Gen. Math. Notes, Vol. 5, No. 1, July 2011, pp.34-45
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Composite Implicit Random Iterations for
Approximating Common Random Fixed Point
for a Finite Family of Strictly Pseudocontractive
Random Operators
Shrabani Banerjee1 and Binayak S. Choudhury2
1
Department of Mathematics
Bengal Engineering and Science University, Shibpur
Howrah-711103, India
E-mail: banerjee.shrabani@yahoo.com
2
Department of Mathematics
Bengal Engineering and Science University, Shibpur
Howrah-711103, India.
E-mail: binayak12@yahoo.co.in
(Received: 21-4-11/Accepted:24-6-11)
Abstract
In the present work we construct a composite implicit random iterative
process for a finite family of strictly pseudocontractive random operators and
study the necessary and sufficient condition for the convergence of this process
to a common random fixed point of a finite family strictly pseudocontractive
random operators.
Keywords: Banach spaces, Condition(B), Composite implicit random iterative process, measurable spaces, strictly pseudocontractive random operators.
2000 MSC No: 47H10.
Composite Implicit Random Iterations for Approximating...
1
35
Introduction
Random fixed point theory was initiated by the Prague school of probabilists
in the works of Hans [14] and Spacek [20] as stochastic generalization of deterministic fixed point theory. After that till recent times a large number of
research papers focussing on various aspects of random fixed point theory have
appeared in recent literatures. Some of these references are noted in [1], [9]
and references therein. Fixed point iteration schemes for nonlinear operators
on Banach and Hilbert spaces have been developed and studied by many authors in recent times. The development of random fixed point iterations was
initiated by Choudhury in [8] where random Ishikawa iteration scheme was
defined and its strong convergence to a random fixed point in Hilbert spaces
was discussed. After that several authors have worked on random fixed point
iterations some of which are noted in [2], [10], [11], [12], [13], [17] and [18].
In 2005 Beg and Abbas [2] constructed and studied different random iterative
algorithms for weakly contractive and asymptotically nonexpansive random
operators on arbitrary Banach spaces. They also established the convergence
of an implicit random iteration process to a common random fixed point for
a finite family of asymptotically quasi-nonexpansive random operators. Very
recently Plubtieng et al. [17] constructed and established the convergence of
an implicit random iteration process with errors for a common random fixed
point of a finite family of asymptotically quasi-nonexpansive random operators
in the setting of uniformly convex Banach spaces. Very recently Kumam et al.
[16] have established the convergence of an implicit random iteration process
to a common random fixed point for a finite family of strictly pseudocontractive random operators in real separable Banach spaces.
The purpose of this paper is to construct a composite implicit random iterative scheme for a finite family of strictly pseudocontractive random operators
and to study the convergence of this iterative process in real separable Banach
spaces. Our results extend and improve some recent results in the existing
literature.
2
Preliminaries
Throughout this paper, (Ω, Σ) denotes a measurable space and X stands for
a real Banach space. For any function T : Ω × X → X we denote the n-th
iterate T (t, T (t, ..., T (t, x)))) of T by T n (t, x). A function f : Ω → X is said to
be measurable if f −1 (B) ∈ Σ for every Borel subset B of X. A single-valued
operator T : Ω × X → X is called a random operator if T (·, x) : Ω → X
is measurable, for every x ∈ X. The letter I denotes the identity random
operator I : Ω × X → X defined by I(t, x) = x and T 0 = I. A random
operator T : Ω × X → X is continuous if T (t, ·) : X → X is continuous for
36
Shrabani Banerjee et al.
each t ∈ Ω. A measurable function p : Ω → X is said to be a random fixed
point of the random operator T : Ω × X → X if T (t, p(t)) = p(t), ∀t ∈ Ω. The
set of all random fixed points of T is denoted by RF (T ).
?
A mapping J from X to 2X defined by
J(x) = {x? ∈ X ? : hx, x? i = kxk.kx? k, kx? k = kxk}
where h., .i denotes the generalized duality pairing is called the normalized
duality mapping.
Definition 2.1 [16] Let C be a nonempty subset of a real separable Banach
space X and T : Ω × C → C be a random operator. Then T is said to be
(i)strictly pseudocontractive random operator if for some measurable mapping
λ : Ω → (0, 1) and for any x, y ∈ C, there exists j(x − y) ∈ J(x − y) such that
for each t ∈ Ω,
hT (t, x) − T (t, y), j(x − y)i ≤ kx − yk2 − λ(t)kx − y − (T (t, x) − T (t, y))k2
(ii) a L-Lipschitzian random operator if for any x, y ∈ C and for each t ∈ Ω,
kT (t, x) − T (t, y)k ≤ Lkx − yk
where L is a positive constant.
(iii) a semi-compact random operator if for a sequence of measurable mappings
{ξn } from Ω to C with limn→∞ kξn (t) − T (t, ξn (t))k = 0, for all t ∈ Ω, we have
a subsequence {ξnk } of {ξn } such that ξnk (t) → ξ(t), for all t ∈ Ω, where ξ is
a measurable mapping from Ω to C.
Every strictly pseudocontractive random operator is a L-Lipschitzian random
operator.
Definition 2.2 Condition(B) A finite family {Ti : i ∈ {1, 2, ..., N }} of N
T
continuous random operators from Ω × C → C with F = N
i=1 RF (Ti ) 6= ∅ is
said to satisfy Condition(B) if there is a nondecreasing function f : [0, ∞) →
[0, ∞) with f (0) = 0 and f (r) > 0 for all r ∈ (0, ∞) such that for all t ∈ Ω
f (d(x(t), F )) ≤ max {kx(t) − Ti (t, x(t))k} for all x
1≤i≤N
where x : Ω → C is a measurable function.
Lemma 2.3 ([15]) Let (Ω, Σ) be a measurable space, X be a separable Banach space and T : Ω × X → X be a continuous random operator. Then for
any measurable function x : Ω → X, the function t → T (t, x(t)) is measurable.
We define the composite implicit random iterative process in the following:
Composite Implicit Random Iterations for Approximating...
37
Definition 2.4 Composite implicit random iterative scheme
Let {Ti : i ∈ {1, 2, ...., N }} be a finite family of N continuous random operators
from Ω×C to C where C be a nonempty closed convex subset of a real separable
Banach space X. Let ξ0 : Ω → C be any measurable function. Then Composite
implicit random iteration scheme is defined as follows:

















































ξ1 (t) = α1 ξ0 (t) + (1 − α1 )T1 (t, β1 ξ1 (t) + (1 − β1 )T1 (t, ξ1 (t)))
ξ2 (t) = α2 ξ1 (t) + (1 − α2 )T2 (t, β2 ξ2 (t) + (1 − β2 )T2 (t, ξ2 (t)))
·
·
ξN (t) = αN ξN −1 (t) + (1 − αN )TN (t, βN ξN (t) + (1 − βN )TN (t, ξN (t)))
ξN +1 (t) = αN +1 ξN (t) + (1 − αN +1 )T1 (t, βN +1 ξN +1 (t) + (1 − βN +1 )T1 (t, ξN +1 (t))
·
·
ξ2N (t) = α2N ξ2N −1 (t) + (1 − α2N )TN (t, β2N ξ2N (t) + (1 − β2N )TN (t, ξ2N (t)))
ξ2N +1 (t) = α2N +1 ξ2N (t) + (1 − α2N +1 )T1 (t, β2N +1 ξ2N +1 (t) + (1 − β2N +1 )T1 (t, ξ2N +1 (t))
·
·
which can be written in the compact form as
(
ξn (t) = αn ξn−1 (t) + (1 − αn )Tn (t, ηn (t))
ηn (t) = βn ξn (t) + (1 − βn )Tn (t, ξn (t)), n ≥ 1, ∀t ∈ Ω
(1)
where Tn = TnmodN and {αn }, {βn } are sequences in [0, 1].
Remark 2.5 By Lemma 2.3 the sequence {ξn } defined in (1) is a sequence
of measurable functions.
Lemma 2.6 ([19], Lemma1) Let {an }, {bn } and {δn } be sequences of nonnegative real numbers satisfying the inequality
an+1 ≤ (1 + δn )an + bn , ∀n ≥ 1.
If
P∞
n=1 δn
< ∞ and
P∞
n=1 bn
< ∞, then
(i) limn→∞ an exists,
(ii) limn→∞ an = 0 whenever lim inf n→∞ an = 0.
Lemma 2.7 ([5], Lemma2.1) Let X be a real Banach space and let J : X →
2 be the normalized duality mapping. Then for any given x, y ∈ X, we have
X?
kx + yk2 ≤ kxk2 + 2 < y, j >, ∀j ∈ J(x + y)
38
3
Shrabani Banerjee et al.
Main Results
Lemma 3.1 Let X be a real separable Banach space and C be a nonempty
closed convex subset of X. Let {Ti : i ∈ {1, 2, .., N }} be N strictly pseudoconT
tractive continuous random operators and F = N
i=1 RF (Ti ) 6= ∅. Let {ξn } be
the implicit random iterative sequence defined by (1) satisfying the following
conditions:
(i)0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1,
P
(ii) ∞
n=1 (1 − αn )(1 − βn ) < ∞.
Then for each t ∈ Ω,
(i) limn→∞ kξn (t) − ξ(t)k exists,
(ii) limn→∞ d(ξn (t), F ) exists,
(iii) limn→∞ kξn (t) − Tl (t, ξn (t))k = 0, for all l = 1, 2, ...., N.
Proof: Let t ∈ Ω. Let ξ ∈ F . Then for all n ≥ 1,
=
=
≤
≤
kξn (t) − ξ(t)k2 = hξn (t) − ξ(t), j(ξn (t) − ξ(t))i
αn hξn−1 (t) − ξ(t), j(ξn (t) − ξ(t))i + (1 − αn )hTn (t, ηn (t)) − ξ(t), j(ξn (t) − ξ(t))i
αn kξn−1 (t) − ξ(t)k.kξn (t) − ξ(t)k + (1 − αn )hTn (t, ξn (t)) − ξ(t), j(ξn (t) − ξ(t))i
+(1 − αn )hTn (t, ηn (t)) − Tn (t, ξn (t)), j(ξn (t) − ξ(t))i
αn
[kξn−1 (t) − ξ(t)k2 + kξn (t) − ξ(t)k2 ] + (1 − αn )[kξn (t) − ξ(t)k2 −
2
λ(t)kξn (t) − Tn (t, ξn (t))k2 ] + (1 − αn )kTn (t, ηn (t)) − Tn (t, ξn (t))k.kξn (t) − ξ(t)k
αn
αn
kξn−1 (t) − ξ(t)k2 +
kξn (t) − ξ(t)k2 + (1 − αn )kξn (t) − ξ(t)k2
2
2
−(1 − αn )λ(t)kξn (t) − Tn (t, ξn (t))k2 + (1 − αn )Lkηn (t) − ξn (t)k.kξn (t) − ξ(t)k
(2)
Now
kηn (t) − ξn (t)k = (1 − βn )kTn (t, ξn (t)) − ξn (t)k
≤ (1 − βn )[kTn (t, ξn (t)) − ξ(t)k + kξn (t) − ξ(t)k]
≤ (1 − βn )(L + 1)kξn (t) − ξ(t)k
(3)
From (2) and (3) it follows that for all t ∈ Ω and for all n ≥ 1,
kξn (t) − ξ(t)k2
αn
αn
≤
kξn−1 (t) − ξ(t)k2 +
kξn (t) − ξ(t)k2 + (1 − αn )kξn (t) − ξ(t)k2
2
2
−(1 − αn )λ(t)kξn (t) − Tn (t, ξn (t))k2 + (1 − αn )(1 − βn )(L + 1)Lkξn (t) − ξ(t)k2
αn
αn
=
kξn−1 (t) − ξ(t)k2 + [
+ 1 − αn + (1 − αn )(1 − βn )(L + 1)L]kξn (t) − ξ(t)k2
2
2
−(1 − αn )λ(t)kξn (t) − Tn (t, ξn (t))k2
Composite Implicit Random Iterations for Approximating...
39
which implies that
αn kξn (t) − ξ(t)k2 ≤ αn kξn−1 (t) − ξ(t)k2 + 2(1 − αn )(1 − βn )(L + 1)Lkξn (t) − ξ(t)k2
−2(1 − αn )λ(t)kξn (t) − Tn (t, ξn (t))k2
= αn kξn−1 (t) − ξ(t)k2 + µn kξn (t) − ξ(t)k2 −
2(1 − αn )λ(t)kξn (t) − Tn (t, ξn (t))k2
(4)
( where µn = 2(1 − αn )(1 − βn )(L + 1)L)
Since 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1, there exists a positive integer
n0 and η, η 0 ∈ (0, 1) such that 0 < η < αn < η 0 < 1 for all n ≥ n0 . Therefore
from (4) it follows that for all n ≥ n0 ,
αn
kξn (t) − ξ(t)k2 −
η
2(1 − αn )λ(t)kξn (t) − Tn (t, ξn (t))k2
αn kξn (t) − ξ(t)k2 ≤ αn kξn−1 (t) − ξ(t)k2 + µn .
which implies that for all n ≥ n0 ,
kξn (t) − ξ(t)k2 ≤ kξn−1 (t) − ξ(t)k2 +
µn
1 − αn
kξn (t) − ξ(t)k2 − 2
λ(t)kξn (t) − Tn (t, ξn (t))k2
η
αn
which implies that for all n ≥ n0 ,
kξn (t) − ξ(t)k2
η
1 − αn
η
≤
kξn−1 (t) − ξ(t)k2 − 2
.
.λ(t)kξn (t) − Tn (t, ξn (t))k2
η − µn
η − µn αn
µn
η
1 − αn
= (1 +
)kξn−1 (t) − ξ(t)k2 − 2
.
.λ(t)kξn (t) − Tn (t, ξn (t))k2
η − µn
η − µn αn
(5)
By the condition of the theorem we get that ∞
n=1 µn < ∞. Therefore limn→∞ µn =
0. So there exists n1 (> n0 ) ∈ N such that µn < η2 for all n ≥ n1 . Thus for all
n ≥ n1 , from (5) it follows that
P
kξn (t) − ξ(t)k2
2
η
1 − αn
.
.λ(t)kξn (t) − Tn (t, ξn (t))k2
≤ (1 + µn )kξn−1 (t) − ξ(t)k2 − 2
η
η − µn αn
η
1 − αn
= (1 + σn )kξn−1 (t) − ξ(t)k2 − 2
.
.λ(t)kξn (t) − Tn (t, ξn (t))k2
η − µn αn
2
( where σn = µn )
(6)
η
From (6) it follows that for all n ≥ n1 ,
kξn (t) − ξ(t)k2 ≤ (1 + σn )kξn−1 (t) − ξ(t)k2
(7)
40
Shrabani Banerjee et al.
∞
Since ∞
n=1 σn < ∞. Thus by Lemma2.6 we have that
n=1 µn < ∞, we have
limn→∞ kξn (t) − ξ(t)k exists for all ξ ∈ F and for each t ∈ Ω. From (7) we get
that for all t ∈ Ω and for all n ≥ n1 ,
P
P
1
1
kξn (t) − ξ(t)k ≤ (1 + σn ) 2 kξn−1 (t) − ξ(t)k ≤ (1 + σn )kξn−1 (t) − ξ(t)k
2
√
(as 1 + x ≤ 1 + 12 x)
(8)
Taking infimum over all ξ ∈ F , from (8) it follows that for each t ∈ Ω,
1
d(ξn (t), F ) ≤ (1 + σn )d(ξn−1 (t), F )
2
(9)
Hence by Lemma 2.6 we have limn→∞ d(ξn (t), F ) exists for each t ∈ Ω. Since
for t ∈ Ω, limn→∞ kξn (t)−ξ(t)k exists, there exists M (t) > 0 such that kξn (t)−
ξ(t)k ≤ M (t). Therefore from (6) it follows that for all n ≥ n1 and t ∈ Ω,
1 − η0
.λ(t)kξn (t) − Tn (t, ξn (t))k2
η0
η
1 − αn
≤ 2
.
.λ(t)kξn (t) − Tn (t, ξn (t))k2
η − µn αn
≤ kξn−1 (t) − ξ(t)k2 − kξn (t) − ξ(t)k2 + σn (M (t))2
2
(10)
From (10) it follows that for each t ∈ Ω,
∞
X
kξn (t) − Tn (t, ξn (t))k2 < ∞
n=n1
which implies that
lim kξn (t) − Tn (t, ξn (t))k = 0
n→∞
(11)
Now for each t ∈ Ω,
kηn (t) − ξn (t)k = (1 − βn )kTn (t, ξn (t)) − ξn (t)k → 0 as n → ∞(by(11)) (12)
Now for each t ∈ Ω,
kξn (t) − ξn−1 (t)k = (1 − αn )kTn (t, ηn (t)) − ξn−1 (t)k
≤ (1 − αn )kTn (t, ηn (t)) − Tn (t, ξn (t))k +
(1 − αn )kTn (t, ξn (t)) − ξn (t)k + (1 − αn )kξn (t) − ξn−1 (t)k
which implies that
kξn (t) − ξn−1 (t)k ≤
1 − αn
1 − αn
Lkηn (t) − ξn (t)k +
kTn (t, ξn (t)) − ξn (t)k(13)
αn
αn
Composite Implicit Random Iterations for Approximating...
41
So for each t ∈ Ω and for all n ≥ n0 , from (13) it follows that
1−η
1−η
Lkηn (t) − ξn (t)k +
kTn (t, ξn (t)) − ξn (t)k
η
η
→ 0 as n → ∞(by(11), (12))
(14)
kξn (t) − ξn−1 (t)k ≤
So we have
lim kξn (t) − ξn+l (t)k = 0 for all l ∈ {1, 2, ...., N } and for each t ∈ Ω.
n→∞
(15)
Now for each t ∈ Ω and for all l ∈ {1, 2, ...., N },
kξn (t) − Tn+l (t, ξn (t))k ≤ kξn (t) − ξn+l (t)k + kξn+l (t) − Tn+l (t, ξn+l (t))k +
kTn+l (t, ξn+l (t)) − Tn+l (t, ξn (t))k
≤ (1 + L)kξn (t) − ξn+l (t)k + kξn+l (t) − Tn+l (t, ξn+l (t))k
→ 0 as n → ∞(by(11), (15))
Consequently we have
lim kξn (t) − Tl (t, ξn (t))k = 0 for all l ∈ {1, 2, ...., N } and for each t ∈ Ω. (16)
n→∞
Theorem 3.2 Let X be a real separable Banach space and C be a nonempty
closed convex subset of X. Let {Ti : i ∈ {1, 2, ...., N }} be N strictly pseudoT
contractive continuous random operators and F = N
i=1 RF (Ti ) 6= ∅. Let {ξn }
be the implicit random iterative sequence defined by (1) satisfying the following
conditions:
(i)0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1,
P
(ii) ∞
n=1 (1 − αn )(1 − βn ) < ∞.
Then {ξn } converges strongly to a common random fixed point of the random
operators {Ti , i ∈ {1, 2, ...., N }} if and only if for all t ∈ Ω, lim inf n→∞ d(ξn (t), F ) =
0, where d(ξn (t), F ) = inf{kξn (t) − ξ(t)k : ξ ∈ F }.
Proof: The necessary part is obvious. We only prove the sufficiency part. Let
ξ ∈ F . Now from (8) we have that for each t ∈ Ω and for all n ≥ n1 ,
1
kξn (t) − ξ(t)k ≤ (1 + σn )kξn−1 (t) − ξ(t)k = (1 + λn )kξn−1 (t) − ξ(t)k (17)
2
where λn =
m, n ≥ n1 ,
1
σ .
2 n
Now from (17) we have that for each t ∈ Ω and for all
kξn+m (t) − ξ(t)k ≤ [1 + λn+m ]kξn+m−1 (t) − ξ(t)k ≤ eλn+m kξn+m−1 (t) − ξ(t)k
≤ eλn+m +λn+m−1 kξn+m−2 − ξ(t)k
≤ ·························
≤ e
Pn+m
i=n+1
λi
kξn (t) − ξ(t)k ≤ Rkξn (t) − ξ(t)k
(18)
42
Shrabani Banerjee et al.
P∞
where R = e
that
n=1
λn
< ∞. Therefore for any ξ ∈ F and for t ∈ Ω, we have
kξn+m (t) − ξn (t)k ≤ Rkξn (t) − ξ(t)k + kξn (t) − ξ(t)k
= (R + 1)kξn (t) − ξ(t)k
(19)
Since limn→∞ d(ξn (t), F ) = 0, there exists n2 (≥ n1 ) ∈ N such that for all
n ≥ n2 we have d(ξn (t), F ) < R+1
. So there exists q ∈ F such that kξn (t) −
for all n ≥ n2 . Therefore from (19) we have that for all n ≥ n2 ,
q(t)k < R+1
kξn+m (t) − ξn (t)k ≤ (R + 1)
=
R+1
which in turn implies that {ξn (t)} is a cauchy sequence for each t ∈ Ω. Therefore ξn (t) → p(t) as n → ∞ for each t ∈ Ω where the function p : Ω → X, being
the limit of the sequence of measurable functions is also measurable. Now we
prove that p ∈ F . Since for each t ∈ Ω, ξn (t) → p(t) as n → ∞ there exists n3 ∈
for all n ≥ n3 . Since limn→∞ d(ξn (t), F ) = 0
N such that kξn (t)−p(t)k < 2(1+L)
for each t ∈ Ω, there exists n4 ∈ N such that d(ξn (t), F ) < 2(1+L)
for all
∗
∗
n ≥ n4 . So there exists ξ ∈ F such that kξn (t) − ξ (t)k ≤ 2(1+L) for all
n ≥ n4 , where L is the Lipschitz constant. Let n5 = max {n3 , n4 }. Now for
all l ∈ {1, 2, ...., N }, t ∈ Ω and for all n ≥ n5
kTl (t, p(t)) − ξ ∗ (t)k + kξ ∗ (t) − p(t)k
kTl (t, p(t)) − Tl (t, ξ ∗ (t))k + kξ ∗ (t) − p(t)k
Lkξ ∗ (t) − p(t)k + kξ ∗ (t) − p(t)k
(1 + L)kξ ∗ (t) − p(t)k
(1 + L)kξ ∗ (t) − ξn (t)k + (1 + L)kξn (t) − p(t)k
< (1 + L)
+ (1 + L)
=
2(1 + L)
2(1 + L)
kTl (t, p(t)) − p(t)k ≤
≤
≤
=
≤
which implies that Tl (t, p(t)) = p(t) for all l ∈ {1, 2, ...., N } and for each t ∈ Ω.
Therefore p ∈ F . Thus {ξn } converges strongly to a common random fixed
point of {Ti , i ∈ {1, 2, ...., N }}.
Theorem 3.3 Let X be a real separable Banach space and C be a nonempty
closed convex subset of X. Let {Ti : i ∈ {1, 2, ...., N }} be N strictly pseudoT
contractive continuous random operators and F = N
i=1 RF (Ti ) 6= ∅. Let {ξn }
be the implicit random iterative sequence defined by (1) satisfying the following
conditions:
(i)0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1,
P
(ii) ∞
n=1 (1 − αn )(1 − βn ) < ∞.
If the family {Ti : i ∈ {1, 2, ...., N }} satisfies Condition(B), for each t ∈ Ω,
then {ξn } converges strongly to a common random fixed point of {Ti , i ∈
{1, 2, ...., N }}.
Composite Implicit Random Iterations for Approximating...
43
Proof: By the proof of Lemma 3.1 we have limn→∞ d(ξn (t), F ) exists for each
t ∈ Ω. Again by Lemma 3.1 and Condition(B), we have that for each t ∈ Ω,
limn→∞ f (d(ξn (t), F )) = 0. Since f : [0, ∞) → [0, ∞) is a nondecreasing
function with f (0) = 0 so we have limn→∞ d(ξn (t), F ) = 0. Hence the result
follows by Theorem 3.2.
Theorem 3.4 Let X be a real separable Banach space and C be a nonempty
closed convex subset of X. Let {Ti : i ∈ {1, 2, ...., N }} be N strictly pseudoT
contractive continuous random operators and F = N
i=1 RF (Ti ) 6= ∅. Let {ξn }
be the implicit random iterative sequence defined by (1) satisfying the following
conditions:
(i)0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1,
P
(ii) ∞
n=1 (1 − αn )(1 − βn ) < ∞.
Let one member of the family {Ti : i ∈ {1, 2, ...., N }} to be semi-compact random operator, then {ξn } converges strongly to a common random fixed point
of {Ti , i ∈ {1, 2, ...., N }}.
Proof: From Lemma 3.1 we get that limn→∞ kξn (t) − Tl (t, ξn (t))k = 0 for each
t ∈ Ω and for each l ∈ {1, 2, ...., N }. Let us assume that T1 is semi-compact
random operator. So there exists a subsequence {ξnk (t)} of {ξn (t)} such that
ξnk (t) → ξ(t) for each t ∈ Ω, where ξ is a measurable mapping from Ω to C.
Now for each t ∈ Ω and for each l ∈ {1, 2, ...., N }, we have
kξ(t) − Tl (t, ξ(t))k = lim kξnk (t) − Tl (t, ξnk (t))k = 0
k→∞
From above it follows that ξ ∈ F . Since {ξn (t)} has a subsequence {ξnk (t)}
such that ξnk (t) → ξ(t) for each t ∈ Ω, we have that lim inf n→∞ d(ξn (t), F ) = 0.
Hence the result follows by Theorem 3.2.
Remark 3.5 Our results in this paper extend and improve the corresponding results of [16].
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