Introduction and preliminaries {#Sec1}
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A metric space is a set X together with a function *d* (called a metric or "distance function") which assigns a real number *d*(*x*, *y*) to every pair x, y belonging to X satisfying the properties (or axioms):$\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, y) \ge 0$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, y)=0$$\end{document}$ iff $\documentclass[12pt]{minimal}
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                \begin{document}$$x = y$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, y)=d(y, x)$$\end{document}$,$\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, y) + d(y, z)\ge d(x, z)$$\end{document}$.The pair (*X*, *d*) is called a metric space.

Metric fixed point theory is one of the most important and fundamental area of analysis. Due to this a flood of research work have been generated from this area. As a part of this study generalisation of metric space becomes one of the most interesting topic in which many researchers have devoted and continued working. Since the introduction of metric space by Frachet, there is a lot of generalisation of this space. Some of them which can be mentioned are 2-metric space, *D*-metric space, *G*-metric space, cone metric space, fuzzy metric space, Menger space, probabilitic metric space, partial metric space, quasi metric space, *b*-metric space, multiplicative metric space, modular metric space, cyclic metric space, *S*-metric space, *b*-cone metric space etc.

In a recent paper, Aghajani et al. ([@CR3]) introduced a new generalisation of metric space. They used the concepts of both *G*-metric and *b*-metric and generated the new definition and named it as *G*~*b*~-metric space. They also pointed out that the class of *G*~*b*~-metric space is effectively larger than that of *G*-metric space and *G*-metric space becomes a particular case of *G*~*b*~-metric space. They claimed that every *G*~*b*~-metric space is topologically equivalent to a *b*-metric space. For more results on *b*-metric space one can study the research papers of Malhotra and Bansal ([@CR19]), Czerwik ([@CR9], [@CR10]), Hussain et al. ([@CR12]), Singh and Singh ([@CR30]) and references there in. Results of *G*~*b*~-metric also can be found in the research papers of Mustafa et al. ([@CR22], [@CR23], [@CR24]), Sedghi et al. ([@CR26]), Shahkoohi et al. ([@CR27]), Roshan et al. ([@CR25]) and references there in.

The study of fixed points for more than one dimension is becoming an interest for many researchers for the last many years. This concept was first initiated by Guo and Lakshmikantham ([@CR11]) by introducing the definition of coupled fixed point in the year 1987. After a gap of about twenty years Bhaskar and Lakshmikantham ([@CR8]) in the year 2006 proved a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order. Since then large number of research papers came out about coupled fixed point. This concept is further extended to tripled fixed point by Berinde and Borcut ([@CR7]) and to quadrupled fixed point by Karapinar. For more results on multidimensional fixed point one can see the research papers in Kutbi et al. ([@CR17]), Mustafa et al. ([@CR22], [@CR23], [@CR24]), Sedghi et al. ([@CR26]), Shahkoohi et al. ([@CR27]), Guo and Lakshmikantham ([@CR11]), Berinde and Borcut ([@CR7]), Abbas et al. ([@CR1]), Karapinar ([@CR14]), Long et al. ([@CR18]), Kadelburg and Radenovic ([@CR13]), Batra and Vashistha ([@CR6]), Batra et al. ([@CR5]), Karapinar and Turkoglu ([@CR16]), Shantanawi ([@CR28]), Aghajani et al. ([@CR2]), Mehta and Joshi ([@CR21]), Malhotra and Bansal ([@CR19]), Karapinar, Bhaskar and Lakshmikantham ([@CR8]), Singh and Singh ([@CR29]) and reference therein.

In our present study we prove some unique coupled common fixed point theorems for three mappings satisfying some new rational contractive conditions in $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space. Our result is a new result of this type in the setting of $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space.

Following definition was given by Mustafa et al. ([@CR22], [@CR24])

**Definition 1** {#FPar1}
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(Mustafa et al. [@CR24]) Let X be a nonempty set and $\documentclass[12pt]{minimal}
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                \begin{document}$$G:X^{3}\rightarrow \mathbb {R^{+}}$$\end{document}$ be a function satisfying the following properties:$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,y,z)=0$$\end{document}$ if and only if $\documentclass[12pt]{minimal}
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                \begin{document}$$x=y=z$$\end{document}$;$\documentclass[12pt]{minimal}
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                \begin{document}$$0<G(x, x, y)$$\end{document}$ for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x, y\in X$$\end{document}$ with $\documentclass[12pt]{minimal}
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                \begin{document}$$x \ne y$$\end{document}$;$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x, x, y)\le G(x, y, z)$$\end{document}$ for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z\in X$$\end{document}$ with $\documentclass[12pt]{minimal}
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                \begin{document}$$z \ne y$$\end{document}$;$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x, y, z)=G(x, z, y)=G(y, z, x)=\dots$$\end{document}$ (symmetry in all three variables);$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x, y, z)\le G(x, a, a) + G(a, y, z)$$\end{document}$ for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z,a\in X$$\end{document}$ (rectangle inequality).Then the function *G* is called a $\documentclass[12pt]{minimal}
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                \begin{document}$$G-$$\end{document}$metric on *X* and the pair (*X*, *G*) is called a G-metric space.

Following definition was given by Bakhtin ([@CR4]), Czerwik ([@CR9], [@CR10])

**Definition 2** {#FPar2}
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(Malhotra and Bansal [@CR19]) Let X be a (nonempty) set and $\documentclass[12pt]{minimal}
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                \begin{document}$$s\ge 1$$\end{document}$ a given real number. A function $\documentclass[12pt]{minimal}
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                \begin{document}$$d: X\times X\rightarrow \mathbb {R^{+}}$$\end{document}$ (nonnegative real numbers) is called a $\documentclass[12pt]{minimal}
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                \begin{document}$$b-$$\end{document}$metric provided that, for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z \in X$$\end{document}$, following conditions are satisfied:$\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, y) = 0$$\end{document}$ if and only if $\documentclass[12pt]{minimal}
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                \begin{document}$$x = y$$\end{document}$;$\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, y) = d(y, x);$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$d(x, z) \le s[d(x, y) + d(y, z)]$$\end{document}$The pair (*X*, *d*) is called a b-metric space with parameter s.

Following definition was given by Aghajani et al. ([@CR3])

**Definition 3** {#FPar3}
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(Aghajani et al. [@CR3]) Let *X* be a nonempty set and $\documentclass[12pt]{minimal}
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                \begin{document}$$s\ge 1$$\end{document}$ be a given real number. Suppose that a mapping $\documentclass[12pt]{minimal}
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                \begin{document}$$G:X\times X\times X\rightarrow R^{+}$$\end{document}$ satisfies: $\documentclass[12pt]{minimal}
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                \begin{document}$$(G_{b}1)$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,y,z) =0$$\end{document}$ if $\documentclass[12pt]{minimal}
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                \begin{document}$$x=y=z$$\end{document}$,$\documentclass[12pt]{minimal}
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                \begin{document}$$(G_{b}2)$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$0<G(x,x,y)$$\end{document}$ for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y\in X$$\end{document}$ with $\documentclass[12pt]{minimal}
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                \begin{document}$$x\not =y$$\end{document}$,$\documentclass[12pt]{minimal}
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                \begin{document}$$(G_{b}3)$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,x,y)\le G(x,y,z)$$\end{document}$ for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z\in X$$\end{document}$ with $\documentclass[12pt]{minimal}
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                \begin{document}$$y\not =z$$\end{document}$,$\documentclass[12pt]{minimal}
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                \begin{document}$$(G_{b}4)$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,y,z)=G(p[x,y,z])$$\end{document}$, where *p* is a permutation of *x*, *y*, *z* (symmetry),$\documentclass[12pt]{minimal}
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                \begin{document}$$(G_{b}5)$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,y,z)=s[G(x,a,a)+G(a,y,z)]$$\end{document}$ for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z,a\in X$$\end{document}$ (rectangle inequality). Then *G* is called a generalized *b*-metric and pair (*X*, *G*) is called a generalized *b*-metric space or $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space.

Aghajani et al. ([@CR3]) remarked that the class of $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space is effectively larger than that of *G*-metric spaces given in Mustafa et al. ([@CR23]). Following example given by Aghajani et al. ([@CR3]) shows that a $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric on *X* need not be a *G*-metric on *X*.

*Example 4* {#FPar4}
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(Aghajani et al. [@CR3]) Let (*X*, *G*) be a *G*-metric space, and $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{*}(x,y,z)=G(x,y,z)^{p}$$\end{document}$, where $\documentclass[12pt]{minimal}
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                \begin{document}$$p>1$$\end{document}$ is a real number. Note that $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{*}$$\end{document}$ is a $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric with $\documentclass[12pt]{minimal}
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                \begin{document}$$s=2^{p-1}$$\end{document}$.

Also in the above example, $\documentclass[12pt]{minimal}
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                \begin{document}$$(X, G_{*})$$\end{document}$ is not necessarily a *G*-metric space. For example, let $\documentclass[12pt]{minimal}
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                \begin{document}$$X ={\mathbb {R}}$$\end{document}$ and *G*-metric *G* be defined by$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x,y,z)=\frac{1}{3}(|x-y|+|y-z|+|x-z|), \end{aligned}$$\end{document}$$for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z\in {\mathbb {R}}$$\end{document}$. Then $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{*}(x,y,z)^{2}=\frac{1}{9}(|x-y|+|y-z|+|x-z|)^{2}$$\end{document}$ is a $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric on $\documentclass[12pt]{minimal}
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                \begin{document}$${\mathbb {R}}$$\end{document}$ with $\documentclass[12pt]{minimal}
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                \begin{document}$$s=2^{2-1}=2$$\end{document}$, but it is not a *G*-metric on $\documentclass[12pt]{minimal}
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                \begin{document}$${\mathbb {R}}$$\end{document}$. To see this, let $\documentclass[12pt]{minimal}
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                \begin{document}$$x=3$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$y=5$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$z=7$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$a=\frac{7}{2}$$\end{document}$ we get, $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{*}(3,5,7)=\frac{64}{9}$$\end{document}$, $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{*}(3,\frac{7}{2},\frac{7}{2})=\frac{1}{9}$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{*}\left( \frac{7}{2},5,7\right) =\frac{49}{9}$$\end{document}$, so $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{*}(3,5,7)=\frac{64}{9} \nleq \frac{50}{9}=G_{*}(3,\frac{7}{2},\frac{7}{2})+G_{*}\left( \frac{7}{2},5,7\right)$$\end{document}$.

Following definitions and propositions in $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space are given in Aghajani et al. ([@CR3]).

**Definition 5** {#FPar5}
----------------

(Aghajani et al. [@CR3]) A $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-metric *G* is said to be symmetric if $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x,y,y)=G(y,x,x)$$\end{document}$ for all $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$x,y\in X$$\end{document}$.

**Definition 6** {#FPar6}
----------------

(Aghajani et al. [@CR3]) Let (*X*, *G*) be a $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$G_{b}$$\end{document}$-metric space then for $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x_{0}\in X$$\end{document}$, $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$r>0$$\end{document}$, the $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-ball with center $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$x_{0}$$\end{document}$ and radius *r* is$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} B_{G}(x_{0},r)=\{y\in X|G(x_{0},y,y)< r\} \end{aligned}$$\end{document}$$

For example, let $\documentclass[12pt]{minimal}
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                \begin{document}$$X={\mathbb {R}}$$\end{document}$ and consider the $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric *G* defined by$$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\begin{aligned} G(x,y,z)=\frac{1}{9}(|x-y|+|y-z|+|x-z|)^{2} \end{aligned}$$\end{document}$$for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z \in {\mathbb {R}}$$\end{document}$. Then$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} B_{G}(3,4)& = \{y\in X:\, G(3,y,y)<4\} \\& = \left\{y\in X:\frac{1}{9}(|y-3|+|y-3|)^{2}<4\right\} \\& = \{y\in X:\,\,|y-3|^{2}<9\} \\& = (0,6). \end{aligned}$$\end{document}$$

**Proposition 7** {#FPar7}
-----------------

(Aghajani et al. [@CR3]) *LetXbe a*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space, then for each*$\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,z,a \in X$$\end{document}$*it follows that:*if $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x,y,z)=0$$\end{document}$ then $\documentclass[12pt]{minimal}
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                \begin{document}$$x=y=z$$\end{document}$,$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x,y,z)\le s(G(x,x,y)+G(x,x,z))$$\end{document}$,$\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x,y,y)\le 2sG(y,x,x)$$\end{document}$,$\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x,y,z)\le s(G(x,a,z)+G(a,y,z))$$\end{document}$

**Definition 8** {#FPar8}
----------------

(Aghajani et al. [@CR3]) Let *X* be a $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-metric space, we define $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$d_{G}(x, y)=G(x,y,y)+G(x,x,y)$$\end{document}$, it is easy to see that $\documentclass[12pt]{minimal}
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                \begin{document}$$d_{G}$$\end{document}$ defines a *b*-metric on *X*, which we call it *b*-metric associated with *G*.

**Proposition 9** {#FPar9}
-----------------

(Aghajani et al. [@CR3]) *LetXbe a*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-*metric space, then for any*$\documentclass[12pt]{minimal}
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                \begin{document}$$x_{0}\in X$$\end{document}$*and*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$r>0$$\end{document}$, *if*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$y\in B_{G}(x_{0},r)$$\end{document}$*then there exists a*$\documentclass[12pt]{minimal}
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                \begin{document}$$\delta >0$$\end{document}$*such that*$\documentclass[12pt]{minimal}
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                \begin{document}$$B_{G}(y,\delta )\subseteq B_{G}(x_{0}, r)$$\end{document}$.

**Proposition 10** {#FPar10}
------------------

(Aghajani et al. [@CR3]) *LetXbe a*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-*metric space, then for any*$\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$x_{0}\in X$$\end{document}$*and*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$r>0$$\end{document}$, *we have*$$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\begin{aligned} B_{G}\left( x_{0},\frac{r}{2s+1}\right) \subseteq B_{d_{G}}(x_{0}, r) \subseteq B_{G}(x_{0}, r) \end{aligned}$$\end{document}$$*Thus every*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$*-metric space is topologically equivalent to ab-metric space. This allows us to readily transport many concepts and results fromb-metric spaces into*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$*-metric space setting.*

**Definition 11** {#FPar11}
-----------------

(Aghajani et al. [@CR3]) Let *X* be a $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space. A sequence $\documentclass[12pt]{minimal}
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                \begin{document}$$\{x_{n}\}$$\end{document}$ in *X* is said to be:$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-Cauchy sequence if, for each $\documentclass[12pt]{minimal}
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                \begin{document}$$\varepsilon >0$$\end{document}$, there exists a positive integer $\documentclass[12pt]{minimal}
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                \begin{document}$$n_{0}$$\end{document}$ such that, for all $\documentclass[12pt]{minimal}
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                \begin{document}$$m,n,l\ge n_{0}, G(x_{n}, x_{m}, x_{l})<\varepsilon$$\end{document}$;$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-convergent to a point $\documentclass[12pt]{minimal}
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                \begin{document}$$x\in X$$\end{document}$ if, for each $\documentclass[12pt]{minimal}
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                \begin{document}$$\varepsilon > 0$$\end{document}$, there exists a positive integer $\documentclass[12pt]{minimal}
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                \begin{document}$$n_{0}$$\end{document}$ such that, for all $\documentclass[12pt]{minimal}
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                \begin{document}$$m,n=n_{0}, G(x_{n},x_{m},x)<\varepsilon$$\end{document}$.

**Proposition 12** {#FPar12}
------------------

(Aghajani et al. [@CR3]) *LetXbe a*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space, then following statements are equivalent:the sequence*$\documentclass[12pt]{minimal}
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                \begin{document}$$\{x_{n}\}$$\end{document}$*is*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-*Cauchy*.*for any*$\documentclass[12pt]{minimal}
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                \begin{document}$$\varepsilon >0$$\end{document}$, *there exists*$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$n_{0}\in N$$\end{document}$*such that*$\documentclass[12pt]{minimal}
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                \begin{document}$$G(x_{n},x_{m},x_{m})<\varepsilon$$\end{document}$, *for all*$\documentclass[12pt]{minimal}
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                \begin{document}$$m,n\ge n_{0}$$\end{document}$.

**Proposition 13** {#FPar13}
------------------

(Aghajani et al. [@CR3]) *LetXbe a*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space, then following statements are equivalent*:$\documentclass[12pt]{minimal}
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                \begin{document}$$\{x_{n}\}$$\end{document}$ is $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-*convergent* to *x*.$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x_{n},x_{n},x)\rightarrow 0$$\end{document}$ as $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$n\rightarrow +\infty$$\end{document}$.$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x_{n},x,x)\rightarrow 0$$\end{document}$ as $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$n\rightarrow +\infty$$\end{document}$.

**Definition 14** {#FPar14}
-----------------

(Aghajani et al. [@CR3]) A $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-metric space *X* is called $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-complete if every $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-Cauchy sequence is $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-convergent in *X*.

**Definition 15** {#FPar15}
-----------------

(Kutbi et al. [@CR17]) Let *X* be a nonempty set. Then $\documentclass[12pt]{minimal}
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                \begin{document}$$(X,G,\preceq )$$\end{document}$ is called partially ordered $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-metric space if *G* is a $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G_{b}$$\end{document}$-metric on a partially ordered set $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$(X,\preceq )$$\end{document}$.

Main results {#Sec2}
============

Now we prove the following theorem.

**Theorem 16** {#FPar16}
--------------

*Let* (*X*, *G*) *be a complete symmetric*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space with parameter*$\documentclass[12pt]{minimal}
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                \begin{document}$$s\ge 1$$\end{document}$*and let the mappings*$\documentclass[12pt]{minimal}
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                \begin{document}$$S,T,R:X^{2}\rightarrow X$$\end{document}$*satisfying*$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(S(x,y),T(u,v),R(a,b))& \le \alpha _{1}\frac{G(x,u,a)+G(y,v,b)}{2} \nonumber \\& \quad +\alpha _{2}\frac{G(S(x,y),T(u,v),R(a,b))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{3}\frac{G(S(x,y),T(u,v),R(a,b))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{4}\frac{G(x,x,S(x,y))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{5}\frac{G(x,x,S(x,y))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{6}\frac{G(u,u,T(u,v))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{7}\frac{G(u,u,T(u,v))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{8}\frac{G(a,a,R(a,b))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \nonumber \\& \quad +\alpha _{9}\frac{G(a,a,R(a,b))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} \end{aligned}$$\end{document}$$*for all*$\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,u,v,a,b \in X$$\end{document}$*and*$\documentclass[12pt]{minimal}
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                \begin{document}$$\alpha _{1},\alpha _{2},...,\alpha _{9}\ge 0$$\end{document}$*with*$\documentclass[12pt]{minimal}
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                \begin{document}$$\alpha _{1}+\alpha _{2}+\alpha _{3}+2(\alpha _{4}+\alpha _{5})+\alpha _{6}+\alpha _{7}+\alpha _{8}+\alpha _{9}<1$$\end{document}$. *ThenS*, *TandRhave a unique common coupled fixed point inX*.

*Proof* {#FPar17}
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Let $\documentclass[12pt]{minimal}
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                \begin{document}$$x_{0},y_{0}\in X$$\end{document}$ be arbitrary points.

Define$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} x_{3k+1}& = S(x_{3k},y_{3k}),y_{3k+1} = S(y_{3k},x_{3k}) \\ x_{3k+2}& = T(x_{3k+1},y_{3k+1}),y_{3k+2} = T(y_{3k+1},x_{3k+1}) \\ x_{3k+3}& = R(x_{3k+2},y_{3k+2}),y_{3k+3} = R(y_{3k+2},x_{3k+2}) \end{aligned}$$\end{document}$$for $\documentclass[12pt]{minimal}
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                \begin{document}$$k=0,1,2,3,...$$\end{document}$. Then$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x_{3k+1},x_{3k+2},x_{3k+3})& = G(S(x_{3k},y_{3k}),T(x_{3k+1},y_{3k+1}),R(x_{3k+2},y_{3k+2}))\\& \le \alpha _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2} \\& \quad +\alpha _{2}\frac{G(S(x_{3k},y_{3k}),T(x_{3k+1},y_{3k+1}),R(x_{3k+2},y_{3k+2}))G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{3}\frac{G(S(x_{3k},y_{3k}),T(x_{3k+1},y_{3k+1}),R(x_{3k+2},y_{3k+2}))G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{4}\frac{G(x_{3k},x_{3k},S(x_{3k},y_{3k}))G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{5}\frac{G(x_{3k},x_{3k},S(x_{3k},y_{3k}))G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{6}\frac{G(x_{3k+1},x_{3k+1},T(x_{3k+1},y_{3k+1}))G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{7}\frac{G(x_{3k+1},x_{3k+1},T(x_{3k+1},y_{3k+1}))G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{8}\frac{G(x_{3k+2},x_{3k+2},R(x_{3k+2},y_{3k+2}))G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{9}\frac{G(x_{3k+2},x_{3k+2},R(x_{3k+2},y_{3k+2}))G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& = \alpha _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2} \\& \quad +\alpha _{2}\frac{G(x_{3k+1},x_{3k+2},x_{3k+3})G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{3}\frac{G(x_{3k+1},x_{3k+2},x_{3k+3})G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{4}\frac{G(x_{3k},x_{3k},x_{3k+1})G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{5}\frac{G(x_{3k},x_{3k},x_{3k+1})G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{6}\frac{G(x_{3k+1},x_{3k+1},x_{3k+2})G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{7}\frac{G(x_{3k+1},x_{3k+1},x_{3k+2})G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{8}\frac{G(x_{3k+2},x_{3k+2},x_{3k+3})G(x_{3k},x_{3k+1},x_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \\& \quad +\alpha _{9}\frac{G(x_{3k+2},x_{3k+2},x_{3k+3})G(y_{3k},y_{3k+1},y_{3k+2})}{1+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})} \end{aligned}$$\end{document}$$which implies that$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&(1-\alpha _{2}-\alpha _{3})G(x_{3k+1},x_{3k+2},x_{3k+3}) \le \frac{\alpha _{1}}{2}G(x_{3k},x_{3k+1},x_{3k+2}) \\ & \quad +\frac{\alpha _{1}}{2}G(y_{3k},y_{3k+1},x_{3k+2}) \\&\quad +(\alpha _{4}+\alpha _{5})G(x_{3k},x_{3k+1},x_{3k+2})+(\alpha _{6}+\alpha _{7})G(x_{3k+1},x_{3k+2},x_{3k+3})\\&\quad +(\alpha _{8}+\alpha _{9})G(x_{3k+1},x_{3k+2},x_{3k+3})\\&\Rightarrow (1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7} -\alpha _{8}-\alpha _{9})G(x_{3k+1},x_{3k+2},x_{3k+3})\\ & \le \left( \frac{\alpha _{1}}{2}+\alpha _{4}+\alpha _{5}\right) G(x_{3k},x_{3k+1},x_{3k+2})\\&\quad +\frac{\alpha _{1}}{2}G(y_{3k},y_{3k+1},y_{3k+2}) \end{aligned}$$\end{document}$$

Therefore,$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x_{3k+1},x_{3k+2},x_{3k+3})& \le \frac{\left( \frac{\alpha _{1}}{2}+\alpha _{4}+\alpha _{5}\right) G(x_{3k},x_{3k+1},x_{3k+2})}{1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7}-\alpha _{8}-\alpha _{9}} \nonumber \\& \quad + \frac{\frac{\alpha _{1}}{2}G(y_{3k},y_{3k+1},y_{3k+2})}{1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7}-\alpha _{8}-\alpha _{9}} \end{aligned}$$\end{document}$$Proceeding similarly one can prove that$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(y_{3k+1},y_{3k+2},y_{3k+3})& \le \frac{\left( \frac{\alpha _{1}}{2}+\alpha _{4}+\alpha _{5}\right) G(y_{3k},y_{3k+1},y_{3k+2})}{1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7}-\alpha _{8}-\alpha _{9}} \nonumber \\& \quad + \frac{\frac{\alpha _{1}}{2}G(x_{3k},x_{3k+1},x_{3k+2})}{1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7}-\alpha _{8}-\alpha _{9}} \end{aligned}$$\end{document}$$Adding ([2](#Equ2){ref-type=""}) and ([3](#Equ3){ref-type=""}) we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x_{3k+1},x_{3k+2},x_{3k+3})+G(y_{3k+1},y_{3k+2},y_{3k+3}))\\&\quad \le \frac{\alpha _{1}+2(\alpha _{4}+\alpha _{5})}{1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7}-\alpha _{8}-\alpha _{9}}[G(x_{3k},x_{3k+1},x_{3k+2}) +G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$Therefore$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x_{3k+1},x_{3k+2},x_{3k+3})+G(y_{3k+1},y_{3k+2},y_{3k+3}))\\&\quad \le h[G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$where$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} h=\frac{\alpha _{1}+2(\alpha _{4}+\alpha _{5})}{1-\alpha _{2}-\alpha _{3}-\alpha _{6}-\alpha _{7}-\alpha _{8}-\alpha _{9}}<1 \end{aligned}$$\end{document}$$Also, we can show that$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x_{3k+2},x_{3k+3},x_{3k+4})+G(y_{3k+2},y_{3k+3},y_{3k+4}))\\&\quad \le h[G(x_{3k+1},x_{3k+2},x_{3k+3})+G(y_{3k+1},y_{3k+2},y_{3k+3})] \\&\quad \le h^{2} [G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$Continuing this way, we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x_{n},x_{n+1},x_{n+2})+G(y_{n},y_{n+1},y_{n+2})\\&\quad \le h[G(x_{n-1},x_{n},x_{n+1})+G(y_{n-1},y_{n},y_{n+1})]\\&\quad \le h^{2}[G(x_{n-2},x_{n-1},x_{n})+G(y_{n-2},y_{n-1},y_{n})]\\&\quad \le \cdots \le h^{n}[G(x_{0},x_{1},x_{2})+G(y_{0},y_{1},y_{2})] \end{aligned}$$\end{document}$$If $\documentclass[12pt]{minimal}
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                \begin{document}$$G(x_{n},x_{n+1},x_{n+2})+G(y_{n},y_{n+1},y_{n+2})=G_{n}$$\end{document}$, then $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{n}\le hG_{n-1}\le h^{2}G_{n-2}\le \cdots \le h^{n}G_{0}$$\end{document}$.

By property ([3](#Equ3){ref-type=""}) of Definition 1, we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x_{n},x_{n+1},x_{n+1})+G(y_{n},y_{n+1},y_{n+1}) \le G_{n} \le h^{n}G_{0} \end{aligned}$$\end{document}$$For $\documentclass[12pt]{minimal}
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                \begin{document}$$m>n$$\end{document}$,$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x_{n},x_{m},x_{m})+G(y_{n},y_{m},y_{m})\\&\quad \le s[G(x_{n},x_{n+1},x_{n+1})+G(x_{n+1},x_{m},x_{m})+G(y_{n},y_{n+1},y_{n+1})+G(y_{n+1},y_{m},y_{m})]\\&\quad \le s[G(x_{n},x_{n+1},x_{n+1})+G(y_{n},y_{n+1},y_{n+1})]+s^{2}[G(x_{n+1},x_{n+2},x_{n+2})+G(y_{n+1},y_{n+2},y_{n+2})]+ \cdots \\&\qquad +s^{m-n}[G(x_{m-1},x_{m},x_{m})+G(y_{m-1},y_{m},y_{m})]\\&\quad \le sh^{n}G_{0}+s^{2}h^{n+1}G_{0}+ \cdots +s^{m-n}h^{m-1}G_{0}\\&\quad <sh^{n}[1+sh+(sh)^{2}+ \cdots ]G_{0}\\&\quad =\frac{sh^{n}}{1-sh} G_{0}\rightarrow 0 \quad {\text{as }} n\rightarrow \infty \end{aligned}$$\end{document}$$which shows that $\documentclass[12pt]{minimal}
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                \begin{document}$$\{x_{n}\}$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$\{y_{n}\}$$\end{document}$ are Cauchy sequences in *X*. As *X* is complete $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space, so there exists $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y\in X$$\end{document}$ such that $\documentclass[12pt]{minimal}
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                \begin{document}$$x_{n}\rightarrow x$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y_{n}\rightarrow y$$\end{document}$ as $\documentclass[12pt]{minimal}
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                \begin{document}$$n\rightarrow \infty$$\end{document}$.

Now we will prove that $\documentclass[12pt]{minimal}
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                \begin{document}$$x=S(x,y)$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y=S(y,x)$$\end{document}$. On contrary suppose that $\documentclass[12pt]{minimal}
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                \begin{document}$$x\ne S(x,y)$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y\ne S(y,x)$$\end{document}$. Then $\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,S(x,y),S(x,y))=l_{1}>0$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$G(y,y,S(y,x))=l_{2}>0$$\end{document}$.

Using inequality ([1](#Equ1){ref-type=""}) we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} l_{1}& = G(x,S(x,y),S(x,y)) \\& \le s[G(x,x_{n+1},x_{n+1})+G(x_{n+1},S(x,y),S(x,y))] \\& \le sG(x,x_{n+1},x_{n+1})+s\left[ \alpha _{1}\frac{G(x_{n},x,x)+G(y_{n},y,y)}{2}\right. \\& \quad \left. +\;\alpha _{2}\frac{G(S(x_{n},y_{n}),S(x,y),S(x,y))G(x_{n},x,x)}{1+G(x_{n},x,x)+G(y_{n},y,y)} +\alpha _{3}\frac{G(S(x_{n},y_{n}),S(x,y),S(x,y))G(y_{n},y,y)}{1+G(x_{n},x,x)+G(y_{n},y,y)} \right. \\& \quad \left. +\;\alpha _{4}\frac{G(x_{n},x_{n},S(x_{n},y_{n}))G(x_{n},x,x)}{1+G(x_{n},x,x)+G(y_{n},y,y)} +\alpha _{5}\frac{G(x_{n},x_{n},S(x_{n},y_{n}))G(y_{n},y,y)}{1+G(x_{n},x,x)+G(y_{n},y,y)}\right. \\& \quad \left. +\;\alpha _{6}\frac{G(x,x,S(x,y))G(x_{n},x,x)}{1+G(x_{n},x,x)+G(y_{n},y,y)} +\alpha _{7}\frac{G(x,x,S(x,y))G(y_{n},y,y)}{1+G(x_{n},x,x)+G(y_{n},y,y)}\right. \\& \quad \left. +\;\alpha _{8}\frac{G(x,x,S(x,y))G(x_{n},x,x)}{1+G(x_{n},x,x)+G(y_{n},y,y)} +\alpha _{9}\frac{G(x,x,S(x,y))G(y_{n},y,y)}{1+G(x_{n},x,x)+G(y_{n},y,y)} \right] . \end{aligned}$$\end{document}$$Since $\documentclass[12pt]{minimal}
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                \begin{document}$$\{x_{n}\}$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$\{y_{n}\}$$\end{document}$ are convergent to x and y, therefore by taking limits as $\documentclass[12pt]{minimal}
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                \begin{document}$$n\rightarrow \infty$$\end{document}$ we get $\documentclass[12pt]{minimal}
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                \begin{document}$$l_{1}\le 0$$\end{document}$, which is a contradiction, so $\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,S(x,y),S(x,y))=0$$\end{document}$ which gives $\documentclass[12pt]{minimal}
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                \begin{document}$$x=S(x,y)$$\end{document}$.

Similarly, we can prove that $\documentclass[12pt]{minimal}
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                \begin{document}$$y=S(y,x)$$\end{document}$. Also, we can prove that $\documentclass[12pt]{minimal}
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                \begin{document}$$x=T(x,y)$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y=T(y,x)$$\end{document}$. Similarly $\documentclass[12pt]{minimal}
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                \begin{document}$$x=R(x,y)$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y=R(y,x)$$\end{document}$. Then (*x*, *y*) is a Common coupled fixed point of *S*, *T* and *R*.

In order to prove the uniqueness of the coupled fixed point, if possible let (*p*, *q*) be the second common coupled fixed point of *S*, *T* and *R*.

Then by using inequality ([1](#Equ1){ref-type=""}), we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x,p,p)& = G(S(x,y),T(p,q),R(p,q)) \nonumber \\& \le \alpha _{1}\frac{G(x,p,p)+G(y,q,q)}{2}+\alpha _{2}\frac{G(S(x,y),T(p,q),R(p,q))G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{3}\frac{G(S(x,y),T(p,q),R(p,q))G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{4}\frac{G(x,x,S(x,y))G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{5}\frac{G((x,x,S(x,y)))G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{6}\frac{G(p,p,T(p,q))G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{7}\frac{G(p,p,T(p,q))G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{8}\frac{G(p,p,R(p,q))G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{9}\frac{G(p,p,R(p,q))G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& = \frac{\alpha _{1}}{2}[G(x,p,p)+G(y,q,q)]\nonumber \\& \quad +\alpha _{2}\frac{G(x,p,p)G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{3}\frac{G(x,p,p)G(y,p,p)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{4}\frac{G(x,x,x)G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{5}\frac{G(x,x,x)G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{6}\frac{G(p,p,p)G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{7}\frac{G(p,p,p)G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\& \quad +\alpha _{8}\frac{G(p,p,p)G(x,p,p)}{1+G(x,p,p)+G(y,q,q)}+\alpha _{9}\frac{G(p,p,p)G(y,q,q)}{1+G(x,p,p)+G(y,q,q)}\nonumber \\ & \Rightarrow G(x,p,p) \le \frac{\alpha _{1}}{2}[G(x,p,p)+G(y,q,q)]+\alpha _{2}G(x,p,p)+\alpha _{3}G(x,p,p)\nonumber \\& \Rightarrow (1-\frac{\alpha _{1}}{2}-\alpha _{2}-\alpha _{3})G(x,p,p) \le \frac{\alpha _{1}}{2}G(y,q,q)\nonumber \\ & \Rightarrow G(x,p,p) \le \frac{\alpha _{1}}{2-\alpha _{1}-2\alpha _{2}-2\alpha _{3}}G(y,q,q) \end{aligned}$$\end{document}$$Similarly,$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(y,q,q) \le \frac{\alpha _{1}}{2-\alpha _{1}-2\alpha _{2}-2\alpha _{3}}G(x,p,p) \end{aligned}$$\end{document}$$

Adding ([4](#Equ4){ref-type=""}) and ([5](#Equ5){ref-type=""}) we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x,p,p)+G(y,q,q) \le \frac{\alpha _{1}}{2-\alpha _{1}-2\alpha _{2}-2\alpha _{3}}[G(x,p,p)+G(y,q,q)]\\&\quad \Rightarrow \left[ 1-\frac{\alpha _{1}}{2-\alpha _{1}-2\alpha _{2}-2\alpha _{3}}\right] [G(x,p,p)+G(y,q,q)] \le 0 \\&\quad \Rightarrow \frac{2(1-\alpha _{1}-\alpha _{2}-\alpha _{3})}{2-\alpha _{1}-2\alpha _{2}-2\alpha _{3}}[G(x,p,p)+G(y,q,q)] \le 0 \end{aligned}$$\end{document}$$Since $\documentclass[12pt]{minimal}
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                \begin{document}$$\alpha _{1}+\alpha _{2}+\alpha _{3}<1$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$\frac{2(1-\alpha _{1}-\alpha _{2}-\alpha _{3})}{2-\alpha _{1}-2\alpha _{2}-2\alpha _{3}}>0$$\end{document}$.

Hence $\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,p,p)+G(y,q,q)=0$$\end{document}$, which implies that $\documentclass[12pt]{minimal}
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                \begin{document}$$x=p$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y=q \Rightarrow (x,y)=(p,q)$$\end{document}$.

Thus *S*, *T*, *R* have unique coupled common fixed point.

This completes the proof. $\documentclass[12pt]{minimal}
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                \begin{document}$$\square$$\end{document}$

**Corollary 17** {#FPar18}
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*Let* (*X*, *G*) *be a complete symmetric*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space with parameter*$\documentclass[12pt]{minimal}
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                \begin{document}$$s\ge 1$$\end{document}$*and let the mapping*$\documentclass[12pt]{minimal}
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                \begin{document}$$S:X^{2}\rightarrow X$$\end{document}$*satisfying*$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(S(x,y),S(u,v),S(a,b))& \le \alpha _{1}\frac{G(x,u,a)+G(y,v,b)}{2} \\& \quad + \alpha _{2}\frac{G(S(x,y),S(u,v),S(a,b))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)}\\& \quad + \alpha _{3}\frac{G(S(x,y),S(u,v)),S(a,b))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)}\\& \quad +\alpha _{4}\frac{G(x,x,S(x,y))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)}+\alpha _{5}\frac{G(x,x,S(x,y))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)}\\& \quad +\alpha _{6}\frac{G(u,u,S(u,v))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)}+\alpha _{7}\frac{G(u,u,S(u,v))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)}\\& \quad +\alpha _{8}\frac{G(a,a,S(a,b))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)}+\alpha _{9}\frac{G(a,a,S(a,b))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} \end{aligned}$$\end{document}$$*for all*$\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,u,v,a,b \in X$$\end{document}$*and*$\documentclass[12pt]{minimal}
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                \begin{document}$$\alpha _{1},\alpha _{2},\dots ,\alpha _{9}\ge 0$$\end{document}$*with*$\documentclass[12pt]{minimal}
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                \begin{document}$$\alpha _{1}+\alpha _{2}+\alpha _{3}+2(\alpha _{4}+\alpha _{5})+\alpha _{6}+\alpha _{7}+\alpha _{8}+\alpha _{9}<1$$\end{document}$. *ThenShas a unique coupled fixed point inX*.

**Theorem 18** {#FPar19}
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*Let* (*X*, *G*) *be a complete symmetric*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space with parameter*$\documentclass[12pt]{minimal}
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                \begin{document}$$s\ge 1$$\end{document}$*and let the mappings*$\documentclass[12pt]{minimal}
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                \begin{document}$$S,T,R:X\times X\rightarrow X$$\end{document}$*satisfy*$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(S(x,y),T(u,v),R(a,b)) \le \beta _{1}\frac{G(x,u,a)+G(y,v,b)}{2} \nonumber \\&\qquad +\beta _{2}\frac{G(x,x,S(x,y))G(u,u,T(u,v))}{1+s[G(x,x,T(u,v))+G(u,u,S(x,y))+G(x,u,a)+G(y,v,b)]} \nonumber \\&\qquad +\beta _{3}\frac{G(u,u,T(u,v)),G(a,a,R(a,b))}{1+s[G(u,u,R(a,b))+G(a,a,T(u,v))+G(x,u,a)+G(y,v,b)]} \nonumber \\&\qquad +\beta _{4}\frac{G(a,a,R(a,b))G(x,x,S(x,y))}{1+s[G(a,b,S(x,y))+G(x,x,R(a,b))+G(x,u,a)+G(y,v,b)]} \end{aligned}$$\end{document}$$*for all*$\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,u,v,a,b \in X$$\end{document}$*and*$\documentclass[12pt]{minimal}
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                \begin{document}$$\beta _{1},\beta _{2},\beta _{3},\beta _{4}$$\end{document}$*are non-negative real numbers with*$\documentclass[12pt]{minimal}
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                \begin{document}$$\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}<1$$\end{document}$. *ThenS*, *T*, *Rhave unique coupled common fixed point*.

*Proof* {#FPar20}
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Let $\documentclass[12pt]{minimal}
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                \begin{document}$$x_{0},y_{0}\in X$$\end{document}$ be arbitrary points. Define$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} x_{3k+1}& = S(x_{3k},y_{3k}), y_{3k+1}=S(y_{3k},x_{3k})\\ x_{3k+2}& = T(x_{3k+1},y_{3k+1}), y_{3k+2}=T(y_{3k+1},x_{3k+1})\\ x_{3k+3}& = R(x_{3k+2},y_{3k+2}), y_{3k+3}=R(y_{3k+2},x_{3k+2}) \end{aligned}$$\end{document}$$for $\documentclass[12pt]{minimal}
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                \begin{document}$$k=0,1,2,3 \ldots$$\end{document}$

Then$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x_{3k+1},x_{3k+2},x_{3k+3})= G(S(x_{3k},y_{3k}),T(x_{3k+1},y_{3k+1}),R(x_{3k+1},y_{3k+1})) \\&\quad \le \beta _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2} \\&\qquad +\beta _{2}\frac{G(x_{3k},x_{3k},S(x_{3k},y_{3k}))G(x_{3k+1},x_{3k+1},T(x_{3k+1},y_{3k+1}))}{1+s[G(x_{3k},x_{3k},T(x_{3k+1},y_{3k+1}))+G(x_{3k+1},x_{3k+1},S(x_{3k},y_{3k}))+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\qquad +\beta _{3}\frac{G(x_{3k+1},x_{3k+1},T(x_{3k+1},y_{3k+1}))G(x_{3k+2},x_{3k+2},R(x_{3k+2},y_{3k+2}))}{1+s[G(x_{3k+1},x_{3k+1},R(x_{3k+2},y_{3k+2}))+G(x_{3k+2},x_{3k+2},T(x_{3k+1},y_{3k+1}))+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\qquad +\beta _{4}\frac{G(x_{3k+2},x_{3k+2},R(x_{3k+2},y_{3k+2}))G(x_{3k},x_{3k},S(x_{3k},y_{3k}))}{1+s[G(x_{3k+2},x_{3k+2},S(x_{3k},y_{3k}))+G(x_{3k},x_{3k},R(x_{3k+2},y_{3k+2}))+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]}\\&\quad =\beta _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2}\\&\qquad +\beta _{2}\frac{G(x_{3k},x_{3k},x_{3k+1})G(x_{3k+1},x_{3k+1},x_{3k+2})}{1+s[G(x_{3k},x_{3k},x_{3k+2})+G(x_{3k+1},x_{3k+1},x_{3k+1})+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\qquad +\beta _{3}\frac{G(x_{3k+1},x_{3k+1},x_{3k+2})G(x_{3k+2},x_{3k+2},x_{3k+3})}{1+s[G(x_{3k+1},x_{3k+1},x_{3k+3})+G(x_{3k+2},x_{3k+2},x_{3k+2})+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\qquad +\beta _{4}\frac{G(x_{3k+2},x_{3k+2},x_{3k+3})G(x_{3k},x_{3k},x_{3k+1})}{1+s[G(x_{3k+2},x_{3k+2},x_{3k+1})+G(x_{3k},x_{3k},x_{3k+3})+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\quad \le \beta _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2}\\&\qquad +\beta _{2}\frac{G(x_{3k},x_{3k+1},x_{3k+2})G(x_{3k+1},x_{3k+2},x_{3k+3})}{1+s[G(x_{3k},x_{3k},x_{3k+2})+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\qquad +\beta _{3}\frac{G(x_{3k},x_{3k+1},x_{3k+2})G(x_{3k+1},x_{3k+2},x_{3k+3})}{1+s[G(x_{3k+1},x_{3k+1},x_{3k+3})+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\qquad +\beta _{4}\frac{G(x_{3k+1},x_{3k+2},x_{3k+3})G(x_{3k},x_{3k+1},x_{3k+2})}{1+s[G(x_{3k+2},x_{3k+2},x_{3k+1})+G(x_{3k},x_{3k},x_{3k+3})+G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]} \\&\quad \le \beta _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2}\\&\qquad +\beta _{2} G(x_{3k},x_{3k+1},x_{3k+2})+\beta _{3} G(x_{3k},x_{3k+1},x_{3k+2})+\beta _{4} G(x_{3k},x_{3k+1},x_{3k+2}) \\&\quad \le \beta _{1}\frac{G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})}{2} \\&\qquad +(\beta _{2}+\beta _{3}+\beta _{4})G(x_{3k+1},x_{3k+2},x_{3k+3}) \end{aligned}$$\end{document}$$which implies that$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} (1-\beta _{2}-\beta _{3}-\beta _{4})G(x_{3k+1},x_{3k+2},x_{3k+3})& \le \frac{\beta _{1}}{2} G(x_{3k},x_{3k+1},x_{3k+2}) +\frac{\beta _{1}}{2} G(y_{3k},y_{3k+1},y_{3k+2}) \nonumber \\ \Rightarrow G(x_{3k+1},x_{3k+2},x_{3k+3})& \le \frac{\beta _{1}}{2(1-\beta _{2}-\beta _{3}-\beta _{4})} [G(x_{3k},x_{3k+1},x_{3k+2}) \nonumber \\&\quad +G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$Similarly we can show that$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} (1-\beta _{2}-\beta _{3}-\beta _{4})G(y_{3k+1},y_{3k+2},y_{3k+3})& \le \frac{\beta _{1}}{2} G(x_{3k},x_{3k+1},x_{3k+2}) +\frac{\beta _{1}}{2} G(y_{3k},y_{3k+1},y_{3k+2}) \nonumber \\ \Rightarrow G(y_{3k+1},y_{3k+2},y_{3k+3})& \le \frac{\beta _{1}}{2(1-\beta _{2}-\beta _{3}-\beta _{4})} [G(x_{3k},x_{3k+1},x_{3k+2}) \nonumber \\&\quad +G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$

Adding ([7](#Equ7){ref-type=""}) and ([8](#Equ8){ref-type=""}) we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x_{3k+1},x_{3k+2},x_{3k+3}) + & G(y_{3k+1},y_{3k+2},y_{3k+3}) \\& \le\; \frac{\beta _{1}}{1-\beta _{2}-\beta _{3}-\beta _{4}}[G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})]\\& =\; k [G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$where $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$k=\frac{\beta _{1}}{1-\beta _{2}-\beta _{3}-\beta _{4}}$$\end{document}$.

Also, we can show that$$\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\begin{aligned} & G(x_{3k+2},x_{3k+3},x_{3k+4}) + G(y_{3k+2},y_{3k+3},y_{3k+4}) \\& \quad \le k [G(x_{3k+1},x_{3k+2},x_{3k+3})+G(y_{3k+1},y_{3k+2},y_{3k+3})] \\& \quad \le k^{2}[G(x_{3k},x_{3k+1},x_{3k+2})+G(y_{3k},y_{3k+1},y_{3k+2})] \end{aligned}$$\end{document}$$Now if $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$G(x_{n},x_{n+1},x_{n+2})+G(y_{n},y_{n+1},y_{n+2})=G_{n}$$\end{document}$ then $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$G_{n}\le kG_{n-1}\le \cdots \le k^{n}G_{0}$$\end{document}$.

By property ([3](#Equ3){ref-type=""}) of Definition 1, we have$$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\begin{aligned} G(x_{n},x_{n+1},x_{n+1})+G(y_{n},y_{n+1},y_{n+1}) \le G_{n} \le k^{n}G_{0} \end{aligned}$$\end{document}$$

So, for $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$m>n$$\end{document}$ we have$$\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\begin{aligned} G(x_{n},x_{m},x_{m}) & + G(y_{n},y_{m},y_{m}) \\& \le s[G(x_{n},x_{n+1},x_{n+1})+G(x_{n+1},x_{m},x_{m})+G(y_{n},y_{n+1},y_{n+1})+G(y_{n+1},y_{m},y_{m})]\\& \le s[G(x_{n},x_{n+1},x_{n+1})+G(y_{n},y_{n+1},y_{n+1})] \\& \quad +s^{2}[G(x_{n+1},x_{n+2},x_{n+2})+G(y_{n+1},y_{n+2},y_{n+2})] \\& \quad + \cdots +s^{m-n}[G(x_{m-1},x_{m},x_{m})+G(y_{m-1},y_{m},y_{m})] \\& \le sk^{n}G_{0}+s^{2}k^{n+1}G_{0}+ \cdots +s^{m-n}k^{m-1}G_{0} \\& < sk^{n}[1+(sk)+(sk)^{2}+ \cdots ]G_{0} \\& = \frac{sk^{n}}{1-sk}G_{0} \rightarrow 0 \quad {\rm as}\;n \rightarrow \infty \end{aligned}$$\end{document}$$

Therefore $\documentclass[12pt]{minimal}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\{x_{n}\}$$\end{document}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$\{y_{n}\}$$\end{document}$ are Cauchy sequences in *X*. Since *X* is complete $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space, there exist $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y \in X$$\end{document}$ such that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x_{n} \rightarrow x$$\end{document}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$y_{n} \rightarrow y$$\end{document}$ as $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$n \rightarrow \infty$$\end{document}$.

Now, we will show that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x=S(x,y)$$\end{document}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$y=S(y,x)$$\end{document}$. Suppose on contrary that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x\not =S(x,y)$$\end{document}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$y\not =S(y,x)$$\end{document}$, so that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$G(x,S(x,y),S(x,y))=l_{1}>0$$\end{document}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$G(y,y,S(y,x))=l_{2}>0$$\end{document}$. Consider the following and using inequality ([6](#Equ6){ref-type=""}), we get$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} l_{1}& = G(x,S(x,y),S(x,y)) \\& \le s[G(x,x_{n+1},x_{n+1})+G(x_{n+1},S(x,y),S(x,y))] \\& = s[G(x,x_{n+1},x_{n+1})+G(S(x_{n},y_{n}),S(x,y),S(x,y))] \\& \le sG(x,x_{n+1},x_{n+1})+s\left[ \beta _{1}\frac{G(x_{n},x,x)+G(y_{n},y,y)}{2} \right. \\& \quad \left. +\beta _{2}\frac{G(x_{n},x_{n},S(x_{n},y_{n}))G(x,x,S(x,y))}{1+s[G(x_{n},x_{n},S(x,y))+G(x,x,S(x_{n},y_{n}))+G(x_{n},x,x)+G(y_{n},y,y)]} \right. \\& \quad \left. +\beta _{3}\frac{G(x,x,S(x,y))G(x,x,S(x,y))}{1+s[G(x,x,S(x,y))+G(x,x,S(x,y))+G(x_{n},x,x)+G(y_{n},y,y)]} \right. \\& \quad \left. +\beta _{4}\frac{G(x,x,S(x,y))G(x_{n},x_{n},S(x_{n},y_{n}))}{1+s[G(x,x,S(x_{n},y_{n}))+G(x_{n},x_{n},S(x,y))+G(x_{n},x,x)+G(y_{n},y,y)]}\right] \\& = sG(x,x_{n+1},x_{n+1})+\frac{s\beta _{1}}{2}G(x_{n},x,x)+\frac{s\beta _{1}}{2}G(y_{n},y,y) \\& \quad +s\beta _{2}\frac{G(x_{n},x_{n},x_{n+1})G(x,x,S(x,y))}{1+s[G(x_{n},x_{n},S(x,y))+G(x,x,x_{n+1})+G(x_{n},x,x)+G(y_{n},y,y)]} \\& \quad +s\beta _{3}\frac{G(x,x,S(x,y))G(x,x,S(x,y))}{1+s[G(x,x,S(x,y))+G(x,x,S(x,y))+G(x_{n},x,x)+G(y_{n},y,y)]} \\& \quad +s\beta _{4}\frac{G(x,x,S(x,y))G(x_{n},x_{n},x_{n+1})}{1+s[G(x,x,x_{n+1})+G(x_{n},x_{n},S(x,y))+G(x_{n},x,x)+G(y_{n},y,y)]} \end{aligned}$$\end{document}$$Taking limit as $\documentclass[12pt]{minimal}
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                \begin{document}$$n \rightarrow \infty$$\end{document}$ we get$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x,S(x,y),S(x,y)) \le 0+0+0+s.\beta _{2}.0+\beta _{3}G(x,x,S(x,y))+s.\beta _{4}.0 \\&\quad \Rightarrow G(x,S(x,y),S(x,y)) \le 2s\beta _{3}G(x,S(x,y),S(x,y)) \\&\quad \Rightarrow (1-2s\beta _{3})G(x,S(x,y),S(x,y)) \le 0 \end{aligned}$$\end{document}$$

Therefore $\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,S(x,y),S(x,y))=0$$\end{document}$.

Which implies that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x=S(x,y)$$\end{document}$. Similarly we can prove that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$y=S(y,x)$$\end{document}$.

Also, we can prove that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x=T(x,y)$$\end{document}$, $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$y=T(y,x)$$\end{document}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
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                \begin{document}$$x=R(x,y)$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$y=R(y,x)$$\end{document}$.

Hence, (*x*, *y*) is a common coupled fixed point of *S*, *T* and *R*.

In order to prove the uniqueness of the common coupled fixed point of *S*, *T* and *R*, if possible let (*p*, *q*) be the second common copuled fixed point of *S*, *T* and *R*.

Then by using inequality ([6](#Equ6){ref-type=""}) we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x,p,p) = G(S(x,y),T(p,q),R(p,q)) \nonumber \\&\quad \le \beta _{1}\frac{G(x,p,p)+G(y,q,q)}{2} \nonumber \\&\qquad +\beta _{2}\frac{G(x,x,S(x,y))G(p,p,T(p,q))}{1+s[G(x,x,T(p,q))+G(p,p,S(x,y))+G(x,p,p)+G(y,q,q)]} \nonumber \\&\qquad +\beta _{3}\frac{G(p,p,T(p,q))G(p,p,R(p,q))}{1+s[G(p,p,R(p,q))+G(p,p,T(p,q))+G(x,p,p)+G(y,q,q)]} \nonumber \\&\qquad +\beta _{4}\frac{G(p,p,R(p,q))G(x,x,S(x,y))}{1+s[G(p,p,S(x,y))+G(x,x,R(p,q))+G(x,p,p)+G(y,q,q)]} \nonumber \\&\quad \Rightarrow G(x,p,p) \le \beta _{1}\frac{G(x,p,p)+G(y,q,q)}{2}+\beta _{2}.0+\beta _{3}.0+\beta _{4}.0 \nonumber \\&\quad \Rightarrow \left( 1-\frac{\beta _{1}}{2}\right) G(x,p,p) \le \frac{\beta _{1}}{2}G(y,q,q) \nonumber \\&\quad \Rightarrow G(x,p,p) \le \frac{\beta _{1}}{2-\beta _{1}}G(y,q,q) \end{aligned}$$\end{document}$$Similarly$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(y,q,q) \le \frac{\beta _{1}}{2-\beta _{1}}G(x,p,p) \end{aligned}$$\end{document}$$Adding ([9](#Equ9){ref-type=""}) and ([10](#Equ10){ref-type=""}) we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned}&G(x,p,p)+G(y,q,q) \le \frac{\beta _{1}}{2-\beta _{1}}[G(x,p,p)+G(y,q,q)] \\&\quad \Rightarrow \left( 1-\frac{\beta _{1}}{2-\beta _{1}}\right) [G(x,p,p)+G(y,q,q)] \le 0 \\&\quad \Rightarrow \frac{2-2\beta _{1}}{2-\beta _{1}}[G(x,p,p)+G(y,q,q)] \le 0 \end{aligned}$$\end{document}$$

But $\documentclass[12pt]{minimal}
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                \begin{document}$$\frac{2-2\beta _{1}}{2-\beta _{1}} > 0$$\end{document}$. Therefore $\documentclass[12pt]{minimal}
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                \begin{document}$$G(x,p,p)+G(y,q,q)=0$$\end{document}$

Which implies that $\documentclass[12pt]{minimal}
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                \begin{document}$$x=p$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$y=q$$\end{document}$$\documentclass[12pt]{minimal}
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                \begin{document}$$\Rightarrow (x,y)=(p,q)$$\end{document}$.

Thus *S*, *T*, *R* have a unique common coupled fixed point.

This completes the proof. $\documentclass[12pt]{minimal}
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                \begin{document}$$\square$$\end{document}$

**Corollary 19** {#FPar21}
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*Let* (*X*, *G*) *be a complete symmetric*$\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-*metric space with parameter*$\documentclass[12pt]{minimal}
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                \begin{document}$$s\ge 1$$\end{document}$*and let the mapping*$\documentclass[12pt]{minimal}
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                \begin{document}$$S:X^{2} \rightarrow X$$\end{document}$*satisfying*$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(S(x,y),S(u,v),S(a,b))& \le \beta _{1}\frac{G(x,u,a)+G(y,v,b)}{2} \\& \quad +\beta _{2}\frac{G(x,x,S(x,y))G(u,u,S(u,v))}{1+s[G(x,x,S(u,v))+G(u,u,S(x,y))+G(x,u,a)+G(y,v,b)]} \\& \quad +\beta _{3}\frac{G(u,u,S(u,v))G(a,a,S(a,b))}{1+s[G(u,u,S(a,b))+G(a,a,S(u,v))+G(x,u,a)+G(y,v,b)]} \\& \quad +\beta _{4}\frac{G(a,a,S(a,b))G((x,x,S(x,y))}{1+s[G(a,a,S(x,y))+G(x,x,S(a,b))+G(x,u,a)+G(y,v,b)]} \\ \end{aligned}$$\end{document}$$*for all*$\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,u,v,a,b \in X$$\end{document}$*and*$\documentclass[12pt]{minimal}
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                \begin{document}$$\beta _{1},\beta _{2},\beta _{3},\beta _{4}$$\end{document}$*are non-negative realnumbers with*$\documentclass[12pt]{minimal}
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                \begin{document}$$\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}< 1$$\end{document}$. *ThenShas a unique coupled fixed point*.

*Example 20* {#FPar22}
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Let $\documentclass[12pt]{minimal}
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                \begin{document}$$X=[0,1]$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$G:X^{3} \rightarrow R$$\end{document}$ be defined by$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(x,y,z)=\max \{d(x,y),d(y,z),d(z,x)\}\;{\text{with }} d(x,y)=|x-y|^{2} {\text { where }} s=2. \end{aligned}$$\end{document}$$

Define$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} S(x,y)& = \frac{x-y}{6}\\ T(u,v)& = \frac{u-v}{12}\\ R(a,b)& = \frac{a-b}{24} \end{aligned}$$\end{document}$$for all $\documentclass[12pt]{minimal}
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                \begin{document}$$x,y,u,v,a,b \in X$$\end{document}$. Suppose $\documentclass[12pt]{minimal}
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                \begin{document}$$\alpha =\frac{1}{4}$$\end{document}$ and $\documentclass[12pt]{minimal}
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                \begin{document}$$0< \alpha _{2},\alpha _{3},\alpha _{4},\alpha _{5},\alpha _{6},\alpha _{7},\alpha _{8},\alpha _{9}<1$$\end{document}$. Now we have$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G(S(x,y),T(u,v),R(a,b))& = G\left( \frac{x-y}{6},\frac{u-v}{12},\frac{a-b}{24}\right) \\& = \max \left\{ \left| \frac{2(x-y)-(u-v)}{12}\right| ^{2},\left| \frac{2(u-v)-(a-b)}{24}\right| ^{2}, \quad \left| \frac{4(x-y)-(a-b)}{24}\right| ^{2}\right\} \end{aligned}$$\end{document}$$Let $\documentclass[12pt]{minimal}
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                \begin{document}$$\max =\frac{2(x-y)-(u-v)}{12}$$\end{document}$, then$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} G\left( \frac{x-y}{6},\frac{u-v}{12},\frac{a-b}{24}\right)& = \left| \frac{2(x-y)-(u-v)}{12}\right| ^{2} \\& \le \frac{1}{4}\left\{ \frac{G(x,u,a)+G(y,v,b)}{2}\right\} \\& \le \frac{1}{4}\left\{ \frac{G(x,u,a)+G(y,v,b)}{2}\right\} +\alpha _{2}\frac{G(S(x,y),T(u,v),R(a,b))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)}\\& \quad +\alpha _{3}\frac{G(S(x,y),T(u,v),R(a,b))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} +\alpha _{4}\frac{G(x,x,S(x,y))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \\& \quad +\alpha _{5}\frac{G(x,x,S(x,y))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} +\alpha _{6}\frac{G(u,u,T(u,v))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \\& \quad +\alpha _{7}\frac{G(u,u,T(u,v))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} +\alpha _{8}\frac{G(a,a,R(a,b))G(x,u,a)}{1+G(x,u,a)+G(y,v,b)} \\& \quad +\alpha _{9}\frac{G(a,a,R(a,b))G(y,v,b)}{1+G(x,u,a)+G(y,v,b)} \end{aligned}$$\end{document}$$Obviously all the conditions of Theorem 16 are satisfied. Also, (0, 0) is the unique common coupled fixed point of *S*, *T* and *R*.

Conclusion {#Sec3}
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We prove the existence and uniqueness of common coupled fixed point theorems for three mappings with a new rational contractive conditions in $\documentclass[12pt]{minimal}
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                \begin{document}$$G_{b}$$\end{document}$-metric space. Our results improve and genaralise the similar results in *b*-metric and *G*-metric spaces. These results may be extended to other spaces.

Bulbul Khomdram, Yumnam Rohen and Thokchom Chhatrajit Singh contributed equally to this work

All authors contributed equally in preparation of the final manuscript. All the authors read and approved the final manuscript.

Acknowlegdements {#FPar23}
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Authors would like to thank the referees for their suggestions.

Competing interests {#FPar24}
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The authors declare that they have no competing interests.
