Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction

Abstract

In this paper we utilize the notion of Ω-distance in the sense of Saadati et al. (Math. Comput. Model. 52:797-801, 2010) to construct and prove some fixed and coupled fixed point theorems in a complete G-metric space for a nonlinear contraction. Also, we provide an example to support our results.

MSC:47H10, 54H25.

1 Introduction

The concept of G-metric space was introduced by Mustafa and Sims [1]. After that, many authors constructed fixed point theorems in G-metric spaces. In [2] and [3], common fixed points results for mappings which satisfy the generalized $(\phi ,\psi )$-weak contraction are obtained. In [4], the author proves a common fixed point theorem for two self-mappings verifying a contractive condition of integral type in G-metric spaces. In [5, 6] and [7], tripled coincidence point results for a mixed monotone mapping in G-metric spaces are established; also see [8]. Some common fixed point results for two self-mappings, one of them being a generalized weakly G-contraction of type A and B with respect to the other mapping, are stated in [9]. Fixed point theorems for mappings with a contractive iterate at a point are formulated in [10] and in [11]. Papers [12] and [13] refer to common fixed point theorems for single-valued and multi-valued mappings which satisfy contractive conditions on G-metric spaces. In [14] and [15], theorems from G-metric spaces are used to obtain several results on complete D-metric spaces. Various contractive conditions on G-metric spaces which lead to fixed point results are stated in [16]. Paper [17] deals with the existence of fixed point results in G-metric spaces. In [18], common fixed point theorems with ϕ-maps on G-cone metric spaces are established. In [19], a general fixed point theorem for mappings satisfying an ϕ-implicit relation is proved. Paper [20] states fixed point theorems for mappings satisfying ϕ-maps in G-metric spaces. Mohamed Jleli and Bessem Samet [21] in their nice paper pointed out that the quasi-metric spaces play a major role to construct some known fixed point theorems in a G-metric space. For other recent results in G-metric spaces, please see [22–24].

The coupled fixed point is one of the most interesting subjects in metric spaces. The notion of coupled fixed point was introduced by Bhaskar and Lakshmikantham [25], and the notion of coincidence coupled fixed point was introduced by Lakshmikantham and Ćirić [26]. In recent years many authors established many nice coupled and coincidence coupled fixed point theorems in metric spaces, partial metric spaces and G-metric spaces. For some works on this subject, we refer the reader to [27–38].

2 Preliminaries

It is fundamental to recall the definition of G-metric spaces.

Definition 2.1 ([1])

Let X be a nonempty set. $G:X\times X\times X\to X$ is called G-metric if the following axioms are fulfilled:

• (1) $G(x,y,z)=0$ if $x=y=z$ (the coincidence);

• (2) $G(x,x,y)>0$ for all $x,y\in X$, $x\ne y$;

• (3) $G(x,x,z)\le G(x,y,z)$ for each triple $(x,y,z)$ from $X\times X\times X$ with $z\ne y$;

• (4) $G(x,y,z)=G(p\{x,y,z\})$ for each permutation of $\{x,y,z\}$ (the symmetry);

• (5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for each x, y, z and a in X (the rectangle inequality).

Definition 2.2 ([1])

Consider X to be a G-metric space and $(x_n)$ to be a sequence in G.

• (1) $(x_n)$ is called a G-Cauchy sequence if for each $\epsilon >0$, there is a positive integer $n_0$ so that for all $m,n,l\ge n_0$, $G(x_n,x_m,x_l)<\epsilon$.

• (2) $(x_n)$ is said to be G-convergent to $x\in X$ if for each $\epsilon >0$, there is a positive integer $n_0$ such that $G(x_m,x_n,x)<\epsilon$ for each $m,n\ge n_0$.

Now, we recall the definitions of coupled and coincidence coupled fixed points.

Definition 2.3 ([25])

Consider X to be a nonempty set. The pair $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $F:X\times X\to X$ if

$$F(x,y)=x,F(y,x)=y.$$

Definition 2.4 ([26])

Let X be a nonempty set. The element $(x,y)\in X\times X$ is a coupled coincidence point of mappings $F:X\times X\to X$ and $g:X\to X$ if

$$F(x,y)=gx,F(y,x)=gy.$$

In 2010, Saadati et al. [39] utilized the notion of G-metric spaces to introduce the concept of Ω-distance. Moreover, Saadati et al. [40] constructed some fixed point theorem in G-metric spaces by using the notion of Ω-distance.

Definition 2.5 ([39])

Consider $(X,G)$ to be a G-metric space and $\mathrm{\Omega }:X\times X\times X\to [0,+\mathrm{\infty })$. Ω is called an Ω-distance on X if it satisfies the three conditions as follows:

• (1) $\mathrm{\Omega }(x,y,z)\le \mathrm{\Omega }(x,a,a)+\mathrm{\Omega }(a,y,z)$ for all x, y, z, a from X.

• (2) For each x, y from X, $\mathrm{\Omega }(x,y,\cdot ),\mathrm{\Omega }(x,\cdot ,y):X\to [0,+\mathrm{\infty })$ are lower semi-continuous.

• (3) For each $\epsilon >0$, there is $\delta >0$, so that $\mathrm{\Omega }(x,a,a)\le \delta$ and $\mathrm{\Omega }(a,y,z)\le \delta$ imply $G(x,y,z)\le \epsilon$.

The following lemma is very useful in this paper.

Lemma 2.1 ([39, 40])

Let X be a metric space endowed with metric G, and let Ω be an Ω-distance on X. $(x_n)$, $(y_n)$ are sequences in X, $({\alpha }_n)$ and $({\beta }_n)$ are sequences in $[0,+\mathrm{\infty })$, with ${lim}_{n\to +\mathrm{\infty }}{\alpha }_n={lim}_{n\to +\mathrm{\infty }}{\beta }_n=0$. If x, y, z and $a\in X$, then

• (1) If $\mathrm{\Omega }(y,x_n,x_n)\le {\alpha }_n$ and $\mathrm{\Omega }(x_n,y,z)\le {\beta }_n$, for $n\in \mathbb{N}$, then $G(y,y,z)<\epsilon$, and, by consequence, $y=z$.

• (2) Inequalities $\mathrm{\Omega }(y_n,x_n,x_n)\le {\alpha }_n$ and $\mathrm{\Omega }(x_n,y_m,z)\le {\beta }_n$, for $m>n$, imply $G(y_n,y_m,z)\to 0$, hence $y_n\to z$.

• (3) If $\mathrm{\Omega }(x_n,x_m,x_l)\le {\alpha }_n$ for $l,m,n\in \mathbb{N}$ with $n\le m\le l$, then $(x_n)$ is a G-Cauchy sequence.

• (4) If $\mathrm{\Omega }(x_n,a,a)\le {\alpha }_n$, $n\in \mathbb{N}$, then $(x_n)$ is a G-Cauchy sequence.

The following two sets are very useful to build our nonlinear contraction in this paper:

$$\begin{array}{c}\mathrm{\Phi }=\{\phi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })|\phi \text{ is continuous, increasing},\phi (t)=0\text{ if and only if }t=0\},\hfill \\ \mathrm{\Psi }=\{\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })|\psi \text{ is lower semi-continuous},\hfill \\ \phantom{\mathrm{\Psi }=}\psi (t)=0\text{ if and only if }t=0\}.\hfill \end{array}$$

For some works on fixed point theorems based on the above sets, see, for example, [1, 7, 14–20, 23, 33–39, 41–44].

In the present paper, we utilize the concept of Ω-distance and the sets Φ, Ψ to establish some fixed and coupled fixed point theorems. Also, we introduce an example as an application of our results.

3 Main results

In the first part of the section, we introduce and prove the following fixed point theorem.

Theorem 3.1 Let $(X,G)$ be a G-metric space and Ω be an Ω-distance on X. Consider $\phi \in \mathrm{\Phi }$, $\psi \in \mathrm{\Psi }$ and $T:X\to X$ such that

$$\phi \mathrm{\Omega }(Tx,Ty,Tz)\le \phi \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z)$$

(1)

holds for each $(x,y,z)\in X\times X\times X$.

Suppose that if $u\ne Tu$, then

$$inf\{\mathrm{\Omega }(x,Tx,u):x\in X\}>0.$$

Then T has a unique fixed point.

Proof Let $x_0\in X$ and $x_{n+1}=T{x}_n$ for each $n\in \mathbb{N}$. If there is $n\in \mathbb{N}$ for which $x_{n+1}=x_n$, then $x_n$ is a fixed point of T.

In the following, we assume $x_{n+1}\ne x_n$ for each $n\in \mathbb{N}$.

First we shall prove that ${lim}_{n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_{n+1},x_{n+1})=0$.

For $n\in \mathbb{N}$, $n\ge 1$, we have

$$\begin{array}{rl}\phi \mathrm{\Omega }(x_n,x_{n+1},x_{n+1})& =\phi \mathrm{\Omega }(T{x}_{n-1},T{x}_n,T{x}_n)\\ \le \phi \mathrm{\Omega }(x_{n-1},x_n,x_n)-\psi \mathrm{\Omega }(x_{n-1},x_n,x_n)\\ \le \phi \mathrm{\Omega }(x_{n-1},x_n,x_n).\end{array}$$

(2)

φ is a nondecreasing function, hence $\mathrm{\Omega }(x_n,x_{n+1},x_{n+1})\le \mathrm{\Omega }(x_{n-1},x_n,x_n)$, $n\ge 1$. It follows that $(\mathrm{\Omega }(x_n,x_{n+1},x_{n+1}))$ is a nondecreasing sequence, therefore there exists ${lim}_{n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_{n+1},x_{n+1})=r\ge 0$.

Taking $n\to +\mathrm{\infty }$ in inequality (2) and using the continuity of φ and the lower semi-continuity of ψ, we get

$$\phi r\le \phi r-\underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\psi \mathrm{\Omega }(x_{n-1},x_n,x_n)\le \phi r-\psi r,$$

imposing $\psi r=0$, that is, $r=0$.

Analogously, it can be proved that ${lim}_{n\to +\mathrm{\infty }}\mathrm{\Omega }(x_{n+1},x_n,x_n)=0$ and also that

$$\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(x_n,x_n,x_{n+1})=0.$$

The next step is to prove that ${lim}_{m,n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_m,x_m)=0$, $m>n$.

By reductio ad absurdum, suppose the contrary. Hence, there exist $\epsilon >0$ and two sequences $(n_k)$ and $(m_k)$ such that

$$\mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})\ge \epsilon ,\mathrm{\Omega }(x_{n_k},x_{m_k-1},x_{m_k-1})<\epsilon ,m_k>n_k.$$

As ${lim}_{n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_{n+1},x_{n+1})=0$, it follows

$$\begin{array}{rcl}\epsilon & \le & \mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})\le \mathrm{\Omega }(x_{n_k},x_{m_k-1},x_{m_k-1})+\mathrm{\Omega }(x_{m_k-1},x_{m_k},x_{m_k})\\ <& \epsilon +\mathrm{\Omega }(x_{m_k-1},x_{m_k},x_{m_k})\to \epsilon \text{as }k\to +\mathrm{\infty }.\end{array}$$

Therefore, ${lim}_{k\to +\mathrm{\infty }}\mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})=\epsilon$.

On the other hand,

$$\begin{array}{rl}\epsilon & \le \mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})\le \mathrm{\Omega }(x_{n_k},x_{n_k+1},x_{n_k+1})+\mathrm{\Omega }(x_{n_k+1},x_{m_k},x_{m_k})\\ \le \mathrm{\Omega }(x_{n_k},x_{n_k+1},x_{n_k+1})+\mathrm{\Omega }(x_{n_k+1},x_{m_k+1},x_{m_k+1})+\mathrm{\Omega }(x_{m_k+1},x_{m_k},x_{m_k}).\end{array}$$

(3)

The contraction condition (1) yields

$$\begin{array}{rcl}\phi \mathrm{\Omega }(x_{n_k+1},x_{m_k+1},x_{m_k+1})& \le & \phi \mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})-\psi \mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})\\ \le & \phi \mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k}),\end{array}$$

so $\mathrm{\Omega }(x_{n_k+1},x_{m_k+1},x_{m_k+1})\le \mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})$, and relation (3) becomes

$$\begin{array}{rcl}\epsilon & \le & \mathrm{\Omega }(x_{n_k},x_{n_k+1},x_{n_k+1})+\mathrm{\Omega }(x_{n_k+1},x_{m_k+1},x_{m_k+1})+\mathrm{\Omega }(x_{m_k+1},x_{m_k},x_{m_k})\\ \le & \mathrm{\Omega }(x_{n_k},x_{n_k+1},x_{n_k+1})+\mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})+\mathrm{\Omega }(x_{m_k+1},x_{m_k},x_{m_k}).\end{array}$$

Letting $k\to +\mathrm{\infty }$, we get ${lim}_{k\to +\mathrm{\infty }}\mathrm{\Omega }(x_{n_k+1},x_{m_k+1},x_{m_k+1})=\epsilon$.

Having in mind the continuity of φ and the lower semi-continuity of ψ, we obtain

$$\phi \epsilon \le \phi \epsilon -\underset{k\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\mathrm{\Omega }(x_{n_k},x_{m_k},x_{m_k})\le \phi \epsilon -\psi \epsilon ,$$

which is impossible, since $\epsilon >0$.

It follows that ${lim}_{m,n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_m,x_m)=0$, $m>n$.

In a similar manner, it can be proved that ${lim}_{m,n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_n,x_m)=0$, $m>n$.

Consider now $l>m>n$, $l,m,n\in \mathbb{N}$. Since

$$\mathrm{\Omega }(x_n,x_m,x_l)\le \mathrm{\Omega }(x_n,x_m,x_m)+\mathrm{\Omega }(x_m,x_m,x_l)\to 0$$

as $l,m,n\to +\mathrm{\infty }$, we conclude that ${lim}_{l,m,n\to +\mathrm{\infty }}\mathrm{\Omega }(x_n,x_m,x_l)=0$. By Lemma 2.1, $(x_n)$ is a G-Cauchy sequence in the G-complete space $(X,G)$, so it converges to $u\in X$.

Suppose $u\ne Tu$. Consider $\epsilon >0$. As $(x_n)$ is a Cauchy sequence, there is $n_0\in \mathbb{N}$ such that

$$\mathrm{\Omega }(x_n,x_m,x_l)<\epsilon ,\mathrm{\forall }n,m,l\ge n_0.$$

Thus

$$\underset{l\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\mathrm{\Omega }(x_n,x_m,x_l)\le \underset{l\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\epsilon =\epsilon ,\mathrm{\forall }n,m\ge n_0.$$

From the lower semi-continuity of Ω in its third variables, we have

$$\mathrm{\Omega }(x_n,x_m,u)\le \underset{l\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\mathrm{\Omega }(x_n,x_m,x_l)\le \epsilon ,\mathrm{\forall }n,m\ge n_0.$$

(4)

Considering $m=n+1$ in inequality (4), we get

$$\mathrm{\Omega }(x_n,x_{n+1},u)\le \epsilon .$$

On the other hand, we have

$$\begin{array}{rcl}0& <& inf\{\mathrm{\Omega }(x,Tx,u):x\in X\}\\ \le & inf\{\mathrm{\Omega }(x_n,x_{n+1},u):n\ge n_0\}<\epsilon ,\end{array}$$

which contradicts the hypotheses.

Therefore, $u=Tu$ and hence u is a fixed point of T.

We shall deal now with the uniqueness of the fixed point of T.

Suppose that there are u and v in X fixed points of the mapping T.

It follows that

$$\phi \mathrm{\Omega }(v,u,u)=\phi \mathrm{\Omega }(Tv,Tu,Tu)\le \phi \mathrm{\Omega }(v,u,u)-\psi \mathrm{\Omega }(v,u,u),$$

which is possible only for $\psi \mathrm{\Omega }(v,u,u)=0$, that is, $\mathrm{\Omega }(v,u,u)=0$.

Similarly, it can be proved that $\mathrm{\Omega }(u,v,u)=0$.

According to the definition of an Ω-distance, $\mathrm{\Omega }(v,u,u)=0$ and $\mathrm{\Omega }(u,v,u)=0$ imply $u=v$. Hence, T has a unique fixed point. □

Haghi et al. [45] in their interesting paper showed that some common fixed point theorems can be obtained from the known fixed point theorems; for other interesting article by Haghi et al., please see [46]. By using the same method of Haghi et al. [45], we get the following result.

Theorem 3.2 Let $(X,G)$ be a G-metric space and Ω be an Ω-distance on X. Consider $\phi \in \mathrm{\Phi }$, $\psi \in \mathrm{\Psi }$ and $T,S:X\to X$ such that

$$\phi \mathrm{\Omega }(Tx,Ty,Tz)\le \phi \mathrm{\Omega }(Sx,Sy,Sz)-\psi \mathrm{\Omega }(Sx,Sy,Sz)$$

holds for each $(x,y,z)\in X\times X\times X$.

Suppose the following hypotheses:

• (1) $TX\subseteq SX$.

• (2) If $Su\ne Tu$, then

  $$inf\{\mathrm{\Omega }(Sx,Tx,Su):x\in X\}>0.$$

Then T and S have a unique common fixed point.

As consequent results of Theorem 3.1 and Theorem 3.2, we have the following.

Corollary 3.1 Let $(X,G)$ be a G-metric space and Ω be an Ω-distance on X. Consider $\psi \in \mathrm{\Psi }$ and $T:X\to X$ such that

$$\mathrm{\Omega }(Tx,Ty,Tz)\le \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z)$$

holds for each $(x,y,z)\in X\times X\times X$.

Suppose that if $u\ne Tu$, then

$$inf\{\mathrm{\Omega }(x,Tx,u):x\in X\}>0.$$

Then T has a unique fixed point.

Corollary 3.2 Let $(X,G)$ be a G-metric space and Ω be an Ω-distance on X. Consider $\psi \in \mathrm{\Psi }$ and $T,S:X\to X$ such that

$$\mathrm{\Omega }(Tx,Ty,Tz)\le \mathrm{\Omega }(Sx,Sy,Sz)-\psi \mathrm{\Omega }(Sx,Sy,Sz)$$

holds for each $(x,y,z)\in X\times X\times X$.

Suppose the following hypotheses:

• (1) $TX\subseteq SX$.

• (2) If $Su\ne Tu$, then

  $$inf\{\mathrm{\Omega }(Sx,Tx,Su):x\in X\}>0.$$

Then T and S have a unique common fixed point.

In the second part of the section, we introduce and prove the following coincidence coupled fixed point theorem.

Theorem 3.3 Consider $(X,G)$ to be a G-metric space endowed with an Ω-distance called  Ω. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings with the properties $F(X\times X)\subseteq gX$, and gX is a complete subspace of X with respect to the topology induced by G.

Suppose that there exist $\phi \in \mathrm{\Phi }$ and $\psi \in \mathrm{\Psi }$ such that

$$\begin{array}{r}\phi (\mathrm{\Omega }(F(x,y),F(x^{\ast },y^{\ast }),F(z,z^{\ast }))+\mathrm{\Omega }(F(y,x),F(y^{\ast },x^{\ast }),F(z^{\ast },z)))\\ \le \phi (\mathrm{\Omega }(gx,g{x}^{\ast },gz)+\mathrm{\Omega }(gy,g{y}^{\ast },g{z}^{\ast }))-\psi (\mathrm{\Omega }(gx,g{x}^{\ast },gz)+\mathrm{\Omega }(gy,g{y}^{\ast },g{z}^{\ast }))\end{array}$$

(5)

for each $(x,y),(x^{\ast },y^{\ast }),(z,z^{\ast })\in X\times X$.

Additionally, suppose that if $F(u,v)\ne gu$ or $F(v,u)\ne gv$, then

$$inf\{\mathrm{\Omega }(gx,F(x,y),gu)+\mathrm{\Omega }(gy,F(y,x),gv):x,y\in X\}>0.$$

Then F and g have a unique coupled coincidence point $(u,v)$, with $F(u,v)=gu=gv=F(v,u)$.

Proof Let $(x_0,y_0)\in X\times X$. Having in mind that $F(X\times X)\subseteq gX$, for each $n\in \mathbb{N}$, there is a pair $(x_{n+1},y_{n+1})\in X\times X$ such that

$$g{x}_{n+1}=F(x_n,y_n),g{y}_{n+1}=F(y_n,x_n).$$

First, we prove that

$$\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})=0\text{and}\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1})=0.$$

Using inequality (5), we get

$$\begin{array}{r}\phi (\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})+\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1}))\\ =\phi (\mathrm{\Omega }(F(x_{n-1},y_{n-1}),F(x_n,y_n),F(x_n,y_n))\\ +\mathrm{\Omega }(F(y_{n-1},x_{n-1}),F(y_n,x_n),F(y_n,x_n)))\\ \le \phi (\mathrm{\Omega }(g{x}_{n-1},g{x}_n,g{x}_n)+\mathrm{\Omega }(g{y}_{n-1},g{y}_n,g{y}_n))\\ -\psi (\mathrm{\Omega }(g{x}_{n-1},g{x}_n,g{x}_n)+\mathrm{\Omega }(g{y}_{n-1},g{y}_n,g{y}_n))\\ \le \phi (\mathrm{\Omega }(g{x}_{n-1},g{x}_n,g{x}_n)+\mathrm{\Omega }(g{y}_{n-1},g{y}_n,g{y}_n)).\end{array}$$

(6)

Since φ is a nondecreasing function, we obtain

$$\begin{array}{r}\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})+\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1})\\ \le \mathrm{\Omega }(g{x}_{n-1},g{x}_n,g{x}_n)+\mathrm{\Omega }(g{y}_{n-1},g{y}_n,g{y}_n),n\in \mathbb{N},n\ge 1,\end{array}$$

that is, $(\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})+\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1}))$ is a nondecreasing sequence. Denote by $r\ge 0$ its limit.

Letting $n\to +\mathrm{\infty }$ in relation (6), the continuity of φ and the lower semi-continuity of ψ imply

$$\phi r\le \phi r-\underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\psi (\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})+\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1}))\le \phi r-\psi r,$$

which forces $\phi r=0$, that is, $r=0$.

Since Ω takes nonnegative values,

$$\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})=0\text{and}\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1})=0.$$

A similar procedure leads us to

$$\begin{array}{c}\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{x}_{n+1},g{x}_n,g{x}_n)=0,\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{y}_{n+1},g{y}_n,g{y}_n)=0;\hfill \\ \underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{x}_n,g{x}_n,g{x}_{n+1})=0,\underset{n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{y}_n,g{y}_n,g{y}_{n+1})=0.\hfill \end{array}$$

Now, our purpose is to show that

$$\underset{m,n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_m)=0\text{and}\underset{m,n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_m)=0,m>n.$$

Supposing the contrary, there exist $\epsilon >0$ and two subsequences $(n_k)$ and $(m_k)$ for which

$$\begin{array}{c}\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k})\ge \epsilon ,\hfill \\ \mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k-1},g{x}_{m_k-1})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k-1},g{y}_{m_k-1})<\epsilon ,m_k>n_k.\hfill \end{array}$$

We obtain

$$\begin{array}{rcl}\epsilon & \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k})\\ \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k-1},g{x}_{m_k-1})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k-1},g{y}_{m_k-1})\\ +\mathrm{\Omega }(g{x}_{m_k-1},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{m_k-1},g{y}_{m_k},g{y}_{m_k})\\ <& \epsilon +\mathrm{\Omega }(g{x}_{m_k-1},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{m_k-1},g{y}_{m_k},g{y}_{m_k}).\end{array}$$

As $k\to +\mathrm{\infty }$ and ${lim}_{n\to +\mathrm{\infty }}(\mathrm{\Omega }(g{x}_n,g{x}_{n+1},g{x}_{n+1})+\mathrm{\Omega }(g{y}_n,g{y}_{n+1},g{y}_{n+1}))=0$, we get

$$\underset{k\to +\mathrm{\infty }}{lim}(\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k}))=0.$$

Also, using the properties of Ω, we have

$$\begin{array}{rl}\epsilon \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k})\\ \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{n_k+1},g{x}_{n_k+1})+\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k},g{x}_{m_k})\\ +\mathrm{\Omega }(g{y}_{n_k},g{y}_{n_k+1},g{y}_{n_k+1})+\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k},g{y}_{m_k})\\ \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{n_k+1},g{x}_{n_k+1})+\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k+1},g{x}_{m_k+1})\\ +\mathrm{\Omega }(g{x}_{m_k+1},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{n_k+1},g{y}_{n_k+1})\\ +\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k+1},g{y}_{m_k+1})+\mathrm{\Omega }(g{y}_{m_k+1},g{y}_{m_k},g{y}_{m_k}).\end{array}$$

(7)

Taking advantage of the contraction condition, it follows

$$\begin{array}{c}\phi (\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k+1},g{x}_{m_k+1})+\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k+1},g{y}_{m_k+1}))\hfill \\ \le \phi (\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k}))\hfill \\ -\psi (\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k}))\hfill \\ \le \phi (\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k})).\hfill \end{array}$$

Hence

$$\begin{array}{c}\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k+1},g{x}_{m_k+1})+\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k+1},g{y}_{m_k+1})\hfill \\ \le \mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k}),\hfill \end{array}$$

and relation (7) becomes

$$\begin{array}{rcl}\epsilon & \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{n_k+1},g{x}_{n_k+1})+\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k+1},g{x}_{m_k+1})\\ +\mathrm{\Omega }(g{x}_{m_k+1},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{n_k+1},g{y}_{n_k+1})\\ +\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k+1},g{y}_{m_k+1})+\mathrm{\Omega }(g{y}_{m_k+1},g{y}_{m_k},g{y}_{m_k})\\ \le & \mathrm{\Omega }(g{x}_{n_k},g{x}_{n_k+1},g{x}_{n_k+1})+\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})\\ +\mathrm{\Omega }(g{x}_{m_k+1},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{n_k+1},g{y}_{n_k+1})\\ +\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k})+\mathrm{\Omega }(g{y}_{m_k+1},g{y}_{m_k},g{y}_{m_k}).\end{array}$$

For $k\to +\mathrm{\infty }$, ${lim}_{k\to +\mathrm{\infty }}(\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k+1},g{x}_{m_k+1})+\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k+1},g{y}_{m_k+1}))=\epsilon$.

The properties of φ, ψ lead us to

$$\begin{array}{rcl}\phi \epsilon & =& \underset{k\to +\mathrm{\infty }}{lim}\phi (\mathrm{\Omega }(g{x}_{n_k+1},g{x}_{m_k+1},g{x}_{m_k+1})+\mathrm{\Omega }(g{y}_{n_k+1},g{y}_{m_k+1},g{y}_{m_k+1}))\\ \le & \phi \epsilon -\underset{k\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\psi (\mathrm{\Omega }(g{x}_{n_k},g{x}_{m_k},g{x}_{m_k})+\mathrm{\Omega }(g{y}_{n_k},g{y}_{m_k},g{y}_{m_k}))\le \phi \epsilon -\psi \epsilon .\end{array}$$

Since $\epsilon >0$, we obtain a contradiction. Therefore, ${lim}_{m,n\to +\mathrm{\infty }}\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_m)=0$ and ${lim}_{m,n\to +\mathrm{\infty }}\mathrm{\Omega }(g{y}_n,g{y}_m,g{y}_m)=0$, $m>n$.

Analogously, it can be proved that ${lim}_{m,n\to +\mathrm{\infty }}\mathrm{\Omega }(g{x}_n,g{x}_n,g{x}_m)=0$ and also

$$\underset{m,n\to +\mathrm{\infty }}{lim}\mathrm{\Omega }(g{y}_n,g{y}_n,g{y}_m)=0,m>n.$$

Consider $l>m>n$. Then

$$\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_l)\le \mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_m)+\mathrm{\Omega }(g{x}_m,g{x}_m,g{x}_l)\to 0\text{as }n,m,l\to +\mathrm{\infty }.$$

By Lemma 2.1, we get ${lim}_{n,m,l\to +\mathrm{\infty }}\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_l)=0$, $l>m>n$. Hence, $(g{x}_n)$ is a G-Cauchy sequence in gX, which is complete. Similarly, $(g{y}_n)$ converges in gX. Let $gu={lim}_{n\to +\mathrm{\infty }}g{x}_n$ and $gv={lim}_{n\to +\mathrm{\infty }}g{y}_n$, $u,v\in X$.

Let us show now that $(u,v)$ is a coupled coincidence point of F and g. In that respect, consider $\epsilon >0$. Since $(g{x}_n)$ is a Cauchy sequence, then there exists $n_0\in \mathbb{N}$ such that for each $n,m,l\ge n_0$, $\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_l)<\epsilon$. The properties of lower semi-continuity of Ω imply

$$\mathrm{\Omega }(g{x}_n,g{x}_m,gu)\le \underset{p\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\mathrm{\Omega }(g{x}_n,g{x}_m,g{x}_p)\le \epsilon ,$$

(8)

$$\mathrm{\Omega }(g{y}_n,g{y}_m,gv)\le \underset{p\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\mathrm{\Omega }(g{y}_n,g{y}_m,g{y}_p)\le \epsilon .$$

(9)

Considering $m=n+1$ in (8) and (9), we obtain

$$\mathrm{\Omega }(g{x}_n,F(x_n,y_n),gu)+\mathrm{\Omega }(g{y}_n,F(y_n,x_n),gv)\le 2\epsilon .$$

On the other hand, we get

$$\begin{array}{rcl}0& <& inf\{\mathrm{\Omega }(gx,F(x,y),gu)+\mathrm{\Omega }(gy,F(y,x),gv):x,y\in X\}\\ \le & inf\{\mathrm{\Omega }(g{x}_n,F(x_n,y_n),gu)+\mathrm{\Omega }(g{y}_n,F(y_n,x_n),gv):n\ge n_0\}\le 2\epsilon ,\end{array}$$

which is a contradiction.

Therefore, $F(u,v)=gu$ and $F(v,u)=gv$.

In the following, we refer to the uniqueness of the coupled coincidence point of F and g.

Consider $(u,v)$ and $(u^{\ast },v^{\ast })$ to be two coupled coincidence points of F and g.

By using the contraction condition, we obtain

$$\begin{array}{c}\phi (\mathrm{\Omega }(g{u}^{\ast },gu,gu)+\mathrm{\Omega }(g{v}^{\ast },gv,gv))\hfill \\ =\phi (\mathrm{\Omega }(F(u^{\ast },v^{\ast }),F(u,v),F(u,v))+\mathrm{\Omega }(F(v^{\ast },u^{\ast }),F(v,u),F(v,u)))\hfill \\ \le \phi (\mathrm{\Omega }(g{u}^{\ast },gu,gu)+\mathrm{\Omega }(g{v}^{\ast },gv,gv))-\psi (\mathrm{\Omega }(g{u}^{\ast },gu,gu)+\mathrm{\Omega }(g{v}^{\ast },gv,gv))\hfill \\ \le \phi (\mathrm{\Omega }(g{u}^{\ast },gu,gu)+\mathrm{\Omega }(g{v}^{\ast },gv,gv)),\hfill \end{array}$$

which leads us to $\psi (\mathrm{\Omega }(g{u}^{\ast },gu,gu)+\mathrm{\Omega }(g{v}^{\ast },gv,gv))=0$, or $\mathrm{\Omega }(g{u}^{\ast },gu,gu)=\mathrm{\Omega }(g{v}^{\ast },gv,gv)=0$.

In a similar manner, we prove that $\mathrm{\Omega }(gu,g{u}^{\ast },gu)=\mathrm{\Omega }(gv,g{v}^{\ast },gv)=0$.

Lemma 2.1 implies that $gu=g{u}^{\ast }$ and $gv=g{v}^{\ast }$.

Having in mind that $gu=F(u,v)$ and $gv=F(v,u)$, we get

$$\begin{array}{c}\phi (\mathrm{\Omega }(gu,gv,gv)+\mathrm{\Omega }(gv,gu,gv))\hfill \\ =\phi (\mathrm{\Omega }(F(u,v),F(v,u),F(v,u))+\mathrm{\Omega }(F(v,u),F(u,v),F(u,v)))\hfill \\ \le \phi (\mathrm{\Omega }(gu,gv,gv)+\mathrm{\Omega }(gv,gu,gv))-\psi (\mathrm{\Omega }(gu,gv,gv)+\mathrm{\Omega }(gv,gu,gv)),\hfill \end{array}$$

hence $\psi (\mathrm{\Omega }(gu,gv,gv)+\mathrm{\Omega }(gv,gu,gv))=0$, or $\mathrm{\Omega }(gu,gv,gv)=0$ and $\mathrm{\Omega }(gv,gu,gv)=0$. Applying Lemma 2.1, it follows that $gu=gv$. □

Taking $g=I{d}_X$, the identity mapping, in Theorem 3.3 we obtain a theorem of coupled fixed points.

Corollary 3.3 Consider $(X,G)$ to be a complete G-metric space endowed with an Ω-distance called Ω. Let $F:X\times X\to X$ be a mapping.

Suppose that there exist $\phi \in \mathrm{\Phi }$ and $\psi \in \mathrm{\Psi }$ such that

$$\begin{array}{c}\phi (\mathrm{\Omega }(F(x,y),F(x^{\ast },y^{\ast }),F(z,z^{\ast }))+\mathrm{\Omega }(F(y,x),F(y^{\ast },x^{\ast }),F(z^{\ast },z)))\hfill \\ \le \phi (\mathrm{\Omega }(x,x^{\ast },z)+\mathrm{\Omega }(y,y^{\ast },z^{\ast }))-\psi (\mathrm{\Omega }(x,x^{\ast },z)+\mathrm{\Omega }(y,y^{\ast },z^{\ast }))\hfill \end{array}$$

holds for each $(x,y),(x^{\ast },y^{\ast }),(z,z^{\ast })\in X\times X$.

Additionally, suppose that if $F(u,v)\ne u$ or $F(v,u)\ne v$, then

$$inf\{\mathrm{\Omega }(x,F(x,y),u)+\mathrm{\Omega }(y,F(y,x),v):x,y\in X\}>0.$$

Then F and g have a unique coupled coincidence point $(u,v)$, with $F(u,v)=u=v=F(v,u)$.

Taking $\phi =i_{[0,+\mathrm{\infty })}$, the identity function, in Theorem 3.3 and Corollary 3.3, we get the following results.

Corollary 3.4 Consider $(X,G)$ to be a G-metric space endowed with an Ω-distance called  Ω. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings with the properties $F(X\times X)\subseteq gX$, and gX is a complete subspace of X with respect to the topology induced by G.

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

$$\begin{array}{c}\mathrm{\Omega }(F(x,y),F(x^{\ast },y^{\ast }),F(z,z^{\ast }))+\mathrm{\Omega }(F(y,x),F(y^{\ast },x^{\ast }),F(z^{\ast },z))\hfill \\ \le \mathrm{\Omega }(gx,g{x}^{\ast },gz)+\mathrm{\Omega }(gy,g{y}^{\ast },g{z}^{\ast })-\psi (\mathrm{\Omega }(gx,g{x}^{\ast },gz)+\mathrm{\Omega }(gy,g{y}^{\ast },g{z}^{\ast }))\hfill \end{array}$$

holds for each $(x,y),(x^{\ast },y^{\ast }),(z,z^{\ast })\in X\times X$.

Additionally, suppose that if $F(u,v)\ne gu$ or $F(v,u)\ne gv$, then

$$inf\{\mathrm{\Omega }(gx,F(x,y),gu)+\mathrm{\Omega }(gy,F(y,x),gv):x,y\in X\}>0.$$

Then F and g have a unique coupled coincidence point $(u,v)$, with $F(u,v)=gu=gv=F(v,u)$.

Corollary 3.5 Consider $(X,G)$ to be a complete G-metric space endowed with an Ω-distance called Ω. Let $F:X\times X\to X$ be a mapping.

Suppose that there exist $\phi \in \mathrm{\Phi }$ and $\psi \in \mathrm{\Psi }$ such that

$$\begin{array}{c}\mathrm{\Omega }(F(x,y),F(x^{\ast },y^{\ast }),F(z,z^{\ast }))+\mathrm{\Omega }(F(y,x),F(y^{\ast },x^{\ast }),F(z^{\ast },z))\hfill \\ \le \mathrm{\Omega }(x,x^{\ast },z)+\mathrm{\Omega }(y,y^{\ast },z^{\ast })-\psi (\mathrm{\Omega }(x,x^{\ast },z)+\mathrm{\Omega }(y,y^{\ast },z^{\ast }))\hfill \end{array}$$

holds for each $(x,y),(x^{\ast },y^{\ast }),(z,z^{\ast })\in X\times X$.

Additionally, suppose that if $F(u,v)\ne u$ or $F(v,u)\ne v$, then

$$inf\{\mathrm{\Omega }(x,F(x,y),u)+\mathrm{\Omega }(y,F(y,x),v):x,y\in X\}>0.$$

Then F and g have a unique coupled coincidence point $(u,v)$, with $F(u,v)=u=v=F(v,u)$.

Corollary 3.6 Consider $(X,G)$ to be a G-metric space endowed with an Ω-distance called  Ω. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings with the properties $F(X\times X)\subseteq gX$, and gX is a complete subspace of X with respect to the topology induced by G.

Suppose that there exists $k\in [0,1)$ such that

$$\begin{array}{c}\mathrm{\Omega }(F(x,y),F(x^{\ast },y^{\ast }),F(z,z^{\ast }))+\mathrm{\Omega }(F(y,x),F(y^{\ast },x^{\ast }),F(z^{\ast },z))\hfill \\ \le k(\mathrm{\Omega }(gx,g{x}^{\ast },gz)+\mathrm{\Omega }(gy,g{y}^{\ast },g{z}^{\ast }))\hfill \end{array}$$

holds for each $(x,y),(x^{\ast },y^{\ast }),(z,z^{\ast })\in X\times X$.

Additionally, suppose that if $F(u,v)\ne gu$ or $F(v,u)\ne gv$, then

$$inf\{\mathrm{\Omega }(gx,F(x,y),gu)+\mathrm{\Omega }(gy,F(y,x),gv):x,y\in X\}>0.$$

Then F and g have a unique coupled coincidence point $(u,v)$, with $F(u,v)=gu=gv=F(v,u)$.

Proof The proof follows from Corollary 3.4 by defining $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ via $\psi (t)=(1-k)t$. □

Corollary 3.7 Consider $(X,G)$ to be a complete G-metric space endowed with an Ω-distance called Ω. Let $F:X\times X\to X$ be a mapping.

Suppose that there exists $k\in [0,1)$ such that

$$\begin{array}{c}\mathrm{\Omega }(F(x,y),F(x^{\ast },y^{\ast }),F(z,z^{\ast }))+\mathrm{\Omega }(F(y,x),F(y^{\ast },x^{\ast }),F(z^{\ast },z))\hfill \\ \le k(\mathrm{\Omega }(x,x^{\ast },z)+\mathrm{\Omega }(y,y^{\ast },z^{\ast }))\hfill \end{array}$$

holds for each $(x,y),(x^{\ast },y^{\ast }),(z,z^{\ast })\in X\times X$.

Additionally, suppose that if $F(u,v)\ne u$ or $F(v,u)\ne v$, then

$$inf\{\mathrm{\Omega }(x,F(x,y),u)+\mathrm{\Omega }(y,F(y,x),v):x,y\in X\}>0.$$

Then F and g have a unique coupled coincidence point $(u,v)$, with $F(u,v)=u=v=F(v,u)$.

Proof The proof follows from Corollary 3.5 by defining $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ via $\psi (t)=(1-k)t$. □

The following example supports our results.

Example 3.1 Take $X=\{0,1,2,3,\dots \}$. Define $G:X\times X\times X\to [0,+\mathrm{\infty })$ by the formula

$$G(x,y,z)=\{\begin{array}{cc}0\hfill & \text{if }x=y=z;\hfill \\ x+y+z\hfill & \text{if }x\ne y,\text{ or }x\ne z,\text{ or }y\ne z.\hfill \end{array}$$

Define

$$\mathrm{\Omega }:X\times X\times X\to X,\mathrm{\Omega }(x,y,z)=x+2max\{y,z\}$$

and

$$T:X\to X,Tx=\{\begin{array}{cc}0\hfill & \text{if }x=0,1;\hfill \\ x-1\hfill & \text{if }x\ge 2.\hfill \end{array}$$

Also, define $\phi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ via $\phi (t)=t^2$ and $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ via $\psi (t)=t$. Then:

• (1) $(X,G)$ is a complete G-metric space.

• (2) $\phi \in \mathrm{\Phi }$ and $\psi \in \mathrm{\Psi }$.

• (3) Ω is an Ω-distance function.

• (4) If $u\ne Tu$, then

  $$inf\{\mathrm{\Omega }(x,Tx,u):x\in X\}>0.$$

• (5) The following inequality:

  $$\phi \mathrm{\Omega }(Tx,Ty,Tz)\le \phi \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z)$$

holds for all $x,y,z\in X$.

Proof The proofs of (1) and (2) are clear. To prove part (3), consider $x,y,z,a\in X$. Since

$$x+2max\{y,z\}\le x+2a+a+2max\{y,z\},$$

we get

$$\mathrm{\Omega }(x,y,z)\le \mathrm{\Omega }(x,a,a)+\mathrm{\Omega }(a,y,z).$$

This finishes the proof of the first item of the definition of Ω-distance.

To prove the second item of the definition of Ω-distance, let $x,y\in X$ and $(z_n)$ be any sequence in X converging to z with respect to the topology induced by G in X. Thus $z_n=z$ for all $n\in \mathbb{N}$ except finitely many terms. Therefore

$$x+2max\{y,z_n\}\to x+2max\{y,x\}\text{as }n\to +\mathrm{\infty }.$$

So, $\mathrm{\Omega }(x,y,z_n)\to \mathrm{\Omega }(x,y,z)$ and hence $\mathrm{\Omega }(x,y,\cdot ):X\to [0,+\mathrm{\infty })$ is lower semi-continuous.

Similarly, we can show that $\mathrm{\Omega }(x,\cdot ,z):X\to [0,+\mathrm{\infty })$ is lower semi-continuous.

To prove the last item of the definition of Ω-distance, consider $\epsilon >0$. Take $\delta =\frac{\epsilon }{2}$. Given $x,y,z\in X$ such that $\mathrm{\Omega }(x,a,a)\le \delta$ and $\mathrm{\Omega }(a,y,z)\le \delta$, by the definition of a G-metric space, we have

$$\begin{array}{rcl}G(x,y,z)& \le & G(x,a,a)+G(a,y,z)\\ \le & x+2a+a+y+z\\ \le & x+2a+a+2max\{y,z\}\\ =& \mathrm{\Omega }(x,a,a)+\mathrm{\Omega }(a,y,z)\\ \le & \epsilon .\end{array}$$

This completes the proof of an Ω-distance.

To prove part (4), given $u\in X$ such that $u\ne Tu$, then $u\ne 0$. Note that

$$\begin{array}{c}inf\{\mathrm{\Omega }(x,Tx,u):x\in X\}\hfill \\ \ge inf\{x+2u+:x\in X\}\hfill \\ \ge 2u>0.\hfill \end{array}$$

To prove part (5), given $x,y,z\in X$, we divide the proof into the following four cases.

Case 1: $x=y=z=0$. Here, $\mathrm{\Omega }(x,y,z)=0$ and $\mathrm{\Omega }(Tx,Ty,Tz)=0$. Thus

$$\phi \mathrm{\Omega }(Tx,Ty,Tz)\le \phi \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z).$$

Case 2: $x>0$ and $y=z=0$. Here, $\mathrm{\Omega }(x,y,z)=x$ and $\mathrm{\Omega }(Tx,Ty,Tz)=x-1$. Since ${(x-1)}^2\le x^2-x$, we have

$$\phi \mathrm{\Omega }(Tx,Ty,Tz)\le \phi \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z).$$

Case 3: $x=0$ and y or z are not equal to 0. Without loss of generality, we may assume that $y\ge z$. Thus $y\ne 0$. Here, $\mathrm{\Omega }(x,y,z)=2y$ and $\mathrm{\Omega }(Tx,Ty,Tz)=2(y-1)$. Since $4{(y-1)}^2\le 4y^2-2y$, we have

$$\phi \mathrm{\Omega }(Tx,Ty,Tz)\le \phi \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z).$$

Case 4: x, y, z are all different from 0. Without loss of generality, we assume that $y\ge z$. Then $\mathrm{\Omega }(x,y,z)=x+2y$ and $\mathrm{\Omega }(Tx,Ty,Tz)=x-1+2(y-1)$. Since ${(x-1)}^2\le x^2-x$ and $4{(y-1)}^2\le 4y^2-2y$, we have

$$\begin{array}{rcl}\phi \mathrm{\Omega }(Tx,Ty,Tz)& =& {[x-1+2(y-1)]}^2\\ =& {(x-1)}^2+4(x-1)(y-1)+4{(y-1)}^2\\ \le & x^2-x+4xy+4y^2-2y\\ =& {(x+2y)}^2-(x+2y)\\ =& \phi \mathrm{\Omega }(x,y,z)-\psi \mathrm{\Omega }(x,y,z).\end{array}$$

Note that Example 3.1 satisfies all the hypotheses of Theorem 3.1. Thus T has a unique fixed point. Here, 0 is the unique fixed point of T. □