Existence of a tripled coincidence point in ordered $G_b$-metric spaces and applications to a system of integral equations

Abstract

In this paper, tripled coincidence points of mappings satisfying some nonlinear contractive conditions in the framework of partially ordered $G_b$-metric spaces are obtained. Our results extend the results of Aydi et al. (Fixed Point Theory Appl., 2012:101, 2012, doi:10.1186/1687-1812-2012-101). Moreover, some examples of the main result are given. Finally, some tripled coincidence point results for mappings satisfying some contractive conditions of integral type in complete partially ordered $G_b$-metric spaces are deduced.

MSC: 47H10, 54H25.

1 Introduction and preliminaries

The concepts of mixed monotone mapping and coupled fixed point were introduced in [1] by Bhaskar and Lakshmikantham. Also, they established some coupled fixed point theorems for a mixed monotone mapping in partially ordered metric spaces. For more details on coupled fixed point theorems and related topics in different metric spaces, we refer the reader to [2–13] and [14–25].

Also, Berinde and Borcut [26] introduced a new concept of tripled fixed point and obtained some tripled fixed point theorems for contractive-type mappings in partially ordered metric spaces. For a survey of tripled fixed point theorems and related topics, we refer the reader to [26–32].

Definition 1.1 [26]

An element $(x,y,z)\in X^3$ is called a tripled fixed point of $F:X^3\to X$ if $F(x,y,z)=x$, $F(y,x,y)=y$ and $F(z,y,x)=z$.

Definition 1.2 [27]

An element $(x,y,z)\in X^3$ is called a tripled coincidence point of the mappings $F:X^3\to X$ and $g:X\to X$ if $F(x,y,z)=g(x)$, $F(y,x,y)=gy$ and $F(z,y,x)=gz$.

Definition 1.3 [27]

An element $(x,y,z)\in X^3$ is called a tripled common fixed point of $F:X^3\to X$ and $g:X\to X$ if $x=g(x)=F(x,y,z)$, $y=g(y)=F(y,x,y)$ and $z=g(z)=F(z,y,x)$.

Definition 1.4 [29]

Let X be a nonempty set. We say that the mappings $F:X^3\to X$ and $g:X\to X$ are commutative if $g(F(x,y,z))=F(gx,gy,gz)$ for all $x,y,z\in X$.

The notion of altering distance function was introduced by Khan et al. [10] as follows.

Definition 1.5 The function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is called an altering distance function if

• 1. ψ is continuous and nondecreasing.

• 2. $\psi (t)=0$ if and only if $t=0$.

The concept of generalized metric space, or G-metric space, was introduced by Mustafa and Sims [33]. Mustafa and others studied several fixed point theorems for mappings satisfying different contractive conditions (see [33–45]).

Definition 1.6 (G-metric space, [33])

Let X be a nonempty set and $G:X^3\to R^{+}$ be a function satisfying the following properties:

• (G1) $G(x,y,z)=0$ iff $x=y=z$;

• (G2) $0<G(x,x,y)$ for all $x,y?X$ with $x?y$;

• (G3) $G(x,x,y)=G(x,y,z)$ for all $x,y,z?X$ with $z?y$;

• (G4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=?$ (symmetry in all three variables);

• (G5) $G(x,y,z)=G(x,a,a)+G(a,y,z)$ for all $x,y,z,a?X$ (rectangle inequality).

Then the function G is called a G-metric on X and the pair $(X,G)$ is called a G-metric space.

Example 1.7 If we think that $G(x,y,z)$ is measuring the perimeter of the triangle with vertices at x, y and z, then (G5) can be interpreted as

$$[x,y]+[x,z]+[y,z]\le 2[x,a]+[a,y]+[a,z]+[y,z],$$

where $[x,y]$ is the ‘length’ of the side x, y. If we take $y=z$, we have

$$2[x,y]\le 2[x,a]+2[a,y].$$

Thus, (G5) embodies the triangle inequality. And so (G5) can be sharp.

In [46], Aydi et al. established some tripled coincidence point results for mappings $F:X^3\to X$ and $g:X\to X$ involving nonlinear contractions in the setting of ordered G-metric spaces.

Theorem 1.8 [46]

Let $(X,⪯)$ be a partially ordered set and $(X,G)$ be a G-metric space such that $(X,G)$ is G-complete. Let $F:X^3\to X$ and $g:X\to X$. Assume that there exist $\psi ,\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that ψ is an altering distance function and ϕ is a lower-semicontinuous and nondecreasing function with $\varphi (t)=0$ if and only if $t=0$ and for all $x,y,z,u,v,w,r,s,t\in X$, with $gx⪯gu⪯gr$, $gy⪰gv⪰gs$ and $gz⪯gw⪯gt$, we have

$$\begin{array}{r}\psi \left(G(F(x,y,z),F(u,v,w),F(r,s,t))\right)\\ \le \psi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\})\\ -\varphi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}).\end{array}$$

Assume that F and g satisfy the following conditions:

• (1) $F(X^3)\subseteq g(X)$,

• (2) F has the mixed g-monotone property,

• (3) F is continuous,

• (4) g is continuous and commutes with F.

Let there exist $x_0,y_0,z_0\in X$ such that $g{x}_0⪯F(x_0,y_0,z_0)$, $g{y}_0⪰F(y_0,x_0,y_0)$ and $g{z}_0⪯F(z_0,y_0,x_0)$. Then F and g have a tripled coincidence point in X, i.e., there exist $x,y,z\in X$ such that $F(x,y,z)=gx$, $F(y,x,y)=gy$ and $F(z,y,x)=gz$.

Also, they proved that the above theorem is still valid for F not necessarily continuous assuming the following hypothesis (see Theorem 19 of [46]).

• (I) If $\{x_n\}$ is a nondecreasing sequence with $x_n\to x$, then $x_n⪯x$ for all $n\in \mathbb{N}$.

• (II) If $\{y_n\}$ is a nonincreasing sequence with $y_n\to y$, then $y_n⪰y$ for all $n\in \mathbb{N}$.

A partially ordered G-metric space $(X,G)$ with the above properties is called regular.

In this paper, we obtain some tripled coincidence point theorems for nonlinear $(\psi ,\phi )$-weakly contractive mappings in partially ordered $G_b$-metric spaces. This results generalize and modify several comparable results in the literature. First, we recall the concept of generalized b-metric spaces, or $G_b$-metric spaces.

Definition 1.9 [47]

Let X be a nonempty set and $s\ge 1$ be a given real number. Suppose that a mapping $G:X^3\to {\mathbb{R}}^{+}$ satisfies:

• (G _b 1) $G(x,y,z)=0$ if $x=y=z$,

• (G _b 2) $0<G(x,x,y)$ for all $x,y?X$ with $x?y$,

• (G _b 3) $G(x,x,y)=G(x,y,z)$ for all $x,y,z?X$ with $y?z$,

• (G _b 4) $G(x,y,z)=G(p\{x,y,z\})$, where p is a permutation of x, y, z (symmetry),

• (G _b 5) $G(x,y,z)=s[G(x,a,a)+G(a,y,z)]$ for all $x,y,z,a?X$ (rectangle inequality).

Then G is called a generalized b-metric and the pair $(X,G)$ is called a generalized b-metric space or a $G_b$-metric space.

Obviously, each G-metric space is a $G_b$-metric space with $s=1$. But the following example shows that a $G_b$-metric on X need not be a G-metric on X (see also [48]).

Example 1.10 If we think that $G_b(x,y,z)$ is the maximum of the squares of length sides of a triangle with vertices at x, y and z such that:

If $x\ne y\ne z$, then $G_b(x,y,z)=max\{{([x,y])}^2,{([y,z])}^2,{([z,x])}^2\}$.

If $x\ne y=z$, then $G_b(x,y,y)={([x,y])}^2$,

where $[x,y]$ is the ‘length’ of the side x, y. Then it is easy to see that $G_b(x,y,z)$ is a $G_b$ function with $s=2$.

Since by the triangle inequality we have

$$[x,y]\le [x,a]+[a,y],[z,x]\le [z,a]+[a,x],$$

hence

$$\begin{array}{rcl}{G}_b(x,y,z)& =& max\{{([x,y])}^2,{([y,z])}^2,{([z,x])}^2\}\\ \le & max\{{([x,a]+[a,y])}^2,{([y,z])}^2,{([z,a]+[a,x])}^2\}\\ \le & max\{2({([x,a])}^2+{([a,y])}^2),{([y,z])}^2,2({([z,a])}^2+{([a,x])}^2)\}\\ \le & 2{([x,a])}^2+max\{2{([a,y])}^2,{([y,z])}^2,2{([z,a])}^2\}\\ \le & 2{([x,a])}^2+max\{2{([a,y])}^2,2{([y,z])}^2,2{([z,a])}^2\}\\ =& 2(G_b(x,a,a)+G_b(a,y,z)).\end{array}$$

Example 1.11 [47]

Let $(X,G)$ be a G-metric space and $G_{\ast }(x,y,z)=G{(x,y,z)}^p$, where $p>1$ is a real number. Then $G_{\ast }$ is a $G_b$-metric with $s=2^{p-1}$.

Also, in the above example, $(X,G_{\ast })$ is not necessarily a G-metric space. For example, let $X=\mathbb{R}$ and G-metric G be defined by

$$G(x,y,z)=\frac{1}{3}(|x-y|+|y-z|+|x-z|)$$

for all $x,y,z\in \mathbb{R}$ (see [33]). Then $G_{\ast }(x,y,z)=G{(x,y,z)}^2=\frac{1}{9}{(|x-y|+|y-z|+|x-z|)}^2$ is a $G_b$-metric on ℝ with $s=2^{2-1}=2$, but it is not a G-metric on ℝ.

Example 1.12 [47]

Let $X=\mathbb{R}$ and $d(x,y)=|x-y{|}^2$. We know that $(X,d)$ is a b-metric space with $s=2$. Let $G(x,y,z)=d(x,y)+d(y,z)+d(z,x)$, then $(X,G)$ is not a $G_b$-metric space. Indeed, (G _b 3) is not true for $x=0$, $y=2$ and $z=1$. To see this, we have

$$G(0,0,2)=d(0,0)+d(0,2)+d(2,0)=2d(0,2)=8$$

and

$$G(0,2,1)=d(0,2)+d(2,1)+d(1,0)=4+1+1=6.$$

So, $G(0,0,2)>G(0,2,1)$.

However, $G(x,y,z)=max\{d(x,y),d(y,z),d(z,x)\}$ is a $G_b$-metric on ℝ with $s=2$. Similarly, if $d(x,y)=|x-y{|}^p$ is selected with $p\ge 1$, then $G(x,y,z)=max\{d(x,y),d(y,z),d(z,x)\}$ is a $G_b$-metric on ℝ with $s=2^{p-1}$.

Now we present some definitions and propositions in a $G_b$-metric space.

Definition 1.13 [47]

A $G_b$-metric G is said to be symmetric if $G(x,y,y)=G(y,x,x)$ for all $x,y\in X$.

Definition 1.14 Let $(X,G)$ be a $G_b$-metric space. Then, for $x_0\in X$ and $r>0$, the $G_b$-ball with center $x_0$ and radius r is

$$B_G(x_0,r)=\{y\in X\mid G(x_0,y,y)<r\}.$$

By some straight forward calculations, we can establish the following.

Proposition 1.15 [47]

Let X be a $G_b$-metric space. Then, for each $x,y,z,a\in X$, it follows that:

• (1) if $G(x,y,z)=0$, then $x=y=z$,

• (2) $G(x,y,z)\le s(G(x,x,y)+G(x,x,z))$,

• (3) $G(x,y,y)\le 2sG(y,x,x)$,

• (4) $G(x,y,z)\le s(G(x,a,z)+G(a,y,z))$.

Definition 1.16 [47]

Let X be a $G_b$-metric space. We define $d_G(x,y)=G(x,y,y)+G(x,x,y)$ for all $x,y\in X$. It is easy to see that $d_G$ defines a b-metric d on X, which we call the b-metric associated with G.

Proposition 1.17 [47]

Let X be a $G_b$-metric space. Then, for any $x_0\in X$ and $r>0$, if $y\in B_G(x_0,r)$, then there exists $\delta >0$ such that $B_G(y,\delta )\subseteq B_G(x_0,r)$.

From the above proposition, the family of all $G_b$-balls

$$Ϝ=\{B_G(x,r)\mid x\in X,r>0\}$$

is a base of a topology $\tau (G)$ on X, which we call the $G_b$-metric topology.

Now, we generalize Proposition 5 in [34] for a $G_b$-metric space as follows.

Proposition 1.18 [47]

Let X be a $G_b$-metric space. Then, for any $x_0\in X$ and $r>0$, we have

$$B_G(x_0,\frac{r}{2s+1})\subseteq B_{d_G}(x_0,r)\subseteq B_G(x_0,r).$$

Thus every $G_b$-metric space is topologically equivalent to a b-metric space. This allows us to readily transport many concepts and results from b-metric spaces into $G_b$-metric space setting.

Definition 1.19 [47]

Let X be a $G_b$-metric space. A sequence $\{x_n\}$ in X is said to be:

• (1) $G_b$-Cauchy if for each $\epsilon >0$, there exists a positive integer $n_0$ such that, for all $m,n,l\ge n_0$, $G(x_n,x_m,x_l)<\epsilon$;

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

Proposition 1.20 [47]

Let X be a $G_b$-metric space. Then the following are equivalent:

• (1) the sequence $\{x_n\}$ is $G_b$-Cauchy;

• (2) for any $\epsilon >0$, there exists $n_0\in \mathbb{N}$ such that $G(x_n,x_m,x_m)<\epsilon$ for all $m,n\ge n_0$.

Proposition 1.21 [47]

Let X be a $G_b$-metric space. The following are equivalent:

• (1) $\{x_n\}$ is $G_b$-convergent to x;

• (2) $G(x_n,x_n,x)\to 0$ as $n\to +\mathrm{\infty }$;

• (3) $G(x_n,x,x)\to 0$ as $n\to +\mathrm{\infty }$.

Definition 1.22 [47]

A $G_b$-metric space X is called complete if every $G_b$-Cauchy sequence is $G_b$-convergent in X.

Definition 1.23 [47]

Let $(X,G)$ and $(X^{\prime },G^{\prime })$ be two $G_b$-metric spaces. Then a function $f:X\to X^{\prime }$ is $G_b$-continuous at a point $x\in X$ if and only if it is $G_b$-sequentially continuous at x, that is, whenever $\{x_n\}$ is $G_b$-convergent to x, $\{f(x_n)\}$ is $G_b^{\prime }$-convergent to $f(x)$.

Mustafa and Sims proved that each G-metric function $G(x,y,z)$ is jointly continuous in all three of its variables (see Proposition 8 in [33]). But, in general, a $G_b$-metric function $G(x,y,z)$ for $s>1$ is not jointly continuous in all its variables. Now, we recall an example of a discontinuous $G_b$-metric.

Example 1.24 [49]

Let $X=\mathbb{N}\cup \{\mathrm{\infty }\}$ and let $D:X\times X\to \mathbb{R}$ be defined by

$$D(m,n)=\{\begin{array}{cc}0\hfill & \text{if }m=n,\hfill \\ |\frac{1}{m}-\frac{1}{n}|\hfill & \text{if one of }m,n\text{ is even and the other is even or }\mathrm{\infty },\hfill \\ 5\hfill & \text{if one of }m,n\text{ is odd and the other is odd }(\text{and }m\ne n)\text{ or }\mathrm{\infty },\hfill \\ 2\hfill & \text{otherwise}.\hfill \end{array}$$

Then it is easy to see that for all $m,n,p\in X$, we have

$$D(m,p)\le \frac{5}{2}(D(m,n)+D(n,p)).$$

Thus, $(X,D)$ is a b-metric space with $s=\frac{5}{2}$ (see corrected Example 3 in [9]).

Let $G(x,y,z)=max\{D(x,y),D(y,z),D(z,x)\}$. It is easy to see that G is a $G_b$-metric with $s=5/2$. In [49], it is proved that $G(x,y,z)$ is not a continuous function.

So, from the above discussion, we need the following simple lemma about the $G_b$-convergent sequences in the proof of our main result.

Lemma 1.25 [49]

Let $(X,G)$ be a $G_b$-metric space with $s>1$ and suppose that $\{x_n\}$, $\{y_n\}$ and $\{z_n\}$ are $G_b$-convergent to x, y and z, respectively. Then we have

$$\begin{array}{rl}\frac{1}{s^3}G(x,y,z)& \le \underset{n⟶\mathrm{\infty }}{lim\hspace{0.17em}inf}G(x_n,y_n,z_n)\\ \le \underset{n⟶\mathrm{\infty }}{lim\hspace{0.17em}sup}G(x_n,y_n,z_n)\le s^{3}G(x,y,z).\end{array}$$

In particular, if $x=y=z$, then we have ${lim}_{n⟶\mathrm{\infty }}G(x_n,y_n,z_n)=0$.

In this paper, we present some tripled coincidence point results in ordered $G_b$-metric spaces. Our results extend and generalize the results in [46].

2 Main results

Let $(X,⪯,G)$ be an ordered $G_b$-metric space and $F:X^3\to X$ and $g:X\to X$. In the rest of this paper, unless otherwise stated, for all $x,y,z,u,v,w,r,s,t\in X$, let

$$\begin{array}{rl}{M}_F(x,y,z,u,v,w,r,s,t)=& max\{G(F(x,y,z),F(u,v,w),F(r,s,t)),\\ G(F(y,x,y),F(v,u,v),F(s,r,s)),\\ G(F(z,y,x),F(w,v,u),F(t,s,r))\}\end{array}$$

and

$$M_g(x,y,z,u,v,w,r,s,t)=max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}.$$

Now, the main result is presented as follows.

Theorem 2.1 Let $(X,⪯,G)$ be a partially ordered $G_b$-metric space and $F:X^3\to X$ and $g:X\to X$ be such that $F(X^3)\subseteq g(X)$. Assume that

$$\begin{array}{r}\psi (s{M}_F(x,y,z,u,v,w,r,s,t))\\ \le \psi (M_g(x,y,z,u,v,w,r,s,t))-\phi (M_g(x,y,z,u,v,w,r,s,t))\end{array}$$

(2.1)

for every $x,y,z,u,v,w,r,s,t\in X$ with $gx⪯gu⪯gr$, $gy⪰gv⪰gs$ and $gz⪯gw⪯gt$, or $gr⪯gu⪯gx$, $gs⪰gv⪰gy$ and $gt⪯gw⪯gz$, where $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are altering distance functions.

Assume that

• (1) F has the mixed g-monotone property.

• (2) g is $G_b$-continuous and commutes with F.

Also suppose that

• (a) either F is $G_b$-continuous and $(X,G)$ is $G_b$-complete, or

• (b) $(X,G)$ is regular and $(g(X),G)$ is $G_b$-complete.

If there exist $x_0,y_0,z_0\in X$ such that $g{x}_0⪯F(x_0,y_0,z_0)$, $g{y}_0⪰F(y_0,x_0,y_0)$ and $g{z}_0⪯F(z_0,y_0,x_0)$, then F and g have a tripled coincidence point in X.

Proof Let $x_0,y_0,z_0\in X$ be such that $g{x}_0⪯F(x_0,y_0,z_0)$, $g{y}_0⪰F(y_0,x_0,y_0)$ and $g{z}_0⪯F(z_0,y_0,x_0)$. Define $x_1,y_1,z_1\in X$ such that $g{x}_1=F(x_0,y_0,z_0)$, $g{y}_1=F(y_0,x_0,y_0)$ and $g{z}_1=F(z_0,y_0,x_0)$. Then $g{x}_0⪯g{x}_1$, $g{y}_0⪰g{y}_1$ and $g{z}_0⪯g{z}_1$. Similarly, define $g{x}_2=F(x_1,y_1,z_1)$, $g{y}_2=F(y_1,x_1,y_1)$ and $g{z}_2=F(z_1,y_1,x_1)$. Since F has the mixed g-monotone property, we have $g{x}_0⪯g{x}_1⪯g{x}_2$, $g{y}_0⪰g{y}_1⪰g{y}_2$ and $g{z}_0⪯g{z}_1⪯g{z}_2$.

In this way, we construct the sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ as

$$\begin{array}{c}{a}_n=g{x}_n=F(x_{n-1},y_{n-1},z_{n-1}),\hfill \\ b_n=g{y}_n=F(y_{n-1},x_{n-1},y_{n-1})\hfill \end{array}$$

and

$$c_n=g{z}_n=F(z_{n-1},y_{n-1},x_{n-1})$$

for all $n\ge 1$.

We will finish the proof in two steps.

Step I. We shall show that $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are $G_b$-Cauchy.

Let

$${\delta }_n=max\{G(a_{n-1},a_n,a_n),G(b_{n-1},b_n,b_n),G(c_{n-1},c_n,c_n)\}.$$

So, we have

$${\delta }_n=M_F(x_{n-2},y_{n-2},z_{n-2},x_{n-1},y_{n-1},z_{n-1},x_{n-1},y_{n-1},z_{n-1})$$

and

$${\delta }_n=M_g(x_{n-1},y_{n-1},z_{n-1},x_n,y_n,z_n,x_n,y_n,z_n).$$

As $g{x}_{n-1}⪯g{x}_n$, $g{y}_{n-1}⪰g{y}_n$ and $g{z}_{n-1}⪯g{z}_n$, using (2.1) we obtain that

$$\begin{array}{rcl}\psi (s{\delta }_{n+1})& =& \psi (s{M}_F(x_{n-1},y_{n-1},z_{n-1},x_n,y_n,z_n,x_n,y_n,z_n))\\ \le & \psi (M_g(x_{n-1},y_{n-1},z_{n-1},x_n,y_n,z_n,x_n,y_n,z_n))\\ -\phi (M_g(x_{n-1},y_{n-1},z_{n-1},x_n,y_n,z_n,x_n,y_n,z_n))\\ =& \psi ({\delta }_n)-\phi ({\delta }_n)\\ \le & \psi (s{\delta }_n)-\phi ({\delta }_n).\end{array}$$

(2.2)

Since ψ is an altering distance function, by (2.2) we deduce that

$${\delta }_{n+1}\le {\delta }_n,$$

that is, $\{{\delta }_n\}$ is a nonincreasing sequence of nonnegative real numbers. Thus, there is $r\ge 0$ such that

$$\underset{n\to \mathrm{\infty }}{lim}{\delta }_n=r.$$

Letting $n\to \mathrm{\infty }$ in (2.2), from the continuity of ψ and φ, we obtain that

$$\psi (sr)\le \psi (sr)-\phi (r),$$

which implies that $\phi (r)=0$ and hence $r=0$.

Next, we claim that $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are $G_b$-Cauchy.

We shall show that for every $\epsilon >0$, there exists $k\in \mathbb{N}$ such that if $m,n\ge k$,

$$max\{G(a_m,a_n,a_n),G(b_m,b_n,b_n),G(c_m,c_n,c_n)\}<\epsilon .$$

Suppose that the above statement is false. Then there exists $\epsilon >0$ for which we can find subsequences $\{a_{m(k)}\}$ and $\{a_{n(k)}\}$ of $\{a_n\}$, $\{b_{m(k)}\}$ and $\{b_{n(k)}\}$ of $\{b_n\}$ and $\{c_{m(k)}\}$ and $\{c_{n(k)}\}$ of $\{c_n\}$ such that $n(k)>m(k)>k$ and

$$max\{G(a_{m(k)},a_{n(k)},a_{n(k)}),G(b_{m(k)},b_{n(k)},b_{n(k)}),G(c_{m(k)},c_{n(k)},c_{n(k)})\}\ge \epsilon ,$$

(2.3)

where $n(k)$ is the smallest index with this property, i.e.,

$$max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\}<\epsilon .$$

(2.4)

From (2.4), we have

$$\begin{array}{r}\underset{k⟶\mathrm{\infty }}{lim\hspace{0.17em}sup}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\}\le \epsilon .\end{array}$$

(2.5)

From the rectangle inequality,

$$G(a_{m(k)},a_{n(k)},a_{n(k)})\le s[G(a_{m(k)},a_{n(k)-1},a_{n(k)-1})+G(a_{n(k)-1},a_{n(k)},a_{n(k)})].$$

(2.6)

Similarly,

$$G(b_{m(k)},b_{n(k)},b_{n(k)})\le s[G(b_{m(k)},b_{n(k)-1},b_{n(k)-1})+G(b_{n(k)-1},b_{n(k)},b_{n(k)})]$$

(2.7)

and

$$G(c_{m(k)},c_{n(k)},c_{n(k)})\le s[G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})+G(c_{n(k)-1},c_{n(k)},c_{n(k)})].$$

(2.8)

So,

$$\begin{array}{r}max\{G(a_{m(k)},a_{n(k)},a_{n(k)}),G(b_{m(k)},b_{n(k)},b_{n(k)}),G(c_{m(k)},c_{n(k)},c_{n(k)})\}\\ \le smax\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\}\\ +smax\{G(a_{n(k)-1},a_{n(k)},a_{n(k)}),G(b_{n(k)-1},b_{n(k)},b_{n(k)}),G(c_{n(k)-1},c_{n(k)},c_{n(k)})\}.\end{array}$$

(2.9)

Letting $k\to \mathrm{\infty }$ as ${lim}_{n\to \mathrm{\infty }}{\delta }_n=0$, by (2.3) and (2.4), we can conclude that

$$\begin{array}{rl}\frac{\epsilon }{s}\le & \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\}.\end{array}$$

(2.10)

Since

$$G(a_{m(k)},a_{n(k)},a_{n(k)})\le sG(a_{m(k)},a_{m(k)+1},a_{m(k)+1})+sG(a_{m(k)+1},a_{n(k)},a_{n(k)})$$

(2.11)

and

$$G(b_{m(k)},b_{n(k)},b_{n(k)})\le sG(b_{m(k)},b_{m(k)+1},b_{m(k)+1})+sG(b_{m(k)+1},b_{n(k)},b_{n(k)}),$$

(2.12)

and

$$G(c_{m(k)},c_{n(k)},c_{n(k)})\le sG(c_{m(k)},c_{m(k)+1},c_{m(k)+1})+sG(c_{m(k)+1},c_{n(k)},c_{n(k)}),$$

(2.13)

we obtain that

$$\begin{array}{r}max\{G(a_{m(k)},a_{n(k)},a_{n(k)}),G(b_{m(k)},b_{n(k)},b_{n(k)}),G(c_{m(k)},c_{n(k)},c_{n(k)})\}\\ \le smax\{G(a_{m(k)},a_{m(k)+1},a_{m(k)+1}),G(b_{m(k)},b_{m(k)+1},b_{m(k)+1}),G(c_{m(k)},c_{m(k)+1},c_{m(k)+1})\}\\ +smax\{G(a_{m(k)+1},a_{n(k)},a_{n(k)}),G(b_{m(k)+1},b_{n(k)},b_{n(k)}),\\ G(c_{m(k)+1},c_{n(k)},c_{n(k)})\}.\end{array}$$

(2.14)

If in the above inequality $k\to \mathrm{\infty }$ as ${lim}_{n\to \mathrm{\infty }}{\delta }_n=0$, from (2.3) we have

$$\begin{array}{rl}\frac{\epsilon }{s}\le & \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}max\{G(a_{m(k)+1},a_{n(k)},a_{n(k)}),G(b_{m(k)+1},b_{n(k)},b_{n(k)}),\\ G(c_{m(k)+1},c_{n(k)},c_{n(k)})\}.\end{array}$$

(2.15)

As $n(k)>m(k)$, we have $g{x}_{m(k)}⪯g{x}_{n(k)-1}$, $g{y}_{m(k)}⪰g{y}_{n(k)-1}$ and $g{z}_{m(k)}⪯g{z}_{n(k)-1}$. Putting $x=x_{m(k)}$, $y=y_{m(k)}$, $z=z_{m(k)}$, $u=x_{n(k)-1}$, $v=y_{n(k)-1}$, $w=z_{n(k)-1}$, $r=x_{n(k)-1}$, $s=y_{n(k)-1}$ and $t=z_{n(k)-1}$ in (2.1), we have

$$\begin{array}{r}\psi (s\cdot max\{G(a_{m(k)+1},a_{n(k)},a_{n(k)}),G(b_{m(k)+1},b_{n(k)},b_{n(k)}),G(c_{m(k)+1},c_{n(k)},c_{n(k)})\})\\ =\psi (s\cdot M_F(x_{m(k)},y_{m(k)},z_{m(k)},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1}))\\ \le \psi (M_g(x_{m(k)},y_{m(k)},z_{m(k)},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1}))\\ -\phi (M_g(x_{m(k)},y_{m(k)},z_{m(k)},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1}))\\ =\psi (max\{G(g{x}_{m(k)},g{x}_{n(k)-1},g{x}_{n(k)-1}),G(g{y}_{m(k)},g{y}_{n(k)-1},g{y}_{n(k)-1}),\\ G(g{z}_{m(k)},g{z}_{n(k)-1},g{z}_{n(k)-1})\left\}\right)\\ -\phi (max\{G(g{x}_{m(k)},g{x}_{n(k)-1},g{x}_{n(k)-1}),G(g{y}_{m(k)},g{y}_{n(k)-1},g{y}_{n(k)-1}),\\ G(g{z}_{m(k)},g{z}_{n(k)-1},g{z}_{n(k)-1})\left\}\right)\\ =\psi (max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\left\}\right)\\ -\phi (max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\left\}\right).\end{array}$$

(2.16)

Letting $k\to \mathrm{\infty }$ in (2.16),

$$\begin{array}{rcl}\psi (s\cdot \frac{\epsilon }{s})& \le & \psi (s\cdot \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}max\{G(a_{m(k)+1},a_{n(k)},a_{n(k)}),G(b_{m(k)+1},b_{n(k)},b_{n(k)}),\\ G(c_{m(k)+1},c_{n(k)},c_{n(k)})\left\}\right)\\ \le & \psi (\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\left\}\right)\\ -\phi (\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\left\}\right)\\ \le & \psi (\epsilon )-\phi (\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\left\}\right).\end{array}$$

(2.17)

From (2.17), we have

$$\begin{array}{r}\phi (\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),\\ G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\left\}\right)\le 0.\end{array}$$

Therefore,

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}max\{G(a_{m(k)},a_{n(k)-1},a_{n(k)-1}),G(b_{m(k)},b_{n(k)-1},b_{n(k)-1}),G(c_{m(k)},c_{n(k)-1},c_{n(k)-1})\}=0,$$

which is a contradiction to (2.10). Consequently, $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are $G_b$-Cauchy.

Step II. We shall show that F and g have a tripled coincidence point.

First, let (a) hold, that is, F is $G_b$-continuous and $(X,G)$ is $G_b$-complete.

Since X is $G_b$-complete and $\{a_n\}$ is $G_b$-Cauchy, there exists $a\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}G(a_n,a_n,a)=\underset{n\to \mathrm{\infty }}{lim}G(g{x}_n,g{x}_n,a)=0.$$

(2.18)

Similarly, there exist $b,c\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}G(b_n,b_n,b)=\underset{n\to \mathrm{\infty }}{lim}G(g{y}_n,g{y}_n,b)=0$$

(2.19)

and

$$\underset{n\to \mathrm{\infty }}{lim}G(c_n,c_n,c)=\underset{n\to \mathrm{\infty }}{lim}G(g{z}_n,g{z}_n,c)=0.$$

(2.20)

Now, we prove that $(a,b,c)$ is a tripled coincidence point of F and g.

Continuity of g and Lemma 1.25 yields that

$$\begin{array}{rcl}0& =& \frac{1}{s^3}G(ga,ga,ga)\le lim\underset{n\to \mathrm{\infty }}{inf}G(g(g{x}_n),g(g{x}_n),ga)\\ \le & lim\underset{n\to \mathrm{\infty }}{sup}G(g(g{x}_n),g(g{x}_n),ga)\le s^{3}G(ga,ga,ga)=0.\end{array}$$

Hence,

$$\underset{n\to \mathrm{\infty }}{lim}G(g(g{x}_n),g(g{x}_n),ga)=0$$

(2.21)

and similarly,

$$\underset{n\to \mathrm{\infty }}{lim}G(g(g{y}_n),g(g{y}_n),gb)=0$$

(2.22)

and

$$\underset{n\to \mathrm{\infty }}{lim}G(g(g{z}_n),g(g{z}_n),gc)=0.$$

(2.23)

Since $g{x}_{n+1}=F(x_n,y_n,z_n)$, $g{y}_{n+1}=F(y_n,x_n,y_n)$ and $g{z}_{n+1}=F(z_n,y_n,x_n)$, the commutativity of F and g yields that

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

(2.24)

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

(2.25)

and

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

(2.26)

From the continuity of F and (2.24), (2.25) and (2.26) and Lemma 1.25, $\{g(g{x}_{n+1})\}$ is $G_b$-convergent to $F(a,b,c)$, $\{g(g{y}_{n+1})\}$ is $G_b$-convergent to $F(b,a,b)$ and $\{g(g{z}_{n+1})\}$ is $G_b$-convergent to $F(c,b,a)$. From (2.21), (2.22) and (2.23) and uniqueness of the limit, we have $F(a,b,c)=ga$, $F(b,a,b)=gb$ and $F(c,b,a)=gc$, that is, g and F have a tripled coincidence point.

In what follows, suppose that assumption (b) holds.

Following the proof of the previous step, there exist $u,v,w\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}G(g{x}_n,g{x}_n,gu)=0,$$

(2.27)

$$\underset{n\to \mathrm{\infty }}{lim}G(g{y}_n,g{y}_n,gv)=0$$

(2.28)

and

$$\underset{n\to \mathrm{\infty }}{lim}G(g{z}_n,g{z}_n,gw)=0,$$

(2.29)

as $(g(X),G)$ is $G_b$-complete.

Now, we prove that $F(u,v,w)=gu$, $F(v,u,v)=gv$ and $F(w,v,u)=gw$. From regularity of X and using (2.1), we have

$$\begin{array}{r}\psi (s{M}_F(x_n,y_n,z_n,u,v,w,u,v,w))\\ \le \psi (max\{G(g{x}_n,gu,gu),G(g{y}_n,gv,gv),G(g{z}_n,gw,gw)\})\\ -\phi (max\{G(g{x}_n,gu,gu),G(g{y}_n,gv,gv),G(g{z}_n,gw,gw)\}).\end{array}$$

(2.30)

As $\{g{x}_n\}$ is $G_b$-convergent to gu, from Lemma 1.25, we have ${lim}_{n\to \mathrm{\infty }}G(g{x}_n,gu,gu)=0$. Analogously, ${lim}_{n\to \mathrm{\infty }}G(g{y}_n,gv,gv)={lim}_{n\to \mathrm{\infty }}G(g{z}_n,gw,gw)=0$.

As ψ and φ are continuous, from (2.30) we have

$$\underset{n\to \mathrm{\infty }}{lim}{M}_F(x_n,y_n,z_n,u,v,w,u,v,w)=0,$$

or, equivalently,

$$\underset{n\to \mathrm{\infty }}{lim}G(g{x}_{n+1},F(u,v,w),F(u,v,w))=0.$$

(2.31)

Similarly,

$$\underset{n\to \mathrm{\infty }}{lim}G(g{y}_{n+1},F(v,u,v),F(v,u,v))=\underset{n\to \mathrm{\infty }}{lim}G(g{z}_{n+1},F(w,v,u),F(w,v,u))=0.$$

(2.32)

On the other hand,

$$\begin{array}{r}G(gu,F(u,v,w),F(u,v,w)\\ \le sG(gu,g{x}_{n+1},g{x}_{n+1})+sG(g{x}_{n+1},F(u,v,w),F(u,v,w).\end{array}$$

(2.33)

Taking the limit when $n\to \mathrm{\infty }$ and using (2.27) and (2.31), we get

$$\begin{array}{rl}G(gu,F(u,v,w),F(u,v,w))\le & s\underset{n\to \mathrm{\infty }}{lim}G(gu,g{x}_{n+1},g{x}_{n+1})\\ +s\underset{n\to \mathrm{\infty }}{lim}G(g{x}_{n+1},F(u,v,w),F(u,v,w)=0,\end{array}$$

(2.34)

that is, $gu=F(u,v,w)$.

Analogously, we can show that $gv=F(v,u,v)$ and $gw=F(w,v,u)$.

Thus, we have proved that g and F have a tripled coincidence point. This completes the proof of the theorem. □

Let

$$M(x,y,z,u,v,w,r,s,t)=max\{G(x,u,r),G(y,v,s),G(z,w,t)\}.$$

Taking $g=I_X$ (the identity mapping on X) in Theorem 2.1, we obtain the following tripled fixed point result.

Corollary 2.2 Let $(X,⪯,G)$ be a $G_b$-complete partially ordered $G_b$-metric space, and let $F:X^3\to X$ be a mapping with the mixed monotone property. Assume that

$$\begin{array}{r}\psi (s{M}_F(x,y,z,u,v,w,r,s,t))\\ \le \psi (M(x,y,z,u,v,w,r,s,t))-\phi (M(x,y,z,u,v,w,r,s,t))\end{array}$$

(2.35)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$, where $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are altering distance functions.

Also suppose that

• (a) either F is $G_b$-continuous, or

• (b) $(X,G)$ is regular.

If there exist $x_0,y_0,z_0\in X$ such that $x_0⪯F(x_0,y_0,z_0)$, $y_0⪰F(y_0,x_0,y_0)$ and $z_0⪯F(z_0,y_0,x_0)$, then F has a tripled fixed point in X.

Taking $\psi (t)=t$ and $\phi (t)=\frac{t^2}{1+t}$ for all $t\in [0,\mathrm{\infty })$ in Corollary 2.2, we obtain the following tripled fixed point result.

Corollary 2.3 Let $(X,⪯,G)$ be a $G_b$-complete partially ordered $G_b$-metric space and $F:X^3\to X$ with the mixed monotone property. Assume that

$$s{M}_F(x,y,z,u,v,w,r,s,t)\le \frac{M(x,y,z,u,v,w,r,s,t)}{1+M(x,y,z,u,v,w,r,s,t)}$$

(2.36)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

• (a) either F is $G_b$-continuous, or

• (b) $(X,G)$ is regular.

If there exist $x_0,y_0,z_0\in X$ such that $x_0⪯F(x_0,y_0,z_0)$, $y_0⪰F(y_0,x_0,y_0)$ and $z_0⪯F(z_0,y_0,x_0)$, then F has a tripled fixed point in X.

Remark 2.4 Theorem 1.8 is a special case of Theorem 2.1.

Remark 2.5 Theorem 2.1 part (a) holds if we replace the commutativity assumption of F and g by compatibility assumption (also see Remark 2.2 of [30]).

The following corollary can be deduced from our previously obtained results.

Corollary 2.6 Let $(X,⪯)$ be a partially ordered set and $(X,G)$ be a $G_b$-complete $G_b$-metric space. Let $F:X^3\to X$ be a mapping with the mixed monotone property such that

$$\begin{array}{rl}\psi (s{M}_F(x,y,z,u,v,w,r,s,t))\le & \psi \left(\frac{G(x,u,r)+G(y,v,s)+G(z,w,t)}{3}\right)\\ -\phi (max\{G(x,u,r),G(y,v,s),G(z,w,t)\})\end{array}$$

(2.37)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

• (a) either F is $G_b$-continuous, or

• (b) $(X,G)$ is regular.

If there exist $x_0,y_0,z_0\in X$ such that $x_0⪯F(x_0,y_0,z_0)$, $y_0⪰F(y_0,x_0,y_0)$ and $z_0⪯F(z_0,y_0,x_0)$, then F has a tripled fixed point in X.

Proof If F satisfies (2.37), then F satisfies (2.35). So, the result follows from Theorem 2.1. □

In Theorem 2.1, if we take $\psi (t)=t$ and $\phi (t)=(1-k)t$ for all $t\in [0,\mathrm{\infty })$, where $k\in [0,1)$, we obtain the following result.

Corollary 2.7 Let $(X,⪯)$ be a partially ordered set and $(X,G)$ be a $G_b$-complete $G_b$-metric space. Let $F:X^3\to X$ be a mapping having the mixed monotone property and

$$M_F(x,y,z,u,v,w,r,s,t)\le \frac{k}{s}M(x,y,z,u,v,w,r,s,t)$$

(2.38)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

• (a) either F is $G_b$-continuous, or

• (b) $(X,G)$ is regular.

If there exist $x_0,y_0,z_0\in X$ such that $x_0⪯F(x_0,y_0,z_0)$, $y_0⪰F(y_0,x_0,y_0)$ and $z_0⪯F(z_0,y_0,x_0)$, then F has a tripled fixed point in X.

Corollary 2.8 Let $(X,⪯)$ be a partially ordered set and $(X,G)$ be a $G_b$-complete $G_b$-metric space. Let $F:X^3\to X$ be a mapping with the mixed monotone property such that

$$M_F(x,y,z,u,v,w,r,s,t)\le \frac{k}{3s}[G(x,u,r)+G(y,v,s)+G(z,w,t)]$$

(2.39)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

• (a) either F is $G_b$-continuous, or

• (b) $(X,G)$ is regular.

If there exist $x_0,y_0,z_0\in X$ such that $x_0⪯F(x_0,y_0,z_0)$, $y_0⪰F(y_0,x_0,y_0)$ and $z_0⪯F(z_0,y_0,x_0)$, then F has a tripled fixed point in X.

Proof If F satisfies (2.39), then F satisfies (2.38). □

Note that if $(X,⪯)$ is a partially ordered set, then we can endow $X^3$ with the following partial order relation:

$$(x,y,z)⪯(u,v,w)⟺x⪯u,y⪰v,z⪯w$$

for all $(x,y,z),(u,v,w)\in X^3$ (see [26]).

In the following theorem, we give a sufficient condition for the uniqueness of the common tripled fixed point (also see, e.g., [4, 46, 50] and [51]).

Theorem 2.9 In addition to the hypotheses of Theorem 2.1, suppose that for every $(x,y,z)$ and $(x^{\ast },y^{\ast },z^{\ast })\in X\times X\times X$, there exists $(u,v,w)\in X^3$ such that $(F(u,v,w),F(v,u,v),F(w,v,u))$ is comparable with $(F(x,y,z),F(y,x,y),F(z,y,x))$ and $(F(x^{\ast },y^{\ast },z^{\ast }),F(y^{\ast },x^{\ast },y^{\ast }),F(z^{\ast },y^{\ast },x^{\ast }))$. Then F and g have a unique common tripled fixed point.

Proof From Theorem 2.1 the set of tripled coincidence points of F and g is nonempty. We shall show that if $(x,y,z)$ and $(x^{\ast },y^{\ast },z^{\ast })$ are tripled coincidence points, that is,

$$g(x)=F(x,y,z),g(y)=F(y,x,y),g(z)=F(z,y,x)$$

and

$$g\left(x^{\ast }\right)=F(x^{\ast },y^{\ast },z^{\ast }),g\left(y^{\ast }\right)=F(y^{\ast },x^{\ast },y^{\ast }),g\left(z^{\ast }\right)=F(z^{\ast },y^{\ast },x^{\ast }),$$

then $gx=g{x}^{\ast }$ and $gy=g{y}^{\ast }$ and $gz=g{z}^{\ast }$.

Choose an element $(u,v,w)\in X^3$ such that $(F(u,v,w),F(v,u,v),F(w,v,u))$ is comparable with

$$(F(x,y,z),F(y,x,y),F(z,y,x))$$

and

$$(F(x^{\ast },y^{\ast },z^{\ast }),F(y^{\ast },x^{\ast },y^{\ast }),F(z^{\ast },y^{\ast },x^{\ast })).$$

Let $u_0=u$, $v_0=v$ and $w_0=w$ and choose $u_1$, $v_1$ and $w_1\in X$ so that $g{u}_1=F(u_0,v_0,w_0)$, $g{v}_1=F(v_0,u_0,v_0)$ and $g{w}_1=F(w_0,v_0,u_0)$. Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences $\{g{u}_n\}$, $\{g{v}_n\}$ and $\{g{w}_n\}$ such that $g{u}_{n+1}=F(u_n,v_n,w_n)$, $g{v}_{n+1}=F(v_n,u_n,v_n)$ and $g{w}_{n+1}=F(w_n,v_n,u_n)$. Since $(gx,gy,gz)=(F(x,y,z),F(y,x,y),F(w,y,x))$ and $(F(u,v,w),F(v,u,v),F(w,v,u))=(g{u}_1,g{v}_1,g{w}_1)$ are comparable, we may assume that $(gx,gy,gz)⪯(g{u}_1,g{v}_1,g{w}_1)$. Then $gx⪯g{u}_1$, $gy⪰g{v}_1$ and $gz⪯g{w}_1$. Using the mathematical induction, it is easy to prove that $gx⪯g{u}_n$, $gy⪰g{v}_n$ and $gz⪯g{w}_n$ for all $n\ge 0$.

Applying (2.1), as $gx⪯g{u}_n$, $gy⪰g{v}_n$ and $gz⪯g{w}_n$, one obtains that

$$\begin{array}{r}\psi (smax\{G(gx,g{u}_{n+1},g{u}_{n+1}),G(gy,g{v}_{n+1},g{v}_{n+1}),G(gz,g{w}_{n+1},g{w}_{n+1})\})\\ =\psi (s{M}_F(x,y,z,u_n,v_n,w_n,u_n,v_n,w_n))\\ \le \psi (M_g(x,y,z,u_n,v_n,w_n,u_n,v_n,w_n))-\phi (M_g(x,y,z,u_n,v_n,w_n,u_n,v_n,w_n))\\ =\psi (max\{G(gx,g{u}_n,g{u}_n),G(gy,g{v}_n,g{v}_n),G(gz,g{w}_n,g{w}_n)\})\\ -\phi (max\{G(gx,g{u}_n,g{u}_n),G(gy,g{v}_n,g{v}_n),G(gz,g{w}_n,g{w}_n)\}).\end{array}$$

(2.40)

From the properties of ψ, we deduce that

$$\{max\{G(gx,g{u}_n,g{u}_n),G(gy,g{v}_n,g{v}_n),G(gz,g{w}_n,g{w}_n)\}\}$$

is nonincreasing.

Hence, if we proceed as in Theorem 2.1, we can show that

$$\underset{n\to \mathrm{\infty }}{lim}max\{G(gx,g{u}_n,g{u}_n),G(gy,g{v}_n,g{v}_n),G(gz,g{w}_n,g{w}_n)\}=0,$$

that is, $\{g{u}_n\}$, $\{g{v}_n\}$ and $\{g{w}_n\}$ are $G_b$-convergent to gx, gy and gz, respectively.

Similarly, we can show that

$$\underset{n\to \mathrm{\infty }}{lim}max\{G(g{x}^{\ast },g{u}_n,g{u}_n),G(g{y}^{\ast },g{v}_n,g{v}_n),G(g{z}^{\ast },g{w}_n,g{w}_n)\}=0,$$

that is, $\{g{u}_n\}$, $\{g{v}_n\}$ and $\{g{w}_n\}$ are $G_b$-convergent to $g{x}^{\ast }$, $g{y}^{\ast }$ and $g{z}^{\ast }$, respectively. Finally, since the limit is unique, $gx=g{x}^{\ast }$, $gy=g{y}^{\ast }$ and $gz=g{z}^{\ast }$.

Since $gx=F(x,y,z)$, $gy=F(y,x,y)$ and $gz=F(z,y,x)$, by commutativity of F and g, we have $g(gx)=g(F(x,y,z))=F(gx,gy,gz)$, $g(gy)=g(F(y,x,y))=F(gy,gx,gy)$ and $g(gz)=g(F(z,y,x))=F(gz,gy,gx)$. Let $gx=a$, $gy=b$ and $g(z)=c$. Then $ga=F(a,b,c)$, $gb=F(b,a,b)$ and $gc=F(c,b,a)$. Thus, $(a,b,c)$ is another tripled coincidence point of F and g. Then $a=gx=ga$, $b=gy=gb$ and $c=gz=gc$. Therefore, $(a,b,c)$ is a tripled common fixed point of F and g.

To prove the uniqueness, assume that $(p,q,r)$ is another tripled common fixed point of F and g. Then $p=gp=F(p,q,r)$, $q=gq=F(q,p,q)$ and $r=gr=F(r,p,q)$. Since $(p,q,r)$ is a tripled coincidence point of F and g, we have $gp=gx$, $gq=gy$ and $gr=gz$. Thus, $p=gp=ga=a$, $q=gq=gb=b$ and $r=gr=gc=c$. Hence, the tripled common fixed point is unique. □

3 Examples

The following examples support our results.

Example 3.1 Let $X=(-\mathrm{\infty },\mathrm{\infty })$ be endowed with the usual ordering and the $G_b$-complete $G_b$-metric

$$G(x,y,z)={(|x-y|+|y-z|+|z-x|)}^2,$$

where $s=2$.

Define $F:X^3\to X$ as

$$F(x,y,z)=\frac{x-2y+4z}{96}$$

for all $x,y,z\in X$ and $g:X\to X$ with $g(x)=\frac{x}{2}$ for all $x\in X$.

Let $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be defined by $\phi (t)=ln(t+1)$, and let $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be defined by $\psi (t)=ln(\frac{4t+4}{t+4})$.

Now, from the fact that for $\alpha ,\beta ,\gamma \ge 0$, ${(\alpha +\beta +\gamma )}^p\le 2^{2p-2}{\alpha }^p+2^{2p-2}{\beta }^p+2^{p-1}{\gamma }^p$, we have

$$\begin{array}{r}\psi \left(sG(F(x,y,z),F(u,v,w),F(r,s,t))\right)\\ =ln\left(\begin{array}{c}2(\frac{1}{96}[|(x-2y+4z)-(u-2v+4w)|]+\frac{1}{96}[|(u-2v+4w)-(r-2s+4t)|]\\ {+\frac{1}{96}[|(r-2s+4t)-(x-2y+4z)|])}^2+1\end{array}\right)\\ \le ln\left(\begin{array}{c}2(\frac{1}{48}|\frac{x}{2}-\frac{u}{2}|+\frac{1}{24}|\frac{y}{2}-\frac{v}{2}|+\frac{1}{12}|\frac{z}{2}-\frac{w}{2}|+\frac{1}{48}|\frac{u}{2}-\frac{r}{2}|+\frac{1}{24}|\frac{v}{2}-\frac{s}{2}|\\ {+\frac{1}{12}|\frac{w}{2}-\frac{t}{2}|+\frac{1}{48}|\frac{r}{2}-\frac{x}{2}|+\frac{1}{24}|\frac{s}{2}-\frac{y}{2}|+\frac{1}{12}|\frac{t}{2}-\frac{z}{2}|)}^2+1\end{array}\right)\\ =ln\left(\begin{array}{c}2(\frac{1}{48}[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|]+\frac{1}{24}[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|]\\ {+\frac{1}{12}[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|])}^2+1\end{array}\right)\\ \le ln\left(\begin{array}{c}\frac{8}{{48}^2}({[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|]}^2+\frac{8}{{24}^2}{[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|]}^2\\ +\frac{4}{{12}^2}{[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|]}^2)+1\end{array}\right)\\ \le ln\left(\begin{array}{c}\frac{1}{12}({[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|]}^2+\frac{1}{12}{[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|]}^2\\ +\frac{1}{12}{[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|]}^2)+1\end{array}\right)\\ \le ln\left(\begin{array}{c}\frac{1}{4}max\{{[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|]}^2,{[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|]}^2,\\ {[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|]}^2\}+1\end{array}\right)\\ \le ln(\frac{1}{4}max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}+1)\\ =ln(max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}+1)\\ -ln\left(\frac{4max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}+4}{max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}+4}\right)\\ =\psi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\})\\ -\phi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}).\end{array}$$

Analogously, we can show that

$$\begin{array}{r}\psi \left(G(F(y,x,y),F(v,u,v),F(s,r,s))\right)\\ \le \psi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\})\\ -\phi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\})\end{array}$$

and

$$\begin{array}{r}\psi \left(G(F(z,y,x),F(w,v,u),F(t,s,r))\right)\\ \le \psi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\})\\ -\phi (max\{G(gx,gu,gr),G(gy,gv,gs),G(gz,gw,gt)\}).\end{array}$$

Thus,

$$\begin{array}{rl}\psi (s{M}_F(x,y,z,u,v,w,r,s,t))\le & \psi (M_g(x,y,z,u,v,w,r,s,t))\\ -\phi (M_g(x,y,z,u,v,w,r,s,t)).\end{array}$$

Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, $(0,0,0)$ is the unique common tripled fixed point of F and g.

The following example has been constructed according to Example 2.12 of [2].

Example 3.2 Let $X=\{(x,0,x)\}\cup \{(0,x,0)\}\subset R^3$, where $x\in [0,\mathrm{\infty }]$ with the order ⪯ defined as

$$(x_1,y_1,z_1)⪯(x_2,y_2,z_2)⟺x_1\le x_2,y_1\le y_2,z_1\le z_2.$$

Let d be given as

$$d(x,y)=max\{{|x_1-x_2|}^2,{|y_1-y_2|}^2,{|z_1-z_2|}^2\}$$

and

$$G(x,y,z)=max\{d(x,y),d(y,z),d(z,x)\},$$

where $x=(x_1,y_1,z_1)$ and $y=(x_2,y_2,z_2)$. $(X,G)$ is, clearly, a $G_b$-complete $G_b$-metric space.

Let $g:X\to X$ and $F:X^3\to X$ be defined as follows:

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

and

$$g((x,0,x))=(0,x,0)\text{and}g((0,x,0))=(x,0,x).$$

Let $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be as in the above example.

According to the order on X and the definition of g, we see that for any element $x\in X$, $g(x)$ is comparable only with itself.

By a careful computation, it is easy to see that all of the conditions of Theorem 2.1 are satisfied. Finally, Theorem 2.1 guarantees the existence of a unique common tripled fixed point for F and g, i.e., the point $((0,0,0),(0,0,0),(0,0,0))$.

4 Applications

In this section, we obtain some tripled coincidence point theorems for a mapping satisfying a contractive condition of integral type in a complete ordered $G_b$-metric space.

We denote by Λ the set of all functions $\mu :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ verifying the following conditions:

• (I) μ is a positive Lebesgue integrable mapping on each compact subset of $[0,+\mathrm{\infty })$.

• (II) For all $\epsilon >0$, ${\int }_0^{\epsilon }\mu (t)dt>0$.

Corollary 4.1 Replace the contractive condition (2.1) of Theorem 2.1 by the following condition:

There exists $\mu \in \mathrm{\Lambda }$ such that

$$\begin{array}{r}{\int }_0^{\psi (s{M}_F(x,y,z,u,v,w,r,s,t))}\mu (t)dt\\ \le {\int }_0^{\psi (M_g(x,y,z,u,v,w,r,s,t))}\mu (t)dt-{\int }_0^{\phi (M_g(x,y,z,u,v,w,r,s,t))}\mu (t)dt.\end{array}$$

(4.1)

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof Consider the function $\mathrm{\Gamma }(x)={\int }_0^x\mu (t)dt$. Then (4.1) becomes

$$\begin{array}{r}\mathrm{\Gamma }\left(\psi (s{M}_F(x,y,z,u,v,w,r,s,t))\right)\\ \le \mathrm{\Gamma }\left(\psi (M_g(x,y,z,u,v,w,r,s,t))\right)-\mathrm{\Gamma }\left(\phi (M_g(x,y,z,u,v,w,r,s,t))\right).\end{array}$$

Taking ${\psi }_1=\mathrm{\Gamma }o\psi$ and ${\phi }_1=\mathrm{\Gamma }o\phi$ and applying Theorem 2.1, we obtain the proof (it is easy to verify that ${\psi }_1$ and ${\phi }_1$ are altering distance functions). □

Corollary 4.2 Substitute the contractive condition (2.1) of Theorem 2.1 by the following condition:

There exists $\mu \in \mathrm{\Lambda }$ such that

$$\begin{array}{r}\psi ({\int }_0^{s{M}_F(x,y,z,u,v,w,r,s,t)}\mu (t)dt)\\ \le \psi ({\int }_0^{M_g(x,y,z,u,v,w,r,s,t)}\mu (t)dt)-\phi ({\int }_0^{M_g(x,y,z,u,v,w,r,s,t)}\mu (t)dt).\end{array}$$

(4.2)

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof Again, as in Corollary 4.1, define the function $\mathrm{\Gamma }(x)={\int }_0^x\mu (t)dt$. Then (4.2) changes to

$$\begin{array}{rl}\psi \left(\mathrm{\Gamma }(s{M}_F(x,y,z,u,v,w,r,s,t))\right)\le & \psi \left(\mathrm{\Gamma }(M_g(x,y,z,u,v,w,r,s,t))\right)\\ -\phi \left(\mathrm{\Gamma }(M_g(x,y,z,u,v,w,r,s,t))\right).\end{array}$$

Now, if we define ${\psi }_1=\psi o\mathrm{\Gamma }$ and ${\phi }_1=\phi o\mathrm{\Gamma }$ and apply Theorem 2.1, then the proof is completed. □

Corollary 4.3 Replace the contractive condition (2.1) of Theorem 2.1 by the following condition:

There exists $\mu \in \mathrm{\Lambda }$ such that

$$\begin{array}{r}{\psi }_1({\int }_0^{{\psi }_2(s{M}_F(x,y,z,u,v,w,r,s,t))}\mu (t)dt)\\ \le {\psi }_1({\int }_0^{{\psi }_2(M_g(x,y,z,u,v,w,r,s,t))}\mu (t)dt)-{\phi }_1({\int }_0^{{\phi }_2(M_g(x,y,z,u,v,w,r,s,t))}\mu (t)dt)\end{array}$$

(4.3)

for altering distance functions ${\psi }_1$, ${\psi }_2$, ${\phi }_1$ and ${\phi }_2$. If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Similar to [52], let $N\in \mathbb{N}$ be fixed. Let ${\{{\mu }_i\}}_{1\le i\le N}$ be a family of N functions which belong to Λ. For all $t\ge 0$, we define

$$\begin{array}{r}{I}_1(t)={\int }_0^t{\mu }_1(s)ds,\\ I_2(t)={\int }_0^{I_{1}t}{\mu }_2(s)ds={\int }_0^{{\int }_0^t{\mu }_1(s)ds}{\mu }_2(s)ds,\\ I_3(t)={\int }_0^{I_{2}t}{\mu }_3(s)ds={\int }_0^{{\int }_0^{{\int }_0^t{\mu }_1(s)ds}{\mu }_2(s)ds}{\mu }_3(s)ds,\\ \cdots ,\\ I_N(t)={\int }_0^{I_{(N-1)}t}{\mu }_N(s)ds.\end{array}$$

We have the following result.

Corollary 4.4 Replace inequality (2.1) of Theorem 2.1 by the following condition:

$$\begin{array}{rl}\psi \left(I_N(s{M}_F(x,y,z,u,v,w,r,s,t))\right)\le & \psi \left(I_N(M_g(x,y,z,u,v,w,r,s,t))\right)\\ -\phi \left(I_N(M_g(x,y,z,u,v,w,r,s,t))\right).\end{array}$$

(4.4)

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof Consider $\hat{\mathrm{\Psi }}=\psi o{I}_N$ and $\hat{\mathrm{\Phi }}=\phi o{I}_N$. Then the above inequality becomes

$$\begin{array}{rl}\hat{\mathrm{\Psi }}(s{M}_F(x,y,z,u,v,w,r,s,t))\le & \hat{\mathrm{\Psi }}(M_g(x,y,z,u,v,w,r,s,t))\\ -\hat{\mathrm{\Phi }}(M_g(x,y,z,u,v,w,r,s,t)).\end{array}$$

Applying Theorem 2.1, we obtain the desired result (it is easy to verify that $\hat{\mathrm{\Psi }}$ and $\hat{\mathrm{\Phi }}$ are altering distance functions). □

Another consequence of the main theorem is the following result.

Corollary 4.5 Substitute contractive condition (2.1) of Theorem 2.1 by the following condition:

There exist ${\mu }_1,{\mu }_2\in \mathrm{\Lambda }$ such that

$$\begin{array}{r}{\int }_0^{s{M}_F(x,y,z,u,v,w,r,s,t)}{\mu }_1(t)dt\\ \le {\int }_0^{M_g(x,y,z,u,v,w,r,s,t)}{\mu }_1(t)dt-{\int }_0^{M_g(x,y,z,u,v,w,r,s,t)}{\mu }_2(t)dt.\end{array}$$

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof It is clear that the function $s\to {\int }_0^s{\mu }_i(t)dt$ for $i=1,2$ is an altering distance function. □

Motivated by [46], we study the existence of solutions for nonlinear integral equations using the results proved in the previous section.

Consider the integral equations in the following system.

$$\begin{array}{r}x(t)=\omega (t)+{\int }_0^{T}S(t,r)[f(r,x(r))+k(r,y(r))+h(r,z(r))]dr,\\ y(t)=\omega (t)+{\int }_0^{T}S(t,r)[f(r,y(r))+k(r,x(r))+h(r,y(r))]dr,\\ z(t)=\omega (t)+{\int }_0^{T}S(t,r)[f(\lambda ,z(r))+k(r,y(r))+h(r,x(r))]dr.\end{array}$$

(4.5)

We will consider system (4.5) under the following assumptions:

• (i) $f,k,h:[0,T]\times R\to R$ are continuous,

• (ii) $\omega :[0,T]\to R$ is continuous,

• (iii) $S:[0,T]\times R\to [0,\mathrm{\infty })$ is continuous,

• (iv) there exists $q>0$ such that for all $x,y\in R$,

  $$\begin{array}{c}0\le f(r,y)-f(r,x)\le q(y-x),\hfill \\ 0\le k(r,x)-k(r,y)\le q(y-x)\hfill \end{array}$$

  and

  $$0\le h(r,y)-h(r,x)\le q(y-x).$$

• (v) We suppose that

  $$2^{3p-3}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p<1.$$

• (vi) There exist continuous functions $\alpha ,\beta ,\gamma :[0,T]\to R$ such that

  $$\begin{array}{c}\alpha (t)\le \omega (t)+{\int }_0^{T}S(t,r)[f(r,\alpha (r))+k(r,\beta (r))+h(r,\gamma (r))]dr,\hfill \\ \beta (t)\ge \omega (t)+{\int }_0^{T}S(t,r)[f(r,\beta (r))+k(r,\alpha (r))+h(r,\beta (r))]dr\hfill \end{array}$$

and

$$\gamma (t)\le \omega (t)+{\int }_0^{T}S(t,r)[f(r,\gamma (r))+k(r,\beta (r))+h(r,\alpha (r))]dr.$$

We consider the space $X=C([0,T],R)$ of continuous functions defined on $[0,T]$ endowed with the $G_b$-metric given by

$$G(\theta ,\phi ,\psi )=\left(\underset{t\in [0,T]}{max}\right|\theta (t)-\phi (t){|}^p,\underset{t\in [0,T]}{max}|\phi (t)-\psi (t){|}^p,\underset{t\in [0,T]}{max}|\psi (t)-\theta (t){|}^p)$$

for all $\theta ,\phi ,\psi \in X$, where $s=2^{p-1}$ and $p\ge 1$ (see Example 1.12).

We endow X with the partial ordered ⪯ given by

$$x⪯y⟺x(t)\le y(t),\text{for all }t\in [0,T].$$

On the other hand, $(X,d)$ is regular [53].

Our result is the following.

Theorem 4.6 Under assumptions (i)-(vi), system (4.5) has a solution in $X^3$ where $X=(C[0,T],\mathbb{R})$.

Proof As in [46], we consider the operators $F:X^3\to X$ and $g:X\to X$ defined by

$$F(x_1,x_2,x_3)(t)=\omega (t)+{\int }_0^{T}S(t,r)[f(r,x_1(r))+k(r,x_2(r))+h(r,x_3(r))]dr$$

and

$$g(x)=x$$

for all $t\in [0,T]$ and $x_1,x_2,x_3,x\in X$.

F has the mixed monotone property (see Theorem 25 of [46]).

Let $x,y,z,u,v,w\in X$ be such that $x\ge u$, $y\le v$ and $z\ge w$. Since F has the mixed monotone property, we have

$$F(u,v,w)\le F(x,y,z).$$

On the other hand,

$$G(F(x,y,z),F(u,v,w),F(a,b,c))=max\left\{\begin{array}{c}{max}_{t\in [0,T]}{|F(x,y,z)(t)-F(u,v,w)(t)|}^p,\\ {max}_{t\in [0,T]}{|F(u,v,w)(t)-F(a,b,c)(t)|}^p,\\ {max}_{t\in [0,T]}{|F(a,b,c)(t)-F(x,y,z)(t)|}^p\end{array}\right\}.$$

Now, for all $t\in [0,T]$ from (iv) and the fact that for $\alpha ,\beta ,\gamma \ge 0$, ${(\alpha +\beta +\gamma )}^p\le 2^{2p-2}{\alpha }^p+2^{2p-2}{\beta }^p+2^{p-1}{\gamma }^p$, we have

$$\begin{array}{r}{|F(x,y,z)(t)-F(u,v,w)(t)|}^p\\ {=\left|\begin{array}{c}{\int }_0^{T}S(t,r)[f(r,x(r))-f(r,u(r))]dr\\ +{\int }_0^{T}S(t,r)[k(r,y(r))-k(r,v(r))]dr\\ +{\int }_0^{T}S(t,r)[h(r,z(r))-h(r,w(r))]dr\end{array}\right|}^p\\ {\le \left(\begin{array}{c}|{\int }_0^{T}S(t,r)[f(r,x(r))-f(r,u(r))]dr|\\ +|{\int }_0^{T}S(t,r)[k(r,y(r))-k(r,v(r))]dr|\\ +|{\int }_0^{T}S(t,r)[h(r,z(r))-h(r,w(r))]dr|\end{array}\right)}^p\\ \le \left(\begin{array}{c}{2}^{2p-2}|{\int }_0^{T}S(t,r)[f(r,x(r))-f(r,u(r))]dr{|}^p\\ +2^{2p-2}|{\int }_0^{T}S(t,r)[k(r,y(r))-k(r,v(r))]dr{|}^p\\ +2^{p-1}|{\int }_0^{T}S(t,r)[h(r,z(r))-h(r,w(r))]dr{|}^p\end{array}\right)\\ \le 2^{2p-2}\left[\begin{array}{c}{({\int }_0^T|S(t,r)[f(r,x(r))-f(r,u(r))]|dr)}^p\\ +{({\int }_0^T|S(t,r)[k(r,y(r))-k(r,v(r))]|dr)}^p\\ +{({\int }_0^T|S(t,r)[h(r,z(r))-h(r,w(r))]|dr)}^p\end{array}\right]\\ \le 2^{2p-2}{q}^p[{\left(\underset{r\in [0,T]}{max}|x(r)-u(r)|\right)}^p+{\left(\underset{r\in [0,T]}{max}|y(r)-v(r)|\right)}^p\\ +{\left(\underset{r\in [0,T]}{max}|z(r)-w(r)|\right)}^p]{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ =2^{2p-2}{q}^p\left[\begin{array}{c}{max}_{r\in [0,T]}{|x(r)-u(r)|}^p\\ +{max}_{r\in [0,T]}{|y(r)-v(r)|}^p\\ +{max}_{r\in [0,T]}{|z(r)-w(r)|}^p\end{array}\right]{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p.\end{array}$$

Thus,

$$\begin{array}{r}\underset{t\in [0,T]}{max}|F(x,y,z)(t)-F(u,v,w)(t){|}^p\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\{\underset{r\in [0,T]}{max}{|x(r)-u(r)|}^p,\underset{r\in [0,T]}{max}{|y(r)-v(r)|}^p,\underset{r\in [0,T]}{max}{|z(r)-w(r)|}^p\}.\end{array}$$

(4.6)

Repeating this idea and using the definition of the $G_b$-metric G, we obtain

$$\begin{array}{r}\underset{t\in [0,T]}{max}|F(u,v,w)(t)-F(a,b,c)(t){|}^p\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\{\underset{r\in [0,T]}{max}{|u(r)-a(r)|}^p,\underset{r\in [0,T]}{max}{|v(r)-b(r)|}^p,\underset{r\in [0,T]}{max}{|w(r)-c(r)|}^p\}\end{array}$$

(4.7)

and

$$\begin{array}{r}\underset{t\in [0,T]}{max}|F(a,b,c)(t)-F(x,y,z)(t){|}^p\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\{\underset{r\in [0,T]}{max}{|a(r)-x(r)|}^p,\underset{r\in [0,T]}{max}{|b(r)-y(r)|}^p,\underset{r\in [0,T]}{max}{|c(r)-z(r)|}^p\}.\end{array}$$

(4.8)

So, from (4.6), (4.7) and (4.8), we have

$$\begin{array}{r}G(F(x,y,z),F(u,v,w),F(a,b,c))\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\left\{\begin{array}{c}max\{{max}_{r\in [0,T]}{|x(r)-u(r)|}^p,{max}_{r\in [0,T]}{|y(r)-v(r)|}^p,\\ {max}_{r\in [0,T]}{|z(r)-w(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|u(r)-a(r)|}^p,{max}_{r\in [0,T]}{|v(r)-b(r)|}^p,\\ {max}_{r\in [0,T]}{|w(r)-c(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|a(r)-x(r)|}^p,{max}_{r\in [0,T]}{|b(r)-y(r)|}^p,\\ {max}_{r\in [0,T]}{|c(r)-z(r)|}^p\}\end{array}\right\}.\end{array}$$

(4.9)

Similarly, we can obtain

$$\begin{array}{r}G(F(y,x,y),F(v,u,v),F(b,a,b))\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\left\{\begin{array}{c}max\{{max}_{r\in [0,T]}{|y(r)-v(r)|}^p,{max}_{r\in [0,T]}{|x(r)-u(r)|}^p,\\ {max}_{r\in [0,T]}{|y(r)-v(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|v(r)-b(r)|}^p,{max}_{r\in [0,T]}{|u(r)-a(r)|}^p,\\ {max}_{r\in [0,T]}{|v(r)-b(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|b(r)-y(r)|}^p,{max}_{r\in [0,T]}{|a(r)-x(r)|}^p,\\ {max}_{r\in [0,T]}{|b(r)-y(r)|}^p\}\end{array}\right\}\end{array}$$

(4.10)

and

$$\begin{array}{r}G(F(z,y,x),F(w,v,u),F(c,b,a))\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\left\{\begin{array}{c}max\{{max}_{r\in [0,T]}{|z(r)-w(r)|}^p,{max}_{r\in [0,T]}{|y(r)-v(r)|}^p,\\ {max}_{r\in [0,T]}{|x(r)-u(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|w(r)-c(r)|}^p,{max}_{r\in [0,T]}{|v(r)-b(r)|}^p,\\ {max}_{r\in [0,T]}{|u(r)-a(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|c(r)-z(r)|}^p,{max}_{r\in [0,T]}{|b(r)-y(r)|}^p,\\ {max}_{r\in [0,T]}{|a(r)-x(r)|}^p\}\end{array}\right\}.\end{array}$$

(4.11)

Now, from (4.9), (4.10) and (4.11), we have

$$\begin{array}{r}max\left\{\begin{array}{c}G(F(x,y,z),F(u,v,w),F(a,b,c)),\\ G(F(y,x,y),F(v,u,v),F(b,a,b)),\\ G(F(z,y,x),F(w,v,u),F(c,b,a))\end{array}\right\}\\ \le 2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p\\ \times max\left\{\begin{array}{c}max\{{max}_{r\in [0,T]}{|x(r)-u(r)|}^p,{max}_{r\in [0,T]}{|u(r)-a(r)|}^p,\\ {max}_{r\in [0,T]}{|x(r)-a(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|y(r)-v(r)|}^p,{max}_{r\in [0,T]}{|v(r)-b(r)|}^p,\\ {max}_{r\in [0,T]}{|y(r)-b(r)|}^p\},\\ max\{{max}_{r\in [0,T]}{|z(r)-w(r)|}^p,{max}_{r\in [0,T]}{|w(r)-c(r)|}^p,\\ {max}_{r\in [0,T]}{|z(r)-c(r)|}^p\}\end{array}\right\}\\ =2^{2p-2}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^{p}max\{G(x,u,a),G(y,v,b),G(z,w,c)\}\\ =\frac{2^{3p-3}3q^p{sup}_{t\in [0,T]}{({\int }_0^T|S(t,r)|dr)}^p}{2^{p-1}}max\{G(x,u,a),G(y,v,b),G(z,w,c)\}.\end{array}$$

But from (v), we have

$$2^{3p-3}3q^p\underset{t\in [0,T]}{sup}{\left({\int }_0^T\right|S(t,r)\left|dr\right)}^p<1.$$

This proves that the operator F satisfies the contractive condition appearing in Corollary 2.7.

Let α, β, γ be the functions appearing in assumption (vi), then by (vi), we get

$$\alpha \le F(\alpha ,\beta ,\gamma ),\beta \ge F(\beta ,\alpha ,\beta ),\gamma \le F(\gamma ,\beta ,\alpha ).$$

Applying Corollary 2.7, we deduce the existence of $x_1,x_2,x_3\in X$ such that $x_1=F(x_1,x_2,x_3)$, $x_2=F(x_2,x_1,x_2)$ and $x_3=F(x_3,x_2,x_1)$. □