International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 1 (2014), 26-35
http://www.etamaths.com

ON RANDOM COINCIDENCE & FIXED POINTS FOR A PAIR

OF MULTI-VALUED & SINGLE-VALUED MAPPINGS

PANKAJ KUMAR JHADE1,∗ AND A. S. SALUJA2

Abstract. Let (X,d) be a Polish space, CB(X) the family all nonempty

closed and bounded subsets of X and (Ω, Σ) be a measurable space. In this
paper a pair of hybrid measurable mappings f : Ω×X → X and T : Ω×X →
CB(X), satisfying the inequality (2.1) below are introduced and investigated.

It is proved that if X is complete, T(ω, ·), f(ω, ·) are continuous for all ω ∈ Ω,
T(·,x), f(·,x) are measurable for all x ∈ X and T(ω,ξ(ω)) ⊆ f(ω × X) and
f(ω×X) = X for each ω ∈ Ω, then there is a measurable mapping ξ : Ω → X
such that f(ω,ξ(ω)) ∈ T(ω,ξ(ω)) for all ω ∈ Ω.

1. Introduction

Random nonlinear analysis is an important mathematical discipline which is
mainly concerned with the study of random nonlinear operators and their proper-
ties and is much needed for the study of various classes of random equations. Of
course famously random methods have revolutionized the financial markets. Ran-
dom fixed point theorems for random contraction mappings on separable complete
metric spaces were first proved by Špaček[24] and Hanš[7,8]. The survey article
by Bharucha-Reid [1] in 1976 attracted the attention of several mathematicians
(see Chang and Huang[2], Hanš[7],[8], Špaček[24], Huang[10], Itoh [11], Liu [14],
Papageorgiou [15],[16], Shahzad and Hussain [21],Shahzad and Latif [22], Tan and
Yuan [25]) and give wings to this theory. Itoh [11] extended Špaček’s and Hanš’s
theorem to multi-valued contraction mapping . The stochastic version of the well-
known Schauder’s fixed point theorem was proved by Sehgal and Singh [20].
Let (X,d) be a metric space and T : X → X a mapping. The class of mappings T
satisfying the following contractive conditions:

d(Tx,Ty) ≤ a max{d(x,y),d(x,Tx),d(y,Ty),
d(x,Ty) + d(y,Tx)

2
}

+ b max{d(x,Tx),d(y,Ty)} + c[d(x,Ty) + d(y,Tx)]
(1.1)

for all x,y ∈ X, where a,b,c are non-negative real numbers such that b > 0 c > 0
and a + b + 2c = 1, was introduced and investigated by Ćirić [3]. Ćirić proved
that in a complete metric space such mappings have a unique fixed point. This
class of mappings was further studied by many authors (Ćirić[4],[5], Singh and

Mishra[23], and Rhoades et al. [18]).Sehgal and Singh [20] have generalized Ćirić’s

2010 Mathematics Subject Classification. 47H10; 54H25.
Key words and phrases. Separable metric space; random fixed point; random coincidence point

;random multi-function.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

26



ON RANDOM COINCIDENCE & FIXED POINTS 27

[4] fixed point theorem to a common fixed point theorem of a pair of mappings and
presented some application of such theorems to dynamic programming.
In this paper we introduced a new class of nonexpansive type mappings for a pair
of multi-valued and single valued mappings which is a stochastic version of Ćirić’s
[3] fixed point theorem to find the coincidence and fixed points for such class of
mappings.

2. Preliminaries

Let (Ω, Σ) be a measurable space with Σ a sigma algebra of subsets of Ω and let
(X,d) be a metric space. We denote by 2X the family of all subsets of X, by CB(X)
the family of all nonempty closed and bounded subsets of X and by H the Haus-
dorff metric on CB(X), induced by the metric d. For any x ∈ X and A ⊆ X, by
d(x,A) we denote the distance between xandA, i.e., d(x,A) = inf{d(x,a) : a ∈ A}.
A mapping T : Ω → 2X is called Σ−measurable if for any open subset U of X,
T−1(U) = {ω : T(ω)∩U 6= φ}∈ Σ.In what follows, when we speak of measurability
we will mean Σ− measurability. A mapping f : Ω × X → X is called a random
operator if for any x ∈ X,f(·,x) is measurable. A mapping T : Ω×X → CB(X) is
called a multi-valued random operator if for every x ∈ X, T(·,x) is measurable.A
mapping s : Ω → X is called a measurable selector of a measurable multifunction
T : Ω → 2X if s is measurable and s(ω) ∈ T(ω) for all ω ∈ Ω. A measurable
mapping ξ : Ω → X is called a random fixed point of a random multifunction
T : Ω ×X → CB(X) if ξ(ω) ∈ T(ω,ξ(ω)) for every ω ∈ Ω. A mapping ξ : Ω → X
is called a random coincidence of T : Ω × X → CB(X) and f : Ω × X → X if
f(ω,ξ(ω)) ∈ T(ω,ξ(ω)) for all ω ∈ Ω.
The aim of this paper is to prove a stochastic analogue of the Ćirić’s [3] fixed point
theorem for single valued mappings, extended to a coincidence point theorem for
a pair of a random operator f : Ω ×X → X and a multi-valued random operator
T : Ω × X → CB(X), satisfying the following nonexpansive type condition: for
each ω ∈ Ω,

H(T(ω,x),T(ω,y)) ≤ a(ω) max{d(f(ω,x),f(ω,y)),d(f(ω,y),T(ω,y))}
+ b(ω) max{d(f(ω,x),T(ω,x)),d(f(ω,y),T(ω,y)),
d(f(ω,y),T(ω,x))}

+ c(ω)[d(f(ω,x),T(ω,y)) + d(f(ω,y),T(ω,x))]

(2.1)

for every x,y ∈ X, where a,b,c : Ω → [0, 1) are measurable mappings such that for
all ω ∈ Ω

(2.2) b(ω) > 0 c(ω) > 0

(2.3) a(ω) + b(ω) + 2c(ω) = 1

3. Main Results

Now, we give our main results.

Theorem 3.1. Let (X,d) be a complete metric space, (Ω, Σ) be a measurable space
and T : Ω ×X → CB(X) & f : Ω ×X → X be mappings such that



28 PANKAJ KUMAR JHADE AND A. S. SALUJA

(1) T(ω, ·) and f(ω, ·) are continuous for all ω ∈ Ω,
(2) T(·,x) and f(·,x) are measurable for all x ∈ X,
(3) They satisfy (2.1), where a(ω),b(ω),c(ω) : Ω → X satisfy (2.2) and (2.3).

If T(ω,ξ(ω)) ⊆ f(ω × X) and f(ω × X) = X for each ω ∈ Ω, then there is a
measurable mapping ξ : Ω → X such that f(ω,ξ(ω)) ∈ T(ω,ξ(ω)) for all ω ∈ Ω
(i.e. T and f have a random coincidence point).

Proof. Let Ψ = {ξ : Ω → X} be a family of measurable mappings. Define a func-
tion g : Ω ×X → R+ as follows:

g(ω,x) = d(x,T(ω,x)).

Since x → T(ω,x) is continuous for all ω ∈ Ω, we conclude that g(ω, ·) is continuous
for all ω ∈ Ω. Also, since ω → T(ω,x) is measurable for all x ∈ X, we conclude
that g(·,x) is measurable(see Wagner [26], p 868) for all ω ∈ Ω.Thus g(ω,x) is
the Caratheodory function.Therefore, if ξ : Ω → X is a measurable mapping, then
ω → g(ω,ξ(ω)) is also measurable (see [19]).
Now we shall construct a sequence of measurable mappings {ξn} in Ψ and a se-
quence {f(ω,ξn(ω))} in X as follows.Let ξ0 ∈ Ψ be arbitrary. Then the multifunc-
tion G : Ω → CB(X) defined by G(ω) = T(ω,ξ0(ω)) is measurable.
From the Kuratowski-Nardzewski [13] selector theorem there is a measurable se-
lector µ1 : Ω → X such that µ1(ω) ∈ T(ω,ξ0(ω)) for all ω ∈ Ω Since µ1(ω) ∈
T(ω,ξ0(ω)) ⊆ X = f(ω × X), let ξ1 ∈ Ψ be such that f(ω,ξ1(ω)) = µ1.Thus
f(ω,ξ1(ω)) ∈ T(ω,ξ0(ω)) for all ω ∈ Ω.
Let k : Ω → (1,∞) defined by

k(ω) = 1 +
b(ω)c(ω)

2
for all ω ∈ Ω.Then k(ω) is measurable.Since k(ω) > 1 and f(ω,ξ1(ω)) is a selector
of T(ω,ξ0(ω)), from Lemma 2.1 of Papageorgiou [15] there is a measurable selector
µ2(ω) = f(ω,ξ2(ω)); ξ2 ∈ Ψ, such that for all ω ∈ Ω:

f(ω,ξ2(ω)) ∈ T(ω,ξ1(ω))
and

d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) ≤ k(ω)H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))
Similarly, as f(ω,ξ2(ω)) is a selector of T(ω,ξ1(ω)), there is a measurable selector
µ3(ω) = f(ω,ξ3(ω)) of T(ω,ξ2(ω)) ⊆ f(Ω ×X) such that

d(f(ω,ξ2(ω)),f(ω,ξ3(ω))) ≤ k(ω)H(T(ω,ξ1(ω)),T(ω,ξ2(ω)))
Continuing in this way we can construct a sequence of measurable mappings µn :
Ω → X, defined by µn(ω) = f(ω,ξn(ω)); ξn ∈ Ψ, such that for all ω ∈ Ω:

f(ω,ξn+1(ω)) ∈ T(ω,ξn(ω))
and

(3.1) d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) ≤ k(ω)H(T(ω,ξn−1(ω)),T(ω,ξn(ω)))



ON RANDOM COINCIDENCE & FIXED POINTS 29

Now from (2.1)

H(T(ω,ξ0(ω)),T(ω,ξ1(ω))) ≤ a(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))),d(f(ω,ξ1(ω)),T(ω,ξ1(ω)))}
+ b(ω) max{d(f(ω,ξ0(ω)),T(ω,ξ0(ω))),d(f(ω,ξ1(ω)),T(ω,ξ1(ω)))
,d(f(ω,ξ1(ω)),T(ω,ξ0(ω)))}

+ c(ω)[d(f(ω,ξ0(ω)),T(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),T(ω,ξ0(ω)))]

(3.2)

Since f(ω,ξ1(ω)) ∈ T(ω,ξ0(ω)), then

d(f(ω,ξ1(ω)),T(ω,ξ0(ω))) = 0

d(f(ω,ξ0(ω)),T(ω,ξ0(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
d(f(ω,ξ1(ω)),T(ω,ξ1(ω))) ≤ H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))

Thus from (3.2)

H(T(ω,ξ0(ω)),T(ω,ξ1(ω))) ≤ a(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))),H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))}
+ b(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))),H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))}
+ c(ω)[d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))]

(3.3)

If we assume that H(T(ω,ξ0(ω)),T(ω,ξ1(ω))) > d(f(ω,ξ0(ω)),f(ω,ξ1(ω))), then
from (3.3) and (2.3), we get

H(T(ω,ξ0(ω)),T(ω,ξ1(ω))) < a(ω)H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))

+ b(ω)H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))

+ 2c(ω)H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))

= (a(ω) + b(ω) + 2c(ω))H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))

= H(T(ω,ξ0(ω)),T(ω,ξ1(ω)))

a contradiction. Therefore, we have

H(T(ω,ξ0(ω)),T(ω,ξ1(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
Since d(f(ω,ξ1(ω)),T(ω,ξ1(ω))) ≤ H(T(ω,ξ0(ω)),T(ω,ξ1(ω))), we have

d(f(ω,ξ1(ω)),T(ω,ξ1(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
Thus by induction we can show that

(3.4) H(T(ω,ξn(ω)),T(ω,ξn+1(ω))) ≤ d(f(ω,ξn(ω)),f(ω,ξn+1(ω)))

(3.5) d(f(ω,ξn(ω)),T(ω,ξn(ω))) ≤ d(f(ω,ξn−1(ω)),f(ω,ξn(ω)))

for all n ≥ 1 and for all ω ∈ Ω

From (3.1) and (3.4), we have

(3.6) d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) ≤ k(ω)d(f(ω,ξn−1(ω)),f(ω,ξn(ω)))



30 PANKAJ KUMAR JHADE AND A. S. SALUJA

From (3.6), we get

d(f(ω,ξ0(ω)),f(ω,ξ2(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),f(ω,ξ2(ω)))
≤ (1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

(3.7)

From (2.1)

H(T(ω,ξ0(ω)),T(ω,ξ2(ω))) ≤ a(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ2(ω))),d(f(ω,ξ2(ω)),T(ω,ξ2(ω)))}
+ b(ω) max{d(f(ω,ξ0(ω)),T(ω,ξ0(ω))),d(f(ω,ξ2(ω)),T(ω,ξ2(ω)))
,d(f(ω,ξ2(ω)),T(ω,ξ0(ω)))}

+ c(ω)[d(f(ω,ξ0(ω)),T(ω,ξ2(ω))) + d(f(ω,ξ2(ω)),T(ω,ξ0(ω)))]

(3.8)

Using (3.4), (3.5), (3.6) and (3.7) and by triangle inequality, we get

d(f(ω,ξ2(ω)),T(ω,ξ0(ω))) ≤ H(T(ω,ξ1(ω)),T(ω,ξ0(ω)))
≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

d(f(ω,ξ0(ω)),T(ω,ξ2(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
+ d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) + d(f(ω,ξ2(ω)),T(ω,ξ2(ω)))

≤ (1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),f(ω,ξ2(ω)))
≤ (1 + 2k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

Now from (3.8), (3.7),(3.6) and (2.3), we have

H(T(ω,ξ0(ω)),T(ω,ξ2(ω))) ≤ a(ω)(1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
+ b(ω)k(ω)d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

+ 2c(ω)(1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

= [1 + k(ω)(a(ω) + b(ω) + 2c(ω)) − b(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
= [1 + k(ω) − b(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

As 1 + k(ω) < 2k(ω), we have

(3.9) H(T(ω,ξ0(ω)),T(ω,ξ2(ω))) ≤ [2k(ω) − b(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

From (2.3) and (2.1), we have, as f(ω,ξ2(ω)) ∈ T(ω,ξ1(ω))

H(T(ω,ξ1(ω)),T(ω,ξ2(ω))) ≤ a(ω) max{d(f(ω,ξ1(ω)),f(ω,ξ2(ω))),d(f(ω,ξ2(ω)),T(ω,ξ2(ω)))}
+ b(ω) max{d(f(ω,ξ1(ω)),T(ω,ξ1(ω))),d(f(ω,ξ2(ω)),T(ω,ξ2(ω)))
,d(f(ω,ξ2(ω)),T(ω,ξ1(ω)))}

+ c(ω)[d(f(ω,ξ1(ω)),T(ω,ξ2(ω))) + d(f(ω,ξ2(ω)),T(ω,ξ1(ω)))]

≤ [a(ω) + b(ω)] max{d(f(ω,ξ1(ω)),f(ω,ξ2(ω))),d(f(ω,ξ2(ω)),T(ω,ξ2(ω)))}
+ c(ω)d(f(ω,ξ1(ω)),T(ω,ξ2(ω)))

(3.10)



ON RANDOM COINCIDENCE & FIXED POINTS 31

Also by (3.9) since f(ω,ξ1(ω)) ∈ T(ω,ξ0(ω)), we have

d(f(ω,ξ1(ω)),T(ω,ξ2(ω))) ≤ H(T(ω,ξ0(ω)),T(ω,ξ2(ω)))
≤ (2k(ω) − b(ω)))d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

Thus from (3.10) and (3.6), we have

H(T(ω,ξ1(ω)),T(ω,ξ2(ω))) ≤ [a(ω) + b(ω)]k(ω)d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
+ c(ω)(2k(ω) − b(ω)))d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))
= [k(ω)(a(ω) + b(ω) + 2c(ω)) − b(ω)c(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

implies that

(3.11) H(T(ω,ξ1(ω)),T(ω,ξ2(ω))) ≤ [k(ω) − b(ω)c(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

as a(ω) + b(ω) + 2c(ω) = 1

From (3.1) and (3.11), we have

d(f(ω,ξ2(ω)),f(ω,ξ3(ω))) ≤ k(ω)H(T(ω,ξ1(ω)),T(ω,ξ2(ω)))
≤ k(ω)[k(ω) − b(ω)c(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

(3.12)

As k(ω) = 1 +
b(ω)c(ω)

2
, we have

k(ω)[k(ω) − b(ω)c(ω)] =
(

1 +
b(ω)c(ω)

2

)(
1 +

b(ω)c(ω)

2
− b(ω)c(ω)

)
= 1 +

b2(ω)c2(ω)

4

Thus from (3.12)

d(f(ω,ξ2(ω)),f(ω,ξ3(ω))) ≤
(

1 +
b2(ω)c2(ω)

4

)
d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

Similarly

d(f(ω,ξ3(ω)),f(ω,ξ4(ω))) ≤
(

1 +
b2(ω)c2(ω)

4

)
d(f(ω,ξ1(ω)),f(ω,ξ2(ω)))

Hence by induction

d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) ≤
(

1 +
b2(ω)c2(ω)

4

)[ n
2
]

max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω)))

,d(f(ω,ξ1(ω)),f(ω,ξ2(ω)))}

(3.13)

where [ n
2

] stands for the greatest integer not exceeding n
2

. Also ,since b(ω)c(ω) > 0
for all ω ∈ Ω, from (3.13), we have {f(ω,ξn(ω))} is a Cauchy sequence in f(ω×X).
Since f(ω × X) = X is complete, there is a measurable mapping f(ω,ξ(ω)) ∈
f(ω ×X) such that

(3.14) lim
n→∞

f(ω,ξn(ω)) = f(ω,ξ(ω))



32 PANKAJ KUMAR JHADE AND A. S. SALUJA

Again by triangle inequality and (2.1), we get

d(f(ω,ξ(ω)),T(ω,ξ(ω))) ≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω))) + d(f(ω,ξn+1(ω)),T(ω,ξ(ω)))
≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω))) + H(T(ω,ξn(ω)),T(ω,ξ(ω)))
≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω)))
+ a(ω) max{d(f(ω,ξn(ω)),f(ω,ξ(ω))),d(f(ω,ξ(ω)),T(ω,ξ(ω)))}
+ b(ω) max{d(f(ω,ξn(ω)),T(ω,ξn(ω))),d(f(ω,ξ(ω)),T(ω,ξ(ω)))

,d(f(ω,ξ(ω)),T(ω,ξn(ω)))}
+ c(ω)[d(f(ω,ξn(ω)),T(ω,ξ(ω))) + d(f(ω,ξ(ω)),T(ω,ξn(ω)))]

Thus

d(f(ω,ξ(ω)),T(ω,ξ(ω))) ≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω)))
+ a(ω) max{d(f(ω,ξn(ω)),f(ω,ξ(ω))),d(f(ω,ξ(ω)),T(ω,ξ(ω)))}
+ b(ω) max{d(f(ω,ξn(ω)),f(ω,ξn+1(ω))),d(f(ω,ξ(ω)),T(ω,ξ(ω)))

,d(f(ω,ξ(ω)),f(ω,ξn+1(ω)))}
+ c(ω)[d(f(ω,ξn(ω)),T(ω,ξ(ω))) + d(f(ω,ξ(ω)),f(ω,ξn+1(ω)))]

(3.15)

Taking limit as n →∞, we have

d(f(ω,ξ(ω)),T(ω,ξ(ω))) ≤ [a(ω) + b(ω) + c(ω)]d(f(ω,ξ(ω)),T(ω,ξ(ω)))
= [1 − c(ω)]d(f(ω,ξ(ω)),T(ω,ξ(ω)))

implies that d(f(ω,ξ(ω)),T(ω,ξ(ω))) = 0, as 1−c(ω) < 1 and for ω ∈ Ω. Hence as
T(ω,ξ(ω)) is closed f(ω,ξ(ω)) ∈ T(ω,ξ(ω)), for all ω ∈ Ω. �

Remark 3.2. If in Theorem 3.1., f(ω,x) = x for all (ω,x) ∈ Ω ×X, then we get
the following random fixed point theorem.

Corollary 3.3. Let (X,d) be a separable complete metric space. (Ω, Σ) be a mea-
surable space and let a mapping T : Ω × X → CB(X) be such that T(ω, ·) is
continuous for all ω ∈ Ω, T(·,x) is measurable for all x ∈ X and

H(T(ω,x),T(ω,y)) ≤ a(ω) max{d(x,y),d(x,T(ω,y))}
+ b(ω) max{d(x,T(ω,x)),d(y,T(ω,y)),d(y,T(ω,x))}
+ c(ω)[d(x,T(ω,y)) + d(y,T(ω,x))]

(3.16)

for every x,y ∈ X, where a,b,c : Ω → (0, 1) are measurable mappings satisfying
(2.2) and (2.3). Then there is a measurable mapping ξ : Ω → X such that ξ(ω) ∈
T(ω,ξ(ω)) for all ω ∈ Ω.

Corollary 3.4. ([6], Corollary 1) Let (X,d) be a separable complete metric space.
(Ω, Σ) be a measurable space and let a mapping T : Ω ×X → CB(X) be such that
T(ω, ·) is continuous for all ω ∈ Ω, T(·,x) is measurable for all x ∈ X and

H(T(ω,x),T(ω,y)) ≤ a(ω) max{d(x,y),d(x,T(ω,x)),d(y,T(ω,y))

,
1

2
[d(x,T(ω,y)) + d(y,T(ω,x))]}

+ b(ω) max{d(x,T(ω,x)),d(y,T(ω,y))}
+ c(ω)[d(x,T(ω,y)) + d(y,T(ω,x))]

(3.17)



ON RANDOM COINCIDENCE & FIXED POINTS 33

for every x,y ∈ X, where a,b,c : Ω → (0, 1) are measurable mappings satisfying
(2.2) and (2.3). Then there is a measurable mapping ξ : Ω → X such that ξ(ω) ∈
T(ω,ξ(ω)) for all ω ∈ Ω.

Remark 3.5. The nonexpansive type condition (3.16) includes (3.17) if we set

m(x,y) = max{d(x,y),d(x,T(ω,x)),d(y,T(ω,y)),
1

2
[d(x,T(ω,y)) + d(y,T(ω,x))]}

For each x,y such that m(x,y) = d(x,y) and a(ω),b(ω),c(ω) : Ω → (0, 1)
For each x,y such that m(x,y) = max{d(x,T(ω,x)),d(y,T(ω,y))} ,define a(ω) =
0,b(ω) = a(ω) + b(ω),c(ω) = c(ω).
For each x,y such that m(x,y) = 1

2
[d(x,T(ω,y)) + d(y,T(ω,x))], define a(ω) =

0,b(ω) = b(ω),c(ω) = a(ω) + 2c(ω).
Thus Corollary (3.3) is an extension of Corollary (3.4).

Finally, we give a simple example in support of Theorem 3.1. and Corollary 3.3
which shows that these results are actually an improvement of the result of Itoh[11].

Example 3.6. Let (X,d) be any measurable space and K = {0, 1, 2, 4, 6} be the
subset of the real line. Let the mappings f : Ω ×K → K and T : Ω ×K → K be
defined such that for each ω ∈ Ω:

f(ω, 0) = 2 f(ω, 1) = 4 f(ω, 2) = 6 f(ω, 4) = 0 f(ω, 6) = 1

T(ω, 0) = 1 T(ω, 1) = 2 T(ω, 2) = 4 T(ω, 4) = 0 T(ω, 6) = 0

Then for x = 1 and y = 2, we have

d(T(ω, 1),T(ω, 2)) =
4

5
max{‖4 − 6‖,‖6 − 4‖}

+
1

20
max{‖4 − 6‖,‖6 − 4‖,‖6 − 2‖}

+
1

20
[‖4 − 4‖ + ‖6 − 2‖]

=
4

5
.2 +

1

20
.4 +

1

20
.4

= 2

Thus, for x = 1 and y = 2, f and T satisfy (2.1) with a(ω) = 4
5
,b(ω) = 1

20
and

c(ω) = 1
20

. It is easy to show that f and T satisfy (2.1) for all x,y ∈ K with
the same a(ω),b(ω) and c(ω). Also, the rest of the assumptions of Theorem 3.1 is
satisfied and for ξ(ω) = 4, we have

f(ω,ξ(ω)) = 0 = T(ω,ξ(ω))

Note that T does not satisfy (3.16) either, as for instance, for x = 2 and y = 4, we
have

a(ω) max{‖2 − 4‖,‖2 − 0‖} + b(ω) max{‖2 − 4‖,‖4 − 0‖,‖4 − 4‖}
+c(ω)[‖2 − 0‖ + ‖4 − 4‖] = 2a(ω) + 4b(ω) + 2c(ω)
< 4[a(ω) + b(ω) + 2c(ω)] = 4 = d(T(ω, 2),T(ω, 4))



34 PANKAJ KUMAR JHADE AND A. S. SALUJA

Remark 3.7. Our Theorem 3.1 generalizes and extends the corresponding fixed
point theorems for nonexpansive type single valued mapping of Ćirić [3] and Rhoad-
es[17].

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1Department of Mathematics, NRI Institute of Information Science & Technology,Bhopal-

462021 , INDIA

2Department of Mathematics, J. H. Government (PG) College, Betul 460001, INDIA

∗Corresponding author