Approximation of a Common Random Fixed Point for a Finite Family of Random Operators

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2007, Article ID 69626, 12 pagesdoi:10.1155/2007/69626

Research ArticleApproximation of a Common Random Fixed Point for a FiniteFamily of Random Operators

Somyot Plubtieng, Poom Kumam, and Rabian Wangkeeree

Received 16 November 2006; Revised 18 April 2007; Accepted 24 June 2007

Recommended by Pavel Drabek

We construct implicit random iteration process with errors for a common random fixedpoint of a finite family of asymptotically quasi-nonexpansive random operators in uni-formly convex Banach spaces. The results presented in this paper extend and improve thecorresponding results of Beg and Abbas in 2006 and many others.

Copyright © 2007 Somyot Plubtieng et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Random approximations and random fixed point theorems are stochastic generalizationsof classical approximations and random fixed point theorems. The study of random fixedpoints forms a central topic in this area. Random fixed point theorems for random con-traction mappings on separable complete metric spaces were first proved by Spaccek [1]and Hans [2, 3]. Subsequently, Bharucha-Reid [4] had given sufficient conditions for astochastic analogue of Schauder’s fixed point theorem for a random operator. Randomfixed point theorems for multivalued random contraction mappings on separable com-plete metric spaces were first proved by Itoh [5]. Now, this theory has become a full-fledged research area and various ideas associated with random fixed point theory areused to obtain the solution of nonlinear system (cf. Itoh [5]). In an attempt to constructiterations for finding fixed points of random operators defined on linear spaces, randomIshikawa iteration was introduced in [6]. This iteration and some other random itera-tions based on the same idea have been applied for finding solutions of random operatorequations and fixed points of random operators (see [6]).

Recently, Beg [7], Beg and Shahzad [8], Choudhury [9], Duan and Li [10], Li andDuan [11], Yuan et al. [12], and many others have studied fixed point of random op-erators. In 2005, Beg and Abbas [13] studied different random iterative algorithms for

2 International Journal of Mathematics and Mathematical Sciences

weakly contractive and asymptotically nonexpansive random operators on arbitrary Ba-nach spaces. They also established the convergence of an implicit random iterative processto a common random fixed point for a finite family of asymptotically quasi-nonexpansiveoperators.

In 2005, Fukhar-Ud-Din and Khan [14] proved weak and strong convergence of animplicit iterative process with errors, in the sense of Xu [15], for a finite family of asymp-totically quasi-nonexpansive mappings on a closed convex unbounded set in a real uni-formly convex Banach space.

It is our purpose in this paper to construct an implicit random iteration process witherrors which converges strongly to a common random fixed point of a finite family ofasymptotically quasi-nonexpansive random operators on an unbounded set in uniformlyconvex Banach spaces. Our results extend and improve the corresponding onesannounced by Beg and Abbas [13], and many others.

2. Preliminaries

Let (Ω,Σ) be a measurable space with Σ being a sigma-algebra of subsets of Ω and letC be a nonempty subset of a Banach space X . A mapping ξ : Ω→ X is measurable ifξ−1(U) ∈ Σ for each open subset U of X . The mapping T : Ω×C→ C is a random mapif and only if for each fixed x ∈ C, the mapping T(·,x) : Ω→ C is measurable, and it iscontinuous if for each ω ∈Ω, the mapping T(ω,·) : C→ X is continuous. A measurablemapping ξ : Ω→ X is a random fixed point of the random map T : Ω×C→ X if and onlyif T(ω,ξ(ω)) = ξ(ω) for each ω ∈ Ω. We denote by RF(T) the set of all random fixedpoints of a random map T and Tn(ω,x) the nth iteration T(ω,T(ω,T(, . . . ,T(ω,x)))) ofT . The letter I denotes the random mapping I : Ω×C → C defined by I(ω,x) = x andT0 = I .

Definition 2.1. Let C be a nonempty subset of a separable Banach space X and let T :Ω×C→ C be a random map. The map T is said to be

(a) a nonexpansive random operator if for arbitrary x, y ∈ C, one has∥∥T(ω,x)−T(ω, y)

∥∥≤ ‖x− y‖ (2.1)

for each ω ∈Ω;(b) an asymptotically nonexpansive random operator if there exists a sequence of mea-

surable mappings rn : Ω→ [0,∞) with for each ω ∈ Ω, limn→∞ rn(ω) = 0, suchthat for arbitrary x, y ∈ C, one has

∥∥Tn(ω,x)−Tn(ω, y)

∥∥≤ (1 + rn(ω)

)‖x− y‖ for each ω ∈Ω; (2.2)

(c) an asymptotically quasi-nonexpansive random operator if there exists a sequenceof measurable mappings rn : Ω→ [0,∞) with for each ω ∈Ω, limn→∞ rn(ω)= 0,such that∥∥Tn

(

ω,η(ω))− ξ(ω)

∥∥≤ (1 + rn(ω)

)∥∥η(ω)− ξ(ω)

∥∥ for each ω ∈Ω, (2.3)

where ξ : Ω→ C is a random fixed point of T and η : Ω→ C is any measurablemap;

Somyot Plubtieng et al. 3

(d) a completely continuous random operator if the sequence {xn} in C convergesweakly to x0 implies that {T(ω,xn)} converges strongly to T(ω,x0) for each ω ∈Ω;

(e) a uniformly L-Lipschitzian random operator if for arbitrary x, y ∈ C, one has∥∥Tn(ω,x)−Tn(ω, y)

∥∥≤ L‖x− y‖, n= 1,2, . . . , (2.4)

where L is a positive constant;(f) a semicompact random operator if for any sequence of measurable mappings {ξn}

from Ω to C, with limn→∞‖ξn(ω)−T(ω,ξn(ω))‖ = 0, for every ω ∈Ω, there ex-ists a subsequence {ξnk} of {ξn} which converges pointwise to ξ, where ξ : Ω→ Cis a measurable mapping.

Definition 2.2. A family {Ti : i∈ I} of N-mappings on C with F =⋂Ni=1F(Ti) �= ∅ is said

to satisfy condition (B) on C if there is a nondecreasing function f : [0,∞)→ [0,∞) withf (0)= 0 and f (r) > 0 for all r ∈ (0,∞) such that

max1�l�N

{∥∥x−Tlx

∥∥}

� f(

d(x,F))

for all x ∈ C. (2.5)

Definition 2.3. Let {T1,T2,T3, . . . ,TN} be a family of N-random operators from Ω×C toC, where C is a nonempty closed convex subset of a separable Banach space X satisfyingC +C ⊂ C for each ω ∈Ω and let { fn} be a sequence of measurable mappings from Ωto C. Let ξ0 : Ω→ C be a measurable mapping. Following Sun [16], define the randomiteration process with errors {ξn}, in the sense of Liu [17], as follows:

ξ1(ω)= α1ξ0(ω) +(

1−α1)

T1(

ω,ξ1(ω))

+ f1(ω),

ξ2(ω)= α2ξ1(ω) +(

1−α2)

T2(

ω,ξ2(ω))

+ f2(ω),

...

ξN (ω)= αNξN−1(ω) +(

1−αN)

TN(

ω,ξN (ω))

+ fN (ω),

ξN+1(ω)= αN+1ξN (ω) +(

1−αN+1)

T21

(

ω,ξN+1(ω))

+ fN+1(ω),

...

ξ2N (ω)= α2Nξ2N−1(ω) +(

1−α2N)

T2N

(

ω,ξ2N (ω))

+ f2N (ω),

ξ2N+1(ω)= α2N+1ξ2N (ω) +(

1−α2N+1)

T31

(

ω,ξ2N+1(ω))

+ f2N+1(ω),

...

(2.6)

where {αn} is an appropriate real sequence in [0,1]. In the compact form, we have

ξn(ω)= αnξn−1(ω) +(

1−αn)

Tk(n)i(n)

(

ω,ξn(ω))

+ fn(ω), (2.7)

where n= (k− 1)N + i, k = k(n), i= i(n), and each { fn(ω)} is summable sequence in C,that is,

∑∞n=1‖ fn(ω)‖ <∞.


Remark 2.4. Let {T1,T2,T3, . . . ,TN} be a family of N asymptotically quasi-nonexpansivecontinuous random operators with sequences of measurable mappings {rin(ω)} for i =1,2, . . . ,N . If n = (k− 1)N + i, i ∈ {1,2, . . . ,N} = J , then there exists a measurable map-ping rk(ω)=max{r1

k (ω),r2k (ω), . . . ,rNk (ω)}with for all ω ∈Ω, limk→∞ rk(ω)= 0, such that

∥∥Tk(n)

i(n)

(

ω,η(ω))− ξ(ω)

∥∥≤ (1 + rk(ω)

)∥∥η(ω)− ξ(ω)

∥∥ for each ω ∈Ω, (2.8)

where ξ : Ω→ C is a random fixed point of T and η : Ω→ C is any measurable map.

In the sequel, we will need the following lemmas.

Lemma 2.5 (see [18]). Let the nonnegative number sequences {an},{bn}, and {dn} satisfythe following:

an+1 ≤(

1 + bn)

an +dn, ∀n= 1,2, . . . ,∞∑

n=1

bn <∞,∞∑

n=1

dn <∞. (2.9)

Then(1) limn→∞ an exists(2) if liminfn→∞ an = 0, then limn→∞ an = 0.

Lemma 2.6 (see [19]). Let X be a uniformly convex Banach space. Let {xn} and {yn} be thesequences in X , α,β ∈ (0,1), a ≥ 0, and let {αn} be a real sequence number satisfying thefollowing:

(i) 0 < α≤ αn ≤ β < 1, for all n≥ n0, and for some n0 ∈N;(ii) limsupn→∞‖xn‖ ≤ a, limsupn→∞‖yn‖ ≤ a;

(iii) limn→∞‖αnxn + (1−αn)yn‖ = a.Then limn→∞‖xn− yn‖ = 0.

3. An implicit random iterative process

In this section, we present an implicit random iterative process with errors for a finitefamily of asymptotically quasi-nonexpansive mappings. We also establish the necessaryand sufficient condition for the convergence of this process to the common random fixedpoint of the finite family mentioned before. Our results can also be seen as an extensionto the results of Beg and Abbas [13] and Chang et al. [20].

Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly convex sep-arable Banach space X . Let {Ti : i ∈ J} be N asymptotically quasi-nonexpansive randomoperators from Ω×C to C with sequence of measurable mappings rn(ω) : Ω→ [0,+∞) sat-isfying

∑∞n=1 rn(ω) <∞ for each ω ∈Ω and for all i∈ J and F =⋂N

i=1 RF(Ti) �= ∅. Let ξ0 bea measurable mapping from Ω to C, then the sequence of random implicit iteration with er-rors (2.7) converges to a common random fixed point of random operators {Ti : i∈ J} in C ifand only if liminfn→∞d(ξn(ω),F)= 0, where {αn} is a sequence of real numbers in (s,1− s)for some s∈ (0,1) and

∑∞n=1‖ fn(ω)‖ <∞.


Proof. The sufficient condition is obvious. Conversely, for any measurable mapping ξ ∈F, we have

∥∥ξn(ω)− ξ(ω)

∥∥= ∥∥αnξn−1(ω) +

(

1−αn)

Tkn

(

ω,ξ(ω))

+ fn(ω)− ξ(ω)∥∥, (3.1)

where n= (k− 1)N + i, k = k(n), and Tn = Ti( mod N) = Ti. This implies that∥∥ξn(ω)− ξ(ω)

∥∥≤ αn

∥∥ξn−1(ω)− ξ(ω)

∥∥+

(

1−αn)∥∥Tk

i

(

ω,ξn(ω))− ξ(ω)

∥∥+

∥∥ fn(ω)

∥∥

≤ αn∥∥ξn−1(ω)− ξ(ω)

∥∥+

(

1−αn)(

1 + rk(ω))∥∥ξn(ω)− ξ(ω)

∥∥+

∥∥ fn(ω)

∥∥

≤ αn∥∥ξn−1(ω)− ξ(ω)

∥∥+

(

1−αn + rk(ω))∥∥ξn(ω)− ξ(ω)

∥∥+

∥∥ fn(ω)

∥∥.

(3.2)

Thus, we have

∥∥ξn(ω)− ξ(ω)

∥∥≤ ∥∥ξn−1(ω)− ξ(ω)

∥∥+

rk(ω)αn

∥∥ξn(ω)− ξ(ω)

∥∥+

∥∥ fn(ω)

∥∥

αn. (3.3)

Since 0 < s < αn < 1− s < 1, it follows from (3.3) that

∥∥ξn(ω)− ξ(ω)

∥∥≤ ∥∥ξn−1(ω)− ξ(ω)

∥∥+

rk(ω)s

∥∥ξn(ω)− ξ(ω)

∥∥+

∥∥ fn(ω)

∥∥

s. (3.4)

Since∑∞

k=1 rk(ω) <∞ for each ω ∈ Ω, there exists a positive integer n0 such that s−rn(ω) > 0 and rn(ω) < s/2 for each ω∈Ω and for all n= (k− 1)N + i≥ n0.

Thus, we have

∥∥ξn(ω)− ξ(ω)

∥∥≤

(

1 +rk(ω)

s− rk(ω)

)∥∥ξn−1(ω)− ξ(ω)

∥∥+

∥∥ fn(ω)

∥∥

s− rk(ω). (3.5)

It follows from (3.5) that, for each n= (k− 1)N + i≥ n0, we have

∥∥ξn(ω)− ξ(ω)

∥∥≤

(

1 +2rn(ω)

s

)

∥∥ξn−1(ω)− ξ(ω)

∥∥+

2∥∥ fn(ω)

∥∥

sfor each ξ ∈ F.

(3.6)

Setting bn(ω)= 2rk(ω)/s, where n= (k− 1)N + i, i∈ J and k ≥ 1, then we obtain

d(

ξn(ω),F)≤ (1 + vk(ω)

)

d(

ξn−1(ω),F)

+2∥∥ fn(ω)

∥∥

sfor each ω ∈Ω and for all n≥ n0.

(3.7)

Taking an+1(ω) = d(ξn(ω),F), dn(ω) = 2‖ fn(ω)‖/s in Lemma 2.5 and using conditions∑∞

n=1‖ fn(ω)‖ <∞ and∑∞

n=1 rn(ω) <∞, it is easy to see that∑∞

n=1‖bn(ω)‖ <∞,∑∞

n=1

‖cn(ω)‖ <∞. It follows from Lemma 2.5 that limn→∞d(ξn(ω),F)= 0 for each ω ∈Ω. Letbn(ω)= vik (ω), and by (3.6) since bn(ω)= 2rk(ω)/s, where n= (k− 1)N + i, i∈ J . Noticethat when x > 0, 1 + x ≤ exp(x), and

∥∥ξn(ω)− ξ(ω)

∥∥≤ (1 + bn(ω)

)∥∥ξn−1(ω)− ξ(ω)

∥∥+

2∥∥ fn(ω)

∥∥

s, ∀ξ ∈ F. (3.8)


Then∥∥ξn+m(ω)− ξ(ω)

∥∥

≤ (1 + bn+m(ω))∥∥ξn+m−1(ω)− ξ(ω)

∥∥+

2∥∥ fn+m(ω)

∥∥

s

≤ exp(

bn+m(ω))∥∥ξn+m−1(ω)− ξ(ω)

∥∥+

2∥∥ fn+m(ω)

∥∥

s

≤ exp(

bn+m(ω))(

1 + bn+m−1(ω))

(

∥∥ξn+m−2(ω)− ξ(ω)

∥∥+

2∥∥ fn+m−1(ω)

∥∥

s

)

+2∥∥ fn+m(ω)

∥∥

s

...

≤ exp

(n+m∑

k=nbk(ω)

)

∥∥ξn(ω)− ξ(ω)

∥∥+

2s

exp

(n+m−1∑

k=nbk∥∥ fk(ω)

∥∥

)

+2∥∥ fn+m(ω)

∥∥

s

≤ exp

( N∑

i=1

∞∑

k=1

vik (ω)

)

∥∥ξn(ω)−ξ(ω)

∥∥+

2s

exp

(N−1∑

i=1

∞∑

k=1

vik∥∥ fn(ω)

∥∥

)

+2∥∥ fn+m(ω)

∥∥

s

(3.9)

for eachω ∈Ω, and for all natural numbersm, n. PutM = exp(∑N

i=1

∑∞k=1 vik (ω)) + 1 <∞.

Since∑∞

n=1‖ fn(ω)‖ <∞, it follows that

∥∥ξn+m(ω)− ξ(ω)

∥∥≤M

∥∥ξn(ω)− ξ(ω)

∥∥, ∀ω ∈Ω. (3.10)

Let ε > 0. Since limn→∞d(ξn(ω),F)= 0 for each ω ∈Ω, there exists a natural number n1

such that d(ξn,F) < ε/2M for each ω ∈Ω and for all n ≥ n1. In particular, there exists apoint ξ(ω)∈ F such that ‖ξn1 (ω)− ξ(ω)‖ ≤ ε/2M. Now, for each n≥ n1 and for all m≥ 1,we have

∥∥ξn+m(ω)− ξn(ω)

∥∥≤ ∥∥ξn+m(ω)− ξ(ω)

∥∥+

∥∥ξn(ω)− ξ(ω)

∥∥

≤M∥∥ξn(ω)− ξ(ω)

∥∥+

∥∥ξn(ω)− ξ(ω)

∥∥

≤ ε.

(3.11)

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 alsomeasurable. Now, limn→∞d(ξn(ω),F) = 0, for each ω ∈ Ω, and the set F is closed; wehave p ∈ F, that is, p is a common random fixed point of {Ti : i∈ J}. �

Lemma 3.2. Let C be a nonempty closed and convex subset of a uniformly convex separa-ble Banach space X . Let {Ti : i ∈ J} be N uniformly L-Lipschitzian, asymptotically quasi-nonexpansive random operators from Ω×C → C, with sequence of measurable mappings


rn(ω) : Ω→ [0,∞) satisfying∑∞

n=1 rn(ω) <∞ for each ω ∈ Ω and for each i ∈ J and F =⋂N

i=1 RF(Ti) �= ∅. Let ξ0 be a measurable mapping from Ω to C and let the implicit randomiterative process with errors be as in (2.7). If liminfn→∞d(ξn(ω),F)= 0, where {αn} is a se-quence of real numbers in an open interval (s,s− 1) for some s∈ (0,1) and

∑∞n=1‖ fn(ω)‖ <

∞, then

limn→∞

∥∥ξn(ω)−Tn

(

ω,ξn(ω))∥∥= 0 (3.12)

for each ω ∈Ω.

Proof. It follows from (3.6), and Lemma 2.6, that limn→∞‖ξn(ω)− ξ(ω)‖ exists for anyξ ∈ F. Since {ξn(ω)− ξ(ω)} is a convergent sequence, without loss of generality, we canassume that

limn→∞

∥∥ξn(ω)− ξ(ω)

∥∥= dω, (3.13)

where dω ≥ 0. Observe that∥∥ξn(ω)− ξ(ω)

∥∥= ∥∥αn

(

ξn−1(ω)− ξ(ω) + fn(ω))

+(

1−αn)(

Tki

(

ω,ξn−1(ω))− ξ(ω) + fn(ω)

)∥∥.

(3.14)

From∑∞

n=1‖ fn(ω)‖ <∞ and (3.13), it follows that

limsupn→∞

∥∥ξn−1(ω)− ξ(ω) + fn(ω)

∥∥

≤ limsupn→∞

∥∥ξn−1(ω)− ξ(ω)

∥∥+ limsup

n→∞

∥∥ fn(ω)

∥∥≤ dω

(3.15)

and hence

limsupn→∞

∥∥Tk

n

(

ω,ξn(ω))− ξ(ω) + fn(ω)

∥∥

≤ limsupn→∞

rn(ω)∥∥ξn(ω)− ξ(ω)

∥∥+ limsup

n→∞

∥∥ fn(ω)

∥∥≤ dω,

(3.16)

where n= (k− 1)N + i.Therefore from (3.13)–(3.16) and Lemma 2.6, we have that

limn→∞

∥∥Tk

n

(

ω,ξn(ω))− ξn−1(ω)

∥∥= 0, ∀ω ∈Ω. (3.17)

Moreover, since∥∥ξn(ω)− ξn−1(ω)

∥∥= ∥∥(1−αn

)

Tkn

(

ω,ξn(ω))− (1−αn

)

ξn−1(ω) + fn(ω)∥∥

≤ (1−αn)∥∥Tk

n

(

ω,ξn(ω))− ξn−1(ω)

∥∥+

∥∥ fn(ω)

∥∥,

(3.18)

hence by (3.17),

limn→∞

∥∥ξn(ω)− ξn−1(ω)

∥∥= 0 (3.19)


for each ω ∈Ω and ‖ξn(ω)− ξn+l(ω)‖ → 0, for each ω ∈Ω, and for all l < 2N . Now, forn > N , we have

∥∥ξn−1(ω)−Tn

(

ω,ξn(ω))∥∥

≤ ∥∥ξn−1(ω)−Tkn

(

ω,ξn(ω))∥∥+

∥∥Tk

n

(

ω,ξn(ω))−Tn

(

ω,ξn(ω))∥∥


(

ω,ξn(ω))∥∥+L

∥∥Tk−1

n

(

ω,ξn(ω))− ξn(ω)

∥∥


(

ω,ξn(ω))∥∥+L

{∥∥Tk−1

n

(

ω,ξn(ω))−Tk−1

n−N(

ω,ξn−N (ω))∥∥}

+L{∥∥Tk−1

n−N(

ω,ξn−N (ω))− ξ(n−N)−1(ω)

∥∥+

∥∥ξ(n−N)−1(ω)− ξn(ω)

∥∥}

.(3.20)

Since for each n > N , n≡ (n−N) mod N . Thus Tn = Tn−N , therefore∥∥ξn−1(ω))−Tn

(

ω,ξn(ω))∥∥


(

ω,ξn(ω))∥∥+L2

∥∥ξn(ω)− ξn−N (ω)

∥∥

+L∥∥Tk−1

n−N(

ω,ξn−N (ω))− ξ(n−N)−1(ω)

∥∥+L

∥∥ξ(n−N)−1(ω)− ξn(ω)

∥∥.

(3.21)

This implies that

limn→∞


(

ω,ξn(ω))∥∥= 0 (3.22)

for each ω ∈Ω. Now

∥∥ξn(ω)−Tn

(

ω,ξn(ω))∥∥≤ ∥∥ξn−1(ω)− ξn(ω)

∥∥+


(

ω,ξn(ω))∥∥. (3.23)

Hence

limn→∞

∥∥ξn(ω)−Tn

(

ω,ξn(ω))∥∥= 0 (3.24)

for each ω ∈Ω. �

Theorem 3.3. Let C be a nonempty closed and convex subset of a uniformly convex sepa-rable Banach space X . Let {Ti : i∈ J} be N uniformly L-Lipschitzian, asymptotically quasi-nonexpansive random operators from Ω×C → C, with sequence of measurable mappingsrn(ω) : Ω→ [0,+∞) satisfying

∑∞n=1 rn(ω) <∞ for each ω ∈Ω and for each i∈ J . Suppose

that F =⋂Ni=1RF(Ti) �= ∅, and there is one member T in the family {Ti : i∈ J} which is a

semicompact random operator. Let ξ0 be a measurable mapping from Ω to C. Then, the im-plicit random iterative process with errors (2.7) converges to a common random fixed pointof random operators {Ti, i ∈ J}, where {αn} is a sequence of real numbers in (s, s− 1) forsome s∈ (0,1) and

∑∞n=1‖ fn(ω)‖ <∞.

Proof. For any given ξ(ω)∈ F, we note that

limn→∞

∥∥ξn(ω)− ξ(ω)

∥∥= dω, (3.25)


where dω ≥ 0. By Lemma 3.2, we have

limn→∞

∥∥ξn(ω)−Tn

(

ω,ξn(ω))∥∥= 0 (3.26)

for each ω ∈Ω. Consequently, for any j = 1,2, . . . ,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))∥∥

−→ 0,(3.27)

as n→∞ for each ω ∈Ω and j ∈ J .Consequently, ‖ξn(ω)−Tj(ω,ξn(ω))‖→ 0 as n→∞ for each ω ∈Ω and j ∈ J . Assume

that T1 is a semicompact 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

limn→∞

∥∥ξnk (ω)−Tj

(

ω,ξnk (ω))∥∥= ∥∥ξ0(ω)−Tj

(

ω,ξ0(ω))∥∥= 0 (3.28)

for eachω ∈Ω, and j ∈ J . It implies that ξ0 ∈ F, and so liminfn→∞d(ξn(ω),F)= 0. Hence,by Theorem 3.1, we obtain that {ξn} converges to a point in F. �

Corollary 3.4 (cf. [13, Theorem 4.2]). Let C be a nonempty closed bounded and convexsubset of a uniformly convex separable Banach space X . Let {Ti : i ∈ J} be N uniformlyL-Lipschitzian, asymptotically quasi-nonexpansive random operator from Ω×C→ C, withsequence of measurable mappings rin(ω) : Ω→ [0,∞) satisfying

∑∞n=1 rin(ω) <∞, for each

ω ∈Ω and for each i∈ J . Let F =⋂Ni=1 RF(Ti) �= ∅, and there is one member T in the family

{Ti : i∈ J} which is a semicompact random operator. Let ξ0 be a measurable mapping fromΩ to C. Then the sequence {ξn} defined by

ξn(ω)= αnξn−1(ω) +(

1−αn)

Tki

(

ω,ξn(ω))

, (3.29)

where n= (k− 1)N + i, i∈ {1,2, . . . ,N} = J , converges to a common random fixed point ofrandom operators {Ti, i∈ J}, where {αn} is a sequence of real numbers in an open interval(s, s− 1) for some s∈ (0,1).


Proof. Taking fn(ω)= 0, for all n≥ 1, for each ω ∈Ω in Theorem 3.3, the conclusion ofthe corollary is immediate. �

Theorem 3.5. Let C be a nonempty closed and convex subset of a uniformly convex sepa-rable Banach space X . Let {Ti : i∈ J} be N uniformly L-Lipschitzian, asymptotically quasi-nonexpansive random operator from Ω× C → C, with sequence of measurable mappingsrin(ω) : Ω→ [0,∞) satisfying

∑∞n=1 rin(ω) <∞, for each ω ∈Ω and for each i ∈ J . Suppose

that F =⋂Ni=1 RF(Ti) �= ∅, and the family {Ti : i ∈ J} satisfies the condition (B). Let ξ0 be

a measurable mapping from Ω to C. Then the implicit random iterative process with errors(2.7) converges to a common random fixed point of random operators {Ti, i ∈ J}, where{αn} is a sequence of real numbers in (s, s− 1) for some s∈ (0,1) and

∑∞n=1‖ fn(ω)‖ <∞.

Proof. Let ξ(ω)∈ F. Then it follows from (3.25) and the condition (B) that

d(

ξn(ω),F)≤ max

1≤l≤N{∥∥ξn(ω)−Tl

(

ω,ξn(ω))∥∥}

(3.30)

for each n ≥ 1, and ω ∈Ω. Since ‖ξn(ω)−Tj(ω,ξn(ω))‖ → 0 as n→∞, for each ω ∈Ωand j ∈ J , we have

limn→∞ f

(

d(

ξn(ω),F))= 0. (3.31)

Since f is nondecreasing on [0,∞) with f (0)= 0 and f (r) > 0, for all r ∈ (0,∞), it followsthat

limn→∞d

(

ξn(ω),F)= 0. (3.32)

Let ε > 0. Since limn→∞d(ξn(ω),F) = 0, for each ω ∈ Ω, there exists a natural numbern1 such for n ≥ n1, d(ξn,F) ≤ ε/2M, for each ω ∈ Ω. In particular, there exists a pointξ(ω)∈ F such that ‖ξn1 (ω)− ξ(ω)‖ ≤ ε/2M. Now for n≥ n1 and for all m≥ 1, consider

∥∥ξn+m(ω)− ξn(ω)

∥∥≤ ∥∥ξn+m(ω)− ξ(ω)

∥∥+

∥∥ξn(ω)− ξ(ω)

∥∥

≤M∥∥ξn(ω)− ξ(ω)

∥∥+

∥∥ξn(ω)− ξ(ω)

∥∥.

(3.33)

This implies that {ξn(ω)} is a Cauchy sequence for each ω ∈Ω. By the completeness ofthe space X , there exists a measurable mapping p : Ω→ C such that

limn→∞ξn(ω)= p(ω), ∀ω ∈Ω. (3.34)

Next, we prove that p(ω)∈ F. To this end, we let ε > 0 be given, there exists n1 ∈N suchthat ‖ξn(ω)− p(ω)‖ < ε/4 for all n ≥ n1. Since limn→∞d(ξn(ω),F) = 0, there is n2 ≥ n1


such that d(ξn2 (ω),F) < ε/4. This implies that there exists q(ω) ∈ F such that ‖ξn2 (ω)−q(ω)‖ < ε/4. Then for each i= 1,2,3, . . . ,N and n≥ n2, we have

∥∥Tl(

ω, p(ω))− p(ω)

∥∥≤ ∥∥Tl

(

ω,q(ω))− p(ω)

∥∥+

∥∥q(ω)− p(ω)

∥∥

≤ 2∥∥q(ω)− p(ω)

∥∥

≤ 2(∥∥q(ω)− ξn(ω)

∥∥+

∥∥ξn(ω)− p(ω)

∥∥)

≤ ε.

(3.35)

Therefore, Ti(ω, p(ω))= p(ω), for all i= 1,2,3, . . . ,N . This completes the proof. �

Remark 3.6. By using the similar method given in the above proof with different approx-imation obtained from many results, we can conclude that Theorems 3.1, 3.3, and 3.5are also valid for the errors considered, in the sense of Xu [15], under the appropriatelycontrollable conditions on the parameters {αn}, {βn}, and {γn}.

Acknowledgments

The authors would like to thank the Commission on Higher Education, Thailand, forfinancial support and also wish to thank anonymous referees for their suggestions whichled to substantial improvements of this paper.

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Somyot Plubtieng: Department of Mathematics, Faculty of Science, Naresuan University,Phitsanulok 65000, ThailandEmail address: [email protected]

Poom Kumam: Department of Mathematics, Faculty of Science, King Mongkut’s Universityof Technology Thonburi (KMUTT), Bangkok 10140, ThailandEmail address: [email protected]

Rabian Wangkeeree: Department of Mathematics, Faculty of Science, Naresuan University,Phitsanulok 65000, ThailandEmail address: [email protected]

mailto:[email protected]



Approximation of a Common Random Fixed Point for a Finite Family of Random Operators

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