪯ x2 ⪯⋯⪯ xn−1 ⪯ xn ⪯ xn+1 ⪯⋯.

Assume that there exists n0∈N such that xn0=xn0+1. Since xn0=xn0+1=Txn0, then xn0 is the fixed point of T. Now suppose that xn⪵xn+1 for all n∈N, thus by inequality (3.38), we have that

(3.50)x0 ≺ x1 ≺ x2 ≺⋯ ≺ xn−1 ≺ xn ≺ xn+1 ≺⋯.

Now for each λ>0, and x0≺Tx0 for all n∈N implies that ωλG(x0, Tx0, Tx0)>0. Again, let x0∈XωG such that ωλG(x0, Tx0, Tx0)<∞ ∀ λ>0.

First, we show that for all n∈N, the sequence ωλG(Tnx0, Tn+1x0, Tn+1x0)=0 for all λ>0, as n→∞.

Assume that, for each n∈N, there exists λn>0 such that ωλnG(Tnx0, Tn+1x0, Tn+1x0)≠0. Otherwise there is nothing to prove. Suppose that for each n≥1, if 0<λ<λn, then we have that ωλG(Tnx0, Tnx0, Tn+1x0)≠0. Since Tnx0⪯Tn+1x0, we have from inequality (3.5) that ψ(ωλnG(Tnx0,Tn+1x0,Tn+1x0))≤ψ(ωλG(Tnx0,Tn+1x0,Tn+1x0))=ψ(ωλG(TTn−1x0, TTnx0,TTnx0)). Take x=Tn−1x0 and y=Tnx0=z, then inequality (3.5) becomes;

(3.51)ψ(ωλnG(Tnx0, Tn+1x0, Tn+1x0))≤ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλ+ν(λ)G(Tn−1x0, Tnx0, Tnx0))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, TTn−1x0, TTn−1x0))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, TTnx0, TTnx0))+δ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, TTnx0, TTnx0))=α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλ+ν(λ)G(Tn−1x0, Tnx0, Tnx0))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, Tnx0, Tnx0))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))+δ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, Tnx0, Tnx0))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tn−1x0, Tnx0, Tnx0))+γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))+δ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))×ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0)),

which implies that

(3.52)ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))≤ρψ(ωλG(Tn−1x0, Tnx0, Tnx0))≤ψ(ωλG(Tn−1x0, Tnx0, Tnx0))⋮≤ψ(ωλG(x0, Tx0, Tx0))<∞,

where

(3.53)ρ:=α(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+β(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))1−(γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+δ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))).

Therefore, {ψ(ωλG(Tnx0, Tn+1x0, Tn+1x0))}n≥1 is nonincreasing and bounded below, hence converges to some real number s≥0. We can also see clearly that by taking θ(.):=γ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0)))+δ(ψ(ωλG(Tn−1x0, Tnx0, Tnx0))), as y=Tnx0=z, then following Theorem 3.1, T has a unique fixed point in XωG. This complete the proof. □

Theorem 3.6.

Let (X, ωG) be a complete modular G-metric space with a preorder, ⪯ and a nondecreasing self-mapping T: XωG→XωG on XωG such that for each λ>0, there is ν(λ)∈[0, λ) such that the following conditions hold:

(1)

(3.54)ψ(ωλG(Tmx, Tmy, Tmz))≤α(ψ(ωλG(x, y, z)))ψ(ωλ+ν(λ)G(x, y, z))+β(ψ(ωλG(x, y, z)))ψ(ωλG(x, Tmx, Tmx))+γ(ψ(ωλG(x, y, z)))ψ(ωλG(y, Tmy, Tmy))+δ(ψ(ωλG(x, y, z)))ψ(ωλG(z, Tmz, Tmz)),

where ψ∈Ψ¯ and {α, β, γ, δ}∈Ger with α(t)+2max{supt≥0β(t), supt≥0γ(t), supt≥0δ(t)}<1, and x, y, z∈XωG. Assuming that if a nondecreasing sequence {xn}n∈N converges to x, then xn⪯x for each n∈N;

(2) if ψ is subadditive and for any x, y, z∈XωG, there exists w∈XωG with w ⪯ Tw and ωλG(w, Tw, Tw) is finite for all λ>0 such that w is comparable to both x, y and z. Then T has a fixed point u∈XωG for some positive integer m≥1 and the sequence define by {Tnx0}n≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: Take y=z and φ(.)=γ(ψ(ωλG(x, y, z)))+δ(ψ(ωλG(x, y, z))), then Theorem 3.5 tells us that Tm has a fixed point say u∈XωG for some positive integer m≥1. Therefore, Theorem 3.4 shows that T has a unique fixed point in XωG. □

4. Applications to nonlinear Volterra-Fredholm-type integral equations

In this section, we construct a system of nonlinear integral equation that satisfies the conditions of Theorem 3.1. We consider the following general nonlinear Volterra-Fredholm-type integral equations.

(4.1)u(t, x)=h(t, x)+∫ 0t∫BF(t, x, s, y, u(s, y), (L*u)(s, y))dyds,

and

(4.2)v(t, x)=e(t, x)+∫ 0t∫BG(t, x, s, y, v(s, y), (L∗v)(s, y))dsdy,

where

(4.3)(L*u)(t, x)=∫0t∫BK(t, x, τ, z, u(τ, z))dzdτ,

and

(4.4)(L*v)(t, x)=∫0t∫BK(t, x, τ, z, v(τ, z))dzdτ,

h, F, K and e, G, L are given functions and u, v are the unknown functions. We assume that h, e∈C(ℝ+×B, ℝn), K∈C(Ω×ℝn, ℝn), F∈C(Ω×ℝn×ℝn, ℝn), G∈C(Ω×ℝn×ℝn,ℝn) and Ω={(t, x, s, y): 0≤s≤t<∞; x, y∈B} , B=∏ j=1n[aj, bj], bj>aj. Take sup(t,x)∈ℝ+×Bg(s, y)≤1t∏ j=1n[aj, bj], where (t, x)∈ℝ+×B.

Let α, β, γ>0 with α(t)+2maxt∈Ω{supt≥0β(t), supt≥0γ(t)}<1 such that‖F(t, x, s, y, u(s, y), (L*u)(s, y))−G(t, x, s, y, v(s, y), (L*v)(s, y))‖≤g(s, y){α(‖u−v‖)‖u−v‖+β(‖u−v‖)m(u, L*u)+γ(‖u−v‖)r(v, L*v)}. Let F, G: C(Ω×ℝn×ℝn, ℝn)→ℝn be such that Fu, Gv∈C(Ω×ℝn×ℝn, ℝn) and let

(4.5)Fu=∫ 0t∫BF(t, x, s, y, u(s, y), (L∗u)(s, y))dyds,

and

(4.6)Gv=∫ 0t∫BG(t, x, s, y, v(s, y), (L*v)(s, y))dsdy,

for F∈C(Ω×ℝn×ℝn, ℝn), G∈C(Ω×ℝn×ℝn, ℝn) and u, v are the unknown functions.

Now for any λ>0, we define

(4.7)ωλG(x, y, z):=12(1+λ)sup(t,x)∈ℝ+×B{‖x(t)−y(t)‖+‖y(t)−z(t)‖+‖x(t)−z(t)‖},

so that

(4.8)ωλG(x, y, y):=1(1+λ)sup(t,x)∈ℝ+×B{‖x(t)−y(t)‖}.

In fact Eqns. (4.7) and (4.8) satisfies all the conditions in Definition 2.6 endowed with XωG=(X, ωG)=C(Ω × ℝn × ℝn, ℝn).

Now, take A=11+λ sup(t,x)∈ℝ+×B{‖Fu−Gv+h(t, x)−e(t, x)‖}, so that

(4.9)A ≤ 11+λsup(t,x)∈ℝ+×B{∫​0t∫B‖F(t, x, s, y, u(s, y),(L∗u)(s, y))−G(t, x, s, y, v(s, y),(L∗v)(s, y))‖ × dyds}≤ 11+λsup(t,x)∈ℝ+×B{∫​0t∫Bg(s, y){α(‖u−v‖)‖u−v‖+β(‖u−v‖)m(u, L∗u)+γ(‖u−v‖)r(v, L∗v)}dyds}≤ 11+λsup(t,x)∈ℝ+×B{{α(‖u−v‖)‖u−v‖+β(‖u−v‖)m(u, L∗u)+ γ(‖u−v‖) × r(v, L∗v)}}≤ 11+λsup(t,x)∈ℝ+×B{α(‖u−v‖)‖u−v‖}+ 11+λsupt∈Ω{β(‖u−v‖)m(u, L∗u)+γ(‖u−v‖) × r(v, L∗v)}≤ 11+λsup(t,x)∈ℝ+×B{α(‖u−v‖)‖u−v‖}+ 11+λsup(t,x)∈ℝ+×B{β(‖u−v‖)m(u, L*u)}+ 11+λsup(t,x)∈ℝ+×B{γ(‖u−v‖)r(v, L*v)},

where m, r∈C(ℝ+ × B × ℝn, ℝn)

(4.10)m(u, L*u)(t, x)=sup(t,x)∈ℝ+×B‖Fu(t, x)+h(t, x)−u(t, x)‖,

(4.11)r(v, L*v)(t, x)=sup(t,x)∈ℝ+×B‖Gv(t, x)+e(t, x)−v(t, x)‖

Theorem 4.1.

Let XωG=C(Ω×ℝn×ℝn, ℝn) be a complete modular G-metric space and ωG: (0, ∞)×XωG×XωG×XωG→ℝ+n∪ {∞} be defined by

(4.12). ωλG(u, v, v):=11+λsup(t,x)∈ℝ+×B\normu(t, x)−v(t, x), λ>0

and u⪯v⇔u(t, x)≤v(t, x)  ∀  (t, x)∈ℝ+×B. Let Fu, Gv: C(Ω×ℝn×ℝn, ℝn)→ℝn are such that Fu, Gv∈XωG for each u, v∈XωG and Fu, Gv satisfying Eqns. (4.5) and (4.6), respectively, for all (t,x)∈ℝ+×B. Suppose that there exists nonnegative reals α, β, γ>0 with α(t)+2maxt∈Ω{supt≥0β(t), supt≥0γ(t)}<1 such that inequality 4 is satisfied for every u, v∈XωG. Moreover if ψ is subadditive and for any u, v∈XωG, there exists w0, w1∈XωG with w0⪯w1 and ωλG(w0, w1, w1) is finite for all λ>0 such that w is comparable to both u and v. Then the system of integral Eqns (4.1) and (4.2) have a unique solution in XωG.

Proof. Define the mapping T: XωG→XωG by Tu=Fu+e and Tv=Gv+h. Then for λ>0, ωλG(Tu, Tv, Tv)=11+λsup(t,x)∈ℝ+×B\normFu(t, x)−Gv(t, x)+e(t, x)−h(t, x), ωλG(u, Tu, Tu)=11+λsup(t,x)∈ℝ+×B\normFu(t, x)+e(t, x)−u(t, x) and ωλG(v, Tv, Tv)=11+λsup(t,x)∈ℝ+×B\normGv(t, x)+h(t, x)−v(t, x). So from inequality 4, we get by noticing that ψ is continuous and subadditive and there exists ν∈[0,λ) such that

(4.13)ψ(ωλG(Tu, Tv, Tv))≤α(ψ(ωλG(u, v, v)))ψ(ωλ+ν(λ)G(u, v, v))+β(ψ(ωλG(u, v, v)))ψ(ωλG(u, Tu, Tu))+γ(ψ(ωλG(u,v,v)))ψ(ωλG(v, Tv, Tv)),

where ψ∈Ψ¯. By Theorem 3.1, we conclude that the system of nonlinear Volterra-Fredholm integral Eqns (4.1) and (4.2) have a unique solution in XωG. □