Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 18, 2005, 139 – 149

© Ghulam Mustafa, Nusrat Anjum Noshi and Abdur Rashid

ABSTRACT. Some new random coincidence point and random fixed point theorems for multifunctions in separable complete metrically convex metric spaces are proved. Our results are stochastic generalizations of classical coincidence and fixed point theorems.

1. Introduction

In order to give stochastic generalizations for classical coincidence point theorems and classical fixed point theorems many authors ([1, 3, 4, 5, 6, 8, 9]) introduced more general contractive inequalities. We consider a class of generalized contractions that includes the classes considered in ([1, 3, 4, 5, 6, 8, 9]) and this enables us to prove more general random fixed point and random coincidence point theorems for multifunctions. The results presented in this paper are stochastic versions of corresponding results in [10].

Throughout this paper $\left(X,d\right)$ is a separable complete metrically convex metric space, $K$ is a nonempty subset of $X=\left(X,d\right)$ and $\left(\Omega ,\sigma \right)$ is measurable space with a $\sigma$-algebra $\sigma$ of subsets of $\Omega$. Let $2^K$ be the family of all subsets of $K$, and $CB\left(X\right)$ the family of all nonempty closed bounded subsets of $X$. For any nonempty subsets $A,B$ of $X$, we write

and $H\left(.,.\right)$ is called the Hausdorff metric on $CB\left(X\right).$

Definition 2.1 A mapping $\mu :\Omega \to 2^K$ is called measurable if for any open subset $C$ of $K$, ${\mu }^{-1}\left(C\right)=\left\{w\in \Omega :\mu \left(w\right)\cap C\ne \varnothing \right\}\in \sigma$.

Definition 2.2 A mapping $z:\Omega \to X$ is said to be a measurable selector of a measurable mapping $\mu :\Omega \to 2^K$ if $z$ is measurable and for any $w\in \Omega ,z\left(w\right)\in \mu \left(w\right)$.
Definition 2.3 A metric space $\left(X,d\right)$ is said to be metrically convex if for any $x,y\in X$ with $x\ne y$, there exists $z\in X$, $x\ne z\ne y$ such that

Definition 2.4 A mapping $T:\Omega \times K\to X$ is called a random operator if for any $x\in K,T\left(.,x\right)$ is measurable. A mapping $F:\Omega \times K\to CB\left(X\right)$ is called a multifunction if for every $x\in K,F\left(.,x\right)$ is measurable.

Definition 2.5 A measurable mapping $z:\Omega \to X$ is called a random fixed point of a multifunction (random operator) $F:\Omega \times K\to CB\left(X\right)$ $\left(T:\Omega \times K\to X\right)$ if for every $w\in \Omega ,z\left(w\right)\in F\left(w,z\left(w\right)\right)$ $\left(T\left(w,z\left(w\right)\right)=z\left(w\right)\right)$.

Definition 2.6 A measurable mapping $z:\Omega \to X$ is a random coincidence point of $F:\Omega \times K\to CB\left(X\right)$ and $T:\Omega \times K\to X$ if for every $w\in \Omega ,T\left(w,z\left(w\right)\right)\in F\left(w,z\left(w\right)\right)$. Let $C\left(T,F\right)$ stand for the set of random coincidence points of the maps $T$ and $F$, that is, $C\left(T,F\right)$ =$\left\{z\left(w\right):T\left(w,z\left(w\right)\right)\in F\left(w,z\left(w\right)\right)\right\}$.

Definition 2.7 Let $T:\Omega \times K\to X$ be a random operator and $F:\Omega \times K\to CB\left(X\right)$ be a multifunction. Then $T$ and $F$ will be called pointwise $R-$weakly commuting on $K$ if given $x\in K$ and $T\left(w,x\right)\in K$, there exists a measurable map $R:\Omega \to \left(0,\infty \right)$ such that for each $y\in K\cap F\left(w,x\right)$,

The maps $T$ and $F$ will be called $R-$weakly commuting on $K$ if for each $x\in K$, $T\left(w,x\right)\in K$ and (*) holds. If $R\left(w\right)=1$ for each $w\in \Omega$, we get the definition of weak commutativity of $F$ and $T$ on $K$. $T$ and $F$ are commuting at a point $x\in K$ if $T\left(w,F\left(w,x\right)\right)\subset F\left(w,T\left(w,x\right)\right)$ whenever $F\left(w,x\right)\subset K$ and $T\left(w,x\right)\in K$. $T$ and $F$ are commuting on $K$ if they are commuting at each point $x\in K$.

2. MAIN RESULTS

Let $F,G:\Omega \times K\to CB\left(X\right)$ be multifunctions and $S,T:\Omega \times K\to X$ be random operators such that

for each $x,y\in K$ and for each $w\in \Omega$, where $\alpha ,\beta ,\gamma :\Omega \to \left(0,\infty \right)$ are measurable mappings such that

Theorem 1. Let $\left(X,d\right)$ be a separable complete metrically convex metric space, $K$ a nonempty closed subset of $X$, and $\delta K$ the boundary of $K$. Let $F,G:\Omega \times K\to CB\left(X\right)$ be continuous multifunctions and $S,T:\Omega \times K\to X$ be random operators, such that
(i) contractive inequalities (2.1) and (2.2);
(ii) $\delta K\subset S\left(w,K\right)\cap T\left(w,K\right)$, $F\left(w,K\right)\cap K\subset S\left(w,K\right)$, $G\left(w,K\right)\cap K\subset T\left(w,K\right)$; and
(iii) $T\left(w,x\right)\in \delta K\Rightarrow F\left(w,x\right)\subset K$, $S\left(w,x\right)\in \delta K\Rightarrow G\left(w,x\right)\subset K$;
are satisfied.
If, either $T\left(w,K\right)$ and $S\left(w,K\right)$ or $F\left(w,K\right)$ and $G\left(w,K\right)$ are closed subspaces of $X$, then
(a) $F$ and $T$ have a random coincidence point;
(b) $G$ and $S$ have a random coincidence point.
Furthermore,
(c) $F$ and $T$ have a common random fixed point $T\left(w,v\left(w\right)\right)$ provided $T\left(w,T\left(w,v\left(w\right)\right)\right)=T\left(w,v\left(w\right)\right)$ and $T$ and $F$ are commuting at $v\left(w\right)\in C\left(T,F\right)$;
(d) $G$ and $S$ have a common random fixed point $S\left(w,\mu \left(w\right)\right)$, provided $S\left(w,S\left(w,\mu \left(w\right)\right)\right)=S\left(w,\mu \left(w\right)\right)$ and $S$ and $G$ are commuting at $\mu \left(w\right)\in C\left(S,G\right)$;
(e) $S,T,F$ and $G$ have a common random fixed point, provided (c) and (d) both are true.

Proof. If the following equality

holds true, then the theorem holds trivially. Next if $t\left(w\right)>0$, then we proceed to construct the sequences $\left\{x_n\left(w\right)\right\}$ and $\left\{y_n\left(w\right)\right\}$, where $x_n,y_n:\Omega \to X$ are measurable mappings.
Let $x_0,x_1:\Omega \to X$ be a measurable mappings such that $y_1:\Omega \to X$ defined by $y_1\left(w\right)=S\left(w,x_1\left(w\right)\right)\in F\left(w,x_0\left(w\right)\right)$, for all $w\in \Omega$ . Indeed, since $F$ is a continuous random operator, we conclude that, for every $v\in X$, the map $\left(w,x\right)\to d\left(v,F\left(w,x\right)\right)$ is a caratheodory function (that is measurable in $w\in \Omega$, continuous in $x\in X$). Thus it is jointly measurable. Hence since $x_0:\Omega \to X$ is measurable, $w\to d\left(v,F\left(w,x_0\left(w\right)\right)\right)$ is measurable. Therefore, $w\to F\left(w,x_0\left(w\right)\right)$ is weakly measurable by Wagner ([4], p 868). By Kuratowski, K ([7], selection theorem 8), there exists a measurable map $x_1:\Omega \to X$ such that $y_1\left(w\right)=S\left(w,x_1\left(w\right)\right)\in F\left(w,x_0\left(w\right)\right)$ for $x_0\left(w\right),x_1\left(w\right)\in K$, for all $w\in \Omega$. It follows from (ii) and (iii) that $F\left(w,x_0\left(w\right)\right)\subset K$. Therefore, $y_1\left(w\right)\in K.$ It further implies by Itoh ([9], Proposition 4), (ii) and (iii) that there exist measurable mappings $x_2,y_2:\Omega \to X$ such that, for each $w\in \Omega$, and for $y_2\left(w\right)\in K$ (suppose), we have $x_2\left(w\right)\in K$ and $y_2\left(w\right)=T\left(w,x_2\left(w\right)\right)\in G\left(w,x_1\left(w\right)\right)$ such that

If $y_2\left(w\right)\notin K$, then there exists a measurable map $p:\Omega \to X$ such that $p\left(w\right)\in \delta K$ and

Since $p\left(w\right)\in \delta K\subset T\left(w,K\right)$, there exists $x_2\left(w\right)\in K$ such that $p\left(w\right)=T\left(w,x_2\left(w\right)\right)$ and so

Thus repeating the above arguments, we obtain two sequences $\left\{x_n\left(w\right)\right\}$ and $\left\{y_n\left(w\right)\right\}$, where $x_n,y_n:\Omega \to X$ are measurable mappings, and $x_n\left(w\right)\in K$ such that

• $y_{2n}\left(w\right)\in G\left(w,x_{2n-1}\left(w\right)\right)$, $y_{2n+1}\left(w\right)\in F\left(w,x_{2n}\left(w\right)\right)$,
• $y_{2n}\left(w\right)\in K\Rightarrow y_{2n}\left(w\right)=T\left(w,x_{2n}\left(w\right)\right)$ or $y_{2n}\left(w\right)\notin K\Rightarrow T\left(w,x_{2n}\left(w\right)\right)\in \delta K$ and $$\begin{array}{c}d\left(S\left(w,x_{2n-1}\left(w\right)\right),T\left(w,x_{2n}\left(w\right)\right)\right)+d\left(T\left(w,x_{2n}\left(w\right)\right),y_{2n}\left(w\right)\right)\\ =d\left(S\left(w,x_{2n-1}\left(w\right)\right),y_{2n}\left(w\right)\right),\end{array}$$

• $y_{2n+1}\left(w\right)\in K$, $y_{2n+1}\left(w\right)=S\left(w,x_{2n+1}\left(w\right)\right)$, or $y_{2n+1}\left(w\right)\notin K$, $S\left(w,x_{2n+1}\left(w\right)\right)\in \delta K$, and $$\begin{array}{c}d\left(T\left(w,x_{2n}\left(w\right)\right),S\left(w,x_{2n+1}\left(w\right)\right)\right)+d\left(S\left(w,x_{2n+1}\left(w\right)\right),y_{2n+1}\left(w\right)\right)\\ =d\left(T\left(w,x_{2n}\left(w\right)\right),y_{2n+1}\left(w\right)\right).\end{array}$$
  $$\begin{array}{c}d\left(y_{2n-1}\left(w\right),y_{2n}\left(w\right)\right)\le H\left(G\left(w,x_{2n-1}\left(w\right)\right),F\left(w,x_{2n-2}\left(w\right)\right)\right)\\ +\left(\left(1-\beta \left(w\right)-\gamma \left(w\right)\right)∕\left(1+\beta \left(w\right)+\gamma \left(w\right)\right)\right)t^{2n-1}\left(w\right),\end{array}$$
  $$\begin{array}{c}d\left(y_{2n}\left(w\right),y_{2n+1}\left(w\right)\right)\le H\left(F\left(w,x_{2n}\left(w\right)\right),G\left(w,x_{2n-1}\left(w\right)\right)\right)\\ +\left(\left(1-\beta \left(w\right)-\gamma \left(w\right)\right)∕\left(1+\beta \left(w\right)+\gamma \left(w\right)\right)\right)t^{2n}\left(w\right).\end{array}$$

Put

Further, as shown in [2], for measurable maps $z_n:\Omega \to X$, $\left\{z_n\left(w\right)\right\}$ is a Cauchy sequence, where

and there exists at least one subsequence

which is contained in $P_0$, or $Q_0$, respectively. First we suppose that there exists a subsequence $\left\{T\left(w,x_{2n_k}\left(w\right)\right)\right\}$ which is contained in $P_0$, and $T\left(w,K\right)$, $S\left(w,K\right)$ are closed subspaces of $X$. Since $\left\{T\left(w,x_{2n_k}\left(w\right)\right)\right\}$ is a Cauchy sequence in $T\left(w,K\right)$. Then there exists a measurable map $u:\Omega \to X$ such that $\left\{T\left(w,x_{2n_k}\left(w\right)\right)\right\}\to u\left(w\right)\in T\left(w,K\right).$ Let $v\left(w\right)\in K$ for a measurable map $v:\Omega \to X$ and $\left(w,v\left(w\right)\right)\in T^{-1}\left(u\left(w\right)\right).$ Then $u\left(w\right)=T\left(w,v\left(w\right)\right).$ Since $\left\{S\left(w,x_{2n_k+1}\left(w\right)\right)\right\}$ is a subsequence of the Cauchy sequence $\left\{z_n\left(w\right)\right\}$, $\left\{S\left(w,x_{2n_k+1}\left(w\right)\right)\right\}$ converges to $u\left(w\right)$ as well. By (2.1), we have

Letting $n\to \infty$, we obtain

proving $u\left(w\right)\in F\left(w,v\left(w\right)\right)$, since $F\left(w,v\left(w\right)\right)$ is closed. This proves (a).
Since the Cauchy sequence $\left\{z_n\left(w\right)\right\}$ converges to $u\left(w\right)\in K$ and $u\left(w\right)\in F\left(w,v\left(w\right)\right)$, $u\left(w\right)\in F\left(w,K\right)\cap K\subset S\left(w,K\right)$, there exists $\mu \left(w\right)\in K$ such that $S\left(w,\mu \left(w\right)\right)=u\left(w\right)$, where $\mu :\Omega \to X$ is a measurable map. By (2.1) again, we have

this proves (b).

If $F\left(w,K\right)$ and $G\left(w,K\right)$ are closed subspaces, then

and the above argument establishes (a) and (b). If we suppose that there exists a subsequence $\left\{S\left(w,x_{2n_k+1}\left(w\right)\right)\right\}$ contained in $Q_0$, and $T\left(w,K\right)$, $S\left(w,K\right)$ are closed subspaces of $X$, then, noting that $\left\{S\left(w,x_{2n_k+1}\left(w\right)\right)\right\}$ is a Cauchy sequence in $S\left(w,K\right)$, an analogous argument establishes (a) and (b).

To prove (c), note that $v\left(w\right)\in C\left(T,F\right)$ and $u\left(w\right)=T\left(w,v\left(w\right)\right)$. From this $T\left(w,u\left(w\right)\right)=T\left(w,T\left(w,v\left(w\right)\right)\right)=T\left(w,v\left(w\right)\right)$, hence $T\left(w,u\left(w\right)\right)=u\left(w\right)$, and from the commutativity of $T$ and $F$, we derive

Thus $u\left(w\right)$ is a common random fixed point of $T$ and $F$. Similar argument yields $u\left(w\right)=S\left(w,u\left(w\right)\right)\in G\left(w,u\left(w\right)\right)$, proving (d). Now e) is immediate.

Corollary 1. Let $\left(X,d\right)$ be a separable complete metrically convex metric space, $K$ a nonempty closed subset of $X$, and $\delta K$ the boundary of $K$. Let $F,G:\Omega \times K\to CB\left(X\right)$ be continuous multifunctions and $T:\Omega \times K\to X$ be a random operator, such that

(i) contractive inequality (2.1) with $S=T$, and inequality (2.2);

(ii) $\delta K\subset T\left(w,K\right)$, $F\left(w,K\right)\cup G\left(w,K\right)\cap K\subset T\left(w,K\right)$;

(iii) $T\left(w,x\right)\in \delta K\Rightarrow F\left(w,x\right)\cup G\left(w,x\right)\subset K$;

(iv) either $T\left(w,K\right)$ or $F\left(w,K\right)$ and $G\left(w,K\right)$ are closed subspaces of $X$.

Then, $F,G$, and $T$ have a common random coincidence point $v\left(w\right)$. Furthermore, $F,G$, and $T$ have a common random fixed point provided $T\left(w,v\left(w\right)\right)$ is a random fixed point of $T$ and $T$ commutes with each of $F$ and $G$ at $v\left(w\right)$.

Theorem 2. Let $\left(X,d\right)$ be a separable complete metrically convex metric space, $K$ a nonempty closed subset of $X$, and $\delta K$ the boundary of $K$. Let $F,G:\Omega \times K\to CB\left(X\right)$ be continuous multifunctions and $S,T:\Omega \times K\to X$ be continuous random operators, such that

(i) contractive inequalities (2.1) and (2.2);

(ii) $\delta K\subset S\left(w,K\right)\cap T\left(w,K\right)$, $F\left(w,K\right)\cap K\subset S\left(w,K\right)$, $G\left(w,K\right)\cap K\subset T\left(w,K\right)$; and

(iii) $T\left(w,x\right)\in \delta K\Rightarrow F\left(w,x\right)\subset K$, $S\left(w,x\right)\in \delta K\Rightarrow G\left(w,x\right)\subset K$;
are satisfied.
Suppose that $\left(T,F\right)$ and $\left(S,G\right)$ are pointwise $R$-weakly commuting pairs, then

(a) There exists a point $z\left(w\right)\in K$ such that $S\left(w,z\left(w\right)\right)\in G\left(w,z\left(w\right)\right)$ and $T\left(w,z\left(w\right)\right)\in F\left(w,z\left(w\right)\right)$.

Furthermore,

(b) $T$ and $F$ have a common random fixed point provided

(c) $S$ and $G$ have a common random fixed point provided

(d) $S,T,F$ and $G$ have a common random fixed point provided (b) and (c) both are true.

Proof. Proceeding as in the proof of Theorem 1, we suppose that there exists a subsequence $\left\{T\left(w,x_{2n_k}\left(w\right)\right)\right\}$ which is contained in $P_0$. Further, subsequences $\left\{T\left(w,x_{2n_k}\left(w\right)\right)\right\}$ and $\left\{S\left(w,x_{2n_k+1}\left(w\right)\right)\right\}$ both converge to a $z\left(w\right)\in K$, since $K$ is closed in complete $X$, where $z:\Omega \to X$ is a measurable map. Since $T\left(w,x_{2n_k}\left(w\right)\right)\in G\left(w,x_{2n_k-1}\left(w\right)\right)\cap K$ and $S\left(w,x_{2n_k-1}\left(w\right)\right)\in K$, the pointwise $R-$weak commutativity of $G$ and $S$ implies

for some measurable map $R_1:\Omega \to \left(0,\infty \right)$. Also,

Letting $k\to \infty$ in (2.3) and (2.4) and using the continuity of $S$ and $T$, we obtain

yielding $S\left(w,z\left(w\right)\right)\in G\left(w,z\left(w\right)\right)$. Since $y_{2n_k+1}\left(w\right)\in F\left(w,x_{2n_k}\left(w\right)\right)\cap K$ and $T\left(w,x_{2n_k}\left(w\right)\right)\in K$, the pointwise $R-$weak commutativity of $F$ and $T$ implies

for some measurable map $R_2:\Omega \to \left(0,\infty \right)$. Therefore, as previously, the continuity of $T$ and $F$ implies

proving $T\left(w,z\left(w\right)\right)\in F\left(w,z\left(w\right)\right)$. This proves (a).
If we suppose that there exists a subsequence $\left\{S\left(w,x_{2n_k+1}\left(w\right)\right)\right\}$ contained in $Q_0$, then analogous argument establishes (a).

If $T\left(w,T\left(w,z\left(w\right)\right)\right)=T\left(w,z\left(w\right)\right)$ then $T\left(w,z\left(w\right)\right)\in K$. Thus $z\left(w\right)\in K$ and $T\left(w,z\left(w\right)\right)\in K\cap F\left(w,z\left(w\right)\right).$ Now using the pointwise $R-$weak commutativity of $T$ and $F$ at $z\left(w\right)$, we get

for some measurable map $R_3:\Omega \to \left(0,\infty \right)$, where $T\left(w,T\left(w,z\left(w\right)\right)\right)\in F\left(w,T\left(w,z\left(w\right)\right)\right).$ This proves (b). A similar argument proves (c). Now (d) is immediate.

Corollary 2. Let $\left(X,d\right)$ be a separable complete metrically convex metric space, $K$ a nonempty closed subset of $X$, and $\delta K$ the boundary of $K$. Let $F,G:\Omega \times K\to CB\left(X\right)$ be continuous multifunctions and $T:\Omega \times K\to X$ be a continuous random operator, such that

(i) contractive inequality (2.1) with $S=T$ and inequality (2.2);

(ii) $\delta K\subset T\left(w,K\right)$, $F\left(w,K\right)\cup G\left(w,K\right)\cap K\subset T\left(w,K\right)$;

(iii) $T\left(w,x\right)\in \delta K\Rightarrow F\left(w,x\right)\cup G\left(w,x\right)\subset K$.

Suppose that $T$ is pointwise $R-$weakly commuting with each of $F$ and $G$.

Then, $F,G$, and $T$ have common random coincidence point $z\left(w\right)$. Furthermore, $F,G,$ and $T$ have a common random fixed point, provided $T\left(w,T\left(w,z\left(w\right)\right)\right)=T\left(w,z\left(w\right)\right)$.

Consider $F,G:\Omega \times K\to CB\left(X\right)$ and $T:\Omega \times K\to X$ satisfying

when $M\left(x,y\right)>0$, $x,y\in K$, where

and $\alpha ,\beta ,\gamma :\Omega \to \left(0,\infty \right)$ are measurable mappings such that