Quadruple fixed point theorems for compatible type mappings without mixedg-monotone property in partially orderedb-metric spaces

Abstract

In this paper, we establish quadruple fixed point theorems for compatible type mappings in partially ordered $b$-metric spaces without the mixed $g$-monotone property under some conditions. Also, an example is given to show our results are real generalization of known results in quadruple fixed point theory.

Introduction and preliminaries

The concept of $b$-metric space was introduced and studied by Bakhtin [7] and later used by Czerwik [13, 14] which is a generalization of the usual metric space. After that, several papers have dealt with fixed point theory for single-valued and multi-valued operators in $b$-metric spaces have been obtained (see, e.g., [2, 5, 11, 12, 26, 28]).

Existence of coupled fixed point was introduced by Guo and Lakshmikantham [16]. In 2006, Gnana-Bhaskar and Lakshmikantham [10] introduced the concept of mixed monotone property in partially ordered metric space. Afterward, Lakshmikantham and Ćirić in [24] extended these results by giving the definition of the $g$-monotone property. Many papers have been reported on coupled fixed point theory (see, e.g., [1, 3, 4, 9, 18, 27]). In 2011, Vasile Berinde and Marin Borcut [8] extended and generalized the results of [10] and introduced the concept of a tripled fixed point and the mixed monotone property of a mapping $F:X^3\to X.$ For more details on tripled fixed point results, we refer to [6, 26]. Recently, Karapinar and Luong [19] introduced the concept of a quadruple fixed point and the mixed monotone property of a mapping $F:X^4\to X$ and they presented some new quadruple fixed point results. For a survey of quadruple fixed point theorems and related fixed points we refer the reader to [20–22].

Definition 1.1

[14]. Let $X$ be a nonempty set and $s\ge 1$ a given real number. A function $d:X^2\to R^{+}$ is called a $b$-metric provided that, for all $x,y,z\in X,$ the following conditions hold:

• ($b_1$) $d(x,y)=0$ if and only if $x=y,$

• ($b_2$) $d(x,y)=d(y,x),$

• ($b_3$) $d(x,z)\le s[d(x,y)+d(y,z)]$.

The pair $(X,d)$ is called a $b$-metric space with parameter $s.$

Remark 1.1

It is obvious that any metric space must be a $b$-metric space where a $b$-metric space is a metric space when $s=1.$ The following example show that in general a $b$-metric need not necessarily be a metric space (see also [29]).

Example 1.1

[2]. Let $(X,d)$ be a metric space and $\mathit{\rho }(x,y)={(d(x,y))}^p,$ where $p>1$ is a real number. Then $\mathit{\rho }$ is a $b$-metric with $s=2^{p-1}.$ However, if $(X,d)$ is a metric space, then $(X,\mathit{\rho })$ is not necessarily a metric space. For example, if $X=R$ is the set of real numbers and $d(x,y)=|x-y|$ is the usual Euclidean metric, then $\mathit{\rho }(x,y)={(x-y)}^2$ is a $b$-metric on $R$ with $s=2,$ but is not a metric on $R.$

Definition 1.2

[11]. Let $(X,d)$ be a $b$-metric space. Then a sequence $\{x_n\}$ in $X$ is called

• (i) $b$-convergent if and only if there exists $x\in X$ such that $d(x_n,x)\to 0$ as $n\to \infty .$ In this case, we write ${lim}_{n\to \infty }{x}_n=x,$

• (ii) $b$-Cauchy if and only if $d(x_n,x_m)\to 0$ as $n,m\to \infty .$

Proposition 1.1

([11] Remark 2.1) In a$b$-metric space the following assertions hold:

• (i) A$b$-convergent sequence has a unique limit.

• (ii) Each$b$-convergent sequence is$b$-Cauchy.

• (iii) In general, a$b$-metric is not continuous.

Definition 1.3

[11]. Let $(X,d)$ and $(\overline{X},\overline{d})$ be two $b$-metric spaces.

• (i) The space $(X,d)$ is $b$-complete if every $b$-Cauchy sequence in $X$$b$-converges.

• (ii) A function $f:X\to \overline{X}$ is $b$-continuous at a point $x\in X$ if it is $b$-sequentially continuous at $x,$ that is, whenever $\{x_n\}$ is $b$-convergent to $x$, $\{f(x_n)\}$ is $b$-convergent to $f(x).$

Definition 1.4

[11]. The $b$-metric space $(X,d)$ is $b$-complete if every $b$-Cauchy sequence in $X$$b$-converges.

It should be noted that, in general a $b$-metric function $d(x,y)$ for $s>1$ is not jointly continuous in all its variables. The following example on a $b$-metric which is not continuous.

Example 1.2

[17]. Let $X=\mathbf{N}\cup \{\infty \}$ and let $d:X\times X\to R$ be defined by

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

Then considering all possible cases, it can be checked that for all $m,n,p\in X,$ we have

$$\begin{array}{c}\hfill d(m,p)\le \frac{5}{2}(d(m,n)+d(n,p)).\end{array}$$

Thus, $(X,d)$ is a $b$-metric space (with $s=\frac{5}{3}$ ). Let $x_n=2n$ for each $n\in \mathbf{N}.$ Then

$$\begin{array}{c}\hfill d(2n,\infty )=\frac{1}{2n}\to 0\text{as}n\to \infty ,\end{array}$$

that is, $x_n\to \infty ,$ but $d(x_n,1)=2\nrightarrow 5=d(\infty ,1)$ as $n\to \infty .$

Since, in general, a $b$-metric is not continuous, we need the following lemma about the $b$-convergent sequences in the proof of our main result.

Lemma 1.1

[2].Let$(X,d)$be a$b$-metric space with$s\ge 1,$and suppose that$\{x_n\}$and $\{y_n\}$$b$-converge to$x,y,$respectively. Then, we have

$$\begin{array}{c}\hfill \frac{1}{s^2}d(x,y)\le \underset{n\to \infty }{lim\; inf}d(x_n,y_n)\le \underset{n\to \infty }{lim\; sup}d(x_n,y_n)\le s^{2}d(x,y).\end{array}$$

In particular, if $x=y$then$\underset{n\to \infty }{lim}d(x_n,y_n)=0.$Moreover, for each$z\in X$we have

$$\begin{array}{c}\hfill \frac{1}{s}d(x,z)\le \underset{n\to \infty }{lim\; inf}d(x_n,z)\le \underset{n\to \infty }{lim\; sup}d(x_n,z)\le sd(x,z).\end{array}$$

Definition 1.5

Let $X$ be a nonempty set and let $F:X^4\to X,$$g:X\to X.$ An element $(x,y,z,w)\in X^4$ is called

• (i) [19] a quadruple fixed point of $F$ if

  $$\begin{array}{c}\hfill F(x,y,z,w)=x,F(y,z,w,x)=y,F(z,w,x,y)=z\mathrm{and}F(w,x,y,z)=w.\end{array}$$

• (ii) [25] a quadruple coincidence point of $F$ and $g$ if

  $$\begin{array}{c}\hfill F(x,y,z,w)=gx,F(y,z,w,x)=gy,F(z,w,x,y)=gz\mathrm{and}F(w,x,y,z)=gw.\end{array}$$

  $(gx,gy,gz,gw)$ is said to be a quadruple point of coincidence of $F$ and $g.$

• (iii) [25] a quadruple common fixed point of $F$ and $g$ if

  $$\begin{array}{cc}& F(x,y,z,w)=gx=x,F(y,z,w,x)=gy=y,F(z,w,x,y)=gz=z\mathrm{and}F(w,x,y,z)=gw=w.\hfill \end{array}$$

Definition 1.6

[25]. Let $(X,⪯)$ be a partially ordered set and let $F:X^4\to X,$$g:X\to X.$ The mapping $F$ is said to have the mixed $g$-monotone property if for any $x,y,z,w\in X,$

$$\begin{array}{c}\hfill x_1,x_2\in X,g{x}_1\le g{x}_2\text{implies}F(x_1,y,z,w)\le F(x_2,y,z,w),\\ \hfill y_1,y_2\in X,g{y}_1\le g{y}_2\text{implies}F(x,y_2,z,w)\le F(x,y_1,z,w),\\ \hfill z_1,z_2\in X,g{z}_1\le g{z}_2\text{implies}F(x,y,z_1,w)\le F(x,y,z_2,w),\\ \hfill \text{and}{w}_1,w_2\in X,g{w}_1\le g{w}_2\text{implies}F(x,y,z,w_2)\le F(x,y,z,w_1).\end{array}$$

In particular, when $g=i_X,$ then from [19] we say that $F$ has the mixed monotone property that is, for any $x,y,z,w\in X,$

$$\begin{array}{c}\hfill x_1,x_2\in X,x_1\le x_2\text{implies}F(x_1,y,z,w)\le F(x_2,y,z,w),\\ \hfill y_1,y_2\in X,y_1\le y_2\text{implies}F(x,y_2,z,w)\le F(x,y_1,z,w),\\ \hfill z_1,z_2\in X,z_1\le z_2\text{implies}F(x,y,z_1,w)\le F(x,y,z_2,w),\\ \hfill \text{and}{w}_1,w_2\in X,w_1\le w_2\text{implies}F(x,y,z,w_2)\le F(x,y,z,w_1).\end{array}$$

The concept of an altering distance function was introduced by Khan et al. [23] as follows.

Definition 1.7

[23]. A function $\mathit{\psi }:[0,\infty )\to [0,\infty )$ is called an altering distance function if

• (i) $\mathit{\psi }$ is non-decreasing and continuous,

• (ii) $\mathit{\psi }(t)=0$ if and only if $t=0.$

Definition 1.8

[1]. The mappings $F:X\times X\to X$ and $f:X\to X$ are called $w$-compatible if $f(F(x,y))=F(fx,fy)$ whenever $f(x)=F(x,y)$ and $f(y)=F(y,x).$

In 2012, Dorić et al. [15] established coupled fixed point results without the mixed monotone property. This property is automatically satisfied in the case of a totally ordered space. Therefore, these results can be applied in a much wider class of problems. Now, we state a property due to Dorić et al. [15].

If elements $x,y$ of a partially ordered set $(X,⪯)$ are comparable (i.e., $x⪯y$ or $y⪰x$ holds) we will write $x\asymp y.$ Let $g:X\to X$ and $F:X\times X\to X.$ We will consider the following condition:

$$\begin{array}{cc}& \mathrm{if}x,y,u,v\in X\text{are;such;that;}gx\asymp F(x,y)=gu\mathrm{then}F(x,y)\asymp F(u,v).\hfill \end{array}$$

(1.1)

In particular, when $g=i_X,$ it reduces to

$$\begin{array}{c}\hfill \mathrm{for}\mathrm{all}x,y,v\mathrm{if}x\asymp F(x,y)\mathrm{then}F(x,y)\asymp F(F(x,y),v).\end{array}$$

(1.2)

The aim of this paper is to extend the property due to Dorić et al. [15] to the case of mappings $g:X\to X,$$F:X^4\to X,$ and show that a mixed monotone property in quadruple fixed point results for mappings in partially ordered $b$-metric spaces can be replaced by another property which is often easy to check in the case of a totally ordered space. We prove the existence of quadruple coincidence and uniqueness quadruple common fixed point theorems for a compatible and $w$-compatible mappings satisfying generalized contraction in partially ordered $b$-metric spaces without the mixed $g$-monotone property. Also, we state an example showing that our results are effective.

Quadruple coincidence point theorems

Let $g:X\to X$ and $F:X^4\to X.$ We consider the following condition:

$$\begin{array}{c}\hfill \text{if}x,y,z,w,u,v,r,t\in X\text{are;such;that;}gx\asymp F(x,y,z,w)=gu\text{then,}F(x,y,z,w)\asymp F(u,v,r,t).\end{array}$$

(2.1)

In particular, when $g=i_X,$ it reduces to

$$\begin{array}{c}\hfill \mathrm{for}\mathrm{all}x,y,v,r,t\mathrm{if}x\asymp F(x,y,z,w)\mathrm{then}F(x,y,z,w)\asymp F(F(x,y,z,w),v,r,t).\end{array}$$

(2.2)

We will show by examples the condition (2.1), ((2.2) resp.) may be satisfied when $F$ does not have the mixed $g$-monotone property, (monotone property resp.).

Example 2.1

Let $X=\{a,b,c,d\},$   $⪯=\{(a,a),(b,b),(c,c),(d,d),(a,b),(c,d)\},$

$$\begin{array}{c}\hfill g:\left(\begin{array}{cccc}a& b& c& d\\ c& d& c& d\end{array}\right),F:\left(\begin{array}{cccc}(a,y,z,w)& (b,y,z,w)& (c,y,z,w)& (d,y,z,w)\\ b& a& c& d\end{array}\right),\end{array}$$

for all $y,z,w\in X.$ Since $ga=c⪯gb=d$ but $F(a,y,z,w)⪰F(b,y,z,w)$ for all $y,z,w\in X,$ the mapping $F$ does not have the mixed $g$-monotone property. But it has property (2.1) where

• (i) For each $y,z,w\in X,$ we get $gc\asymp F(c,y,z,w)$ and $F(c,y,z,w)\asymp F(c,v,r,t)$ for all $v,r,t\in X.$

• (ii) For each $y,z,w\in X,$ we get $gd\asymp F(d,y,z,w)$ and $F(d,y,z,w)\asymp F(d,v,r,t)$ for all $v,r,t\in X.$

• (iii) The other two cases are trivial.

Example 2.2

Let $X=\{a,b,c,d\},$   $⪯=\{(a,a),(b,b),(c,c),(d,d),(a,b),(c,d)\},$

$$\begin{array}{c}\hfill F:\left(\begin{array}{cccc}(a,y,z,w)& (b,y,z,w)& (c,y,z,w)& (d,y,z,w)\\ b& a& c& d\end{array}\right)\end{array}$$

for all $y,z,w\in X.$ Since $a⪯b$ but $F(a,y,z,w)=b⪰a=F(b,y,z,w)$ for all $y,z,w\in X,$ the mapping $F$ does not have the mixed monotone property. But it has property (2.2) since

• (i) For each $y,z,w\in X,$ we get $a\asymp F(a,y,z,w)$ and $F(a,y,z,w)=b\asymp a=F(F(a,y,z,w),v,r,t)$ for all $v,r,t\in X.$

• (ii) For each $y,z,w\in X,$ we get $b\asymp F(b,y,z,w)$ and $F(b,y,z,w)=a\asymp b=F(F(b,y,z,w),v,r,t)$ for all $v,r,t\in X.$

• (iii) The other two cases are trivial.

Definition 2.1

Let $(X,d)$ be a $b$-metric space and let $g:X\to X,$$F:X^4\to X.$ The mappings $g$ and $F$ are said to be compatible if

$$\begin{array}{cc}& \underset{n\to \infty }{lim}d(gF(x_n,y_n,z_n,w_n),F(g{x}_n,g{y}_n,g{z}_n,g{w}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(y_n,z_n,w_n,x_n),F(g{y}_n,g{z}_n,g{w}_n,g{x}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(z_n,w_n,x_n,y_n),F(g{z}_n,g{w}_n,g{x}_n,g{y}_n))=0\hfill \\ \hfill \text{and}& \underset{n\to \infty }{lim}d(gF(w_n,x_n,y_n,z_n),F(g{w}_n,g{x}_n,g{y}_n,g{z}_n))=0,\hfill \end{array}$$

hold whenever $\{x_n\},$$\{y_n\},$$\{z_n\}$ and $\{w_n\}$ are sequences in $X$ such that

$$\begin{array}{cc}& \underset{n\to \infty }{lim}F(x_n,y_n,z_n,w_n)=\underset{n\to \infty }{lim}g{x}_n,\underset{n\to \infty }{lim}F(y_n,z_n,w_n,x_n)=\underset{n\to \infty }{lim}g{y}_n,\hfill \\ \hfill & \underset{n\to \infty }{lim}F(z_n,w_n,x_n,y_n)=\underset{n\to \infty }{lim}g{z}_n\text{and}\underset{n\to \infty }{lim}F(w_n,x_n,y_n,z_n)=\underset{n\to \infty }{lim}g{w}_n.\hfill \end{array}$$

Definition 2.2

The mappings $F:X^4\to X$ and $g:X\to X$ are called $w$-compatible if $g(F(x,y,z,w))=F(gx,gy,gz,gw)$ whenever $gx=F(x,y,z,w),$$gy=F(y,z,w,x),$$gz=F(z,w,x,y)$ and $gw=F(w,x,y,z).$

Remark 2.1

In an altering distance function $\mathit{\psi }:[0,\infty )\to [0,\infty ),$ since $\mathit{\psi }$ is non-decreasing then for any $a,b,c,d\in [0,\infty )$ the following holds.

$$\begin{array}{c}\hfill \mathit{\psi }(max\{a,b,c,d\})=max\{\mathit{\psi }(a),\mathit{\psi }(b),\mathit{\psi }(c),\mathit{\psi }(d)\}.\end{array}$$

The triple $(X,d,⪯)$ is called a partially ordered $b$-metric space if $(X,⪯)$ is a partially ordered set and $(X,d)$ is a $b$-metric space.

Our first result is the following.

Theorem 2.1

Let$(X,d,⪯)$be a partially ordered complete$b$-metric space with parameter$s\ge 1$.Let$F:X^4\to X$and$g:X\to X$be two mappings such that the following hold:

• (i) $g$and$F$are$b$-continuous,

• (ii) $F(X^4)\subseteq g(X),$ $g$ and $F$ are compatible,

• (iii) $g$ and $F$ satisfy property (2.1),

• (iv) there exist $x_0,y_0,z_0,w_0\in X$such that$g{x}_0\asymp F(x_0,y_0,z_0,w_0),$$g{y}_0\asymp F(y_0,z_0,w_0,x_0),$$g{z}_0\asymp F(z_0,w_0,x_0,y_0)$and$g{w}_0\asymp F(w_0,x_0,y_0,z_0),$

• (v) there exist an altering distance function $\mathit{\psi }$ and $\mathit{\varphi }:{[0,\infty )}^4\to [0,\infty )$ is continuous with $\mathit{\varphi }(t_1,t_2,t_3,t_4)=0$ if and only if $t_1=t_2=t_3=t_4=0$ such that

  $$\begin{array}{c}\hfill \mathit{\psi }(sd(F(x,y,z,w),F(u,v,r,t)))\le \mathit{\psi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\})-\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)),\end{array}$$

  (2.3)

  for all $x,y,z,w,u,v,r,t\in X$ and $gx\asymp gu,$ $gy\asymp gv,$ $gz\asymp gr$ and $gw\asymp gt.$

Then,$F$and$g$have a quadruple coincidence point.

Proof

Let $x_0,y_0,z_0,w_0\in X$ be such that condition (iv) holds. Since $F(X^4)\subseteq g(X),$ then we can choose $x_1,y_1,z_1,w_1\in X$ such that

$$\begin{array}{c}\hfill g{x}_1=F(x_0,y_0,z_0,w_0),g{y}_1=F(y_0,z_0,w_0,x_0),g{z}_1=F(z_0,w_0,x_0,y_0)\text{and}g{w}_1=F(w_0,x_0,y_0,z_0).\end{array}$$

(2.4)

By continuing this process, we can construct sequences $\{x_n\},\{y_n\},\{z_n\}$ and $\{w_n\}$ in $X$ such that

$$\begin{array}{c}\hfill g{x}_{n+1}=F(x_n,y_n,z_n,w_n),g{y}_{n+1}=F(y_n,z_n,w_n,x_n),g{z}_{n+1}=F(z_n,w_n,x_n,y_n),\text{and}g{w}_{n+1}=F(w_n,x_n,y_n,z_n)\text{for,all,}n\ge 0.\end{array}$$

(2.5)

We show that

$$\begin{array}{c}\hfill g{x}_n\asymp g{x}_{n+1},g{y}_n\asymp g{y}_{n+1},g{z}_n\asymp g{z}_{n+1}\text{and}g{w}_n\asymp g{w}_{n+1}\text{for}n\ge 0.\end{array}$$

(2.6)

So, we use the mathematical induction. By condition (iv) and using (2.4) we get $g{x}_0\asymp g{x}_1,$$g{y}_0\asymp g{y}_1,$$g{z}_0\asymp g{z}_1,$ and $g{w}_0\asymp g{w}_1.$ So (2.6) holds for $n=0.$ We assume that (2.6) holds for some $n>0,$ that is $g{x}_n\asymp g{x}_{n+1},$$g{y}_n\asymp g{y}_{n+1},$$g{z}_n\asymp g{z}_{n+1},$ and $g{w}_n\asymp g{w}_{n+1},$ we get

$$\begin{array}{c}\hfill g{x}_n=F(x_{n-1},y_{n-1},z_{n-1},w_{n-1})\asymp F(x_n,y_n,z_n,w_n)=g{x}_{n+1},\end{array}$$

$$\begin{array}{c}\hfill g{y}_n=F(y_{n-1},z_{n-1},w_{n-1},x_{n-1})\asymp F(y_n,z_n,w_n,x_n)=g{y}_{n+1},\end{array}$$

$$\begin{array}{c}\hfill g{z}_n=F(z_{n-1},w_{n-1},x_{n-1},y_{n-1})\asymp F(z_n,w_n,x_n,y_n)=g{z}_{n+1},\end{array}$$

$$\begin{array}{c}\hfill g{w}_n=F(w_{n-1},x_{n-1},y_{n-1},z_{n-1})\asymp F(w_n,x_n,y_n,z_n)=g{w}_{n+1}.\end{array}$$

Hence from condition (iii) we conclude that

$$\begin{array}{c}\hfill F(x_n,y_n,z_n,w_n)\asymp F(x_{n+1},y_{n+1},z_{n+1},w_{n+1}),F(y_n,z_n,w_n,x_n)\asymp F(y_{n+1},z_{n+1},w_{n+1},x_{n+1}),\\ \hfill F(z_n,w_n,x_n,y_n)\asymp F(z_{n+1},w_{n+1},x_{n+1},y_{n+1}),F(w_n,x_n,y_n,z_n)\asymp F(w_{n+1},x_{n+1},y_{n+1},z_{n+1}).\end{array}$$

So, $g{x}_{n+1}\asymp g{x}_{n+2},$$g{y}_{n+1}\asymp g{y}_{n+2},$$g{z}_{n+1}\asymp g{z}_{n+2}$ and $g{w}_{n+1}\asymp g{w}_{n+2}.$ Thus (2.6) holds for all $n\in \mathbf{N}.$ Suppose that for some $k\in \mathbf{N},$

$$\begin{array}{c}\hfill g{x}_k=g{x}_{k+1},g{y}_k=g{y}_{k+1},g{z}_k=g{z}_{k+1}\text{and}g{w}_k=g{w}_{k+1},\end{array}$$

then by (2.5) we get $g{x}_k=F(x_k,y_k,z_k,w_k),$$g{y}_k=F(y_k,z_k,w_k,x_k),$$g{z}_k=F(z_k,w_k,x_k,y_k)$ and $g{w}_k=F(w_k,x_k,y_k,z_k).$ Hence $(x_k,y_k,z_k,w_k)$ is a quadruple coincidence point of $F$ and $g.$ So, we assume that for all $n\in \mathbf{N}$ at least $g{x}_n\ne g{x}_{n+1}$ or $g{y}_n\ne g{y}_{n+1}$ or $g{z}_n\ne g{z}_{n+1}$ or $g{w}_n\ne g{w}_{n+1}.$ Since $g{x}_n\asymp g{x}_{n+1},$$g{y}_n\asymp g{y}_{n+1},$$g{z}_n\asymp g{z}_{n+1}$ and $g{w}_n\asymp g{w}_{n+1}$ for all $n\in \mathbf{N},$ then from (2.3) and (2.5) we obtain

$$\begin{array}{c}\hfill \mathit{\psi }(sd(g{x}_n,g{x}_{n+1})=\mathit{\psi }(sd(F(x_{n-1},y_{n-1},z_{n-1},w_{n-1}),F(x_n,y_n,z_n,w_n)))\le \mathit{\psi }(max\{d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n)\})-\mathit{\varphi }(d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n)),\end{array}$$

(2.7)

$$\begin{array}{c}\hfill \mathit{\psi }(sd(g{y}_n,g{y}_{n+1}))=\mathit{\psi }(sd(F(y_{n-1},z_{n-1},w_{n-1},x_{n-1}),F(y_n,z_n,w_n,x_n)))\le \mathit{\psi }(max\{d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n)\})-\mathit{\varphi }(d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n)),\end{array}$$

(2.8)

$$\begin{array}{c}\hfill \mathit{\psi }(sd(g{z}_n,g{z}_{n+1}))=\mathit{\psi }(sd(F(z_{n-1},w_{n-1},x_{n-1},y_{n-1}),F(z_n,w_n,x_n,y_n)))\le \mathit{\psi }(max\{d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n)\})-\mathit{\varphi }(d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n)),\end{array}$$

(2.9)

and

$$\begin{array}{c}\hfill \mathit{\psi }(sd(g{w}_n,g{w}_{n+1}))=\mathit{\psi }(sd(F(w_{n-1},x_{n-1},y_{n-1},z_{n-1}),F(w_n,x_n,y_n,z_n)))\le \mathit{\psi }(max\{d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n)\})-\mathit{\varphi }(d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n)).\end{array}$$

(2.10)

Set

$$\begin{array}{c}\hfill {\mathit{\delta }}_n=max\{d(g{x}_n,g{x}_{n+1}),d(g{y}_n,g{y}_{n+1}),d(g{z}_n,g{z}_{n+1}),d(g{w}_n,g{w}_{n+1})\}.\end{array}$$

From (2.7) to (2.10) and Remark 2.1, it follows that

$$\begin{array}{c}\hfill \mathit{\psi }(s{\mathit{\delta }}_n)=max\{\mathit{\psi }(sd(g{x}_n,g{x}_{n+1})),\mathit{\psi }(sd(g{y}_n,g{y}_{n+1})),\mathit{\psi }(sd(g{z}_n,g{z}_{n+1})),\mathit{\psi }(sd(g{w}_n,g{w}_{n+1}))\}\le \mathit{\psi }({\mathit{\delta }}_{n-1})-min\{\mathit{\varphi }(d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n)),\mathit{\varphi }(d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n)),\\ \hfill & \mathit{\varphi }(d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n)),\hfill \\ \hfill & \mathit{\varphi }(d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n))\}.\hfill \end{array}$$

(2.11)

and since $\mathit{\psi }$ is non-decreasing then from (2.11) we have that

$$\begin{array}{c}\hfill \mathit{\psi }({\mathit{\delta }}_n)\le \mathit{\psi }(s{\mathit{\delta }}_n)\le \mathit{\psi }({\mathit{\delta }}_{n-1})-min\{\mathit{\varphi }(d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n)),\mathit{\varphi }(d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n)),\mathit{\varphi }(d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n)),\\ \hfill & \mathit{\varphi }(d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n))\}.\hfill \end{array}$$

(2.12)

Hence,

$$\begin{array}{c}\hfill \mathit{\psi }({\mathit{\delta }}_n)\le \mathit{\psi }({\mathit{\delta }}_{n-1})\text{for ,all}n\in \mathbf{N}.\end{array}$$

(2.13)

Since $\mathit{\psi }$ is non-decreasing, we have ${\mathit{\delta }}_n\le {\mathit{\delta }}_{n-1}$ for all $n.$ Therefore, $\{{\mathit{\delta }}_n\}$ is a non-increasing sequence, so there is some $\mathit{\delta }\ge 0$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\mathit{\delta }}_n=\mathit{\delta }.\end{array}$$

Letting $n\to \infty ,$ in (2.12), we get

$$\begin{array}{c}\hfill \mathit{\psi }(\mathit{\delta })\le \mathit{\psi }(\mathit{\delta })-min\{\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n)),\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n)),\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n)),\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n))\}\le \mathit{\psi }(\mathit{\delta }).\end{array}$$

Hence,

$$\begin{array}{c}\hfill min\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n)),\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n)),\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n)),\\ \hfill & \underset{n\to \infty }{lim}\mathit{\varphi }(d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n))\}=0.\hfill \end{array}$$

That is

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n))=0\text{or,}\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n))=0\text{or}\underset{n\to \infty }{lim}\mathit{\varphi }(d(g{z}_{n-1},g{z}_n),d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n))=0\text{or}\\ \hfill & \underset{n\to \infty }{lim}\mathit{\varphi }(d(g{w}_{n-1},g{w}_n),d(g{x}_{n-1},g{x}_n),d(g{y}_{n-1},g{y}_n),d(g{z}_{n-1},g{z}_n))=0.\hfill \end{array}$$

So, using the properties of $\mathit{\varphi }$, we have

$$\begin{array}{cc}& \underset{n\to \infty }{lim}d(g{x}_{n-1},g{x}_n)=0,\underset{n\to \infty }{lim}d(g{y}_{n-1},g{y}_n)=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(g{z}_{n-1},g{z}_n)=0\text{and}\underset{n\to \infty }{lim}d(g{w}_{n-1},g{w}_n)=0.\hfill \end{array}$$

Thus, $\underset{n\to \infty }{lim}{\mathit{\delta }}_{n-1}=0.$ Therefore $\mathit{\delta }=0.$ Now, we show that $\{g{x}_n\},\{g{y}_n\},\{g{z}_n\},$ and $\{g{w}_n\}$ are $b$-Cauchy sequences in $(X,d),$ that is, we show that for every $\mathit{\epsilon }>0,$ there exists $k\in \mathbf{N}$ such that for all $m,n\ge k,$

$$\begin{array}{c}\hfill max\{d(g{x}_m,g{x}_n),d(g{y}_m,g{y}_n),d(g{z}_m,g{z}_n),d(g{w}_m,g{w}_n)\}<\mathit{\epsilon }.\end{array}$$

Suppose the contrary, that is at least one of the sequences $\{g{x}_n\},$$\{g{y}_n\},$$\{g{z}_n\}$ and $\{g{w}_n\}$ is not a $b$-Cauchy sequence, so there exists $\mathit{\epsilon }>0$ for which we can find subsequences $\{g{x}_{m(k)}\},$$\{g{x}_{n(k)}\}$ of $\{g{x}_n\},$$\{g{y}_{m(k)}\},\{g{y}_{n(k)}\}$ of $\{g{y}_n\},$$\{g{z}_{m(k)}\},$$\{g{z}_{n(k)}\}$ of $\{g{z}_n\}$ and $\{g{w}_{m(k)}\},$$\{g{w}_{n(k)}\}$ of $\{g{w}_n\}$ with $n(k)>m(k)\ge k$ such that

$$\begin{array}{c}\hfill max\{d(g{x}_{m(k)},g{x}_{n(k)}),d(g{y}_{m(k)},g{y}_{n(k)}),d(g{z}_{m(k)},g{z}_{n(k)}),d(g{w}_{m(k)},g{w}_{n(k)})\}\ge \mathit{\epsilon },\end{array}$$

(2.14)

for every integer $k$, let $n(k)$ be the least positive integer with $n(k)>m(k)\ge k$ satisfying (2.14) and such that

$$\begin{array}{c}\hfill max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\}<\mathit{\epsilon }.\end{array}$$

(2.15)

Using the $b$-triangle inequality, we get

$$\begin{array}{c}\hfill d(g{x}_{m(k)},g{x}_{n(k)})\le s[d(g{x}_{m(k)},g{x}_{n(k)-1})+d(g{x}_{n(k)-1},g{x}_{n(k)})]d(g{y}_{m(k)},g{y}_{n(k)})\le s[d(g{y}_{m(k)},g{y}_{n(k)-1})+d(g{y}_{n(k)-1},g{y}_{n(k)})]d(g{z}_{m(k)},g{z}_{n(k)})\le s[d(g{z}_{m(k)},g{z}_{n(k)-1})+d(g{z}_{n(k)-1},g{z}_{n(k)})]\text{and,}d(g{w}_{m(k)},g{w}_{n(k)})\le s[d(g{w}_{m(k)},g{w}_{n(k)-1})+d(g{w}_{n(k)-1},g{w}_{n(k)})].\end{array}$$

(2.16)

Hence from (2.14) and (2.16), we have

$$\begin{array}{c}\hfill \mathit{\epsilon }\le max\{d(g{x}_{m(k)},g{x}_{n(k)}),d(g{y}_{m(k)},g{y}_{n(k)}),d(g{z}_{m(k)},g{z}_{n(k)}),d(g{w}_{m(k)},g{w}_{n(k)})\}\le s[max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\}+max\{d(g{x}_{n(k)-1},g{x}_{n(k)}),d(g{y}_{n(k)-1},g{y}_{n(k)}),d(g{z}_{n(k)-1},g{z}_{n(k)}),d(g{w}_{n(k)-1},g{w}_{n(k)})\}]=smax\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\}+s{\mathit{\delta }}_{n(k)-1}.\end{array}$$

Taking the upper and lower limits as $k\to \infty$ in the above inequality, from (2.14), (2.15) and as $\underset{n\to \infty }{lim}{\mathit{\delta }}_{n-1}=0$, we conclude that

$$\begin{array}{cc}\hfill \frac{\mathit{\epsilon }}{s}& \le \underset{k\to \infty }{lim\; sup}max\{d(g{x}_{m(k)},g{x}_{n(k)}),d(g{y}_{m(k)},g{y}_{n(k)}),d(g{z}_{m(k)},g{z}_{n(k)}),d(g{w}_{m(k)},g{w}_{n(k)})\}\hfill \\ \hfill & <\mathit{\epsilon },\hfill \end{array}$$

(2.17)

$$\begin{array}{cc}\hfill \frac{\mathit{\epsilon }}{s}& \le \underset{k\to \infty }{lim\; sup}max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\}\hfill \\ \hfill & <\mathit{\epsilon },\hfill \end{array}$$

(2.18)

and

$$\begin{array}{cc}\hfill \frac{\mathit{\epsilon }}{s}& \le \underset{k\to \infty }{lim\; inf}max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\}\hfill \\ \hfill & <\mathit{\epsilon }.\hfill \end{array}$$

(2.19)

Also, from the $b$-triangle inequality we obtain

$$\begin{array}{c}\hfill \mathit{\epsilon }\le max\{d(g{x}_{m(k)},g{x}_{n(k)}),d(g{y}_{m(k)},g{y}_{n(k)}),d(g{z}_{m(k)},g{z}_{n(k)}),d(g{w}_{m(k)},g{w}_{n(k)})\}\\ \hfill \le s[max\{d(g{x}_{m(k)},g{x}_{m(k)+1}),d(g{y}_{m(k)},g{y}_{m(k)+1}),d(g{z}_{m(k)},g{z}_{m(k)+1}),d(g{w}_{m(k)},g{w}_{m(k)+1})\}\\ \hfill +max\{d(g{x}_{m(k)+1},g{x}_{n(k)}),d(g{y}_{m(k)+1},g{y}_{n(k)}),d(g{z}_{m(k)+1},g{z}_{n(k)}),d(g{w}_{m(k)+1},g{w}_{n(k)})\}]\\ \hfill =s{\mathit{\delta }}_{m(k)}+smax\{d(g{x}_{m(k)+1},g{x}_{n(k)}),d(g{y}_{m(k)+1},g{y}_{n(k)}),d(g{z}_{m(k)+1},g{z}_{n(k)}),d(g{w}_{m(k)+1},g{w}_{n(k)})\},\end{array}$$

and

$$\begin{array}{c}\hfill max\{d(g{x}_{m(k)+1},g{x}_{n(k)}),d(g{y}_{m(k)+1},g{y}_{n(k)}),d(g{z}_{m(k)+1},g{z}_{n(k)}),d(g{w}_{m(k)+1},g{w}_{n(k)})\}\\ \hfill \le s[max\{d(g{x}_{m(k)+1},g{x}_{m(k)})+g{y}_{m(k)+1},d(g{y}_{m(k)})+d(g{z}_{m(k)+1},g{z}_{m(k)})+d(g{w}_{m(k)+1},g{w}_{m(k)})\}\\ \hfill +max\{d(g{x}_{m(k)},g{x}_{n(k)}),d(g{y}_{m(k)},g{y}_{n(k)}),d(g{z}_{m(k)},g{z}_{n(k)}),d(g{w}_{m(k)},g{w}_{n(k)})\}]\\ \hfill =s{\mathit{\delta }}_{m(k)}+smax\{d(g{x}_{m(k)},g{x}_{n(k)}),d(g{y}_{m(k)},g{y}_{n(k)}),d(g{z}_{m(k)},g{z}_{n(k)}),d(g{w}_{m(k)},g{w}_{n(k)})\}.\end{array}$$

Taking the upper limit as $k\to \infty$ in the above two inequalities, using (2.14) and as ${lim}_{n\to \infty }{\mathit{\delta }}_{n-1}=0,$ we have

$$\begin{array}{c}\hfill \frac{\mathit{\epsilon }}{s}\le {lim\; sup}_{n\to \infty }max\{d(g{x}_{m(k)+1},g{x}_{n(k)}),d(g{y}_{m(k)+1},g{y}_{n(k)}),d(g{z}_{m(k)+1},g{z}_{n(k)}),d(g{w}_{m(k)+1},g{w}_{n(k)})\}<s\mathit{\epsilon }.\end{array}$$

(2.20)

Since $g{x}_n\asymp g{x}_{n+1},$$g{y}_n\asymp g{y}_{n+1},$$g{z}_n\asymp g{z}_{n+1}$ and $g{w}_n\asymp g{w}_{n+1}$ for all $n\ge 0$, then $g{x}_{m(k)}\asymp g{x}_{n(k)-1},$$g{y}_{m(k)}\asymp g{y}_{n(k)-1},$$g{z}_{m(k)}\asymp g{z}_{n(k)-1}$ and $g{w}_{m(k)}\asymp g{w}_{n(k)-1}.$

Putting $x=x_{m(k)},$$y=y_{m(k)},$$z=z_{m(k)},$$w=w_{m(k)},$$u=x_{n(k)-1},$$v=y_{n(k)-1},$$r=z_{n(k)-1},$$t=w_{n(k)-1},$ in (2.3) for all $k\ge 0,$ we conclude that

$$\begin{array}{c}\mathit{\psi }(sd(g{x}_{m(k)+1},g{x}_{n(k)}))\\ =\mathit{\psi }(sd(F(x_{m(k)},y_{m(k)},z_{m(k)},w_{m(k)}),F(x_{n(k)-1},y_{n(k)-1},z_{n(k)-1},w_{n(k)-1})))\\ \le \mathit{\psi }(max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\})\\ -\mathit{\varphi }(d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})),\end{array}$$

(2.21)

$$\begin{array}{c}\mathit{\psi }(sd(g{y}_{m(k)+1},g{y}_{n(k)}))\\ =\mathit{\psi }(sd(F(y_{m(k)},z_{m(k)},w_{m(k)},x_{m(k)}),F(y_{n(k)-1},z_{n(k)-1},w_{n(k)-1},x_{n(k)-1})))\\ \le \mathit{\psi }(max\{d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1})\})\\ -\mathit{\varphi }(d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1})),\end{array}$$

(2.22)

$$\begin{array}{c}\mathit{\psi }(sd(g{z}_{m(k)+1},g{z}_{n(k)}))\\ =\mathit{\psi }(sd(F(z_{m(k)},w_{m(k)},x_{m(k)},y_{m(k)}),F(z_{n(k)-1},w_{n(k)-1},x_{n(k)-1},y_{n(k)-1})))\\ \le \mathit{\psi }(max\{d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1})\})\\ -\mathit{\varphi }(d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1})),\end{array}$$

(2.23)

and

$$\begin{array}{c}\mathit{\psi }(sd(g{w}_{m(k)+1},g{w}_{n(k)}))\\ =\mathit{\psi }(sd(F(w_{m(k)},x_{m(k)},y_{m(k)},z_{m(k)}),F(w_{n(k)-1},x_{n(k)-1},y_{n(k)-1},z_{n(k)-1})))\\ \le \mathit{\psi }(max\{d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1})\})\\ -\mathit{\varphi }(d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1})).\end{array}$$

(2.24)

From (2.21)–(2.24) and Remark 2.1, it follows that

$$\begin{array}{c}\mathit{\psi }(smax\{d(g{x}_{m(k)+1},g{x}_{n(k)}),d(g{y}_{m(k)+1},g{y}_{n(k)}),d(g{z}_{m(k)+1},g{z}_{n(k)}),d(g{w}_{m(k)+1},g{w}_{n(k)})\})\\ \le \mathit{\psi }(max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\})\\ -min\{\mathit{\varphi }(d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})),\\ \mathit{\varphi }(d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1})),\\ \mathit{\varphi }(d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1})),\\ \mathit{\varphi }(d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}))\}.\end{array}$$

Taking the upper limit as $k\to \infty$ in the above inequality and using (2.18) and (2.20), we have

$$\begin{array}{c}\mathit{\psi }(\mathit{\epsilon })=\mathit{\psi }\left(s\frac{\mathit{\epsilon }}{s}\right)\\ \le \mathit{\psi }(\mathit{\epsilon })-\underset{k\to \infty }{lim\; inf}min\{\mathit{\varphi }(d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})),\\ \mathit{\varphi }(d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1})),\\ \mathit{\varphi }(d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1})),\\ \mathit{\varphi }(d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}))\}.\end{array}$$

Hence,

$$\begin{array}{c}\underset{k\to \infty }{lim\; inf}min\{\mathit{\varphi }(d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})),\\ \mathit{\varphi }(d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1})),\\ \mathit{\varphi }(d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1})),\\ \mathit{\varphi }(d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}))\}=0,\end{array}$$

Therefore,

$$\begin{array}{c}\underset{k\to \infty }{lim\; inf}\mathit{\varphi }(d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}))=0,\text{or}\\ \underset{k\to \infty }{lim\; inf}\mathit{\varphi }(d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}))=0,\text{or}\\ \underset{k\to \infty }{lim\; inf}\mathit{\varphi }(d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}))=0,\text{or}\\ \underset{k\to \infty }{lim\; inf}\mathit{\varphi }(d(g{w}_{m(k)},g{w}_{n(k)-1}),d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}))=0.\end{array}$$

Using the properties of $\mathit{\varphi }$, it follows that

$$\begin{array}{cc}& \underset{k\to \infty }{lim\; inf}d(g{x}_{m(k)},g{x}_{n(k)-1})=0,\underset{k\to \infty }{lim\; inf}d(g{y}_{m(k)},g{y}_{n(k)-1})=0,\hfill \\ \hfill & \underset{k\to \infty }{lim\; inf}d(g{z}_{m(k)},g{z}_{n(k)-1})=0\text{and}\underset{k\to \infty }{lim\; inf}d(g{w}_{m(k)},g{w}_{n(k)-1})=0.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; inf}max\{d(g{x}_{m(k)},g{x}_{n(k)-1}),d(g{y}_{m(k)},g{y}_{n(k)-1}),d(g{z}_{m(k)},g{z}_{n(k)-1}),d(g{w}_{m(k)},g{w}_{n(k)-1})\}=0,\end{array}$$

which is a contradiction to (2.19). Thus, $\{g{x}_n\},$$\{g{y}_n\},$$\{g{z}_n\},$ and $\{g{w}_n\}$ are $b$-Cauchy sequences in $X.$ Now, we show that $F$ and $g$ have a quadruple coincidence point. Since $X$ is $b$-complete and $\{g{x}_n\},$$\{g{y}_n\},$$\{g{z}_n\},$ and $\{g{w}_n\}$ are $b$-Cauchy sequences in $X,$ there exists $x,y,z,w\in X$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(g{x}_n,x)=0,\underset{n\to \infty }{lim}d(g{y}_n,y)=0,\underset{n\to \infty }{lim}d(g{z}_n,z)=0and\underset{n\to \infty }{lim}d(g{w}_n,w)=0.\end{array}$$

(2.25)

From (2.5) and (2.25), we get

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}g{x}_n=\underset{n\to \infty }{lim}F(x_n,y_n,z_n,w_n)=x,\underset{n\to \infty }{lim}g{y}_n=\underset{n\to \infty }{lim}F(y_n,z_n,w_n,x_n)=y,\end{array}$$

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}g{z}_n=\underset{n\to \infty }{lim}F(z_n,w_n,x_n,y_n)=z\underset{n\to \infty }{lim}g{w}_n=\underset{n\to \infty }{lim}F(w_n,x_n,y_n,z_n)=w.\end{array}$$

Hence from the compatibility of $F$ and $g$, we obtain

$$\begin{array}{cc}& \underset{n\to \infty }{lim}d(gF(x_n,y_n,z_n,w_n),F(g{x}_n,g{y}_n,g{z}_n,g{w}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(y_n,z_n,w_n,x_n),F(g{y}_n,g{z}_n,g{w}_n,g{x}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(z_n,w_n,x_n,y_n),F(g{z}_n,g{w}_n,g{x}_n,g{y}_n))=0,\hfill \\ \hfill \text{and}& \underset{n\to \infty }{lim}d(gF(w_n,x_n,y_n,z_n),F(g{w}_n,g{x}_n,g{y}_n,g{z}_n))=0.\hfill \end{array}$$

(2.26)

Further, from the continuity of $F$ and $g$ we get

$$\begin{array}{cc}& \underset{n\to \infty }{lim}gF(x_n,y_n,z_n,w_n)=\underset{n\to \infty }{lim}gg{x}_{n+1}=gx,\underset{n\to \infty }{lim}F(g{x}_n,g{y}_n,g{z}_n,g{w}_n)=F(x,y,z,w),\hfill \\ \hfill & \underset{n\to \infty }{lim}gF(y_n,z_n,w_n,x_n)=\underset{n\to \infty }{lim}gg{y}_{n+1}=gy,\underset{n\to \infty }{lim}F(g{y}_n,g{z}_n,g{w}_n,g{x}_n)=F(y,z,w,x),\hfill \\ \hfill & \underset{n\to \infty }{lim}gF(z_n,w_n,x_n,y_n)=\underset{n\to \infty }{lim}gg{z}_{n+1}=gz,\underset{n\to \infty }{lim}F(g{z}_n,g{w}_n,g{x}_n,g{y}_n)=F(z,w,x,y),\hfill \\ \hfill & \underset{n\to \infty }{lim}gF(w_n,x_n,y_n,z_n)=\underset{n\to \infty }{lim}gg{w}_{n+1}=gw\text{and}\underset{n\to \infty }{lim}F(g{w}_n,g{x}_n,g{y}_n,g{z}_n)=F(w,x,y,z).\hfill \end{array}$$

Thus from (2.26) and using Lemma 1.1, we have that $gx=F(x,y,z,w),$$gy=F(y,z,w,x),$$gz=F(z,w,x,y),$$gw=F(w,x,y,z).$ Hence, $(x,y,z,w)$ is a quadruple coincidence point of $F$ and $g.$$\square$

By removing the continuity and compatibility assumptions of $F$ and $g$ in Theorem 2.1, we prove the following theorem.

Theorem 2.2

Let$(X,d,⪯)$be a partially ordered$b$-metric space with parameter$s\ge 1.$Let$F:X^4\to X$and$g:X\to X$be two mappings satisfying (2.3) for all $x,y,z,w,u,v,r,t\in X,$such that$gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$and$gw\asymp gt,$where$\mathit{\psi }$and$\mathit{\varphi }$are the same as in Theorem2.1.Suppose that

• (i) $F(X^4)\subseteq g(X),$

• (ii) $F$and$g$satisfy property (2.1),

• (iii) there exist $x_0,y_0,z_0,w_0\in X$ such that $g{x}_0\asymp F(x_0,y_0,z_0,w_0),$ $g{y}_0\asymp F(y_0,z_0,w_0,x_0),$ $g{z}_0\asymp F(z_0,w_0,x_0,y_0)$ and $g{w}_0\asymp F(w_0,x_0,y_0,z_0),$

• (iv) $g(X)$is a$b$-complete subspace of$X,$

• (v) if$x_n\to x$when$n\to \infty$in$X,$then$x_n\asymp x$for$n$sufficiently large.

Then there exist$x,y,z,w\in X$such that$gx=F(x,y,z,w),$$gy=F(y,z,w,x),$$gz=F(z,w,x,y),$and$gw=F(w,x,y,z).$Moreover, if$g{x}_0,$$g{y}_0,$$g{z}_0$and$g{w}_0$are comparable, then$gx=gy=gz=gw,$and if$F$and$g$are$w$-compatible, then$F$and$g$have a quadruple coincidence point of the form$(p,p,p,p).$

Proof

From Theorem 2.1, we have that $\{g{x}_n\},\{g{y}_n\},\{g{z}_n\}$ and $\{g{w}_n\}$ are $b$-Cauchy sequences in $X.$ Since $g(X)$ is a $b$-complete subspace of $X$ and $\{g{x}_n\},\{g{y}_n\},\{g{z}_n\},\{g{w}_n\}\subseteq g(X),$ there exist $x,y,z,w\in X$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(g{x}_n,gx)=\underset{n\to \infty }{lim}d(g{y}_n,gy)=\underset{n\to \infty }{lim}d(g{z}_n,gz)=\underset{n\to \infty }{lim}d(g{w}_n,gw)=0.\end{array}$$

Since $g{x}_n\to gx$ when $n\to \infty$ in $X,$ then from condition (v) we obtain $g{x}_n\asymp gx$ for $n$ sufficiently large. Similarly, we may show that $g{y}_n\asymp gy,g{z}_n\asymp gz$ and $g{w}_n\asymp gw$ for $n$ sufficiently large. For such $n,$ using (2.3) we get

$$\begin{array}{c}\hfill \mathit{\psi }(sd(F(x,y,z,w),g{x}_{n+1}))=\mathit{\psi }(sd(F(x,y,z,w),F(x_n,y_n,z_n,w_n)))\\ \hfill \le \mathit{\psi }(max\{d(gx,g{x}_n),d(gy,g{y}_n),d(gz,g{z}_n),d(gw,g{w}_n)\})\\ \hfill -\mathit{\varphi }(d(gx,g{x}_n),d(gy,g{y}_n),d(gz,g{z}_n),d(gw,g{w}_n)).\end{array}$$

From the above inequality, using Lemma 1.1, as $n\to \infty ,$ and using the properties of $\mathit{\varphi }$ we obtain

$$\begin{array}{c}\hfill \mathit{\psi }\left(\frac{1}{s}d(F(x,y,z,w),gx)\right)\le \mathit{\psi }(s\underset{n\to \infty }{lim\; sup}d(F(x,y,z,w),g{x}_{n+1}))=\underset{n\to \infty }{lim\; sup}\mathit{\psi }(sd(F(x,y,z,w),g{x}_{n+1}))\\ \hfill \le \underset{n\to \infty }{lim\; sup}\mathit{\psi }(max\{d(gx,g{x}_n),d(gy,g{y}_n),d(gz,g{z}_n),d(gw,g{w}_n)\})\\ \hfill -\underset{n\to \infty }{lim\; inf}\mathit{\varphi }(d(gx,g{x}_n),d(gy,g{y}_n),d(gz,g{z}_n),d(gw,g{w}_n))\\ \hfill \le \mathit{\psi }(0)-\mathit{\varphi }(0,0,0,0)=0.\end{array}$$

Thus, $F(x,y,z,w)=gx.$ Similarly, we can show that $F(y,z,w,x)=gy,$$F(z,w,x,y)=gz$ and $F(w,x,y,z)=gw.$

Now, assume that $g{x}_0\asymp g{y}_0\asymp g{z}_0\asymp g{w}_0.$ From (2.6), we have

$$\begin{array}{c}\hfill g{x}_n\asymp g{x}_0,g{y}_n\asymp g{y}_0,g{z}_n\asymp g{z}_0\text{and}g{w}_n\asymp g{w}_0.\end{array}$$

Then

$$\begin{array}{cc}& gx\asymp g{x}_n\asymp g{x}_0\asymp g{y}_0\asymp g{y}_n\asymp gy,gy\asymp g{y}_n\asymp g{y}_0\asymp g{z}_0\asymp g{z}_n\asymp gz\hfill \\ \hfill & \text{and}gz\asymp g{z}_n\asymp g{z}_0\asymp g{w}_0\asymp g{w}_n\asymp gw\text{for,}n\text{sufficiently,large.}\hfill \end{array}$$

Hence $gx\asymp gy\asymp gz\asymp gw.$ Therefor by (2.3) we obtain

$$\begin{array}{cc}\hfill \mathit{\psi }(max& \{d(gx,gy),d(gy,gz),d(gz,gw),d(gw,gx)\})\hfill \\ \hfill & \le \mathit{\psi }(smax\{d(gx,gy),d(gy,gz),d(gz,gw),d(gw,gx)\})\hfill \\ \hfill & \le \mathit{\psi }(max\{d(gx,gy),d(gy,gz),d(gz,gw),d(gw,gx)\})-min\{\mathit{\varphi }(d(gx,gy),d(gy,gz),d(gz,gw),d(gw,gx)),\hfill \\ \hfill & \mathit{\varphi }(d(gy,gz),d(gz,gw),d(gw,gx),d(gx,gy)),\mathit{\varphi }(d(gz,gw),d(gw,gx),d(gx,gy),d(gy,gz)),\hfill \\ \hfill & \mathit{\varphi }(d(gw,gx),d(gx,gy),d(gy,gz),d(gz,gw))\}.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}& min\{\mathit{\varphi }(d(gx,gy),d(gy,gz),d(gz,gw),d(gw,gx)),\mathit{\varphi }(d(gy,gz),d(gz,gw),d(gw,gx),d(gx,gy)),\hfill \\ \hfill & \mathit{\varphi }(d(gz,gw),d(gw,gx),d(gx,gy),d(gy,gz)),\mathit{\varphi }(d(gw,gx),d(gx,gy),d(gy,gz),d(gz,gw))\}=0,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \mathit{\varphi }(d(gx,gy),d(gy,gz),d(gz,gw),d(gw,gx))=0\text{or;}\mathit{\varphi }(d(gy,gz),d(gz,gw),d(gw,gx),d(gx,gy))=0\text{or;}\mathit{\varphi }(d(gz,gw),d(gw,gx),d(gx,gy),d(gy,gz))=0\text{or;}\mathit{\varphi }(d(gw,gx),d(gx,gy),d(gy,gz),d(gz,gw))\}=0.\end{array}$$

Then from the properties of $\mathit{\varphi }$ we have $d(gx,gy)=d(gy,gz)=d(gz,gw)=d(gw,gx)=0,$ that is $gx=gy=gz=gw.$ Now, suppose that $gx=gy=gz=gw=p,$ since $F$ and $g$ are $w$-compatible, then

$$\begin{array}{c}\hfill gp=ggx=g(F(x,y,z,w))=F(gx,gy,gz,gw)=F(p,p,p,p).\end{array}$$

So, $F$ and $g$ have a quadruple coincidence point of the form $(p,p,p,p).$$\square$

Corollary 2.1

Replace the contractive condition (2.3)of Theorem2.1 (or Theorem2.2,respectively) by the following condition:

there exist $\mathit{\psi }:[0,\infty )\to [0,\infty )$such that$\mathit{\psi }$is an altering distance function and$\mathit{\phi }:[0,\infty )\to [0,\infty )$is continuous with$\mathit{\phi }(t)=0$if and only if$t=0$such that

$$\begin{array}{c}\hfill \mathit{\psi }(sd(F(x,y,z,w),F(u,v,r,t)))\le \mathit{\psi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\})\\ \hfill -\mathit{\phi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\}),\end{array}$$

for all$x,y,z,w,u,v,r,t\in X$and$gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$and$gw\asymp gt.$Let the other conditions of Theorem2.1(or Theorem2.2,respectively) be satisfied. Then,$F$and$g$have a quadruple coincidence point.

Proof

We replace $\mathit{\varphi }(t_1,t_2,t_3,t_4)=\mathit{\phi }(max(t_1,t_2,t_3,t_4))$ in Theorem 2.1 (or Theorem 2.2, respectively). So $\mathit{\phi }$ is continuous and $\mathit{\phi }(t)=0$ if and only if $t=0.$$\square$

Corollary 2.2

Replace the contractive condition (2.3)of Theorem2.1(or Theorem2.2,respectively) by the following condition:

$$\begin{array}{c}\hfill d(F(x,y,z,w),F(u,v,r,t))\le \frac{k}{4s}max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\},\end{array}$$

for all$x,y,z,w,u,v,r,t\in X,$and$gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$and$gw\asymp gt,$where$k\in [0,1).$Let the other conditions of Theorem2.1( or Theorem2.2) be satisfied. Then,$F$and$g$have a quadruple coincidence point.

Proof

We take $\mathit{\psi }(t)=\frac{t}{4}$ and $\mathit{\varphi }(t_1,t_2,t_3,t_4)=\frac{(1-k)}{4}max(t_1,t_2,t_3,t_4)$ in Theorem 2.1 (or Theorem 2.2, respectively). $\square$

Corollary 2.3

Replace the contractive condition (2.3)of Theorem2.1(or Theorem2.2,respectively) by the following condition:

there exist$\mathit{\varphi }:{[0,\infty )}^4\to [0,\infty )$is continuous with$\mathit{\varphi }(t_1,t_2,t_3,t_4)=0$if and only if$t_1=t_2=t_3=t_4=0$such that

$$\begin{array}{c}\hfill d(F(x,y,z,w),F(u,v,r,t))\le \frac{1}{s}max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\}-\frac{1}{s}\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)),\end{array}$$

for all$x,y,z,w,u,v,r,t\in X$and$gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$and$gw\asymp gt.$Let the other conditions of Theorem2.1(or Theorem2.2,respectively) be satisfied. Then,$F$and$g$have a quadruple coincidence point.

Proof

Taking $\mathit{\psi }(t)=t$ in Theorem 2.1 (or Theorem 2.2, respectively), we have Corollary 2.3. $\square$

Corollary 2.4

Replace the contractive condition(2.3)of Theorem2.1(or Theorem2.2,respectively) by the following condition:

there exist$\mathit{\psi }:[0,\infty )\to [0,\infty )$such that$\mathit{\psi }$is an altering distance function, and$\mathit{\varphi }:{[0,\infty )}^4\to [0,\infty )$is continuous with$\mathit{\varphi }(t_1,t_2,t_3,t_4)=0$if and only if$t_1=t_2=t_3=t_4=0$such that

$$\begin{array}{c}\hfill \mathit{\psi }(sd(F(x,y,z,w),F(u,v,r,t)))\le \mathit{\psi }\left(\frac{d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt)}{4}\right)\\ \hfill -\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)),\end{array}$$

for all$x,y,z,w,u,v,r,t\in X$and$gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$and$gw\asymp gt.$Let the other conditions of Theorem2.1(or Theorem2.2,respectively) be satisfied. Then,$F$and$g$have a quadruple coincidence point.

Proof

Since

$$\begin{array}{c}\hfill \frac{(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt)}{4}\le max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\},\end{array}$$

and since $\mathit{\psi }$ is assumed to be nondecreasing, then we apply Theorem 2.1 (or Theorem 2.2 respectively). $\square$

Corollary 2.5

Replace the contractive condition (2.3)of Theorem2.1(or Theorem2.2respectively) by the following condition

$$\begin{array}{c}\hfill d(F(x,y,z,w),F(u,v,r,t))\le \frac{k}{4s}(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt)),\end{array}$$

for all$x,y,z,w,u,v,r,t\in X$and$gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$and$gw\asymp gt,$where$k\in [0,1).$Let the other conditions of Theorem2.1(or Theorem2.2respectively) be satisfied. Then$F$and$g$have a quadruple coincidence point.

Proof

We take $\mathit{\psi }(t)=t$ and $\mathit{\varphi }(t_1,t_2,t_3,t_4)=\left(\frac{1-k}{4}\right)(t_1+t_2+t_3+t_4)$ in Corollary 2.4. $\square$

Now, we obtain some quadruple coincidence point results for mappings satisfying a contractive condition of integral type. We denote by $\mathrm{\Lambda }$ the set of all functions $\mathit{\alpha }:[0,+\infty )\to [0,+\infty )$ verifying the following conditions:

• (i) $\mathit{\alpha }$ is a positive Lebesgue integrable mapping on each compact subset of $[0,+\infty )$

• (ii) for all $\mathit{\epsilon }>0$, we have ${\int }_0^{\mathit{\epsilon }}\mathit{\alpha }(t)dt>0.$

Let $N\in \mathbf{N}$ be a fixed positive integer. Let ${\{{\mathit{\alpha }}_i\}}_{1\le i\le N}$ be a family of $N$ functions that belong to $\mathrm{\Lambda }.$ For all $t\ge 0,$ we denote ${(I_i)}_{i=1,...,N}$ as follows:

• $I_1(t)={\int }_0^t{\mathit{\alpha }}_1(s)\mathrm{d}s,$

• $I_2(t)={\int }_0^{I_1(t)}{\mathit{\alpha }}_2(s)\mathrm{d}s={\int }_0^{{\int }_0^t{\mathit{\alpha }}_1(s)\mathrm{d}s}{\mathit{\alpha }}_2(s)\mathrm{d}s,$

• $I_3(t)={\int }_0^{I_2(t)}{\mathit{\alpha }}_3(s)\mathrm{d}s={\int }_0^{{\int }_0^{{\int }_0^t{\mathit{\alpha }}_1(s)\mathrm{d}s}{\mathit{\alpha }}_2(s)\mathrm{d}s}{\mathit{\alpha }}_3(s)\mathrm{d}s,$

• $\cdots$

• $I_N(t)={\int }_0^{I_{N-1}(t)}{\mathit{\alpha }}_N(s)\mathrm{d}s.$

We have the following result.

Corollary 2.6

Replace the contractive condition (2.3)of Theorem2.1(or Theorem2.2respectively) by the following condition:

$$\begin{array}{c}\hfill I_N(\mathit{\psi }(sd(F(x,y,z,w),F(u,v,r,t))))\le I_N(\mathit{\psi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\}))\\ \hfill -I_N(\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt))).\end{array}$$

(2.27)

Let the other conditions of Theorem2.1(or Theorem2.2respectively) be satisfied. Then$F$and$g$have a quadruple coincidence point.

Proof

Consider the function $\mathrm{\Psi }=I_N\circ \mathit{\psi }$ and $\mathrm{\Phi }=I_N\circ \mathit{\varphi }.$ Then (2.27) becomes

$$\begin{array}{cc}\hfill \mathrm{\Psi }(sd(F(x,y,z,w),F(u,v,r,t)))& \le \mathrm{\Psi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\})\hfill \\ \hfill & -\mathrm{\Phi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)).\hfill \end{array}$$

It is easy to show that $\mathrm{\Psi }$ is an altering distance function, $\mathrm{\Phi }$ is continuous and $\mathrm{\Phi }(t_1,t_2,t_3,t_4)=0$ if and only if $t_1=t_2=t_3=t_4=0.$ Applying Theorem 2.1 (or Theorem 2.2 respectively) we obtain the proof. $\square$

In the case $N=1$, we have the following corollary.

Corollary 2.7

Replace the contractive condition (2.3)of Theorems2.1(or Theorem2.2respectively) by the following: There exists$\mathit{\alpha }\in \mathrm{\Lambda }$such that

$$\begin{array}{cc}\hfill {\int }_0^{\mathit{\psi }(sd(F(x,y,z,w),F(u,v,r,t)))}\mathit{\alpha }(t)\mathrm{d}t& \le {\int }_0^{\mathit{\psi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\})}\mathit{\alpha }(t)\mathrm{d}t\hfill \\ \hfill & -{\int }_0^{\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt))}\mathit{\alpha }(t)\mathrm{d}t.\hfill \end{array}$$

Let the other conditions of Theorem2.1(or Theorem2.2respectively) be satisfied. Then$F$and$g$have a quadruple coincidence point.

Uniqueness of quadruple fixed point

In this section, we will show the uniqueness of a quadruple common fixed point.

For a product $X^4$ of a partially ordered set $(X,⪯),$ we define a partial ordering in the following way. For all $(x,y,z,w),(u,v,r,t)\in X^4,$

$$\begin{array}{c}\hfill (x,y,z,w)⪯(u,v,r,t)\iff x⪯u,y⪰v,z⪯r,w⪰t.\end{array}$$

(3.1)

We say that $(x,y,z,w)$ and $(u,v,r,t)$ are comparable if

$$\begin{array}{c}\hfill (x,y,z,w)⪯(u,v,r,t)\text{or,}(u,v,r,t)⪰(x,y,z,w).\end{array}$$

Also, we say that $(x,y,z,w)$ is equal to $(u,v,r,t)$ if and only if $x=u,$$y=v,$$z=r,$$w=t.$

Theorem 3.1

In addition to hypotheses of Theorem2.1(or Theorem2.2,respectively) assume that for all quadruple coincidence points$(x,y,z,w),$$(u,v,r,t)\in X^4,$there exists$(a,b,c,d)\in X^4$such that

$(F(a,b,c,d),F(b,c,d,a),F(c,d,a,b),F(d,a,b,c))$ is comparable to both

$(F(x,y,z,w),F(y,z,w,x),F(z,w,x,y),F(w,x,y,z))\text{and}(F(u,v,r,t),F(v,r,t,u),F(r,t,u,v),F(t,u,v,r)).$ Then $F$ and $g$ have a unique quadruple common fixed point $(x,y,z,w)$ such that $x=gx=F(x,y,z,w),$ $y=gy=F(y,z,w,x),$ $z=gz=F(z,w,x,y),$ and $w=gw=F(w,x,y,z).$

Proof

Theorem 2.1 (or Theorem 2.2 respectively) implies that The set of quadruple coincidence points of $F$ and $g$ is not empty. Suppose that $(x,y,z,w)$ and $(u,v,r,t)$ are two quadruple coincidence points of $F$ and $g,$ that is, $F(x,y,z,w)=gx,$$F(u,v,r,t)=gu,$$F(y,z,w,x)=gy,$$F(v,r,t,u)=gv,$$F(z,w,x,y)=gz,$$F(r,t,u,v)=gr,$$F(w,x,y,z)=gw,$$F(t,u,v,r)=gt.$ We show that $(gx,gy,gz,gw)=(gu,gv,gr,gt).$ By assumption, there exists $(a,b,c,d)\in X^4$ such that $(F(a,b,c,d),F(b,c,d,a),F(c,d,a,b),F(d,a,b,c))$ is comparable to both

$$\begin{array}{c}\hfill (F(x,y,z,w),F(y,z,w,x),F(z,w,x,y),F(w,x,y,z))\text{and}(F(u,v,r,t),F(v,r,t,u),F(r,t,u,v),F(t,u,v,r)).\end{array}$$

Since $F(X^4)\subseteq g(X),$ we can define the sequences $\{g{a}_n\},\{g{b}_n\},\{g{c}_n\}$ and $\{g{d}_n\}$ such that $a_0=a,$$b_0=b,$$c_0=c,$$d_0=d,$ and

$$\begin{array}{c}\hfill g{a}_{n+1}=F(a_n,b_n,c_n,d_n),g{b}_{n+1}=F(b_n,c_n,d_n,a_n),\\ \hfill g{c}_{n+1}=F(c_n,d_n,a_n,b_n),g{d}_{n+1}=F(d_n,a_n,b_n,c_n),\end{array}$$

for all $n\ge 0.$ Also, in the same way define the sequences $\{g{x}_n\},$$\{g{y}_n\},$$\{g{z}_n\},$$\{g{w}_n\}$ and $\{g{u}_n\},$$\{g{v}_n\},$$\{g{r}_n\},$$\{g{t}_n\},$ such that $x_0=x,$$y_0=y,$$z_0=z,$$w_0=w,$ and $u_0=u,$$v_0=v,$$r_0=r,$$t_0=t,$ by

$$\begin{array}{cc}& g{x}_{n+1}=F(x_n,y_n,z_n,w_n),g{y}_{n+1}=F(y_n,z_n,w_n,x_n),g{z}_{n+1}=F(z_n,w_n,x_n,y_n),g{w}_{n+1}=F(w_n,x_n,y_n,z_n),\hfill \\ \hfill \text{and}\\ \hfill & g{u}_{n+1}=F(u_n,v_n,r_n,t_n),g{v}_{n+1}=F(v_n,r_n,t_n,u_n),g{r}_{n+1}=F(r_n,t_n,u_n,v_n),g{t}_{n+1}=F(t_n,u_n,v_n,r_n),\hfill \end{array}$$

for all $n\ge 0.$ Since $(x,y,z,w)$ and $(u,v,r,t)$ are quadruple coincidence points of $F$ and $g,$, then $g{x}_n=F(x,y,z,w),$$g{u}_n=F(u,v,r,t),$$g{y}_n=F(y,z,w,x),$$g{v}_n=F(v,r,t,u),$$g{z}_n=F(z,w,x,y),$$g{r}_n=F(r,t,u,v),$$g{w}_n=F(w,x,y,z),$$g{t}_n=F(t,u,v,r),$ for all $n\ge 0.$

Since $(F(x,y,z,w),F(y,z,w,x),F(z,w,x,y),F(w,x,y,z))=(g{x}_1,g{y}_1,g{z}_1,g{w}_1)=(gx,gy,gz,gw)$ is comparable to $(F(a,b,c,d),F(b,c,d,a),F(c,d,a,b),F(d,a,b,c))=(g{a}_1,g{b}_1,g{c}_1,g{d}_1),$ then it is easy to show $gx\asymp g{a}_1,$$gy\asymp g{b}_1,$$gz\asymp g{c}_1,$$gw\asymp g{d}_1.$ In a similar way, we get that

$$\begin{array}{c}\hfill gx\asymp g{a}_n,gy\asymp g{b}_n,gz\asymp g{c}_n,gw\asymp g{d}_n\text{for,all;}n.\end{array}$$

(3.2)

From (2.3) and (3.2), we obtain

$$\begin{array}{c}\hfill \mathit{\psi }(sd(gx,g{a}_{n+1}))\\ \hfill =\mathit{\psi }(sd(F(x,y,z,w),F(a_n,b_n,c_n,d_n)))\\ \hfill \le \mathit{\psi }(max\{d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n)\})\\ \hfill -\mathit{\varphi }(d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n)),\end{array}$$

(3.3)

$$\begin{array}{c}\hfill \mathit{\psi }(sd(gy,g{b}_{n+1}))\\ \hfill =\mathit{\psi }(sd(F(y,z,w,x),F(b_n,c_n,d_n,a_n)))\\ \hfill \le \mathit{\psi }(max\{d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n)\})\\ \hfill -\mathit{\varphi }(d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n)),\end{array}$$

(3.4)

$$\begin{array}{c}\hfill \mathit{\psi }(sd(gz,g{c}_{n+1}))\\ \hfill =\mathit{\psi }(sd(F(z,w,x,y),F(c_n,d_n,a_n,b_n)))\\ \hfill \le \mathit{\psi }(max\{d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n)\})\\ \hfill -\mathit{\varphi }(d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n)),\end{array}$$

(3.5)

and

$$\begin{array}{c}\hfill \mathit{\psi }(sd(gw,g{d}_{n+1}))\\ \hfill =\mathit{\psi }(sd(F(w,x,y,z),F(d_n,a_n,b_n,c_n)))\\ \hfill \le \mathit{\psi }(max\{d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n)\})\\ \hfill -\mathit{\varphi }(d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n)).\end{array}$$

(3.6)

Set

$$\begin{array}{c}\hfill {\mathit{\gamma }}_n=max\{d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n)\}.\end{array}$$

By (3.3)–(3.6) and Remark 2.1, we obtain that

$$\begin{array}{c}\hfill \mathit{\psi }({\mathit{\gamma }}_{n+1})\le \mathit{\psi }(s{\mathit{\gamma }}_{n+1})\\ \hfill \le \mathit{\psi }({\mathit{\gamma }}_n)-min\{\mathit{\varphi }(d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n)),\\ \hfill \mathit{\varphi }(d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n)),\\ \hfill \mathit{\varphi }(d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n)),\\ \hfill \mathit{\varphi }(d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n))\}\le \mathit{\psi }({\mathit{\gamma }}_n).\end{array}$$

(3.7)

Hence,

$$\begin{array}{c}\hfill \mathit{\psi }({\mathit{\gamma }}_{n+1})\le \mathit{\psi }({\mathit{\gamma }}_n)\text{for,all;}n\in \mathbf{N}.\end{array}$$

Since $\mathit{\psi }$ is nondecreasing, then ${\mathit{\gamma }}_{n+1}\le {\mathit{\gamma }}_n$ for all $n.$ This implies that ${\mathit{\gamma }}_n$ is a non-increasing sequence. Therefore, there exists $\mathit{\gamma }\ge 0$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\mathit{\gamma }}_n=\mathit{\gamma }.\end{array}$$

We show that $\mathit{\gamma }=0.$ Letting $n\to \infty ,$ in (3.7), we get

$$\begin{array}{cc}\hfill \mathit{\psi }(\mathit{\gamma })\le \mathit{\psi }(\mathit{\gamma })& -min\{\underset{n\to \infty }{lim}\mathit{\varphi }(d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n)),\hfill \\ \hfill & \underset{n\to \infty }{lim}\mathit{\varphi }(d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n)),\hfill \\ \hfill & \underset{n\to \infty }{lim}\mathit{\varphi }(d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n)),\hfill \\ \hfill & \underset{n\to \infty }{lim}\mathit{\varphi }(d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n))\}\le \mathit{\psi }(\mathit{\gamma }).\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill min\{\underset{n\to \infty }{lim}\mathit{\varphi }(d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n)),\\ \hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n)),\\ \hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n)),\\ \hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n))\}=0.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n))=0\mathrm{or}\\ \hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gy,g{b}_n),d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n))=0\mathrm{or}\\ \hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gz,g{c}_n),d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n))=0\mathrm{or}\\ \hfill \underset{n\to \infty }{lim}\mathit{\varphi }(d(gw,g{d}_n),d(gx,g{a}_n),d(gy,g{b}_n),d(gz,g{c}_n))\}=0.\end{array}$$

Using the properties of $\mathit{\varphi }$, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(gx,g{a}_n)=\underset{n\to \infty }{lim}d(gy,g{b}_n)=\underset{n\to \infty }{lim}d(gz,g{c}_n)=\underset{n\to \infty }{lim}d(gw,g{d}_n)=0.\end{array}$$

(3.8)

Thus $\underset{n\to \infty }{lim}{\mathit{\gamma }}_n=0.$ Similarly, we can show that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(gu,g{a}_n)=\underset{n\to \infty }{lim}d(gv,g{b}_n))=\underset{n\to \infty }{lim}d(gr,g{c}_n)=\underset{n\to \infty }{lim}d(gt,g{d}_n)=0.\end{array}$$

(3.9)

From (3.8) and (3.9), we conclude that $(gx,gy,gz,gw)=(gu,gv,gr,gt),$ That is the quadruple coincidence point of $F$ and $g$ is unique.

Denote $gx=x^{\ast },$$gy=y^{\ast },$$gz=z^{\ast },$$gw=w^{\ast }$ and since $gx=F(x,y,z,w),$$gy=F(y,z,w,x),$$gz=F(z,w,x,y),$$gw=F(w,x,y,z)$ so we have that

$$\begin{array}{cc}\hfill g{x}^{\ast }=ggx=gF(x,y,z,w),& g{y}^{\ast }=ggy=gF(y,z,w,x),\hfill \end{array}$$

(3.10)

$$\begin{array}{c}\hfill g{z}^{\ast }=ggz=gF(z,w,x,y)\text{and;}g{w}^{\ast }=ggw=gF(w,x,y,z).\end{array}$$

(3.11)

By definition of the sequences $\{g{x}_n\},\{g{y}_n\},\{g{z}_n\}$ and $\{g{w}_n\}$, we have

$$\begin{array}{cc}& g{x}_n=F(x,y,z,w)=F(x_{n-1},y_{n-1},z_{n-1},w_{n-1}),g{y}_n=F(y,z,w,x)=F(y_{n-1},z_{n-1},w_{n-1},x_{n-1}),\hfill \\ \hfill & g{z}_n=F(z,w,x,y)=F(z_{n-1},w_{n-1},x_{n-1},y_{n-1}),g{w}_n=F(w,x,y,z)=F(w_{n-1},x_{n-1},y_{n-1},z_{n-1}).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}& \underset{n\to \infty }{lim}g{x}_n=\underset{n\to \infty }{lim}F(x_n,y_n,z_n,w_n)=F(x,y,z,w),\hfill \end{array}$$

(3.12)

$$\begin{array}{cc}& \underset{n\to \infty }{lim}g{y}_n=\underset{n\to \infty }{lim}F(y_n,z_n,w_n,x_n)=F(y,z,w,x),\hfill \end{array}$$

(3.13)

$$\begin{array}{cc}& \underset{n\to \infty }{lim}g{z}_n=\underset{n\to \infty }{lim}F(z_n,w_n,x_n,y_n)=F(z,w,x,y),\hfill \end{array}$$

(3.14)

$$\begin{array}{cc}& \underset{n\to \infty }{lim}g{w}_n=\underset{n\to \infty }{lim}F(w_n,x_n,y_n,z_n)=F(w,x,y,z).\hfill \end{array}$$

(3.15)

Case 1: In Theorem 2.1, from compatibility and continuity of $F$ and $g$ we obtain

$$\begin{array}{cc}& \underset{n\to \infty }{lim}d(gF(x_n,y_n,z_n,w_n),F(g{x}_n,g{y}_n,g{z}_n,g{w}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(y_n,z_n,w_n,x_n),F(g{y}_n,g{z}_n,g{w}_n,g{x}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(z_n,w_n,x_n,y_n),F(g{z}_n,g{w}_n,g{x}_n,g{y}_n))=0,\hfill \\ \hfill & \underset{n\to \infty }{lim}d(gF(w_n,x_n,y_n,z_n),F(g{w}_n,g{x}_n,g{y}_n,g{z}_n)))=0,\hfill \end{array}$$

(3.16)

where

$$\begin{array}{cc}& \underset{n\to \infty }{lim}gF(x_n,y_n,z_n,w_n)=gF(x,y,z,w),\underset{n\to \infty }{lim}F(g{x}_n,g{y}_n,g{z}_n,g{w}_n)=F(gx,gy,gz,gw)),\hfill \\ \hfill & \underset{n\to \infty }{lim}gF(y_n,z_n,w_n,x_n)=gF(y,z,w,x),\underset{n\to \infty }{lim}F(g{y}_n,g{z}_n,g{w}_n,g{x}_n)=F(gy,gz,gw,gx),\hfill \\ \hfill & \underset{n\to \infty }{lim}gF(z_n,w_n,x_n,y_n)=gF(z,w,x,y),\underset{n\to \infty }{lim}F(g{z}_n,g{w}_n,g{x}_n,g{y}_n)=F(gz,gw,gx,gy),\hfill \\ \hfill & \underset{n\to \infty }{lim}gF(w_n,x_n,y_n,z_n)=gF(w,x,y,z),\underset{n\to \infty }{lim}F(g{w}_n,g{x}_n,g{y}_n,g{z}_n))=F(gw,gx,gy,gz).\hfill \end{array}$$

Thus from Lemma 1.1, we conclude that

$$\begin{array}{c}\hfill gF(x,y,z,w)=F(gx,gy,gz,gw),gF(y,z,w,x)=F(gy,gz,gw,gx),\\ \hfill gF(z,w,x,y)=F(gz,gw,gx,gy),gF(w,x,y,z)=F(gw,gx,gy,gz).\end{array}$$

Moreover, from (3.10) implies that

$$\begin{array}{c}\hfill g{x}^{\ast }=F(x^{\ast },y^{\ast },z^{\ast },w^{\ast }),g{y}^{\ast }=F(y^{\ast },z^{\ast },w^{\ast },x^{\ast }),g{z}^{\ast }=F(z^{\ast },w^{\ast },x^{\ast },y^{\ast }),g{w}^{\ast }=F(w^{\ast },x^{\ast },y^{\ast },z^{\ast }).\end{array}$$

Case 2: In Theorem 2.2 since $F$ and $g$ are $w$-compatible, then

$$\begin{array}{cc}& g{x}^{\ast }=ggx=g(F(x,y,z,w))=F(gx,gy,gz,gw)=F(x^{\ast },y^{\ast },z^{\ast },w^{\ast })\hfill \\ \hfill & g{y}^{\ast }=ggy=g(F(y,z,w,x))=F(gy,gz,gw,gx)=F(y^{\ast },z^{\ast },w^{\ast },x^{\ast })\hfill \\ \hfill & g{z}^{\ast }=ggz=g(F(z,w,x,y))=F(gz,gw,gx,gy)=F(z^{\ast },w^{\ast },x^{\ast },y^{\ast })\hfill \\ \hfill & g{w}^{\ast }=ggw=g(F(w,x,y,z))=F(gw,gx,gy,gz)=F(w^{\ast },x^{\ast },y^{\ast },z^{\ast }).\hfill \end{array}$$

Thus, in the two cases we conclude that $(x^{\ast },y^{\ast },z^{\ast },w^{\ast })$ is another quadruple coincidence point of $F$ and $g.$ Hence, $(g{x}^{\ast },g{y}^{\ast },g{z}^{\ast },g{w}^{\ast })=(gx,gy,gz,gw).$ Therefore

$$\begin{array}{c}\hfill g{x}^{\ast }=gx=x^{\ast },g{y}^{\ast }=gy=y^{\ast },g{z}^{\ast }=gz=z^{\ast },\text{and}g{w}^{\ast }=gw=w^{\ast }.\end{array}$$

Hence, $(x^{\ast },y^{\ast },z^{\ast },w^{\ast })$ is a quadruple common fixed point of $F$ and $g.$ The uniqueness of a quadruple common fixed point follows easily from the uniqueness of a quadruple coincidence point. $\square$

Now, we give an example to justify the hypotheses of Theorem 2.1.

Example 3.1

Let $X=[0,\infty )$ be equipped with the $b$-metric $d(x,y)={(x-y)}^2$ for all $x,y\in X,$ where $s=2,$ and suppose that $⪯$ is the usual ordering $\le$ on $X.$ Obviously, $(X,d,⪯)$ is a partially ordered complete $b$-metric space. Let $F:X^4\to X$ and $g:X\to X$ be defined by

$$\begin{array}{c}\hfill F(x,y,z,w)=\frac{x^2+y^2+z^2+w^2}{16}andg(x)=x^2.\end{array}$$

It is easy to see that $g$ and $F$ are compatible. Define $\mathit{\psi }:[0,\infty )\to [0,\infty )$ by $\mathit{\psi }(t)=kt$ and $\mathit{\varphi }:{[0,\infty )}^4\to [0,\infty )$ by $\mathit{\varphi }(t_1,t_2,t_3,t_4)=\frac{k-1}{4}(t_1+t_2+t_3+t_4),$ where $1\le k\le 8.$ Then, $\mathit{\psi }$ and $\mathit{\varphi }$ have the properties mentioned in Theorem 2.1. Further, for all $x,y,z,w,u,v,r,t\in X,$ we have $gx\asymp gu,$$gy\asymp gv,$$gz\asymp gr$ and $gw\asymp gt.$ Hence,

$$\begin{array}{c}\hfill \mathit{\psi }(sd(F(x,y,z,w),F(u,v,r,t)))=\mathit{\psi }\left(2{\left(\frac{x^2+y^2+z^2+w^2}{16}-\frac{u^2+v^2+r^2+t^2}{16}\right)}^2\right)\\ \hfill =\frac{k}{128}{((x^2-u^2)+(y^2-v^2)+(z^2-r^2)+(w^2-t^2))}^2\\ \hfill \le \frac{4k}{128}\left({(x^2-u^2)}^2+{(y^2-v^2)}^2+{(z^2-r^2)}^2+{(w^2-t^2)}^2\right)\\ \hfill =\frac{k}{32}(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt))\\ \hfill \le \frac{8}{32}(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt))\\ \hfill =\frac{1}{4}(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt))\\ \hfill =\frac{k}{4}(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt))-\frac{k-1}{4}(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt))\\ \hfill \le kmax\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\}-\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt))\\ \hfill =\mathit{\psi }(max\{d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)\})-\mathit{\varphi }(d(gx,gu),d(gy,gv),d(gz,gr),d(gw,gt)).\end{array}$$

So, $F$ and $g$ satisfy all the conditions of Theorem and $(0,0,0,0)$ is a quadruple coincidence point of $F$ and $g.$ Moreover, by Theorem 3.1$(0,0,0,0)$ is the unique quadruple common fixed point of $F$ and $g.$

Note that, in this case $F$ does not have the $g$-mixed monotone property, so the results of paper [25] cannot be applied.