J. Nonlinear Sci. Appl. 4 (2011), no. 4, 292–307
The Journal of Nonlinear Sciences and Applications
http://www.tjnsa.com
CONVERGENCE OF IMPLICIT RANDOM ITERATION
PROCESS WITH ERRORS FOR A FINITE FAMILY OF
ASYMPTOTICALLY QUASI-NONEXPANSIVE RANDOM
OPERATORS
GURUCHARAN SINGH SALUJA
Abstract. In this paper, we prove that an implicit random iteration process with errors which is generated by a finite family of asymptotically quasinonexpansive random operators converges strongly to a common random fixed
point of the random operators in uniformly convex Banach spaces.
1. Introduction
Random nonlinear analysis is an important mathematical discipline which is
mainly concerned with the study of random nonlinear operators and their properties and is needed for the study of various classes of random equations. The
study of random fixed point theory was initiated by the Prague school of Probabilities in the 1950s [12, 13, 25]. Common random fixed point theorems are
stochastic generalization of classical common fixed point theorems. The machinery of random fixed point theory provides a convenient way of modeling many
problems arising from economic theory (see e.g. [19]) and references mentioned
therein. Random methods have revolutionized the financial markets. The survey
article by Bharucha-Reid [9] attracted the attention of several mathematicians
and gave wings to the theory. Itoh [15] extended Spacek’s and Hans’s theorem to
multivalued contraction mappings. Now this theory has become the full fledged
research area and various ideas associated with random fixed point theory are
Date: Received: May 21, 2009; Revised: November 12, 2011.
∗
Corresponding author
c 2011 N.A.G.
⃝
2000 Mathematics Subject Classification. 47H09, 47H10, 47J25.
Key words and phrases. Asymptotically quasi nonexpansive random operator, common random fixed point, implicit random iteration scheme with errors, strong convergence, uniformly
convex Banach space.
292
CONVERGENCE OF IMPLICIT RANDOM. . .
293
used to obtain the solution of nonlinear random system (see [4, 5, 8, 14, 21]). Papageorgiou [17, 18], Beg [2, 3] studied common random fixed points and random
coincidence points of a pair of compatible random operators and proved fixed
point theorems for contractive random operators in Polish spaces. Recently, Beg
and Shahzad [7], Choudhury [11] and Badshah and Sayyed [1] used different iteration processes to obtain random fixed points. More recently, Beg and Abbas [6]
studied common random fixed points of two asymptotically nonexpansive random
operators through strong as well as weak convergence of sequence of measurable
functions in the setup of uniformly convex Banach spaces. Also they construct
different random iterative algorithms for asymptotically quasi-nonexpansive random operators on an arbitrary Banach space and established their convergence
to random fixed point of the operators.
In 2001, Xu and Ori [27] have introduced an implicit iteration process for
a finite family of nonexpansive mappings in a Hilbert space H. Let C be a
nonempty subset of H. Let T1 , T2 , . . . , TN be self-mappings of C and suppose that
F = ∩N
i=1 F (Ti ) ̸= ∅, the set of common fixed points of Ti , i = 1, 2, . . . , N . An
implicit iteration process for a finite family of nonexpansive mappings is defined
as follows, with {tn } a real sequence in (0, 1), x0 ∈ C:
x1 = t1 x0 + (1 − t1 )T1 x1 ,
x2 = t2 x1 + (1 − t2 )T2 x2 ,
..
.
xN = tN xN −1 + (1 − tN )TN xN ,
xN +1 = tN +1 xN + (1 − tN +1 )T1 xN +1 ,
..
.
which can be written in the following compact form:
xn = tn xn−1 + (1 − tn )Tn xn , n ≥ 1
(1.1)
where Tk = Tk mod N . (Here the mod N function takes values in N ). And they
proved the weak convergence of the process (1.1).
In 2003, Sun [23] extend the process ( 1.1) to a process for a finite family of
asymptotically quasi-nonexpansive mappings, with {αn } a real sequence in (0, 1)
and an initial point x0 ∈ C, which is defined as follows:
294
GURUCHARAN SINGH SALUJA
x1 =
..
.
xN =
xN +1 =
..
.
x2N =
x2N +1 =
..
.
α1 x0 + (1 − α1 )T1 x1 ,
αN xN −1 + (1 − αN )TN xN ,
αN +1 xN + (1 − αN +1 )T12 xN +1 ,
α2N x2N −1 + (1 − α2N )TN2 x2N ,
α2N +1 x2N + (1 − α2N +1 )T13 x2N +1 ,
which can be written in the following compact form:
xn = αn xn−1 + (1 − αn )Tik xn , n ≥ 1
(1.2)
where n = (k − 1)N + i, i ∈ N .
Sun [23] proved the strong convergence of the process (1.2) to a common fixed
point, requiring only one member T in the family {Ti : i ∈ N } to be semicompact. The result of Sun [23] generalized and extended the corresponding
main results of Wittmann [26] and Xu and Ori [27].
The purpose of this paper is to introduce and study an implicit random iteration
process with errors which converges strongly to a common random fixed point of a
finite family of asymptotically quasi-nonexpansive random operators in uniformly
convex Banach spaces. Our results extend and improve the corresponding results
of Beg and Abbas [5] and many others.
2. Preliminaries
Let (Ω, Σ) be a measurable space (Σ-sigma algebra) and let F be a nonempty
subset of a Banach space X. We will denote by C(X) the family of all compact
subsets of X with Hausdorff metric H induced by the metric of X. A mapping
ξ : Ω → X is measurable if ξ −1 (U ) ∈ Σ, for each open subset U of X. The
mapping T : Ω × X → X is a random map if and only if for each fixed x ∈ X, the
mapping T (., x) : Ω → X is measurable and it is continuous if for each ω ∈ Ω, the
mapping T (ω, .) : X → X is continuous. A measurable mapping ξ : Ω → X is a
random fixed point of a random map T : Ω × X → X if and only if T (ω, ξ(ω)) =
ξ(ω), for each ω ∈ Ω. We denote the set of random fixed points of a random map
T by RF (T ). Let RF (Ti ), (i ∈ N ) be the set of all random fixed points of Ti ,
where N = {1, 2, . . . , N }. The set of common random fixed points of Ti (i ∈ N )
denoted by F, that is, F = ∩N
i=1 RF (Ti ).
Let B(x0 , r) denote the spherical ball centered at x0 with radius r, defined as
the set {x ∈ X : ∥x − x0 ∥ ≤ r}.
CONVERGENCE OF IMPLICIT RANDOM. . .
295
We denote the nth iterate T (ω, T (ω, T (ω, . . . , T (ω, x) . . . , ))) of T by T n (ω, x).
The letter I denotes the random mapping I : Ω × X → X defined by I(ω, x) = x
and T 0 = I.
Definition 2.1. Let C be a nonempty separable subset of a Banach space X and
T : Ω × C → C be a random map. The map T is said to be the following:
(a) A nonexpansive random operator if for x, y ∈ C, one has
∥T (ω, x) − T (ω, y)∥ ≤ ∥x − y∥ ,
(2.1)
for each ω ∈ Ω.
(b) Asymptotically nonexpansive random operator if there exists a sequence of
measurable mapping kn : Ω → [1, ∞) with lim kn (ω) = 1, such that for x, y ∈ C,
n→∞
one has
∥T n (ω, x) − T n (ω, y)∥ ≤ kn (ω) ∥x − y∥ ,
(2.2)
for each ω ∈ Ω.
(c) Asymptotically quasi-nonexpansive random operator if for each ω ∈ Ω,
G(ω) = {x ∈ C : x = T (ω, x)} ̸= ϕ and there exists a sequence of measurable
mapping rn : Ω → [0, ∞) with lim rn (ω) = 0, such that for x ∈ C and y ∈ G(ω),
n→∞
the following inequality holds:
∥T n (ω, x) − y∥ ≤ (1 + rn (ω)) ∥x − y∥ ,
(2.3)
for each ω ∈ Ω.
(d) Uniformly L-Lipschitzian random operator if for x, y ∈ C
∥T n (ω, x) − T n (ω, y)∥ ≤ L ∥x − y∥ ,
(2.4)
for each ω ∈ Ω, where n = 1, 2, . . . and L is a positive constant.
(e) A semicompact random operator if for a sequence of measurable mappings
{ξn } from Ω to C, with lim ∥ξn (ω) − T (ω, ξn (ω))∥ = 0, for every ω ∈ Ω, one
n→∞
has a subsequence {ξnk } of {ξn } and a measurable mapping ξ : Ω → C such that
{ξnk } converges pointwisely to ξ as k → ∞.
Motivated and inspired by Xu and Ori [27], Sun [23] and some others, we
introduce and study an implicit random iteration scheme with errors as follows:
Definition 2.2. Let {T1 , T2 , . . . , TN } be a family of N -random operators from
Ω × C to C, where C is a nonempty closed convex subset of a separable Banach
space X. Let {fn } be a bounded sequence of measurable mappings from Ω to
C and let ξ0 : Ω → C be a measurable mapping. Define the random iteration
process with errors {ξn } as follows:
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GURUCHARAN SINGH SALUJA
ξ1 (ω) = α1 ξ0 (ω) + β1 T1 (ω, ξ1 (ω)) + γ1 f1 (ω),
ξ2 (ω) = α2 ξ1 (ω) + β2 T2 (ω, ξ2 (ω)) + γ2 f2 (ω),
..
.
ξN (ω) = αN ξN −1 (ω) + βN TN (ω, ξN (ω)) + γN fN (ω),
ξN +1 (ω) = αN +1 ξN (ω) + βN +1 T12 (ω, ξN +1 (ω)) + γN +1 fN +1 (ω),
..
.
ξ2N (ω) = α2N ξ2N −1 (ω) + β2N TN2 (ω, ξ2N (ω)) + γ2N f2N (ω),
ξ2N +1 (ω) = α2N +1 ξ2N (ω) + β2N +1 T13 (ω, ξ2N +1 (ω)) + γ2N +1 f2N +1 (ω),
..
.
which can be written in the following compact form:
ξn (ω) = αn ξn−1 (ω) + βn Tik (ω, ξn (ω)) + γn fn (ω),
(2.5)
where n = (k −1)N +i, i ∈ N , and each {fn (ω)} is bounded sequence in C, {αn },
{βn }, {γn } be three appropriate real sequences in [0, 1] such that αn + βn + γn = 1
for n = 1, 2, . . . . Process (2.5) is called the implicit random iteration process with
errors for a finite family of random operators Ti (i = 1, 2, . . . , N ).
Recall that a mapping T : C → C where C is a subset of X with F (T ) ̸= ∅
is said to satisfy condition (A) [22] if there exists a nondecreasing function
f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for all r ∈ (0, ∞) such that
∥x − T x∥ ≥ f (d(x, F (T ))) for all x ∈ C where d(x, F (T )) = inf{∥x − p∥ : p ∈
F (T )}.
Definition 2.3. A family {Ti : i ∈ N } of N -mappings on C, where C be a
nonempty subset of a separable Banach space X with F = ∩N
i=1 F (Ti ) ̸= ∅ is said
to satisfy condition (B) on C if there is a nondecreasing function f : [0, ∞) →
[0, ∞) with f (0) = 0, f (r) > 0 for all r ∈ (0, ∞) such that a1 ∥x − T1 x∥ +
a2 ∥x − T2 x∥ + · · · + aN ∥x − TN x∥ ≥ f (d(x, F)) for all x ∈ C, where d(x, F) =
inf{∥x − p∥ : p ∈ F} and a1 , a2 , . . . , aN are N nonnegative real numbers such
that a1 + a2 + · · · + aN = 1.
Remark 2.4. condition (B) reduces to condition (A) when T1 = T2 = · · · = TN =
T.
In the sequel, we will need the following lemmas.
∞
∞
Lemma 2.5. (Tan and Xu [24]): Let {an }∞
n=1 , {βn }n=1 and {rn }n=1 be sequences
of nonnegative real numbers satisfying
If
an+1 ≤ (1 + rn )an + βn , ∀n ∈ N.
∑∞
n=1 βn < ∞. Then
n=1 rn < ∞,
∑∞
(i) lim an exists.
n→∞
CONVERGENCE OF IMPLICIT RANDOM. . .
297
(ii) If lim inf an = 0, then lim an = 0.
n→∞
n→∞
Lemma 2.6. (Schu [20]) Let E be a uniformly convex Banach space and 0 <
a ≤ tn ≤ b < 1 for all n ≥ 1. Suppose that {xn } and {yn } are sequences in E
satisfying
lim sup ∥xn ∥ ≤ r, lim sup ∥yn ∥ ≤ r,
n→∞
n→∞
lim ∥tn xn + (1 − tn )yn ∥ = r,
n→∞
for some r ≥ 0. Then
lim ∥xn − yn ∥ = 0.
n→∞
3. Main Results
In this section, we study an implicit random iteration scheme with errors to
common random fixed point for a finite family of asymptotically quasi-nonexpansive
random operators in uniformly convex Banach spaces. We also establish the necessary and sufficient condition for the convergence of this process to the common
random fixed point of the above said finite family of random operators. Our results extend the corresponding results of Beg and Abbas [5] and Chang et al. [10].
Theorem 3.1. Let X be a uniformly convex separable Banach space, and let
C be a nonempty closed and convex subset of X. Let {Ti : i ∈ N } be a finite
family of asymptotically quasi-nonexpansive random operators from Ω × C to C
∞
∑
with sequences of measurable mapping rni : Ω → [0, ∞) satisfying
rni (ω) < ∞
n=1
for each ω ∈ Ω and for all i ∈ N and F = ∩N
i=1 RF (Ti ) ̸= ∅. Let ξ0 be a
measurable mapping from Ω to C, then the sequence of random implicit iteration
process with errors defined as by (2.5) converges to a common random fixed point
of random operators {Ti : i ∈ N } in C if and only if lim inf d(ξn (ω), F) = 0,
n→∞
∞
∑
1
where {βn } ⊂ (s, 1 − s) for some s ∈ (0, 2 ),
γn < ∞ and {fn (ω)} is arbitrary
n=1
bounded sequence in C.
Proof. The necessity is obvious. Thus we will only prove the sufficiency. For any
measurable mapping ξ ∈ F, from (2.5), where n = (k −1)N +i, Tn = Tn(mod N ) =
Ti , i ∈ N , it follows that
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GURUCHARAN SINGH SALUJA
∥ξn (ω) − ξ(ω)∥ =∥αn ξn−1 (ω) + βn Tik (ω, ξn (ω)) + γn fn (ω) − ξ(ω)∥
=∥αn (ξn−1 (ω) − ξ(ω)) + βn (Tik (ω, ξn (ω)) − ξ(ω))
+ γn (fn (ω) − ξ(ω))∥
≤αn ∥ξn−1 (ω) − ξ(ω)∥ + βn ∥Tik (ω, ξn (ω)) − ξ(ω)∥
+ γn ∥fn (ω) − ξ(ω)∥
≤αn ∥ξn−1 (ω) − ξ(ω)∥ + βn (1 + rki (ω))∥ξn (ω) − ξ(ω)∥
+ γn ∥fn (ω) − ξ(ω)∥
≤αn ∥ξn−1 (ω) − ξ(ω)∥ + (βn + rki (ω))∥ξn (ω) − ξ(ω)∥
+ γn ∥fn (ω) − ξ(ω)∥
≤αn ∥ξn−1 (ω) − ξ(ω)∥ + (1 − αn + rki (ω))∥ξn (ω) − ξ(ω)∥
+ γn ∥fn (ω) − ξ(ω)∥.
(3.1)
Since lim γn = 0, there exists a natural number n1 such that for n > n1 , γn ≤ 2s .
n→∞
Hence
s
s
αn = 1 − βn − γn ≥ 1 − (1 − s) − = ,
2
2
for n > n1 . Thus, we have from (3.1) that
αn ∥ξn (ω) − ξ(ω)∥ ≤ αn ∥ξn−1 (ω) − ξ(ω)∥ + rki (ω) ∥ξn (ω) − ξ(ω)∥
+γn ∥fn (ω) − ξ(ω)∥
and
∥ξn (ω) − ξ(ω)∥ ≤ ∥ξn−1 (ω) − ξ(ω)∥ +
rki (ω)
∥ξn (ω) − ξ(ω)∥
αn
γn
∥fn (ω) − ξ(ω)∥
αn
2
≤ ∥ξn−1 (ω) − ξ(ω)∥ + rki (ω) ∥ξn (ω) − ξ(ω)∥
s
2
+ γn ∥fn (ω) − ξ(ω)∥ .
s
+
Since
∞
∑
(3.2)
rki (ω) < ∞ for all i ∈ N , lim rni (ω) = 0 for each i ∈ N . Hence there
n=1
exists a natural number n2 , as n >
n→∞
n2
+ 1,
N
that is, n > n2 such that
s
rni (ω) ≤ , ∀ i ∈ N .
4
Then (3.2) becomes
∥ξn (ω) − ξ(ω)∥ ≤
Let
s
2γn
∥ξn−1 (ω) − ξ(ω)∥ +
∥fn (ω) − ξ(ω)∥.
s − 2rki (ω)
s − 2rki (ω)
(3.3)
CONVERGENCE OF IMPLICIT RANDOM. . .
1 + θki (ω) =
299
s
2rki (ω)
=1+
.
s − 2rki (ω)
s − 2rki (ω)
Then
θki (ω) =
2rki (ω)
4
< rki (ω).
s − 2rki (ω)
s
Therefore
∞
∑
4∑
θki (ω) <
rk (ω) < ∞, ∀ i ∈ N
s k=1 i
k=1
∞
and (3.3) becomes
2γn
∥fn (ω) − ξ(ω)∥
s − 2rki (ω)
4
(3.4)
≤(1 + θki (ω))∥ξn−1 (ω) − ξ(ω)∥ + γn M,
s
∥ξn (ω) − ξ(ω)∥ ≤(1 + θki (ω))∥ξn−1 (ω) − ξ(ω)∥ +
where, M = supn≥1 ∥fn (ω) − ξ(ω)∥, since {fn (ω)} is a bounded sequence in C.
This implies that
4
d(ξn (ω), F) ≤ (1 + θki (ω))d(ξn−1 (ω), F) + γn M.
s
Since
∞
∑
k=1
θki (ω) < ∞ and
lim d(ξn (ω), F) = 0.
∞
∑
γn < ∞, it follows from Lemma 2.5, we know that
k=1
n→∞
Next, we will prove that {ξn } is a Cauchy sequence. Notice that when x > 0,
1 + x ≤ ex , from (3.4) we have
300
GURUCHARAN SINGH SALUJA
4M
∥ξn+m (ω) − ξ(ω)∥ ≤(1 + θki (ω))∥ξn+m−1 (ω) − ξ(ω)∥ +
γn+m
s
[
≤(1 + θki (ω)) (1 + θki (ω))∥ξn+m−2 (ω) − ξ(ω)∥
] 4M
4M
+
γn+m−1 +
γn+m
s
s
[
≤(1 + θki (ω))2 (1 + θki (ω))∥ξn+m−3 (ω) − ξ(ω)∥
] 4M
4M
+
γn+m−2 +
(1 + θki (ω))(γn+m−1 + γn+m )
s
s
≤(1 + θki (ω))3 ∥ξn+m−3 (ω) − ξ(ω)∥
(
)
4M
+
(1 + θki (ω))3 γn+m−2 + γn+m−1 + γn+m
s
≤...
{ N ∞
}
∑∑
≤ exp
θki (ω) ∥ξn (ω) − ξ(ω)∥
i=1 k=1
{ N ∞
} n+m
∑∑
∑
4M
+
θki (ω)
exp
γj
s
i=1 k=1
j=n+1
n+m
4M M ′ ∑
γj ,
≤M ∥ξn (ω) − ξ(ω)∥ +
s j=n+1
′
′
for all ξ ∈ F and m, n ∈ N, where M = exp
lim d(ξn (ω), F) = 0 and
n→∞
n1 such that for n ≥ n1 ,
∞
∑
{N ∞
∑∑
(3.5)
}
θki (ω) < ∞. Since
i=1 k=1
rki (ω) < ∞ (i ∈ N ), there exists a natural number
k=1
d(ξn (ω), F) <
ε
and
4M ′
n+m
∑
j=n1 +1
γj ≤
s.ε
.
16M M ′
ε
Thus there exists a point ξ(ω) ∈ F such that d(ξn1 (ω), ξ(ω)) < 4M
′ for each
ω ∈ Ω. It follows from (3.5) that for all n ≥ n1 and m ≥ 1, we have
CONVERGENCE OF IMPLICIT RANDOM. . .
301
∥ξn+m (ω) − ξn (ω)∥ ≤∥ξn+m (ω) − ξ(ω)∥ + ∥ξn (ω) − ξ(ω)∥
≤M ′ ∥ξn1 (ω) − ξ(ω)∥ +
n+m
4M M ′ ∑
γj
s j=n +1
1
n+m
4M M ′ ∑
+ M ∥ξn1 (ω) − ξ(ω)∥ +
γj
s j=n +1
′
1
ε
4M M ′
s.ε
+
.
<M ′ .
4M ′
s
16M M ′
4M M ′
ε
s.ε
+
+ M ′.
.
′
4M
s
16M M ′
=ε.
(3.6)
This implies that {ξn (ω)} is a Cauchy sequence for each ω ∈ Ω. Therefore,
ξn (ω) → p(ω) for each ω ∈ Ω, where p : Ω → F, being the limit of the measurable
mappings, is also measurable. Now, lim d(ξn (ω), F) = 0, for each ω ∈ Ω, and
n→∞
the set F is closed; we have p ∈ F, that is, p is a common random fixed point of
{Ti : i ∈ N }. This completes the proof.
Lemma 3.2. Let X be a uniformly convex separable Banach space, and let C be
a nonempty closed and convex subset of X. Let {Ti : i ∈ N } be a finite family
of uniformly L-Lipschitzian, asymptotically quasi-nonexpansive random operators
from Ω×C to C with sequences of measurable mapping rni : Ω → [0, ∞) satisfying
∞
∑
rni (ω) < ∞ for each ω ∈ Ω and for all i ∈ N and F = ∩N
i=1 RF (Ti ) ̸= ∅. Let
n=1
ξ0 be a measurable mapping from Ω to C, and let the sequence of random implicit
iteration process with errors defined as by (2.5). If lim inf d(ξn (ω), F) = 0, where
n→∞
∞
∑
γn < ∞ and {fn (ω)} is arbitrary bounded
{βn } ⊂ (s, 1−s) for some s ∈ (0, 12 ),
n=1
sequence in C, then
lim ∥ξn (ω) − Tn (ω, ξn (ω))∥ = 0,
n→∞
for each ω ∈ Ω.
Proof. It follows from (3.4), and Lemma 2.5, that lim ∥ξn (ω) − ξ(ω)∥ exists for
n→∞
any ξ ∈ F. Since {ξn (ω) − ξ(ω)} is a convergent sequence, without loss of
generality, we can assume that
lim ∥ξn (ω) − ξ(ω)∥ = dω ,
n→∞
(3.7)
where dω ≥ 0. Observe that
∥ξn (ω) − ξ(ω)∥ =∥αn ξn−1 (ω) + βn Tik (ω, ξn (ω)) + γn fn (ω) − ξ( ω)∥
=∥βn [Tik (ω, ξn (ω)) − ξ(ω) + γn (fn (ω) − ξn−1 (ω))]
+ (1 − βn )[ξn−1 (ω) − ξ(ω) + γn (fn (ω) − ξn−1 (ω))]∥.
(3.8)
302
From
GURUCHARAN SINGH SALUJA
∞
∑
γn < ∞ and (3.7), it follows that
n=1
lim sup ∥ξn−1 (ω) − ξ(ω) + γn (fn (ω) − ξn−1 (ω))∥
n→∞
≤ lim sup[∥ξn−1 (ω) − ξ(ω)∥
n→∞
+ γn ∥fn (ω) − ξn−1 (ω)∥]
≤dω ,
(3.9)
and hence
lim sup ∥Tnk (ω, ξn (ω)) − ξ(ω) + γn (fn (ω) − ξn−1 (ω))∥
n→∞
≤ lim sup[∥Tnk (ω, ξn (ω)) − ξ(ω)∥
n→∞
+ γn ∥fn (ω) − ξn−1 (ω)∥]
≤ lim sup[(1 + rkn (ω))∥ξn (ω)) − ξ(ω)∥
n→∞
+ γn ∥fn (ω) − ξn−1 (ω)∥]
≤dω ,
(3.10)
where n = (k − 1)N + i.
Therefore from (3.7) - (3.10) and Lemma 2.6, we have that
lim Tnk (ω, ξn (ω)) − ξn−1 (ω) = 0,
n→∞
(3.11)
for each ω ∈ Ω.
Moreover, since
∥ξn (ω) − ξn−1 (ω)∥ = αn ξn−1 (ω) + βn Tnk (ω, ξn (ω)) + γn fn (ω) − ξn−1 (ω)
= βn [Tnk (ω, ξn (ω)) − ξn−1 (ω)] + γn [fn (ω) − ξn−1 (ω)]
≤ βn Tnk (ω, ξn (ω)) − ξn−1 (ω)
+γn ∥fn (ω) − ξn−1 (ω)∥ ,
(3.12)
hence by (3.11), we obtain
lim ∥ξn (ω) − ξn−1 (ω)∥ = 0,
n→∞
(3.13)
for each ω ∈ Ω and ∥ξn (ω) − ξn+l (ω)∥ → 0, for each ω ∈ Ω and l < 2N . Now, for
n > N , we have
CONVERGENCE OF IMPLICIT RANDOM. . .
303
∥ξn−1 (ω) − Tn (ω, ξn (ω))∥ ≤∥ξn−1 (ω) − Tnk (ω, ξn (ω))∥
+ ∥Tnk (ω, ξn (ω)) − Tn (ω, ξn (ω))∥
≤∥ξn−1 (ω) − Tnk (ω, ξn (ω))∥
+ L∥Tnk−1 (ω, ξn (ω)) − ξn (ω)∥
≤∥ξn−1 (ω) − Tnk (ω, ξn (ω))∥
k−1
+ L∥Tnk−1 (ω, ξn (ω)) − Tn−N
(ω, ξn−N (ω))∥
[
k−1
+ L ∥Tn−N
(ω, ξn−N (ω)) − ξ(n−N )−1 (ω)∥
]
+ ∥ξ(n−N )−1 (ω) − ξn (ω)∥ .
(3.14)
Since for each n > N , n ≡ (n − N ) mod N . Thus Tn = Tn−N , therefore
∥ξn−1 (ω) − Tn (ω, ξn (ω))∥ ≤∥ξn−1 (ω) − Tnk (ω, ξn (ω))∥ + L2 ∥ξn (ω) − ξn−N (ω)∥
k−1
+ L∥Tn−N
(ω, ξn−N (ω)) − ξ(n−N )−1 (ω)∥
+ L∥ξ(n−N )−1 (ω) − ξn (ω)∥.
(3.15)
This implies that
lim ∥ξn−1 (ω) − Tn (ω, ξn (ω))∥ = 0,
n→∞
(3.16)
for each ω ∈ Ω. Now
∥ξn (ω) − Tn (ω, ξn (ω))∥ ≤∥ξn−1 (ω) − ξn (ω)∥
+ ∥ξn−1 (ω) − Tn (ω, ξn (ω))∥.
(3.17)
Hence
lim ∥ξn (ω) − Tn (ω, ξn (ω))∥ = 0,
n→∞
for each ω ∈ Ω.
(3.18)
Theorem 3.3. Let X be a uniformly convex separable Banach space, and let C
be a nonempty closed and convex subset of X. Let {Ti : i ∈ N } be a finite family
of uniformly L-Lipschitzian, asymptotically quasi-nonexpansive random operators
from Ω×C to C with sequences of measurable mapping rni : Ω → [0, ∞) satisfying
∞
∑
rni (ω) < ∞ for each ω ∈ Ω and for all i ∈ N and F = ∩N
i=1 RF (Ti ) ̸= ∅.
n=1
Suppose there is one member T in the family {Ti : i ∈ N } which is semi-compact
random operator. Let ξ0 be a measurable mapping from Ω to C. Then the sequence
of random implicit iteration process with errors defined as by (2.5) converges to
a common random fixed point of random operators {Ti : i ∈ N }, where {βn } ⊂
∞
∑
γn < ∞ and {fn (ω)} is arbitrary bounded
(s, 1 − s) for some s ∈ (0, 21 ),
n=1
sequence in C.
304
GURUCHARAN SINGH SALUJA
Proof. For any given ξ(ω) ∈ F, we note that
lim ∥ξn (ω) − ξ(ω)∥ = dω ,
(3.19)
n→∞
where dω ≥ 0. By Lemma 3.2, we know that
lim ∥ξn (ω) − Tn (ω, ξn (ω))∥ = 0,
(3.20)
n→∞
for each ω ∈ Ω. Consequently, for any j ∈ N ,
∥ξn (ω) − Tn+j (ω, ξn (ω))∥ ≤ ∥ξn (ω) − ξn+j (ω)∥
+ ∥ξn+j (ω) − Tn+j (ω, ξn+j (ω))∥
+ ∥Tn+j (ω, ξn+j (ω)) − Tn+j (ω, ξn (ω))∥
≤ (1 + L) ∥ξn (ω) − ξn+j (ω)∥
+ ∥ξn+j (ω) − Tn+j (ω, ξn+j (ω))∥
(3.21)
→ 0,
as n → ∞ for each ω ∈ Ω and j ∈ N , where N = {1, 2, . . . , N }.
Consequently, ∥ξn (ω) − Tj (ω, ξn (ω))∥ → 0 as n → ∞ for each ω ∈ Ω and
j ∈ N . Assume that Tj is a semi-compact random operator. Therefore, there
exists a subsequence {ξnk } of {ξn } and a measurable mapping ξ0 : Ω → C such
that ξnk converges pointwise to ξ0 . Now
lim ∥ξnk (ω) − Tj (ω, ξnk (ω))∥ = ∥ξ0 (ω) − Tj (ω, ξ0 (ω))∥ = 0,
n→∞
for each ω ∈ Ω, and j ∈ N . It implies that ξ0 ∈ F, and so lim inf d(ξn (ω), F) = 0.
n→∞
Hence, by Theorem 3.1, we obtain that {ξn } converges to a point in F. This
completes the proof.
Theorem 3.4. Let X be a uniformly convex separable Banach space, and let C
be a nonempty closed and convex subset of X. Let {Ti : i ∈ N } be a finite family
of uniformly L-Lipschitzian, asymptotically quasi-nonexpansive random operators
from Ω×C to C with sequences of measurable mapping rni : Ω → [0, ∞) satisfying
∞
∑
rni (ω) < ∞ for each ω ∈ Ω and for all i ∈ N and F = ∩N
i=1 RF (Ti ) ̸= ∅.
n=1
Suppose the family {Ti : i ∈ N } satisfies the condition (B). Let ξ0 be a measurable
mapping from Ω to C. Then the sequence of random implicit iteration process with
errors defined as by (2.5) converges to a common random fixed point of random
∞
∑
operators {Ti : i ∈ N }, where {βn } ⊂ (s, 1 − s) for some s ∈ (0, 12 ),
γn < ∞
and {fn (ω)} is arbitrary bounded sequence in C.
n=1
Proof. Let ξ(ω) ∈ F. Then it follows from (3.19) and the condition (B) that
d(ξn (ω), F) ≤ a1 ∥ξn (ω) − T1 (ω, ξn (ω))∥ + a2 ∥ξn (ω) − T2 (ω, ξn (ω))∥
+ · · · + aN ∥ξn (ω) − TN (ω, ξn (ω))∥
(3.22)
CONVERGENCE OF IMPLICIT RANDOM. . .
305
for each n ≥ 1, and ω ∈ Ω. Since ∥ξn (ω) − Tj (ω, ξn (ω))∥ → 0 as n → ∞, for each
ω ∈ Ω and j ∈ N , we have
lim f (d(ξn (ω), F)) = 0.
n→∞
(3.23)
Since f is nondecreasing on [0, ∞) with f (0) = 0 and f (r) > 0, for all r ∈ (0, ∞),
it follows that
lim d(ξn (ω), F) = 0.
n→∞
(3.24)
Let ε > 0. Since lim d(ξn (ω), F) = 0, for each ω ∈ Ω, there exists a natural
n→∞
number n1 such that for n ≥ n1 , d(ξn (ω), F) < 3ε , for each ω ∈ Ω. In particular,
there exists a point ξ ∗ (ω) ∈ F such that ∥ξn1 (ω) − ξ ∗ (ω)∥ < 2ε . Now for n ≥ n1
and for all m ≥ 1, we have
∥ξn+m (ω) − ξn (ω)∥ ≤ ∥ξn+m (ω) − ξ ∗ (ω)∥ + ∥ξn (ω) − ξ ∗ (ω)∥
≤ ∥ξn1 (ω) − ξ ∗ (ω)∥ + ∥ξn1 (ω) − ξ ∗ (ω)∥
ε ε
+ = ε.
<
2 2
This implies that {ξn (ω)} is a Cauchy sequence for each ω ∈ Ω. By the completeness of the space X, there exists a measurable mapping p : Ω → C such that
lim ξn (ω) = p(ω), for all ω ∈ Ω. Next, we prove that p(ω) ∈ F. Let ε1 > 0 be
n→∞
given. Then there exists a natural number n2 such that ∥ξn (ω) − p(ω)∥ < ε41
for all n ≥ n2 . Since lim d(ξn (ω), F) = 0, there exists a natural number
n→∞
n3 ≥ n2 such that for all n ≥ n3 we have d(ξn (ω), F) < ε41 and in particular
we have d(ξn3 (ω), F) < ε41 . This implies that there exists q(ω) ∈ F such that
∥ξn3 (ω) − q(ω)∥ < ε41 . Then for each i ∈ N and n ≥ n3 , we have
∥Ti (ω, p(ω)) − p(ω)∥ ≤ ∥Ti (ω, p(ω)) − q(ω)∥ + ∥q(ω) − p(ω)∥
≤ 2 ∥p(ω) − q(ω)∥
[
]
≤ 2 ∥p(ω) − ξn3 (ω)∥ + ∥ξn3 (ω) − q(ω)∥
(ε
ε1 )
1
+
= ε1 .
< 2
4
4
Therefore, Ti (ω, p(ω)) = p(ω), for all i ∈ {1, 2, . . . , N } = N , that is, p(ω) is
a common random fixed point of the random operators {Ti : i ∈ N }. This
completes the proof.
Remark 3.5. Our results extend and improve the corresponding results of Beg
and Abbas [5] to the case of implicit random iteration process with errors for a
finite family of asymptotically quasi-nonexpansive random operators.
Remark 3.6. Theorem 3.1 is a stochastic version of Theorem 2.1 of Kim et al. [16].
306
GURUCHARAN SINGH SALUJA
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Department of Mathematics and Information Technology,
Govt. Nagarjuna P.G. College of Science,
Raipur (C.G.), India.
E-mail address: saluja 1963@rediffmail.com