Random Three-Step Iteration Scheme and Common Random Fixed Point of Three Operators

Hindawi Publishing CorporationJournal of Applied Mathematics and Stochastic AnalysisVolume 2007, Article ID 82517, 10 pagesdoi:10.1155/2007/82517

Research ArticleRandom Three-Step Iteration Scheme and Common RandomFixed Point of Three Operators

Somyot Plubtieng, Poom Kumam, and Rabian Wangkeeree

Received 23 July 2006; Revised 5 November 2006; Accepted 7 November 2006

We construct random iterative processes with errors for three asymptotically nonexpan-sive random operators and study necessary conditions for the convergence of these pro-cesses. The results presented in this paper extend and improve the recent ones announcedby I. Beg and M. Abbas (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

Probabilistic functional analysis has come out as one of the momentous mathematicaldisciplines in view of its requirements in dealing with probabilistic models in appliedproblems. The study of random fixed points forms a central topic in this area. Randomfixed point theorems for random contraction mappings on separable complete metricspaces were first proven by Spacek [1]. Subsequently, Bharucha-Reid [2] has given suffi-cient conditions for a stochastic analog of Schauder’s fixed point theorem for a randomoperator. The study of random fixed point theorems was initiated by Spacek [1] and Hans[3, 4]. In an attempt to construct iterations for finding fixed points of random operatorsdefined on linear spaces, random Ishikawa scheme was introduced in [5]. This iterationand also some other random iterations based on the same ideas have been applied forfinding solutions of random operator equations and fixed points of random operators(see [5]).

Recently, Beg [6], Choudhury [7], Duan and Li [8], Li and Duan [9], Itoh [10], andmany others have studied the fixed point of random operators. Beg and Abbas [11] stud-ied the different random iterative algorithms for weakly contractive and asymptoticallynonexpansive random operators on arbitrary Banach spaces. They also established the

2 Journal of Applied Mathematics and Stochastic Analysis

convergence of an implicit random iterative process to a common random fixed point fora finite family of asymptotically quasi-nonexpansive operators.

More recently, Plubtieng et al. [12] studied weak and strong convergence theorems es-tablished for a modified Noor iterative scheme with errors for three asymptotically non-expansive mappings in Banach spaces.

In this paper, we study the convergence of three-step random iterative processes witherrors for three asymptotically nonexpansive random operators in Banach spaces. Ourresults extend and improve the corresponding ones announced by Beg and Abbas [11],and many others.

2. Preliminaries

Let (Ω,Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω and let C be anonempty subset of a Banach space X . A mapping ξ : Ω→ X is measurable if ξ−1(U)∈ Σfor each open subset U of X . The mapping T : Ω×C → C is a random map if for eachfixed x ∈ C, the mapping T(·,x) : Ω→ C is measurable, and it is continuous if for eachω ∈Ω, the mapping T(ω,·) : C→ X is continuous. A measurable mapping ξ : Ω→ X isthe random fixed point of the random map T : Ω×C→ X if T(ω,ξ(ω))= ξ(ω), for eachω ∈Ω. We denote by RF(T) the set of all random fixed points of a random map T and byTn(ω,x) the nth iterate T(ω,T(ω,T(, . . . ,T(ω,x)))) of T . The letter I denotes the randommapping I : Ω×C→ C defined by I(ω,x)= x and T0 = 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 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 limn→∞ rn(ω) = 0, for each ω ∈ Ω, suchthat for arbitrary x, y ∈ C,

∥∥Tn(ω,x)−Tn(ω, y)

∥∥≤ (1 + rn(ω)

)‖x− y‖, for each ω ∈Ω; (2.2)

(c) a uniformly L-Lipschitzian random operator if arbitrary x, y ∈ C, one has

∥∥Tn(ω,x)−Tn(ω, y)

∥∥≤ L‖x− y‖, (2.3)

where n= 1,2, . . . , and L is a positive constant;

Somyot Plubtieng et al. 3

(d) a semicompact random operator if for a sequence of measurable mappings {ξn}from Ω to C, with limn→∞‖ξn(ω)−T(ω,ξn(ω))‖ = 0, for every ω ∈Ω, one has asubsequence {ξnk} of {ξn} and a measurable mapping ξ : Ω→ C such that {ξnk}converges pointwisely to ξ as k→∞.

Definition 2.2 (three-step random iterative process, cf. [11]). Let T : Ω×C→ C is a ran-dom operator, where C is a nonempty convex subset of a separable Banach space X . Letξ0 : Ω→ C be a measurable mapping from Ω to C. Define sequence of functions {ζn},{ηn}, and {ξn}, as given below:

ζn(ω)= α′′n Tn(

ω,ξn(ω))

+β′′n ξn(ω),

ηn(ω)= α′nTn(

ω,ζn(ω))

+β′nξn(ω),

ξn+1(ω)= αnTn(

ω,ηn(ω))

+βnξn(ω) for each ω∈Ω,

(2.4)

n= 0,1,2, . . . , where {αn}, {α′n}, {α′′n }, {βn}, {β′n}, and {β′′n } are sequences of real num-bers in [0,1]. Obviously {ζn}, {ηn}, and {ξn} are sequences of measurable functions fromΩ to C.

Definition 2.3. Let T1,T2,T3 : Ω × C → C be three random operators, where C is anonempty convex subset of a separable Banach space X . Let ξ0 : Ω→ C be a measurablemapping from Ω to C, let { fn}, { f ′n}, { f ′′n } be bounded sequences of measurable func-tions from Ω to C. Define sequences of functions {ζn}, {ηn}, and {ξn}, as given below:

ζn(ω)= α′′n Tn3

(

ω,ξn(ω))

+β′′n ξn(ω) + γ′′n f ′′n (ω),

ηn(ω)= α′nTn2

(

ω,ζn(ω))

+β′nξn(ω) + γ′n f′n (ω),

ξn+1(ω)= αnTn1

(

ω,ηn(ω))

+βnξn(ω) + γn fn(ω) for each ω∈Ω,

(2.5)

n = 0,1,2, . . . , where {αn}, {α′n}, {α′′n }, {βn}, {β′n}, {β′′n }, {γn}, {γ′n}, and {γ′′n } are se-quences of real numbers in [0,1] with αn +βn + γn = α′n +β′n + γ′nα′′n +β′′n + γ′′n = 1.

Remark 2.4. If we take T1 = T2 = T3 ≡ T , and γn = γ′n = γ′′n ≡ 0, then (2.5) reduces to(2.4).

The purpose of this paper is to establish several convergence results of the three-steprandom iterative process with errors given in (2.5) for three asymptotically nonexpansiverandom operators.

In the sequel, we will need the following lemma.

Lemma 2.5 [13, Lemma 1.3]. Let X be a uniformly convex Banach space with xn, yn ∈ X ,real numbers a≥ 0, α,β ∈ (0,1), and let {αn} be a real sequence of numbers which satisfies

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

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

4 Journal of Applied Mathematics and Stochastic Analysis

3. Main results

In this section, we investigate the convergence of three-step random iterative process witherrors for three asymptotically nonexpansive random operators to obtain the randomsolution of the common random fixed point. This iterative process includes three-steprandom iterative process for a random operator T as special case. Note that the proofgiven below is different form the method of the proof proved by Beg and Abbas [11]. Inorder to prove our main results, we need the following two lemmas.

Lemma 3.1. Let X be a uniformly convex separable Banach space, and let C be a nonemptyclosed and convex subset of X . Let T1, T2, T3 be asymptotically nonexpansive random op-erators from Ω×C to C with sequence of measurable mappings rin(ω) : Ω→ [0,∞) satis-fying

∑∞n=1 rin(ω) <∞, for each ω ∈Ω and for all i= 1,2,3, and F =⋂3

i=1 RF(Ti) = ∅. Let{ξn(ω)} be the sequence as defined by (2.5) with

∑∞n=1 γn<∞,

∑∞n=1 γ

′n <∞, and

∑∞n=1 γ

′′n <∞.

Then limn→∞‖ξn(ω)− ξ(ω)‖ exists for all ξ(ω)∈ F and for each ω ∈Ω.

Proof. Let ξ : Ω→ C be the random common fixed point of {T1,T2,T3}. Since { fn}, { f ′n},and { f ′′n } are bounded sequences of measurable functions from Ω to C, we can put

M(ω)= supn≥1

∥∥ fn(ω)− ξ(ω)

∥∥∨ sup

n≥1

∥∥ f ′n (ω)− ξ(ω)

∥∥∨ sup

n≥1

∥∥ f ′′n (ω)− ξ(ω)

∥∥. (3.1)

Then M(ω) is a finite number for each ω ∈Ω. For each n≥ 1, let rn(ω)=max{rin(ω) | i=1,2,3}. Thus, we have rn(ω)≥ 0, limn→0 rin(ω)= 0, and

∥∥ξn+1(ω)− ξ(ω)

∥∥= ∥∥αnTn

1

(

ω,ηn(ω))

+βnξn(ω) + γn fn(ω)− ξ(ω)∥∥

= αn∥∥Tn

1

(

ω,ηn(ω))−ξ(ω)

∥∥+βn

∥∥ξn(ω)−ξ(ω)

∥∥+ γn

∥∥ fn(ω)− ξ(ω)

∥∥

≤ αn(

1 + rn(ω))∥∥ηn(ω)−ξ(ω)

∥∥+βn

∥∥ξn(ω)−ξ(ω)

∥∥+ γn

∥∥ fn(ω)− ξ(ω)

∥∥.

(3.2)

Similarly, we have

∥∥ηn(ω)− ξ(ω)

∥∥≤ α′n

(

1 + rn(ω))∥∥ζn(ω)− ξ(ω)

∥∥+β′n

∥∥ξn(ω)− ξ(ω)

∥∥+ γ′n

∥∥ f ′n (ω)− ξ(ω)

∥∥,

(3.3)

∥∥ζn(ω)− ξ(ω)

∥∥≤α′′n

(

1+rn(ω))∥∥ξn(ω)− ξ(ω)

∥∥+β′′n

∥∥ξn(ω)− ξ(ω)

∥∥+γ′′n

∥∥ f ′′n (ω)− ξ(ω)

∥∥.

(3.4)

Somyot Plubtieng et al. 5

Substituting (3.4) in (3.3), we get∥∥ηn(ω)− ξ(ω)

∥∥

≤ α′nα′′n

(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥+α′nβ

′′n

(

1 + rn(ω))∥∥ξn(ω)− ξ(ω)

∥∥

+α′nγ′′n

(

1 + rn(ω))∥∥ f ′′n (ω)− ξ(ω)

∥∥+β′n

∥∥ξn(ω)−ξ(ω)

∥∥+ γ′n

∥∥ f ′n (ω)− ξ(ω)

∥∥

= (1−β′n− γ′n)

α′′n(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥+β′n

∥∥ξn(ω)− ξ(ω)

∥∥

+(

1−β′n− γ′n)

β′′n(

1 + rn(ω))∥∥ξn(ω)− ξ(ω)

∥∥+mn(ω)

≤ (1−β′n)

α′′n(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥+β′n

(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥

+(

1−β′n)

β′′n(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥+mn(ω)

≤ (1−β′n)(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥

+β′n(

1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥+mn(ω)

= (1 + rn(ω))2∥∥ξn(ω)− ξ(ω)

∥∥+mn(ω),

(3.5)

where

mn(ω)= α′nγ′′n

(

1 + rn(ω))∥∥ f ′′n (ω)− ξ(ω)

∥∥+ γ′n

∥∥ f ′n (ω)− ξ(ω)

∥∥. (3.6)

Note that∑∞

n=1mn(ω) <∞. Substituting (3.5) in (3.2), we have

∥∥ξn+1(ω)− ξ(ω)

∥∥≤ αn

(

1 + rn(ω))3∥∥ξn(ω)− ξ(ω)

∥∥+αn

(

1 + rn(ω))

mn(ω)

+βn∥∥ξn(ω)− ξ(ω)

∥∥+ γn

∥∥ fn(ω)− ξ(ω)

∥∥

≤ (αn +βn)(

1 + rn(ω))3∥∥ξn(ω)− ξ(ω)

∥∥+ bn(ω)

= (1 + rn(ω))3∥∥ξn(ω)− ξ(ω)

∥∥+ bn(ω),

(3.7)

where

bn(ω)= αn(

1 + rn(ω))

mn(ω) + γn∥∥ fn(ω)− ξ(ω)

∥∥. (3.8)

Since

∞∑

n=1

rn(ω) <∞,∞∑

n=1

bn(ω) <∞, (3.9)

it follows from [10, Lemma 2] that limn→∞‖ξn+1(ω)− ξ(ω)‖ exists for all ω ∈Ω. �

6 Journal of Applied Mathematics and Stochastic Analysis

Lemma 3.2. Let X be a uniformly convex separable Banach space, and let C be a nonemptyclosed and convex subset of X . Let T1, T2, T3 be asymptotically nonexpansive random op-erators from Ω to C with sequence of measurable mappings rin(ω) : Ω → [0,∞) satisfy-ing

∑∞n=1 rin(ω) <∞, for each ω ∈Ω and for all i = 1,2,3, and F =⋂3

i=1 RF(Ti) = ∅. Let{ξn(ω)} be the sequence defined as in (2.5) with the following restrictions:

(1) 0 < α≤ αn, α′n,α′′n ≤ 1−α, for some α∈ (0,1), for all n≥ n0, ∃n0 ∈N,(2)

∑∞n=1 γn <∞,

∑∞n=1 γ

′n <∞, and

∑∞n=1 γ

′′n <∞.

Then

limn→∞

∥∥Tn

1

(

ω,ηn(ω))− ξn(ω)

∥∥= lim

n→∞∥∥Tn

2

(

ω,ζn(ω))− ξn(ω)

∥∥

= limn→∞‖T

n3 (ω,ξn(ω))− ξn(ω)‖ = 0,

(3.10)

for all ω ∈Ω.

Proof. Let ξ(ω)∈ F. It follows from Lemma 3.1 that limn→∞‖ξn+1(ω)− ξ(ω)‖ exists, forall ω ∈ Ω. Let limn→∞‖ξn(ω)− ξ(ω)‖ = a for some a ≥ 0. For each n ≥ 1, let rn(ω) =max{rin(ω) | i= 1,2,3}. Taking the upper limit in inequality (3.5), we obtain that

limsupn→∞

∥∥ηn(ω)− ξ(ω)

∥∥≤ limsup

n→∞

∥∥ξn(ω)− ξ(ω)

∥∥= a. (3.11)

So

limsupn→∞

∥∥Tn

1

(

ω,ηn(ω))− ξ(ω)

∥∥≤ limsup

n→∞

(

1 + rn(ω))∥∥ηn(ω)− ξ(ω)

∥∥≤ a. (3.12)

Next, consider

limsupn→∞

∥∥Tn

1

(

ω,ηn(ω))− ξ(ω) + γn

(

fn(ω)− ξn(ω))∥∥

≤ limsupn→∞

∥∥Tn

1

(

ω,ηn(ω))− ξ(ω)

∥∥+

∥∥γn(

fn(ω)− ξn(ω))∥∥.

(3.13)

It follows from (3.12) that

limsupn→∞

∥∥Tn

1

(

ω,ηn(ω))− ξ(ω) + γn

(

fn(ω)− ξn(ω))∥∥≤ a. (3.14)

By the triangle inequality,

limsupn→∞

∥∥ξn(ω)− ξ(ω) + γn

(

fn(ω)− ξn(ω))∥∥≤ a. (3.15)

Somyot Plubtieng et al. 7

Moreover, we note that

a= limn→∞

∥∥ξn+1(ω)− ξ(ω)

∥∥

= limn→∞

∥∥αnT

n1

(

ω,ηn(ω))

+βnξn(ω) + γn(

fn(ω)− (1−αn)

ξ(ω)−αnξ(ω))∥∥

= limn→∞

∥∥αnT

n1

(

ω,ηn(ω))−αnξ(ω) +αnγn fn(ω)−αnγnξn(ω)

+(

1−αn)

ξn(ω)− (1−αn)

ξ(ω)− γnξn(ω) + γn fn(ω)−αnγn fn(ω) +αnγnξn(ω)∥∥

= limn→∞

∥∥αn

(

Tn1

(

ω,ηn(ω))− ξ(ω) + γn

(

fn(ω)− ξn(ω)))

+(

1−αn)(

ξn(ω)− ξ(ω) + γn(

fn(ω)− ξn(ω)))∥∥.

(3.16)

It follows by (3.14), (3.15), and Lemma 3.2 that limn→∞‖Tn1 (ω,ηn(ω))− ξn(ω)‖ = 0.Next,

we prove that limn→∞‖Tn2 (ω,ζn(ω))− ξn(ω)‖ = 0. For each n≥ 1,

∥∥ξn(ω)− ξ(ω)

∥∥≤ ∥∥Tn

1

(

ω,ηn(ω))− ξn(ω)

∥∥+

∥∥Tn

1

(

ω,ηn(ω))− ξ(ω)

∥∥

≤ ∥∥Tn1

(

ω,η(ω))− ξn(ω)

∥∥+

(

1 + rn(ω))∥∥ηn(ω)− ξn(ω)

∥∥.

(3.17)

Since limn→∞‖Tn1 (ω,ηn(ω))−ξn(ω)‖=0= limn→∞ rn(ω), it follows from (3.11) and (3.17)

that

a= limn→∞

∥∥ξn(ω)− ξ(ω)

∥∥≤ liminf

n→∞∥∥ηn(ω)− ξn(ω)

∥∥≤ limsup

n→∞

∥∥ηn(ω)− ξn(ω)

∥∥≤ a.

(3.18)

Hence, limn→∞‖ηn(ω)− ξ(ω)‖ = a. Observe that ζn(ω)− ξ(ω)‖ ≤ (1 + rn(ω))‖ξn(ω)−ξ(ω)‖+γ′′n ‖ f ′′n (ω)−ξ(ω). By boundedness of { f ′′n (ω)} and limn→∞ rn(ω)=0= limn→∞ γ′′n ,we have limsupn→∞‖ζn(ω) − ξ(ω)‖ ≤ limsupn→∞‖ξn(ω) − ξ(ω)‖ ≤ a and solimsupn→∞‖Tn

2 (ω,ζn(ω))− ξ(ω)‖ ≤ limsupn→∞(1 + rn(ω))‖(ω,ζn(ω))− ξ(ω)‖≤a. Next,we consider

∥∥Tn

2

(

ω,ζn(ω))− ξ(ω) + γ′n

(

f ′n (ω)− ξn(ω))∥∥

≤ ∥∥Tn2

(

ω,ζn(ω))− ξ(ω)

∥∥+ γ′n

∥∥(

f ′n (ω)− ξn(ω))∥∥.

(3.19)

Taking limsupn→∞ in both sides, we have limsupn→∞‖Tn2 (ω,ζn(ω))− ξ(ω) + γ′n( f ′n (ω)−

ξn(ω))‖ ≤ a. By the triangle inequality, we see that limsupn→∞‖ξn(ω)− ξ(ω) + γ′n( f ′n (ω)−ξn(ω))‖ ≤ a. Since limn→∞‖ηn(ω)− ξ(ω)‖ = a, we obtain

a= limn→∞

∥∥ξn(ω)− ξ(ω)

∥∥= lim

n→∞∥∥α′nT

n2

(

ω,ζn(ω))

+β′nξn(ω) + γ′n f′n (ω)− ξ(ω)

∥∥

= limn→∞

∥∥α′n

(

Tn2 (ω,ζn(ω)

)− ξ(ω) + γ′n(

f ′n (ω)− ξn(ω)))

+(

1−α′n)(

ξn(ω)− ξ(ω) + γ′n(

f ′n (ω)− ξn(ω)))∥∥.

(3.20)

8 Journal of Applied Mathematics and Stochastic Analysis

By Lemma 2.5, we obtain limn→∞‖Tn2 (ω,ζn(ω))− ξn(ω)‖ = 0. Similarly, by using the same

argument as in the proof above, we have limn→∞‖Tn3 (ω,ξn(ω))− ξn(ω)‖=0, for allω ∈Ω.

This completes the proof. �

Theorem 3.3. Let C be a nonempty closed and convex subset of a uniformly convex sepa-rable Banach space X . Let T1,T2,T3 : Ω×C→ C be semicompact asymptotically nonexpan-sive random operators with sequence of measurable mappings rin(ω) : Ω→ [0,∞) satisfying∑∞

n=1 rin(ω) <∞, for each ω ∈Ω and for each i= 1,2,3 and F =⋂3i=1 RF(Ti) = ∅. Let ξ0 be

a measurable mapping from Ω to C. Define the sequence of functions {ξn}, {ηn}, and {ζn}by (2.5) with {αn}, {α′n}, {α′′n }, {βn}, {β′n}, {β′n}, {γn}, {γ′n}, and {γ′′n } satisfying

(1) 0 < α≤ αn, α′n,α′′n ≤ 1−α, for some α∈ (0,1), for all n≥ n0, ∃n0 ∈N,(2)

∑∞n=1 γn <∞,

∑∞n=1 γ

′n <∞, and

∑∞n=1 γ

′′n <∞.

Then sequences {ξn}, {ηn}, and {ζn} converge to a common random fixed point of F.

Proof. Let ξ : Ω→ C be the common random fixed point in F. By Lemma 3.2, we have

limn→∞

∥∥Tn

1

(

ω,ζn(ω))− ξn(ω)

∥∥

= limn→∞

∥∥Tn

2

(

ω,ηn(ω))− ξn(ω)

∥∥= lim

n→∞∥∥Tn

3

(

ω,ξn(ω))− ξn(ω)

∥∥= 0

(3.21)

for each ω ∈ Ω. This implies that ‖ξn+1(ω) − ξn(ω)‖ ≤ αn‖Tn1 (ω,ηn(ω)) − ξn(ω)‖ +

γn‖ fn(ω)− ξn(ω)‖→ 0, as n→∞, for each ω ∈Ω. We note that

∥∥Tn

1

(

ω,ξn+1(ω))− ξn+1(ω)

∥∥

≤ ∥∥Tn1

(

ω,ξn+1(ω))−Tn

1

(

ω,ξn(ω))∥∥+

∥∥Tn

1 ξn(ω)− ξn(ω)∥∥+

∥∥ξn(ω)− ξn+1(ω)

∥∥

≤ (1 + γn)∥∥ξn+1(ω)− ξn(ω)

∥∥+

∥∥Tn

1

(

ω,ξn(ω))− ξn(ω)

∥∥

+∥∥ξn(ω)− ξn+1(ω)

∥∥−→ 0, as n−→∞,

(3.22)

for each ω ∈Ω. Using (3.22), we have

∥∥T1

(

ω,ξn+1(ω))− ξn+1(ω)

∥∥

≤ ∥∥T1(

ω,ξn+1(ω))−Tn+1

1

(

ω,ξn(ω))∥∥+

∥∥Tn+1

1

(

ω,ξn+1(ω))− ξn+1(ω)

∥∥

≤ (1 + γ1)∥∥ξn+1(ω)−Tn

1 ξn+1(ω)∥∥

+∥∥Tn+1

1

(

ω,ξn+1(ω))− ξn+1(ω)

∥∥−→ 0, as n−→∞,

(3.23)

for each ω ∈Ω. Thus, we have limn→∞‖T1(ω,ξn(ω))− ξn(ω)‖ = 0 for each ω ∈Ω. Simi-larly, we can show that

limn→∞

∥∥T2

(

ω,ξn(ω))− ξn(ω)

∥∥ limn→∞

∥∥T3

(

ω,ξn(ω))− ξn(ω)

∥∥= 0. (3.24)

Somyot Plubtieng et al. 9

Since T1 is a semicompact continuous random operator and limn→∞‖T1(ω,ξn(ω))−ξn(ω)‖ = 0 for each ω ∈ Ω, there exist a subsequence {ξnk} of {ξn} and a measurablemapping ξ0 : Ω→ C such that ξnk converges pointwisely to ξ0. The mapping ξ0 : Ω→ C,being a pointwise limit of measurable mappings {ξnk}, is measurable. Now,

limk→∞

∥∥ξnk (ω)−T1

(

ω,ξnk (ω))∥∥= ∥∥ξ0(ω)−T1

(

ω,ξ0(ω))∥∥= 0 (3.25)

for each ω ∈Ω. Hence, ξ0(ω) is a random fixed point of T1. Since limn→∞‖ξn(ω)− ξ0(ω)‖exists, limn→∞ ξn(ω) = ξ0(ω) for each ω ∈ Ω. Similarly, we can show that ξ0(ω) is alsoa random fixed point of T2 and T3. Observe that ‖ηn(ω)− ξn(ω)‖ ≤ α′n‖Tn

2 (ω,ζn(ω))−ξn(ω)‖ + γ′n‖ f ′n (ω) − ξn(ω)‖ → 0, and ‖ζn(ω) − ξn(ω)‖ ≤ α′′n ‖Tn

3 (ω,ξn(ω)) − ξn(ω)‖ +γ′′n ‖ f ′′n (ω)− ξn(ω)‖ → 0, as n→∞, for each ω ∈ Ω. Hence, limn→∞ηn(ω) = ξ0(ω) andlimn→∞ ζn(ω)= ξ0(ω) for each ω ∈Ω. Therefore {ξn},{ηn}, and {ζn} converge to a com-mon random fixed point in F. �

If T1 = T2 = T3 := T and γn = γ′n = γ′′n ≡ 0, then Theorem 3.3 reduces to the followingknown result.

Corollary 3.4 (see Beg and Abbas [11, Theorem 3.3]). Let C be a nonempty closedbounded and convex subset of a uniformly convex separable Banach spaceX . LetT : Ω×C→C be completely continuous asymptotically nonexpansive random operator with sequence ofmeasurable mappings rn(ω) : Ω→ [0,∞) satisfying

∑∞n=1 rn(ω) <∞, for each ω ∈Ω. Let ξ0

be a measurable mapping from Ω to C. Define the sequence of functions {ξn}, {ηn}, and{ζn} by (2.4) with {αn} and {βn} satisfying 0 < liminfn→∞αn ≤ limsupn→∞αn < 1, and0 < liminfn→∞βn ≤ limsupn→∞βn < 1. Then sequences {ξn}, {ηn}, and {ζn} converge to arandom fixed point of T .

Proof. By Xu [14] and Ramırez [15], F(T) = ∅. Hence it follows from Theorem 3.3 thatthe sequences {ξn}, {ηn}, and {ζn} converge to a random fixed point of T . �

Remark 3.5. Theorem 3.3 is a generalized stochastic version of the result due to Plubtienget al. [12].

Acknowledgments

The authors would like to thank The Thailand Research Fund for financial support andthe referees for reading this paper carefully, providing valuable suggestions and com-ments, and pointing out a major error in the original version of this paper. This work issupported by Thailand Research Fund under Grant BRG49800018.

References

[1] A. Spacek, “Zufallige Gleichungen,” Czechoslovak Mathematical Journal, vol. 5(80), pp. 462–466,1955.

[2] A. T. Bharucha-Reid, “Fixed point theorems in probabilistic analysis,” Bulletin of the AmericanMathematical Society, vol. 82, no. 5, pp. 641–657, 1976.

10 Journal of Applied Mathematics and Stochastic Analysis

[3] O. Hans, “Random fixed point theorems,” in Transactions of the First Prague Conference on In-formation Theory, Statistical Decision Functions, Random Processes (Liblice, Prague, 1956), pp.105–125, Czechoslovak Academy of Sciences, Prague, Czech Republic, 1957.

[4] O. Hans, “Random operator equations,” in Proceedings 4th Berkeley Symposium Math. Statist.and Prob., Vol. II, pp. 185–202, University California Press, Berkeley, Calif, USA, 1961.

[5] B. S. Choudhury, “Convergence of a random iteration scheme to a random fixed point,” Journalof Applied Mathematics and Stochastic Analysis, vol. 8, no. 2, pp. 139–142, 1995.

[6] I. Beg, “Approximation of random fixed points in normed spaces,” Nonlinear Analysis, vol. 51,no. 8, pp. 1363–1372, 2002.

[7] B. S. Choudhury, “Random Mann iteration scheme,” Applied Mathematics Letters, vol. 16, no. 1,pp. 93–96, 2003.

[8] H. Duan and G. Li, “Random Mann iteration scheme and random fixed point theorems,” Ap-plied Mathematics Letters, vol. 18, no. 1, pp. 109–115, 2005.

[9] G. Li and H. Duan, “On random fixed point theorems of random monotone operators,” AppliedMathematics Letters, vol. 18, no. 9, pp. 1019–1026, 2005.

[10] S. Itoh, “A random fixed point theorem for a multivalued contraction mapping,” Pacific Journalof Mathematics, vol. 68, no. 1, pp. 85–90, 1977.

[11] I. Beg and M. Abbas, “Iterative procedures for solutions of random operator equations in Banachspaces,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 181–201, 2006.

[12] S. Plubtieng, R. Wangkeeree, and R. Punpaeng, “On the convergence of modified Noor iterationswith errors for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis andApplications, vol. 322, no. 2, pp. 1018–1029, 2006.

[13] J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive map-pings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.

[14] H.-K. Xu, “Random fixed point theorems for nonlinear uniformly Lipschitzian mappings,” Non-linear Analysis, vol. 26, no. 7, pp. 1301–1311, 1996.

[15] P. L. Ramırez, “Some random fixed point theorems for nonlinear mappings,” Nonlinear Analysis,vol. 50, no. 2, pp. 265–274, 2002.

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

Poom Kumam: Department of Mathematics, Faculty of Science, Naresuan University,Phitsanulok 65000, Thailand; Department of Mathematics, Faculty of Science, King Mongkut’sUniversity of Technology Thonburi (KMUTT), Bangkok 10140, ThailandEmail addresses: [email protected]; [email protected]

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

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