Lobachevskii Journal of Mathematics

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Vol. 18, 2005, 139–149 \begin{matrix}c \\ ◯ \end{matrix} Ghulam Mustafa, Nusrat Anjum Noshi and Abdur Rashid

Ghulam Mustafa, Nusrat Anjum Noshi and Abdur Rashid

SOME RANDOM COINCIDENCE AND RANDOM FIXED POINT THEOREMS FOR HYBRID CONTRACTIONS

(submitted by D. Kh. Mushtari)

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 (X, d) is a separable complete metrically convex metric space, K is a nonempty subset of X = (X, d) and (Ω, σ) is measurable space with a σ-algebra σ of subsets of Ω. Let 2^{K} be the family of all subsets of K, and CB(X) the family of all nonempty closed

cidence point.

2000 Mathematical Subject Classification. 54H25,47H10.

Key words and phrases. multifunction, random fixed point, random coin-

bounded subsets of X. For any nonempty subsets A, B of X, we write

d(x, A) = inf{d(x, a) : a ∈ A}(x ∈ X),

d(A, B) = inf{d(a, b) : a ∈ A, b ∈ B},

H(A, B) = max {sup_{a∈A} d(a, B), sup_{b∈B} d(b, A)} ,

and H(., .) is called the Hausdorff metric on CB(X). Definition 2.1 A mapping μ : Ω → 2^{K} is called measurable if for any open subset C of K, μ^{−1}(C) = {w ∈ Ω : μ(w) ∩ C ≠ ∅} ∈ σ. Definition 2.2 A mapping z : Ω → X is said to be a measurable selector of a measurable mapping μ : Ω → 2^{K} if z is measurable and for any w ∈ Ω, z(w) ∈ μ(w). Definition 2.3 A metric space (X, d) is said to be metrically convex if for any x, y ∈ X with x ≠ y, there exists z ∈ X, x ≠ z ≠ y such that

d(x, z) + d(z, y) = d(x, y)

Definition 2.4 A mapping T : Ω × K → X is called a random operator if for any x ∈ K, T (., x) is measurable. A mapping F : Ω × K → CB(X) is called a multifunction if for every x ∈ K, F (., x) is measurable. Definition 2.5 A measurable mapping z : Ω → X is called a random fixed point of a multifunction (random operator) F : Ω × K → CB(X) (T : Ω × K → X) if for every w ∈ Ω, z(w) ∈ F (w, z(w)) (T (w, z(w)) = z(w)). Definition 2.6 A measurable mapping z : Ω → X is a random coincidence point of F : Ω × K → CB(X) and T : Ω × K → X if for every w ∈ Ω, T (w, z(w)) ∈ F (w, z(w)). Let C(T, F) stand for the set of random coincidence points of the maps T and F, that is, C(T, F) ={z(w) : T (w, z(w)) ∈ F (w, z(w))}. Definition 2.7 Let T : Ω × K → X be a random operator and F : Ω × K → CB(X) be a multifunction. Then T and F will be called pointwise R−weakly commuting on K if given x ∈ K and T (w, x) ∈ K, there exists a measurable map R : Ω → (0, ∞) such that for each y ∈ K ∩ F (w, x),

d(T (w, y), F (w, T (w, x))) ≤ R(w)d(F (w, x), T (w, x)).  (*)

RANDOM COINCIDENCE AND RANDOM FIXED POINT 141

The maps T and F will be called R−weakly commuting on K if for each x ∈ K, T (w, x) ∈ K and (*) holds. If R(w) = 1 for each w ∈ Ω, we get the definition of weak commutativity of F and T on K. T and F are commuting at a point x ∈ K if T (w, F (w, x)) ⊂ F (w, T (w, x)) whenever F (w, x) ⊂ K and T (w, x) ∈ K. T and F are commuting on K if they are commuting at each point x ∈ K.

2. MAIN RESULTS

Let F, G : Ω × K → CB(X) be multifunctions and S, T : Ω × K → X be random operators such that

H(F (w, x), G(w, y)) ≤ α(w)d(T (w, x), S(w, y))

+ β(w)[d(T (w, x), F (w, x)) + d(S(w, y), G(w, y))] + γ(w)[d(T (w, x), G(w, y)) + d(S(w, y), F (w, x))] (2.1)

for each x, y ∈ K and for each w ∈ Ω, where α, β, γ : Ω → (0, ∞) are measurable mappings such that

(α(w) + β(w) + γ(w))(1 + β(w) + γ(w))/(1 − β(w) − γ(w))^{2} < 1. (2.2)

Theorem 1. Let (X, d) be a separable complete metrically convex metric space, K a nonempty closed subset of X, and δK the boundary of K. Let F, G : Ω × K → CB(X) be continuous multifunctions and S, T : Ω × K → X be random operators, such that

(i) contractive inequalities (2.1) and (2.2); (ii) δK ⊂ S(w, K) ∩ T (w, K), F (w, K) ∩ K ⊂ S(w, K), G(w, K) ∩ K ⊂ T (w, K); and (iii) T (w, x) ∈ δK ⇒ F (w, x) ⊂ K, S(w, x) ∈ δK ⇒ G(w, x) ⊂ K; are satisfied. If, either T (w, K) and S(w, K) or F (w, K) and G(w, K) 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 (w, v(w)) provided T (w, T (w, v(w))) = T (w, v(w)) and T and F are commuting at v(w) ∈ C(T, F);

(d) G and S have a common random fixed point S(w, μ(w)), provided S(w, S(w, μ(w))) = S(w, μ(w)) and S and G are commuting at μ(w) ∈ C(S, G);

(e) S, T, F and G have a common random fixed point, provided (c) and (d) both are true. Proof. If the following equality

t(w) = (α(w) + β(w) + γ(w))(1 + β(w) + γ(w))/(1 − β(w) − γ(w))^{2} = 0 holds true, then the theorem holds trivially. Next if t(w) > 0, then we proceed to construct the sequences {x_{n}(w)} and {y_{n}(w)}, where x_{n}, y_{n} : Ω → X are measurable mappings. Let x_{0}, x_{1} : Ω → X be a measurable mappings such that y_{1} : Ω → X defined by y_{1}(w) = S(w, x_{1}(w)) ∈ F (w, x_{0}(w)), for all w ∈ Ω . Indeed, since F is a continuous random operator, we conclude that, for every v ∈ X, the map (w, x) → d(v, F (w, x)) is a caratheodory function (that is measurable in w ∈ Ω, continuous in x ∈ X). Thus it is jointly measurable. Hence since x_{0} : Ω → X is measurable, w → d(v, F (w, x_{0}(w))) is measurable. Therefore, w → F (w, x_{0}(w)) is weakly measurable by Wagner ([4], p 868). By Kuratowski, K ([7], selection theorem 8), there exists a measurable map x_{1} : Ω → X such that y_{1}(w) = S(w, x_{1}(w)) ∈ F (w, x_{0}(w)) for x_{0}(w), x_{1}(w) ∈ K, for all w ∈ Ω. It follows from (ii) and (iii) that F (w, x_{0}(w)) ⊂ K. Therefore, y_{1}(w) ∈ K. It further implies by Itoh ([9], Proposition 4), (ii) and (iii) that there exist measurable mappings x_{2}, y_{2} : Ω → X such that, for each w ∈ Ω, and for y_{2}(w) ∈ K (suppose), we have x_{2}(w) ∈ K and y_{2}(w) = T (w, x_{2}(w)) ∈ G(w, x_{1}(w)) such that

d(y_{1}(w), y_{2}(w)) ≤ H(F (w, x_{0}(w)), G(w, x_{1}(w)))+ ((1 − β(w) − γ(w))/(1 + β(w) + γ(w)))t(w).

If y_{2}(w) ∉ K, then there exists a measurable map p : Ω → X such that p(w) ∈ δK and

d(S(w, x_{1}(w)), p(w)) + d(p(w), y_{2}(w)) = d(S(w, x_{1}(w)), y_{2}(w)).

Since p(w) ∈ δK ⊂ T (w, K), there exists x_{2}(w) ∈ K such that p(w) = T (w, x_{2}(w)) and so

d(S(w, x_{1}(w)), T (w, x_{2}(w)))+d(T (w, x_{2}(w)), y_{2}(w)) = d(S(w, x_{1}(w)), y_{2}(w)).

Thus repeating the above arguments, we obtain two sequences {x_{n}(w)} and {y_{n}(w)}, where x_{n}, y_{n} : Ω → X are measurable mappings, and x_{n}(w) ∈ K such that

• y_{2n}(w) ∈ G(w, x_{2n−1}(w)), y_{2n+1}(w) ∈ F (w, x_{2n}(w)),

• y_{2n}(w) ∈ K ⇒ y_{2n}(w) = T (w, x_{2n}(w)) or y_{2n}(w) ∉ K

⇒ T (w, x_{2n}(w)) ∈ δK and

d(S(w, x_{2n−1}(w)), T (w, x_{2n}(w))) + d(T (w, x_{2n}(w)), y_{2n}(w)) = d(S(w, x_{2n−1}(w)), y_{2n}(w)), • y_{2n+1}(w) ∈ K, y_{2n+1}(w) = S(w, x_{2n+1}(w)), or y_{2n+1}(w) ∉ K,

S(w, x_{2n+1}(w)) ∈ δK, and

d(T (w, x_{2n}(w)), S(w, x_{2n+1}(w))) + d(S(w, x_{2n+1}(w)), y_{2n+1}(w)) = d(T (w, x_{2n}(w)), y_{2n+1}(w)).

d(y_{2n−1}(w), y_{2n}(w)) ≤ H(G(w, x_{2n−1}(w)), F (w, x_{2n−2}(w))) + ((1 − β(w) − γ(w))/(1 + β(w) + γ(w)))t^{2n−1}(w),

d(y_{2n}(w), y_{2n+1}(w)) ≤ H(F (w, x_{2n}(w)), G(w, x_{2n−1}(w))) + ((1 − β(w) − γ(w))/(1 + β(w) + γ(w)))t^{2n}(w).

Put

P_{0} = {T (w, x_{2i}(w)) ∈ {T (w, x_{2n}(w))} : T (w, x_{2i}(w)) = y_{2i}(w)},
P_{1} = {T (w, x_{2i}(w)) ∈ {T (w, x_{2n}(w))} : T (w, x_{2i}(w)) ≠ y_{2i}(w)},

Q_{0} = {S(w, x_{2i+1}(w)) ∈ {S(w, x_{2n+1}(w))} : S(w, x_{2i+1}(w)) = y_{2i+1}(w)},

Q_{1} = {S(w, x_{2i+1}(w)) ∈ {S(w, x_{2n+1}(w))} : S(w, x_{2i+1}(w)) ≠ y_{2i+1}(w)}.

Further, as shown in [2], for measurable maps z_{n} : Ω → X, {z_{n}(w)} is a Cauchy sequence, where

z_{2n}(w) = T (w, x_{2n}(w)), z_{2n+1}(w) = S(w, x_{2n+1}(w)),

and there exists at least one subsequence

{T (w, x_{2n_{k}}(w))}, or {S(w, x_{2n_{k}+1}(w))},

which is contained in P_{0}, or Q_{0}, respectively. First we suppose that there exists a subsequence {T (w, x_{2n_{k}}(w))} which is contained in P_{0}, and T (w, K), S(w, K) are closed subspaces of X. Since {T (w, x_{2n_{k}}(w))} is a Cauchy sequence in T (w, K). Then there exists a measurable map u : Ω → X such that {T (w, x_{2n_{k}}(w))} → u(w) ∈ T (w, K). Let v(w) ∈ K for a measurable map v : Ω → X and (w, v(w)) ∈ T^{−1}(u(w)). Then u(w) = T (w, v(w)). Since {S(w, x_{2n_{k}+1}(w))} is a subsequence of the Cauchy sequence {z_{n}(w)}, {S(w, x_{2n_{k}+1}(w))} converges to u(w) as well.

By (2.1), we have

d(F (w, v(w)), T (w, x_{2n_{k}}(w))) ≤ H(F (w, v(w)), G(w, x_{2n_{k}−1}(w)))

≤ α(w)d(T (w, v(w)), S(w, x_{2n_{k}−1}(w)))

+β(w)[d(T (w, v(w)), F (w, v(w)))+d(S(w, x_{2n_{k}−1}(w)), G(w, x_{2n_{k}−1}(w)))] +γ(w)[d(T (w, v(w)), G(w, x_{2n_{k}−1}(w)))+d(S(w, x_{2n_{k}−1}(w)), F (w, v(w)))]

≤ α(w)d(u(w), S(w, x_{2n_{k}−1}(w))) + β(w)[d(u(w), F (w, v(w))) + d(S(w, x_{2n_{k}−1}(w)), T (w, x_{2n_{k}}(w)))] + γ(w)[d(u(w), T (w, x_{2n_{k}}(w))) + d(S(w, x_{2n_{k}−1}(w)), F (w, v(w)))].

Letting n → ∞, we obtain

d(F (w, v(w)), u(w)) ≤ (β(w) + γ(w))d(u(w), F (w, v(w))),

proving u(w) ∈ F (w, v(w)), since F (w, v(w)) is closed. This proves (a). Since the Cauchy sequence {z_{n}(w)} converges to u(w) ∈ K and u(w) ∈ F (w, v(w)), u(w) ∈ F (w, K)∩K ⊂ S(w, K), there exists μ(w) ∈ K such that S(w, μ(w)) = u(w), where μ : Ω → X is a measurable map. By (2.1) again, we have

d(S(w, μ(w)), G(w, μ(w))) = d(T (w, v(w)), G(w, μ(w))) ≤ H(F (w, v(w)), G(w, μ(w))) ≤ α(w)d(T (w, v(w)), S(w, μ(w))) + β(w)[d(T (w, v(w)), F (w, v(w))) + d(S(w, μ(w)), G(w, μ(w)))] + γ(w)[d(T (w, v(w)), G(w, μ(w))) + d(S(w, μ(w)), F (w, v(w)))]

= (β(w) + γ(w))d(S(w, μ(w)), G(w, μ(w))),

this proves (b).

If F (w, K) and G(w, K) are closed subspaces, then

u(w) ∈ F (w, K) ∩ K ⊂ S(w, K) or u(w) ∈ G(w, K) ∩ K ⊂ T (w, K),

and the above argument establishes (a) and (b). If we suppose that there exists a subsequence {S(w, x_{2n_{k}+1}(w))} contained in Q_{0}, and T (w, K), S(w, K) are closed subspaces of X, then, noting that {S(w, x_{2n_{k}+1}(w))} is a Cauchy sequence in S(w, K), an analogous argument establishes (a) and (b).

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

u(w) = T (w, u(w)) = T (w, T (w, v(w))) ∈ T (w, F (w, v(w))) ⊂ F (w, T (w, v(w))) = F (w, u(w)).

Thus u(w) is a common random fixed point of T and F. Similar argument yields u(w) = S(w, u(w)) ∈ G(w, u(w)), proving (d). Now e) is immediate. Corollary 1. Let (X, d) be a separable complete metrically convex metric space, K a nonempty closed subset of X, and δK the boundary of K. Let F, G : Ω × K → CB(X) be continuous multifunctions and T : Ω × K → X be a random operator, such that

(i) contractive inequality (2.1) with S = T, and inequality (2.2); (ii) δK ⊂ T (w, K), F (w, K) ∪ G(w, K) ∩ K ⊂ T (w, K); (iii) T (w, x) ∈ δK ⇒ F (w, x) ∪ G(w, x) ⊂ K; (iv) either T (w, K) or F (w, K) and G(w, K) are closed subspaces of

X.

Then, F, G, and T have a common random coincidence point v(w). Furthermore, F, G, and T have a common random fixed point provided T (w, v(w)) is a random fixed point of T and T commutes with each of F and G at v(w). Theorem 2. Let (X, d) be a separable complete metrically convex metric space, K a nonempty closed subset of X, and δK the boundary of K. Let F, G : Ω × K → CB(X) be continuous multifunctions and S, T : Ω × K → X be continuous random operators, such that

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

(ii) δK ⊂ S(w, K)∩T (w, K), F (w, K)∩K ⊂ S(w, K), G(w, K)∩K ⊂ T (w, K); and

(iii) T (w, x) ∈ δK ⇒ F (w, x) ⊂ K, S(w, x) ∈ δK ⇒ G(w, x) ⊂ K; are satisfied. Suppose that (T, F) and (S, G) are pointwise R-weakly commuting pairs, then

(a) There exists a point z(w) ∈ K such that S(w, z(w)) ∈ G(w, z(w)) and T (w, z(w)) ∈ F (w, z(w)).

Furthermore,

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

T (w, T (w, z(w))) = T (w, z(w));

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

S(w, S(w, z(w))) = S(w, z(w));

(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 {T (w, x_{2n_{k}}(w))} which is contained in P_{0}. Further, subsequences {T (w, x_{2n_{k}}(w))} and {S(w, x_{2n_{k}+1}(w))} both converge to a z(w) ∈ K, since K is closed in complete X, where z : Ω → X is a measurable map. Since T (w, x_{2n_{k}}(w)) ∈ G(w, x_{2n_{k}−1}(w)) ∩ K and S(w, x_{2n_{k}−1}(w)) ∈ K, the pointwise R−weak commutativity of G and S implies

d(S(w, T (w, x_{2n_{k}}(w))), G(w, S(w, x_{2n_{k}−1}(w)))) ≤ R_{1}(w)d(T (w, x_{2n_{k}}(w)), S(w, x_{2n_{k}−1}(w))) (2.3)

for some measurable map R_{1} : Ω → (0, ∞). Also,

d(S(w, T (w, x_{2n_{k}}(w))), G(w, z(w)))

≤ d(S(w, T (w, x_{2n_{k}}(w))), G(w, S(w, x_{2n_{k}−1}(w)))) + H(G(w, S(w, x_{2n_{k}−1}(w))), G(w, z(w))) (2.4)

Letting k → ∞ in (2.3) and (2.4) and using the continuity of S and T, we obtain

d(S(w, z(w)), G(w, z(w))) ≤ 0,

yielding S(w, z(w)) ∈ G(w, z(w)). Since y_{2n_{k}+1}(w) ∈ F (w, x_{2n_{k}}(w)) ∩ K and T (w, x_{2n_{k}}(w)) ∈ K, the pointwise R−weak commutativity of F and T implies

d(T (w, y_{2n_{k}+1}(w)), F (w, T (w, x_{2n_{k}}(w)))) ≤ R_{2}(w)d(y_{2n_{k}+1}(w), T (w, x_{2n_{k}}(w)))

for some measurable map R_{2} : Ω → (0, ∞). Therefore, as previously, the continuity of T and F implies

d(T (w, z(w)), F (w, z(w))) ≤ 0,

proving T (w, z(w)) ∈ F (w, z(w)). This proves (a). If we suppose that there exists a subsequence {S(w, x_{2n_{k}+1}(w))} contained in Q_{0}, then analogous argument establishes (a).

If T (w, T (w, z(w))) = T (w, z(w)) then T (w, z(w)) ∈ K. Thus z(w) ∈ K and T (w, z(w)) ∈ K ∩ F (w, z(w)). Now using the pointwise R−weak commutativity of T and F at z(w), we get

d(T (w, T (w, z(w))), F (w, T (w, z(w)))) ≤ R_{3}(w)d(F (w, z(w)), T (w, z(w))),

for some measurable map R_{3} : Ω → (0, ∞), where T (w, T (w, z(w))) ∈ F (w, T (w, z(w))). This proves (b). A similar argument proves (c). Now (d) is immediate. Corollary 2. Let (X, d) be a separable complete metrically convex metric space, K a nonempty closed subset of X, and δK the boundary of K. Let F, G : Ω × K → CB(X) be continuous multifunctions and T : Ω × K → X be a continuous random operator, such that

(i) contractive inequality (2.1) with S = T and inequality (2.2); (ii) δK ⊂ T (w, K), F (w, K) ∪ G(w, K) ∩ K ⊂ T (w, K); (iii) T (w, x) ∈ δK ⇒ F (w, x) ∪ G(w, x) ⊂ 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(w). Furthermore, F, G, and T have a common random fixed point, provided T (w, T (w, z(w))) = T (w, z(w)).

Consider F, G : Ω × K → CB(X) and T : Ω × K → X satisfying

H(F (w, x), G(w, y)) < M(x, y)  (2.5)

when M(x, y) > 0, x, y ∈ K, where

M(x, y) = α(w)d(T (w, x), T (w, y))

+ β(w)[d(T (w, x), F (w, x)) + d(T (w, y), G(w, y))] + γ(w)[d(T (w, x), G(w, y)) + d(T (w, y), F (w, x))]

and α, β, γ : Ω → (0, ∞) are measurable mappings such that

0 < (α(w)+β(w)+γ(w))(1+β(w)+γ(w))/(1−β(w)−γ(w))^{2} ≤ 1. (2.6)

Theorem 3. Let (X, d) be a separable complete metrically convex metric space, K a nonempty compact subset of X, and δK the boundary of K. Let F, G : Ω × K → CB(X) be continuous multifunctions and T : Ω × K → X be a continuous random operator, such that

(i) contractive inequalities (2.5) and (2.6);

(ii) δK ⊂ T (w, K), F (w, K) ∪ G(w, K) ∩ K ⊂ T (w, K);

(iii) T (w, x) ∈ δK ⇒ F (w, x) ∪ G(w, x) ⊂ 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(w). Furthermore, F, G, and T have a common random fixed point provided T (w, z(w)) is a random fixed point of T. Proof. In view of the last part of Corollary 2, it is enough to show that F, G and T have a common random coincidence point. We claim that M(x(w), y(w)) = 0, for some x(w), y(w) ∈ K, where x, y : Ω → X are measurable mappings. Otherwise the function

q(x(w), y(w)) = H(F (w, x(w)), G(w, y(w)))/M(x(w), y(w))

is continuous and satisfies q(x(w), y(w)) < 1 for (x(w), y(w)) ∈ K × K. Since K × K is compact, there exists u(w), v(w) ∈ K such that

q(x(w), y(w)) ≤ q(u(w), v(w)) = ν(w) < 1

for x(w), y(w) ∈ K and for some measurable map ν : Ω → (0, 1). Consequently,

H(F (w, x(w)), G(w, y(w))) ≤ ν(w)M(x(w), y(w)).

Further, in view of (2.6), it is a straightforward verification that

(ν(w)α(w) + ν(w)β(w) + ν(w)γ(w))(1 + ν(w)β(w) + ν(w)γ(w))/

(1 − ν(w)β(w) − ν(w)γ(w))^{2} < 1.

So, by Corollary 2, T (w, z(w)) ∈ F (w, z(w))∩G(w, z(w)) for some z(w) ∈ K, and we have M(z(w), z(w)) = 0. This contradicts M(z(w), z(w)) > 0. Therefore M(x(w), y(w)) = 0 for some x(w), y(w) ∈ K, and this implies T (w, x(w)) ∈ F (w, x(w)) and T (w, x(w)) = T (w, y(w)) ∈ G(w, y(w)). If M(x(w), y(w)) = 0, then T (w, x(w)) ∈ G(w, x(w)), and if M(x(w), x(w)) ≠ 0, then (2.5) implies

(1 − β(w) − γ(w))d(T (w, x(w)), G(w, x(w))) ≤ 0,

yielding T (w, x(w)) ∈ G(w, x(w)). Similarly, in either of the two cases M(y(w), y(w)) = 0 and M(y(w), y(w)) > 0, T (w, y(w)) ∈ F (w, y(w)). This proves that F, G and T have a common random coincidence point.

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RANDOM COINCIDENCE AND RANDOM FIXED POINT 149

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Department of Mathematics, Multan Public School, Multan, Islamia University Bahawal pur, Pakistan, Pakistan E-mail address: mustafa rakib@yahoo.com Department of Mathematics, COMSATS Institute of Information Technology Lahore, Pakistan

Received February 15, 2005; revised version August 29, 2005