Coupled coincidence point results for probabilistic φ-contractions

Abstract

In this paper, we establish a new coupled coincidence point results in partially ordered probabilistic metric spaces by utilizing the Gauge function. We use the compatibility condition between two mappings. We use monotone and mixed monotone properties of functions with respect to the ordering. Our main result has several corollaries. The main result is supported with an example which shows that the corollaries are actually contained in our main theorem. The methodology is a combination of analytic and order theoretic approaches.

Keywords:

• Menger PM-space
• Had z̆ić type t-norm
• $\mathit{\phi }$-function
• partially ordered set
• mixed monotone property
• Cauchy sequence
• compatible mappings
• coupled coincidence point

AMS subject classifications:

• 47H10
• 54H25

Public Interest Statement

Probabilistic analysis is a branch of mathematics which purports to systematically deal with uncertain situations arising in science, engineering, economics, finance, and many other areas of human activities. The present article contains new results on an emerging area of research in probabilistic analysis which has developed during last five years. The work is a part of both theoretical and applied mathematics in which results are obtained by utilizing mathematical control functions. The authors feel that there are large scopes of applications of these types of results deduced here.

1. Introduction

Probabilistic fixed point theory has its origin in the work of Sehgal and Bharucha-Reid in (1972) where they established a probabilistic version of the famous Banach’s contraction mapping principle. The probabilistic contraction mapping defined in the above-mentioned work was extended in many ways by the use of control functions (Choudhury & Das, 2009; Ćirić, 2010; Fang, 2009; Fang, 2015; O’Regan & Saadati, 2008; Xiao, Zhu, & Cao, 2011). Recently such an extension was done by Fang in (2015) in which he introduced a probabilistic $\mathit{\phi }$- contraction where $\mathit{\phi }$ is assumed to satisfy certain conditions making it more general than the class of Gauge functions which were used by some previous authors.

Meanwhile coupled fixed point results occupied a large place in metric fixed point theory. Although the concept of coupled fixed point was introduced by Guo & Lakshmikantham (1987), coupled fixed point problems attracted large attention only after 2006 when a coupled contraction mapping theorem was proved by Bhaskar and Lakshmikantham (2006). Several instances of works on this topic are in (Bhaskar & Lakshmikantham, 2006; Choudhury & Kundu, 2010; Luong & Thuan, 2011; Lakshmikantham & Ćirić, 2009; Mursaleem, Mohiuddine, & Agarwal, 2012; Mohiuddine & Alotaibi, 2012; Samet, 2010).

A probabilistic coupled fixed point result was first successfully introduced by Hu et al. (2011). After that some works have followed in which coupled fixed and coincidence point results have been established in probabilistic metric spaces (Choudhury & Das, 2014; Ćirić, Agarwal, & Samet, 2011; Doric, 2013). Our endeavor here is to establish a new coupled coincidence point results in probabilistic metric spaces by utilizing the Gauge function used by Fang (2015). A compatibility condition between mappings has been used. Our main result has several corollaries. The main result is supported with an example which shows that the corollaries and the works which have been extended are actually contained in our main theorem.

2. Mathematical preliminaries

In this section, we discuss certain definitions and lemmas which will be necessary for establishing the results of the next section.

Throughout this paper $(X,⪯)$ stands for a partially ordered set with partial order $⪯$. By $x⪰y$ we shall mean $y⪯x$ and by $x\prec y$ we shall mean $x⪯y$ with $x\ne y$.

Definition 2.1

(Schweizer & Sklar, 1983)   A mapping $F:R\to R^{+}$  is called a distribution function if it is non-decreasing and left continuous with ${\displaystyle \underset{t\in R}{inf}F(t)=0}$  and  ${\displaystyle \underset{t\in R}{sup}F(t)=1}$, where R is the set of real numbers and  $R^{+}$ denotes the set of all non-negative real numbers.

Definition 2.2

(Had z̆ić & Pap, 2001; Schweizer & Sklar, 1983)   A binary operation $\mathrm{\Delta }:{[0,1]}^2⟶[0,1]$ is called a t-norm if the following properties are satisfied:

(i)	$\mathrm{\Delta }$ is associative and commutative,
(ii)	$\mathrm{\Delta }(a,1)=a$ for all $a\in [0,1],$
(iii)	$\mathrm{\Delta }(a,b)\le \mathrm{\Delta }(c,d)$ whenever $a\le c$ and $b\le d,$ for all $a,b,c,d\in [0,1]$.

Generic examples of t-norm are ${\mathrm{\Delta }}_M(a,b)=min\{a,b\}$, ${\mathrm{\Delta }}_P(a,b)=ab$ etc.

Definition 2.3

(Schweizer & Sklar, Schweizer (1983))   A Menger space is a triplet $(X,F,\mathrm{\Delta })$, where X is a non empty set, F is a function defined on $X\times X$ to the set of distribution functions and $\mathrm{\Delta }$ is a t-norm, such that the following are satisfied:

(i)	$F_{x,y}(0)=0\text{for}\text{all}x,y\in X$,
(ii)	$F_{x,y}(s)=1\text{for}\text{all}s>0\text{if}\text{and}\text{only}\text{if}x=y$,
(iii)	$F_{x,y}(s)=F_{y,x}(s)\text{for}\text{all}s>0,x,y\in X$,
(iv)	$F_{x,y}(u+v)\ge \mathrm{\Delta }(F_{x,z}(u),F_{z,y}(v))\text{for all}u,v\ge 0\text{and}x,y,z\in X.$

Definition 2.4

(Schweizer & Sklar, 1983)   Let $(X,F,\mathrm{\Delta })$ be a Menger space.

(i)	A sequence  $\{x_n\}\subset X$  is said to be converge to a point   $x\in X$  if given   $\mathit{ϵ}>0,\mathit{\lambda }>0$  we can find a positive integer  $N_{\mathit{ϵ},\mathit{\lambda }}$  such that for all  $n>N_{\mathit{ϵ},\mathit{\lambda }}$$$\begin{array}{c}\hfill F_{x_n,x}(\mathit{ϵ})\ge 1-\mathit{\lambda }.\end{array}$$
(ii)	A sequence  $\{x_n\}$  is said to be a Cauchy sequence in X if given $\mathit{ϵ}>0,\mathit{\lambda }>0$  there exists a positive integer   $N_{\mathit{ϵ},\mathit{\lambda }}$  such that $$\begin{array}{c}\hfill F_{x_n,x_m}(\mathit{ϵ})\ge 1-\mathit{\lambda }\text{for all}m,n>N_{\mathit{ϵ},\mathit{\lambda }}.\end{array}$$
(iii)	A Menger space $(X,F,\mathrm{\Delta })$ is said to be complete if every Cauchy sequence is convergent in X.

(i) and (ii) can be equivalently written by replacing “$\ge$" with “$>$”. More often than not, they are written in that way. We have given them in the present form for our convenience in the proof of our results.

Definition 2.5

(Had z̆ić et al., 2001)   A t-norm $\mathrm{\Delta }$ be said to be a Had z̆ić type t-norm if the family ${\{{\mathrm{\Delta }}^m\}}_{m>0}$ of its iterates defined for each $t\in [0,1]$ by

${\mathrm{\Delta }}^0(t)=t$, ${\mathrm{\Delta }}^1(t)=\mathrm{\Delta }(t,t)$ and, in general, for all $m>1$, ${\mathrm{\Delta }}^m(t)=\mathrm{\Delta }(t,{\mathrm{\Delta }}^{m-1}(t))$ is equi-continuous at $t=1$, that is, given $\mathit{\lambda }>0$ there exists $\mathit{\eta }(\mathit{\lambda })\in (0,1)$ such that$$\begin{array}{c}\hfill 1\ge t>\mathit{\eta }(\mathit{\lambda })\Rightarrow {\mathrm{\Delta }}^{(m)}(t)\ge 1-\mathit{\lambda }\text{for all}m>0.\end{array}$$

Lemma 2.6

(Chang, Cho, & Kang, 1994, p. 24)    Let $(X,F,\mathrm{\Delta })$ be a Menger space with a continuous t-norm. Then, for every $t>0$, $x_n\to x$, $y_n\to y$, imply$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}{F}_{x_n,y_n}(t)=F_{x,y}(t).\end{array}$$

In the following lemma, we note a property of a continuous function.

Lemma 2.7

(Choudhury & Das, 2014)    If $f:R^n\to R$ is a continuous function and ${\{a_{i,j}\}}_{i=1}^{\infty }$, $j=1,2,.....,n$ are n number of sequences such that ${lim\; inf}_{i\to \infty }{a}_{ik}=a_k$ for all $k\ne l$ for some l and ${\{a_{ii}\}}_{i=1}^{\infty }$ is bounded. Then$$\begin{array}{c}\hfill \underset{i\to \infty }{lim\; inf}f(a_{i1},a_{i2},......,a_{in})=f(a_1,a_2,.....\underset{i\to \infty }{lim\; inf}{a}_{il},....,a_n).\end{array}$$

Let $(X,⪯)$ be a partially ordered set and $g:X\to X$ be a mapping. The mapping g is said to be non-decreasing if, for all $x_1,x_2\in X$, $x_1⪯x_2$ implies $g(x_1)⪯g(x_2)$ and non-increasing if, for all $x_1,x_2\in X$, $x_1⪯x_2$ implies $g(x_1)⪰g(x_2)$ (Bhaskar & Lakshmikantham, 2006).

Definition 2.8

(Bhaskar & Lakshmikantham, 2006)    Let $(X,⪯)$ be a partially ordered set and $G:X\times X\to X$ be a mapping. The mapping G is said to have the mixed monotone property if G is non-decreasing in its first argument and is non-increasing in its second argument, that is, if, for all $x_1,x_2\in X$, $x_1⪯x_2$ implies $G(x_1,y)⪯G(x_2,y)$ for fixed $y\in X$ and if, for all $y_1,y_2\in X$, $y_1⪯y_2$ implies $G(x,y_1)⪰G(x,y_2)$, for fixed $x\in X$.

Definition 2.9

(Lakshmikantham & Ćirić., 2009)   Let $(X,⪯)$ be a partially ordered set and $G:X\times X\to X$ and $g:X\to X$ be two mappings. The mapping G is said to have the mixed g-monotone property if G is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, if, for all $x_1,x_2\in X$, $g{x}_1⪯g{x}_2$ implies $G(x_1,y)⪯G(x_2,y)$ for fixed $y\in X$ and if, for all $y_1,y_2\in X$, $g{y}_1⪯g{y}_2$ implies $G(x,y_1)⪰G(x,y_2)$, for fixed $x\in X$.

Definition 2.10

(Bhaskar & Lakshmikantham, 2006)    Let X be a nonempty set. An element $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $G:X\times X\to X$ if$$\begin{array}{c}\hfill G(x,y)=x,G(y,x)=y.\end{array}$$

Further Lakshmikantham and Ćirić (2009) had introduced the concept of coupled coincidence point.

Definition 2.11

(Lakshmikantham & Ćirić, 2009)    Let X be a nonempty set. An element $(x,y)\in X\times X$ is called a coupled coincidence point of the mappings $g:X\to X$ and $G:X\times X\to X$ if$$\begin{array}{c}\hfill G(x,y)=gx,G(y,x)=gy.\end{array}$$

Definition 2.12

(Lakshmikantham & Ćirić, 2009)   Let X be a nonempty set. The mappings $g:X\to X$ and $G:X\times X\to X$ are commuting if, for all $x,y\in X$,$$\begin{array}{c}\hfill gG(x,y)=G(gx,gy).\end{array}$$

A compatible pair $\{g,G\}$ in a metric space (X, d), where $g:X\to X$ and $G:X\times X\to X$, was defined by Choudhury et al. in (2010).

Definition 2.13

(Choudhury et al., 2010)   Let (X, d) be a metric space. The mappings g and G where $g:X\to X$ and $G:X\times X\to X$, are said to be compatible if$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(g(G(x_n,y_n)),G(g{x}_n,g{y}_n))=0\end{array}$$

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

whenever $\{x_n\}$ and $\{y_n\}$ are sequences in X such that ${lim}_{n\to \infty }G(x_n,y_n)$ = ${lim}_{n\to \infty }g{x}_n=x$ and ${lim}_{n\to \infty }G(y_n,x_n)$ = ${lim}_{n\to \infty }g{y}_n=y$.

The following is the definition of compatible pairs in Menger spaces.

Definition 2.14

(Doric, 2013)   Let $(X,F,\mathrm{\Delta })$ be a Menger space. The mappings g and G where $g:X\to X$ and $G:X\times X\to X$, are said to be compatible if, for all $t>0$,$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_{g(G(x_n,y_n)),G(g{x}_n,g{y}_n)}(t)=1\end{array}$$

and$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_{g(G(y_n,x_n)),G(g{y}_n,g{x}_n)}(t)=1,\end{array}$$

whenever $\{x_n\}$ and $\{y_n\}$ are sequences in X such that ${lim}_{n\to \infty }G(x_n,y_n)$ = ${lim}_{n\to \infty }g{x}_n=x$ and ${lim}_{n\to \infty }G(y_n,x_n)$ = ${lim}_{n\to \infty }g{y}_n=y$.

Out of several types of gauge functions utilized in extending probabilistic contraction mapping principle, we consider the following two classes.

Definition 2.15

(Jachymski, 2010)    Let $\mathrm{\Phi }$ denote the class of all functions $\mathit{\phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying the following condition:$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\mathit{\phi }}^n(t)=0\text{for all}t>0.\end{array}$$

Definition 2.16

(Fang, 2015)    Let ${\mathrm{\Phi }}_w$ denote the class of all functions $\mathit{\phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying the following condition:

for each $t>0$ there exists $r\ge t$ such that ${lim}_{n\to \infty }{\mathit{\phi }}^n(r)=0$.

Here $\mathrm{\Phi }$ is a proper subclass of ${\mathrm{\Phi }}_w$ [see Fang (2015)].

Lemma 2.17

(Fang, 2015)    Let $\mathit{\phi }\in {\mathrm{\Phi }}_w$, then for each $t>0$ there exists $r\ge t$ such that $\mathit{\phi }(r)<t.$

3. Main results

Theorem 3.1

Let $(X,F,\mathrm{\Delta })$ be a complete Menger space where $\mathrm{\Delta }$ is a continuous Had z̆ić type t-norm on which a partial ordering $⪯$ is defined. Let $g:X\to X$ and $G:X\times X\to X$ be two mappings such that G has the mixed g-monotone property. Let there exist $\mathit{\phi }\in {\mathrm{\Phi }}_w$ such that(3.1) $$\begin{array}{c}\hfill F_{G(x,y),G(u,v)}(\mathit{\phi }(t))\ge {[F_{gx,gu}(t).F_{gy,gv}(t)]}^{\frac{1}{2}},\end{array}$$(3.1)

for all $t>0,x,y,u,v\in X$ with $gx⪯gu$ and $gy⪰gv$. Let g be continuous, monotonic increasing, $G(X\times X)\subseteq g(X)$ and such that $\{g,G\}$ is a compatible pair. Also suppose either

(a)	G is continuous or
(b)	X has the following properties: (i) if a non-decreasing sequence (3.2) $$\begin{array}{c}\hfill \{x_n\}\to x,\text{then}{x}_n⪯x\text{for}\text{all}n\ge 0,\end{array}$$(3.2) (ii) if a non-increasing sequence (3.3) $$\begin{array}{c}\hfill \{y_n\}\to y,\text{then}{y}_n⪰y\text{for all}n\ge 0.\end{array}$$(3.3)

If there exist $x_0,y_0\in X$ such that $g{x}_0⪯G(x_0,y_0)$ and $g{y}_0⪰G(y_0,x_0)$, then g and G have a coupled coincidence point in X, that is, there exist $x,y\in X$ such that $gx=G(x,y)$ and $gy=G(y,x)$.

Proof

By a condition of the theorem, there exist $x_0,y_0\in X$ such that $g{x}_0⪯G(x_0,y_0)$ and $g{y}_0⪰G(y_0,x_0)$. Since $G(X\times X)\subseteq g(X)$, it is possible to define the sequences $\{x_n\}$ and $\{y_n\}$ in X as follows:$$\begin{array}{c}\hfill g{x}_1=G(x_0,y_0)\text{and}g{y}_1=G(y_0,x_0)\\ \hfill g{x}_2=G(x_1,y_1)\text{and}g{y}_2=G(y_1,x_1)\end{array}$$

and, in general, for all $n\ge 0$,(3.4) $$\begin{array}{c}\hfill g{x}_{n+1}=G(x_n,y_n)\text{and}g{y}_{n+1}=G(y_n,x_n).\end{array}$$(3.4)

Next, for all $n\ge 0$, we prove that(3.5) $$\begin{array}{c}\hfill g{x}_n⪯g{x}_{n+1}\end{array}$$(3.5)

and(3.6) $$\begin{array}{c}\hfill g{y}_n⪰g{y}_{n+1}.\end{array}$$(3.6)

Since $g{x}_0⪯G(x_0,y_0)$ and $g{y}_0⪰G(y_0,x_0)$, in view of the facts that $g{x}_1=G(x_0,y_0)$ and $g{y}_1=G(y_0,x_0)$, we have $g{x}_0⪯g{x}_1$ and $g{y}_0⪰g{y}_1$. Therefore, (3.5) and (3.6) hold for $n=0$.

Let (3.5) and (3.6) hold for some $n=m$, that is, $g{x}_m⪯g{x}_{m+1}$ and $g{y}_m⪰g{y}_{m+1}$. As G has the mixed g-monotone property, from (3.4), we get(3.7) $$\begin{array}{c}\hfill g{x}_{m+1}=G(x_m,y_m)⪯G(x_{m+1},y_m)\text{and}G(y_{m+1},x_m)⪯G(y_m,x_m)=g{y}_{m+1}.\end{array}$$(3.7)

Also, for the same reason, we have(3.8) $$\begin{array}{c}\hfill g{x}_{m+2}=G(x_{m+1},y_{m+1})⪰G(x_{m+1},y_m)\text{and}G(y_{m+1},x_m)⪰G(y_{m+1},x_{m+1})=g{y}_{m+2}.\end{array}$$(3.8)

Now, (3.7) and (3.8), imply$$\begin{array}{c}\hfill g{x}_{m+1}⪯g{x}_{m+2}\text{and}g{y}_{m+1}⪰g{y}_{m+2}.\end{array}$$

Then, by induction, it follows that (3.5) and (3.6) hold for all $n\ge 0$.

Now, for all $t>0$, $n\ge 1$, we have(3.9) $$\begin{array}{cc}\hfill F_{g{x}_n,g{x}_{n+1}}(\mathit{\phi }(t))& =F_{G(x_{n-1},y_{n-1}),G(x_n,y_n)}(\mathit{\phi }(t))(\text{by}(3.4))\hfill \\ \hfill & \ge {[F_{g{x}_{n-1},g{x}_n}(t).F_{g{y}_n,g{y}_{n-1}}(t)]}^{\frac{1}{2}}.(\text{by}(3.1),(3.5)\text{and}(3.6))\hfill \end{array}$$(3.9)

Similarly, we have for all $t>0$(3.10) $$\begin{array}{cc}\hfill F_{g{y}_n,g{y}_{n+1}}(\mathit{\phi }(t))& =F_{G(y_{n-1},x_{n-1}),G(y_n,x_n)}(\mathit{\phi }(t))(\text{by}(3.4))\hfill \\ \hfill & \ge {[F_{g{y}_{n-1},g{y}_n}(t).F_{g{x}_n,g{x}_{n-1}}(t)]}^{\frac{1}{2}}.[\text{by}(3.9)\text{and}(3.10)]\hfill \end{array}$$(3.10)

Let(3.11) $$\begin{array}{c}\hfill P_n(t)={[F_{g{x}_{n-1},g{x}_n}(t).F_{g{y}_n,g{y}_{n-1}}(t)]}^{\frac{1}{2}}.\end{array}$$(3.11)

Then $P_n(t).P_n(t)\le F_{g{x}_n,g{x}_{n+1}}(\mathit{\phi }(t)).F_{g{y}_n,g{y}_{n+1}}(\mathit{\phi }(t)),$

which implies that ${[P_n(t)]}^2\le {[P_{n+1}(\mathit{\phi }(t))]}^2,$ that is, $[P_{n+1}(\mathit{\phi }(t))]\ge [P_n(t)]$.

By repeated application of the above inequality, using (3.9) and (3.10), respectively, for all $t>1$, $n>1$, we have that,$$\begin{array}{c}\hfill F_{g{x}_n,g{x}_{n+1}}({\mathit{\phi }}^n(t))\ge P_n({\mathit{\phi }}^{n-1}(t))\ge ..............\ge P_1(t)={[F_{g{x}_0,g{x}_1}(t).F_{g{y}_0,g{y}_1}(t)]}^{\frac{1}{2}},\text{and}\\ \hfill F_{g{y}_n,g{y}_{n+1}}({\mathit{\phi }}^n(t))\ge P_n({\mathit{\phi }}^{n-1}(t))\ge ..............\ge P_1(t)={[F_{g{x}_0,g{x}_1}(t).F_{g{y}_0,g{y}_1}(t)]}^{\frac{1}{2}}.\end{array}$$

Now we prove that$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_{g{x}_n,g{x}_{n+1}}(t)=1\text{for all}t>0\text{and}\underset{n\to \infty }{lim}{F}_{g{y}_n,g{y}_{n+1}}(t)=1,\text{for all}t>0.\end{array}$$

Since $F_{g{x}_0,g{x}_1}(t)\to 1$, $F_{g{y}_0,g{y}_1}(t)\to 1$ as $t\to \infty$, for any $\mathit{ϵ}\in (0,1]$ there exists $t_1>0$ such that $F_{g{x}_0,g{x}_1}(t_1)>1-\mathit{ϵ}$ and $F_{g{y}_0,g{y}_1}(t_1)>1-\mathit{ϵ}$. Since $\mathit{\phi }\in {\mathrm{\Phi }}_w$, there exists $t_0\ge t_1$ such that ${lim}_{n\to \infty }{\mathit{\phi }}^n(t_0)=1$. Thus for each $t>0$, there exists $n_0\ge 1$ such that ${\mathit{\phi }}^n(t_0)<t$ for all $n\ge n_0$. Then from the above, for all $n\ge n_0$,$$\begin{array}{c}\hfill F_{g{x}_n,g{x}_{n+1}}(t)\ge F_{g{x}_n,g{x}_{n+1}}({\mathit{\phi }}^n(t_0)\ge {[F_{g{x}_0,g{x}_1}(t_0).F_{g{y}_0,g{y}_1}(t_0)]}^{\frac{1}{2}}>{[(1-\mathit{ϵ}).(1-\mathit{ϵ})]}^{\frac{1}{2}}>1-\mathit{ϵ}.\end{array}$$

Hence ${lim}_{n\to \infty }{F}_{g{x}_n,g{x}_{n+1}}(t)=1$ for all $t>0$. Similarly, ${lim}_{n\to \infty }{F}_{g{y}_n,g{y}_{n+1}}(t)=1,$ for all $t>0$.

Therefore from (3.11), $P_n(t)\to 1\text{as}n\to \infty$.

Since $\mathit{\phi }\in {\mathrm{\Phi }}_w$, by Lemma 2.17, for any $t>0$ there exists $r\ge t$ such that $\mathit{\phi }(r)<t$. Let $n\ge 1$ be given. We next show by induction that for any $k\ge 1$,(3.12) $$\begin{array}{c}\hfill F_{g{x}_n,g{x}_{n+k}}(t)\ge {\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r))).\end{array}$$(3.12) (3.13) $$\begin{array}{c}\hfill F_{g{y}_n,g{y}_{n+k}}(t)\ge {\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r))).\end{array}$$(3.13)

Since ${\mathrm{\Delta }}^0(s)=s$, this true for $k=1$. Let (3.13) and (3.14) hold for some k.$$\begin{array}{cc}\hfill F_{g{x}_n,g{x}_{n+k+1}}(t)& =F_{g{x}_n,g{x}_{n+k+1}}(t-\mathit{\phi }(r)+\mathit{\phi }(r))\hfill \\ \hfill & \ge \mathrm{\Delta }(F_{g{x}_n,g{x}_{n+1}}(t-\mathit{\phi }(r)),F_{g{x}_{n+1},g{x}_{n+k+1}}(\mathit{\phi }(r)))\hfill \\ \hfill & \ge \mathrm{\Delta }{(F_{g{x}_n,g{x}_{n+1}}(t-\mathit{\phi }(r)),[F_{g{x}_n,g{x}_{n+k}}(r).F_{g{y}_n,g{y}_{n+k}}(r))]}^{\frac{1}{2}})\hfill \\ \hfill (\text{by}(3.1),(3.5)\text{and}(3.6))\\ \hfill & \ge \mathrm{\Delta }{(F_{g{x}_n,g{x}_{n+1}}(t-\mathit{\phi }(r)),[F_{g{x}_n,g{x}_{n+k}}(t).F_{g{y}_n,g{y}_{n+k}}(t))]}^{\frac{1}{2}})(\text{since}r\ge t)\hfill \\ \hfill & \ge \mathrm{\Delta }(P_n(t-\mathit{\phi }(r)),{[{\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r))),{\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r)))]}^{\frac{1}{2}})\hfill \\ \hfill & \ge \mathrm{\Delta }(P_n(t-\mathit{\phi }(r)),{\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r))))\hfill \\ \hfill & \ge {\mathrm{\Delta }}^k(P_n(t-\mathit{\phi }(r))).\hfill \end{array}$$

Similarly, we have $F_{g{y}_n,g{y}_{n+k+1}}(t)\ge {\mathrm{\Delta }}^k(P_n(t-\mathit{\phi }(r)))$.

Therefore, by induction, (3.13) and (3.14) hold for all $k\ge 1$ and $t>0$.

Now, we prove $\{g{x}_n\}$ and $\{g{y}_n\}$ are Cauchy sequences. Since, the t-norm $\mathrm{\Delta }$ is of H- type, the family of iterates $\{{\mathrm{\Delta }}^{(p)}\}$ is equi-continuous at the point $s=1$, that is, there exists $\mathit{\delta }\in (0,1)$ such that(3.14) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{(p)}(s)>1-\mathit{\delta },\end{array}$$(3.14)

whenever $1\ge s>1-\mathit{ϵ}$ and $p\ge 1$.

By (3.11), we have a positive integer $n_0$ such that for all $n\ge n_0$$(P_n(t-\mathit{\phi }(r))>1-\mathit{\delta }$. It follows from (3.15), (3.13) and (3.14) that$$\begin{array}{c}\hfill F_{g{x}_n,g{x}_{n+k}}(t)\ge {\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r)))>1-\mathit{ϵ}\text{for all}n\ge n_0,k\ge 1\end{array}$$

and$$\begin{array}{c}\hfill F_{g{y}_n,g{y}_{n+k}}(t)\ge {\mathrm{\Delta }}^{k-1}(P_n(t-\mathit{\phi }(r)))>1-\mathit{ϵ}\text{for all}n\ge n_0,k\ge 1.\end{array}$$

This shows that $\{g{x}_n\}$ and $\{g{y}_n\}$ are Cauchy sequences. Since X is complete, there exist $x,y\in X$ such that$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}g{x}_n=x\text{and}\underset{n\to \infty }{lim}g{y}_n=y,\end{array}$$

that is,(3.15) $$\begin{array}{c}\hfill \underset{n\to \infty }{lim}G(x_n,y_n)=\underset{)}{lim}n\to \infty g{x}_{n+1}=x\text{and}\underset{n\to \infty }{lim}G(y_n,x_n)=\underset{n\to \infty }{lim}g{y}_{n+1}=y.\end{array}$$(3.15)

Since $\{g,G\}$ is a compatible pair, using continuity of g, we have(3.16) $$\begin{array}{cc}& gx=\underset{n\to \infty }{lim}g(g{x}_{n+1})=\underset{n\to \infty }{lim}g(G(x_n,y_n))=\underset{n\to \infty }{lim}G(g{x}_n,g{y}_n)\hfill \\ \hfill & and\hfill \\ \hfill & gy=\underset{n\to \infty }{lim}g(g{y}_{n+1})=\underset{n\to \infty }{lim}g(G(y_n,x_n))=\underset{n\to \infty }{lim}G(g{y}_n,g{x}_n).\hfill \end{array}$$(3.16)

Now, we show that $gx=G(x,y)$ and $gy=G(y,x)$.

First let us assume that (a) holds.

Then, by continuity of G,$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}G(g{x}_n,g{y}_n)=G(\underset{n\to \infty }{lim}g{x}_n,\underset{n\to \infty }{lim}g{y}_n)=G(x,y)\end{array}$$

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

Then, from (3.17) and (3.18), we conclude that$$\begin{array}{c}\hfill gx=G(x,y)\text{and}gy=G(y,x).\end{array}$$

Next we assume that (b) holds. By (3.5), (3.6), (3.17) and (3.18), we have $\{g{x}_n\}$ is a non-decreasing sequence with $g{x}_n\to x$ and $\{g{y}_n\}$ is a non-increasing sequence with $g{y}_n\to y$ as $n\to \infty$. Then, by (3.11) and (3.12), for all $n\ge 0$, we have(3.17) $$\begin{array}{c}\hfill g{x}_n⪯x\text{and}g{y}_n⪰y.\end{array}$$(3.17)

Since g is monotonic increasing, we have that(3.18) $$\begin{array}{c}\hfill g(g{x}_n)⪯gx\text{and}g(g{y}_n)⪰gy.\end{array}$$(3.18)

Now for all $t>0$ and using the lemma 2.17, we have$$\begin{array}{c}\hfill F_{gx,G(x,y)}(t)\ge \mathrm{\Delta }\{F_{gx,g(g{x}_{(n+1)})}(t-\mathit{\phi }(r)),F_{g(g{x}_{(n+1)}),G(x,y)}(\mathit{\phi }(r))\}.\end{array}$$

Taking limit in the above inequality, for all $t>0$, we have$$\begin{array}{cc}\hfill F_{gx,G(x,y)}(t)& \ge \underset{n\to \infty }{lim\; inf}\mathrm{\Delta }\{F_{gx,g(g{x}_{(n+1)})}(t-\mathit{\phi }(r)),F_{g(g{x}_{(n+1)}),G(x,y)}(\mathit{\phi }(r))\}\hfill \\ \hfill & =\mathrm{\Delta }\{\underset{n\to \infty }{lim}{F}_{gx,g(g{x}_{(n+1)})}(t-\mathit{\phi }(r)),\underset{n\to \infty }{lim\; inf}{F}_{g(g{x}_{(n+1)}),G(x,y)}(\mathit{\phi }(r))\}\hfill \\ \hfill & (\text{by}\text{lemma}2.7\text{and}\text{continuity}\text{of}\mathrm{\Delta })\hfill \\ \hfill & =\mathrm{\Delta }\{1,\underset{n\to \infty }{lim\; inf}{F}_{G(g{x}_n,g{y}_n),G(x,y)}(\mathit{\phi }(r))\}(\text{by}\text{lemma}2.7\text{and}(3.17))\hfill \\ \hfill & =\underset{n\to \infty }{lim\; inf}{F}_{G(g{x}_n,g{y}_n),G(x,y)}(\mathit{\phi }(r)).\hfill \end{array}$$

Now, using (3.1) and (3.20), we have(3.19) $$\begin{array}{c}\hfill F_{G(g{x}_n,g{y}_n),G(x,y)}(\mathit{\phi }(r))\}\ge {[F_{g(g{x}_n),gx}(r).F_{g(g{y}_n),gy}(r)]}^{\frac{1}{2}}.\end{array}$$(3.19)

Taking liminf on both sides of the above inequality, we have(3.20) $$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}{F}_{G(g{x}_n,g{y}_n),G(x,y)}(\mathit{\phi }(r))\}& \ge \underset{n\to \infty }{lim\; inf}{[F_{g(g{x}_n),gx}(r).F_{g(g{y}_n),gy}(r)]}^{\frac{1}{2}}.\hfill \\ \hfill & =\underset{n\to \infty }{lim}[F_{g(g{x}_n),gx}(r).F_{g(g{y}_n),gy}){(r)]}^{\frac{1}{2}}\hfill \\ \hfill & =1.(\text{by}(3.17)\text{and}(3.18))\hfill \end{array}$$(3.20)

From (3.21) and (3.22) we conclude that $F_{gx,G(x,y)}(t)=1$ for all $t>0$, that is, $gx=G(x,y)$. Similarly we can prove that $gy=G(y,x)$, that is, g and G have a coupled coincidence point in X. $\square$

Corollary 3.2

Let $(X,F,\mathrm{\Delta })$ be a complete Menger space where $\mathrm{\Delta }$ is a continuous Had z̆ić type t-norm on which a partial ordering $⪯$ is defined. Let $g:X\to X$ and $G:X\times X\to X$ be two mappings such that G has the mixed g-monotone property. Let there exist $\mathit{\phi }\in {\mathrm{\Phi }}_w$ such that$$\begin{array}{c}\hfill F_{G(x,y),G(u,v)}(\mathit{\phi }(t))\ge {[F_{gx,gu}(t).F_{gy,gv}(t)]}^{\frac{1}{2}},\end{array}$$

for all $t>0,x,y,u,v\in X$ with $gx⪯gu$ and $gy⪰gv$. Let g be continuous, monotonic increasing, $G(X\times X)\subseteq g(X)$ and such that $\{g,G\}$ is a commuting pair. Also suppose either

(a)	G is continuous or
(b)	X has the following properties: (i) if a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\ge 0$, (ii) if a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\ge 0$.

If there are $x_0,y_0\in X$ such that $g{x}_0⪯G(x_0,y_0)$ and $g{y}_0⪰G(y_0,x_0)$, then g and G have a coupled coincidence point in X, that is, there exist $x,y\in X$ such that $gx=G(x,y)$ and $gy=G(y,x)$.

Proof

Since compatibility implies commuting condition, Theorem 3.1 contains Corollary 3.2. Later, with the help of an example we show that the above corollary is properly contained in the above Theorem 3.1.

$\square$

Corollary 3.3

Let $(X,F,\mathrm{\Delta })$ be a complete Menger space where $\mathrm{\Delta }$ is a continuous Had z̆ić type t-norm on which a partial ordering $⪯$ is defined. Let $g:X\to X$ and $G:X\times X\to X$ be two mappings such that G has the mixed g-monotone property. Let there exist $\mathit{\phi }\in \mathrm{\Phi }$ such that$$\begin{array}{c}\hfill F_{G(x,y),G(u,v)}(\mathit{\phi }(t))\ge {[F_{gx,gu}(t).F_{gy,gv}(t)]}^{\frac{1}{2}},\end{array}$$

for all $t>0,x,y,u,v\in X$ with $gx⪯gu$ and $gy⪰gv$. Let g be continuous, monotonic increasing, $G(X\times X)\subseteq g(X)$ and such that $\{g,G\}$ is a compatible pair. Also suppose either

(a)	G is continuous or
(b)	X has the following properties: (i) if a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\ge 0$, (ii) if a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\ge 0$.

If there are $x_0,y_0\in X$ such that $g{x}_0⪯G(x_0,y_0)$ and $g{y}_0⪰G(y_0,x_0)$, then g and G have a coupled coincidence point in X, that is, there exist $x,y\in X$ such that $gx=G(x,y)$ and $gy=G(y,x)$.

Proof

Since $\mathrm{\Phi }$ is a proper subclass of ${\mathrm{\Phi }}_w$, Theorem 3.1 contains Corollary 3.3.

Example 3.5 discussed in the following shows that the above corollary is properly contained in Theorem 3.1. Combining the above two corollaries we have partly a result obtained in (Xiao, Zhu, & Cao, 2011) which we describe in the following corollary.

$\square$

Corollary 3.4

(Xiao, Zhu, & Cao, 2011) Let $(X,F,\mathrm{\Delta })$ be a complete Menger space where $\mathrm{\Delta }$ is a continuous Had z̆ić type t-norm on which a partial ordering $⪯$ is defined. Let $g:X\to X$ and $G:X\times X\to X$ be two mappings such that G has the mixed g-monotone property. Let there exist $\mathit{\phi }\in \mathrm{\Phi }$ such that$$\begin{array}{c}\hfill F_{G(x,y),G(u,v)}(\mathit{\phi }(t))\ge {[F_{gx,gu}(t).F_{gy,gv}(t)]}^{\frac{1}{2}},\end{array}$$

for all $t>0,x,y,u,v\in X$ with $gx⪯gu$ and $gy⪰gv$. Let g be continuous, monotonic increasing, $G(X\times X)\subseteq g(X)$ and such that $\{g,G\}$ is a commuting pair. Also suppose either

(a)	G is continuous or
(b)	X has the following properties: (i) if a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\ge 0$, item[(ii)] if a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\ge 0$.

If there are $x_0,y_0\in X$ such that $g{x}_0⪯G(x_0,y_0)$ and $g{y}_0⪰G(y_0,x_0)$, then g and G have a coupled coincidence point in X, that is, there exist $x,y\in X$ such that $gx=G(x,y)$ and $gy=G(y,x)$.

Proof

Since compatibility implies commuting, Corollary 3.3 contains Corollary 3.4. $\square$

Example 3.5

Let $X=[0,1]$. Let for all $t>0$, $x,y\in X$,$$\begin{array}{c}\hfill F_{x,y}(t)=e^{-{\displaystyle \frac{|x-y|}{t}}}.\end{array}$$

Let $\mathrm{\Delta }(a,b)=min\{a,b\}$ for all $a,b\in [0,1]$. Then $(X,F,\mathrm{\Delta })$ is a complete Menger space.

Let $\mathit{\phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be defined by$$\begin{array}{c}\hfill \mathit{\phi }(t)=\left\{\begin{array}{cc}\frac{t}{2},\hfill & \text{if}t\in [0,1),\hfill \\ -\frac{t}{3}+\frac{4}{3},\hfill & \text{if}t\in [1,2],\hfill \\ t+\frac{4}{3},\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

It is obvious $\mathit{\phi }\in {\mathrm{\Phi }}_w$ but $\mathit{\phi }\notin \mathrm{\Phi }$. From the definition of $\mathit{\phi }$, we have $\mathit{\phi }(t)\ge \frac{t}{2}$ for all $t\ge 0$.

Let the mapping $g:X\to X$ be defined as$$\begin{array}{c}\hfill g(x)=\frac{5}{6}{x}^2\text{for}\text{all}x\in X\end{array}$$

and the mapping $G:X\times X\to X$ be defined as$$\begin{array}{c}\hfill G(x,y)=\frac{x^2-y^2}{9}.\end{array}$$

Then $G(X\times X)\subseteq g(X)$ and G satisfies the mixed g-monotone property.

Let $\{x_n\}$ and $\{y_n\}$ be two sequences in X such that$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}G(x_n,y_n)=a,\underset{n\to \infty }{lim}g(x_n)=a,\\ \hfill \underset{n\to \infty }{lim}G(y_n,x_n)=b\text{and}\underset{n\to \infty }{lim}g(y_n)=b.\end{array}$$

Now, for all $n\ge 0$,$$\begin{array}{c}\hfill g(x_n)=\frac{5}{6}{x}_n^2,g(y_n)=\frac{5}{6}{y}_n^2,\\ \hfill G(x_n,y_n)=\frac{x_n^2-y_n^2}{4}\end{array}$$

and$$\begin{array}{c}\hfill G(y_n,x_n)=\frac{y_n^2-x_n^2}{4}.\end{array}$$

Then necessarily $a=0$ and $b=0$.

It then follows from lemma 2.6 that, for all $t>0,$$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_{g(G(x_n,y_n)),G(g(x_n),g(y_n)}(t)=1\end{array}$$

and$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_{g(G(y_n,x_n)),G(g{y}_n),g(x_n)}(t)=1.\end{array}$$

Therefore, the mappings G and g are compatible in X.

Now we show that the condition (3.1) holds.$$\begin{array}{c}\hfill |G(x,y)-G(u,v)|\le {\displaystyle \frac{1}{4}}[|g(x)-g(u)|+|g(y)-g(v)|],x\ge u,y\le v.\end{array}$$

Therefore, $\frac{|G(x,y)-G(u,v)|}{\mathit{\phi }(t)}\le {\displaystyle \frac{1}{4}}[\frac{|g(x)-g(u)|+|g(y)-g(v)|}{\frac{t}{2}}]$ (since $\mathit{\phi }(t)\ge \frac{t}{2}$ for all $t\ge 0$),

that is, $-\frac{|G(x,y)-G(u,v)|}{\mathit{\phi }(t)}\ge -\frac{|g(x)-g(u)|+|g(y)-g(v)|}{2t}$

Now$$\begin{array}{cc}\hfill F_{G(x,y),G(u,v)}(\mathit{\phi }(t))& =e^{-\frac{|G(x,y)-G(u,v)|}{\mathit{\phi }(t)}}\hfill \\ \hfill & \ge e^{-\frac{[|g(x)-g(u)|+|g(y)-g(v)|]}{2t}}\hfill \\ \hfill & \ge e^{-\frac{|g(x)-g(u)|}{2t}}.e^{-\frac{|g(y)-g(v)|}{2t}}\hfill \\ \hfill & \ge \sqrt{e^{-\frac{|g(x)-g(u)|}{t}}.e^{-\frac{|g(y)-g(v)|}{t}}}\hfill \\ \hfill & ={(F_{g(x),g(u)}(t).F_{g(y),g(v)}(t))}^{\frac{1}{2}}\hfill \end{array}$$

Hence (3.1) holds.

Thus all the conditions of Theorem 3.1 are satisfied. Then, by an application of the Theorem 3.1, we conclude that g and F have a coupled coincidence point. Here, (0, 0) is a coupled coincidence point of g and F in X.

Remark 3.6

 

(i)	$\{g,G\}$ are not commuting but compatible, so Corollary 3.2 is properly contained in Theorem 3.1.
(ii)	$\mathit{\phi }\in \mathrm{\Phi }$, so the Corollary 3.4 is properly contained in Corollary 3.3.
(iii)	For the reason mentioned in (ii), the Corollary 3.4 in properly contained in Theorem 3.1.

Open problems. The gauge function used in this paper is of a very general type. It appears possible to use this function to extend probabilistic contractions as in (Fang, 2015) and to define a new probabilistic contractions as in the present work. The study of such contractions, especially relating to the existence of their fixed point, is a class of problems which will be worthy of investigation.

Acknowledgement

The suggestions of the learned referees have been acknowledged.

Additional information

Funding

The first author gratefully acknowledges the support from DST, West Bengal, India [grant-in-Aid number 624 (sanc.)/ST/P/S]; [grant-in-Aid number T/ Misc-5/2012].

Notes on contributors

Binayak S. Choudhury

Binayak S. Choudhury is a professor of Mathematics at Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India. His research interests are in areas of pure mathematics, applied mathematics, and theoretical physics in which he has published about 200 research articles. He has also acted as Principal Investigators of several projects from Government Agencies of India.

Pradyut Das

Pradyut Das is a senior research student pursuing his PhD work in the Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India. He has 20 research publications to his credit.

P. Saha

P. Saha is a professor of Mathematics at Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India. Her research areas are fuzzy functional analysis and fuzzy logic.