1. Introduction and preliminaries

Available online at http://jnfa.scik.org J. Nonlinear Funct. Anal. 2014, 2014:17 ISSN: 2052-532X

CONVERGENCE AND STABILITY OF TWO RANDOM ITERATION ALGORITHMS ISMAT BEG1,∗ , DEBASHIS DEY2 , MANTU SAHA3 1 Centre

for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore-53200, Pakistan 2 Koshigram

3 Department

Union Institution, Koshigram-713150, Burdwan, West Bengal, India

of Mathematics,The University of Burdwan, Burdwan-713104, West Bengal, India

c 2014 Beg, Dey and Saha. This is an open access article distributed under the Creative Commons Attribution License, which Copyright permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. We study the almost sure T -stability and strong convergence of two random iterative algorithms namely random Halpern iteration scheme and random Xu-Mann iteration scheme for a general class of random operators in a separable Banach space. Our results complement/generalize the known stability results in stochastic verse. Keywords: almost sure T-stabilty; separable Banach space; random Mann iteration; Xu-Mann-type random iteration; random Halpern iteration. 2010 AMS Subject Classification: 47H10, 60H25.

1. Introduction and preliminaries The study of random fixed point theorems was initiated by the Prague school of probabilists in 1950’s. The introduction of randomness leads to several new questions of measurability of solutions, probabilistic and statistical aspects of random solutions. Random fixed point theorems for random contraction mappings on separable complete metric spaces were first proved ∗ Corresponding

author

E-mail addresses: [email protected] (I. Beg), [email protected] (D. Dey), [email protected] (M. Saha) Received October 8, 2014 1

2

I. BEG, D. DEY, M. SAHA

ˇ cek [33] and Hanˇs (see [14]-[15]). The survey article by Bharucha-Reid [7] in 1976 by Spaˇ attracted the attention of several mathematicians and gave wings to this theory. Itoh [17] exˇ cek’s and Hanˇs’s theorems to multivalued contraction mappings. Mukherjee [22] tended Spaˇ gave a random version of Schauder’s fixed point theorem on an atomic probability measure space. Random fixed point theorems with an application to Random differential equations in Banach spaces are obtained by Itoh [17]. Sehgal and Waters [32] had obtained several random fixed point theorems including random analogue of the classical results due to Rothe [26]. Recently Papageorgiou [24], Xu [35], Beg ([1]-[3]), Beg and Shahazad [6], Xu and Beg [36], Liu [20], Saha and Dey [27], Saha and Debnath [28], Saha [29] and Saha and Ganguly [30] have studied the fixed points of random maps over separable Hilbert space and Banach space. Existence of random fixed point using various iterative techniques for approximating fixed points of random nonexpansive mappings have been studied by Beg and Abbas [4], Choudhury [10], Choudhury and Ray [11], Saha and Dey [31]. The aim of this paper is to propose a random Halpern iteration scheme over a separable Banach space, which is the modified random Mann iteration procedure. Then we use this scheme to study the convergence to the random fixed point for a general class of random operator T (ω, x) over a separable Banach space. The study also includes the almost sure T -stability of Xu-Mann type random iterative algorithms and random Halpern iteration algorithm for a general class of random operators in a separable Banach space. First we recall some basic ideas and definitions. Let (X, β X ) be a separable Banach space where β X is a σ -algebra of Borel subsets of X, and let (Ω, β , µ) denote a complete probability measure space with measure µ, and β be a σ -algebra of subsets of Ω. For more details one can see Joshi and Bose [18]. Definition 1.1. A mapping x : Ω → X is said to be an X-valued random variable, if the inverse image under the mapping x of every Borel subset B of X belongs to β ; that is, x−1 (B) ∈ β for all B ∈ β X . Definition 1.2. A mapping x : Ω → X is said to be a finitely valued random variable, if it!is constant on each of finite number of disjoint sets Ai ∈ β and is equal to 0 on Ω −

n [

i=1

Ai .

RANDOM ITERATION ALGORITHMS

3

Mapping x is called a simple random variable if it is finitely valued and µ {ω : kx (ω)k > 0} < ∞. Definition 1.3. A mapping x : Ω → X is said to be a strong random variable, if there exists a sequence {xn (ω)} of simple random variables which converges to x (ω) almost surely, i.e., there exists a set A0 ∈ β with µ (A0 ) = 0 such that lim xn (ω) = x (ω), ω ∈ Ω − A0 . n→∞

Definition 1.4.

A mapping x : Ω → X is said to be a weak random variable, if the function

x∗ (x (ω)) is a real valued random variables for each x∗ ∈ X ∗ , the space X ∗ denoting the first normed dual space of X. In a separable Banach space X, the notions of strong and weak random variables x : Ω → X [see Joshi and Bose [18, Corollary 1]] coincide and in respect of such a space X, x is called as a random variable. We recall the following results. Theorem 1.1. [18, Theorem 6.1.2] Let x, y : Ω → X be strong random variables and α, β be constants. Then the following statement holds:

(a): α x (ω) + β y (ω) is a strong random variable. (b): If f (ω) is a real valued random variable and x (ω) is a strong random variable, then f (ω) x (ω) is a strong random variable. (c): If {xn (ω)} is a sequence of strong random variables converging strongly to x (ω) almost surely, i.e., if there exists a set A0 ∈ β with µ (A0 ) = 0 such that lim kxn (ω) − x (ω)k = n→∞

0 for every ω ∈ / A0 , then x (ω) is a strong random variable.

Remark 1.1. If X is a separable Banach space, then the σ -algebra generated by the class of all spherical neighourhoods of X is equal to the σ -algebra of all Borel subsets of X . Hence every strong and also weak random variable is measurable in the sense of Definition 1.1. Let Y be another Banach space. We also need the following definitions from Joshi and Bose [18] for subsequent use. Definition 1.5. A mapping F : Ω × X → Y is said to be a random mapping if F (ω, x) = y (ω) is a Y -valued random variable for every x ∈ X. Definition 1.6. A mapping F : Ω × X → Y is said to be a continuous random mapping if the set of all ω ∈ Ω for which F (ω, x) is a continuous function of x has measure one.

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Definition 1.7. A random mapping F : Ω × X → Y is said to be demi-continuous at x ∈ X if weakly kxn − xk → 0 implies F (ω, xn )

→

F (ω, x)

almost surely.

Theorem 1.2. [18, Theorem 6.2.2.] Let F : Ω × X → Y be a demi-continuous random mapping where Banach space Y is separable. Then for any X-valued random variable x, the function F (ω, x (ω)) is a Y -valued random variable. Remark 1.2.

(see [29]) Since a continuous random mapping is a demi-continuous random

mapping, Theorem 1.10 is also true for a continuous random mapping. Following definitions are due to Joshi and Bose [18]. Definition 1.8. An equation of the type F (ω, x (ω)) = x (ω) where F : Ω × X → X is a random mapping is called a random fixed point equation. Definition 1.9. Any mapping x : Ω → X which satisfies random fixed point equation F (ω, x (ω)) = x (ω) almost surely is said to be a wide sense solution of the fixed point equation. Definition 1.10. Any X-valued random variable x (ω) which satisfies µ {ω : F (ω, x (ω)) = x (ω)} = 1 is said to be a random solution of the fixed point equation or a random fixed point of F. Remark 1.3.A random solution is a wide sense solution of the fixed point equation. But the converse is not necessarily true. This is evident from following example given by Joshi and Bose [18]. Example 1.1. Let X be the set of all real numbers and let E be a non measurable subset of X. Let F : Ω × X → Y be a random mapping defined as F (ω, x) = x2 + x − 1 for all ω ∈ Ω. In this case, the real valued function x (ω), defined as x (ω) = 1 for all ω ∈ Ω is a random fixed point of F. However the real valued function y (ω) defined as

y (ω) =

   −1,  

ω∈ /E

    1,

ω ∈E


5

is a wide sense solution of the fixed point equation F (ω, x (ω)) = x (ω), without being a random fixed point of F.

2. Random stability and random iteration Zhang et al. [37] proposed almost sure T -stability for some kind of contractive type random operators and also convergence of Ishikawa-type and Mann-type random iterative algorithms for such type of random operators in a separable Banach space. Let (Ω, β , µ) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. Let T : Ω × E → E be a random operator with a random fixed point x∗ (ω) ∈ E. For any given random variable x0 (ω) ∈ E, define an iteration scheme {xn (ω)}∞ n=0 ⊂ E by xn+1 (ω) = f (T ; xn (ω)); n = 0, 1, 2, ... where f is a measurable function in the second variable. Let {yn (ω)}∞ n=0 ⊂ E be an arbitrary sequence of a random variable. Denote ε n (ω) = kyn+1 (ω) − f (T ; yn (ω))k . Then the random iteration scheme (2.1) is said to be T -stable almost surely (or, the random iteration scheme (2.1) is stable with respect to T almost surely) if for ω ∈ Ω, ε n (ω) → 0 as strongly

n → ∞ implies that yn (ω) → x∗ (ω) ∈ E almost surely. Now we recall two random iteration scheme namely Random Mann iteration Scheme and Random Xu-Mann iteration Scheme. The Mann iteration scheme was obtained in [21]. Random analogue of this iteration was investigated by Choudhury [9]. Definition 2.1. (Random Mann iteration scheme) Let E be a nonempty subset of a separable Banach space X and T : Ω × E → E be a random operator. Let x0 : Ω → E be any measurable function. Then the random iteration scheme is defined as the following sequences of functions: (1)

xn+1 (ω) = (1 − bn ) xn (ω) + bn T (ω, xn (ω)) , n ≥ 0, ω ∈ Ω

where {bn } is a sequence in [0, 1].

6


The consideration of error terms is an important part of the convergence of the iteration process in fixed point theory. For this reason, random analogue of Xu-Mann [34] iteration scheme (see [31]) in separable Banach space is needed. Definition 2.2.

(Random Xu-Mann iteration scheme) Let E be a nonempty subset of a sep-

arable Banach space X and T : Ω × E → E be a random operator. Let x0 : Ω → E be any measurable function. Then the random iteration scheme is defined as the following sequences of functions: (2)

xn+1 (ω) = an xn (ω) + bn T (ω, xn (ω)) + cn un , n ≥ 0, ω ∈ Ω

where {an },{bn } and {cn } are sequences in [0, 1] such that an + bn + cn = 1 and {un } is a bounded sequence in E. Clearly, this iteration process contains the first one (Mann Iteration) as it’s special case. Osilike[23] established some deterministic stability results considering the contractive operator T satisfying: (3)

d(T x, Ty) ≤ Ld(x, T x) + ad(x, y)

for all x and y, where L ≥ 0 and 0 ≤ a < 1. He established T -stability for such maps with respect to Picard, Kirk, Mann, and Ishikawa iterations. Imoru and Olatinwo [16] generalized the condition of Osilike by replacing (2.3) with d(T x, Ty) ≤ φ (d(x, T x)) + ad(x, y) where 0 ≤ a < 1 and φ : R+ → R+ is monotone increasing with φ (0) = 0 while Bosede and Rhoades [8] considered the map T having a fixed point p and satisfying the condition: d(p, Ty) ≤ ad(p, y) for some 0 ≤ a < 1 and for each y ∈ X. In the following sections, we consider a general class of random mapping which generalize the contractive operators due to Osilike [23], Imoru and Olatinwo [16], Bosede and Rhoades [8] in stochastic verse. Now before going into our main results we need following well-known lemma.


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Lemma 2.1. [19] Suppose {α n }, {β n }, are sequences of nonnegative numbers satisfying the following inequality α n+1 ≤ (1 − sn ) α n + β n + γ n for all n ≥ 0, ∞

where {sn }∞ n=0 ⊂ [0, 1],

∑ sn = ∞, β n = o(sn) and

n=1

∞

lim α n = 0. ∑ γ n < ∞, then n→∞

n=1

3. Random Halpern iteration schemes Halpern [13] introduced an iterative scheme for a non expansive mapping T on a subset C of a Hilbert space by taking any points u, x1 ∈ C and defined the iterative sequence {xn } by xn+1 = bn u + (1 − bn )xn for all n ≥ 1, where an ∈ [0, 1]. He has also shown that {xn } converges ∞

to a fixed point of T under some control conditions: (H1 ) lim bn = 0 and (H2 ) n→∞

∑ bn = ∞. We

n=1

propose random analogue of this scheme on a nonempty subset E of a separable Banach space X as following: Let E be a nonempty subset of a separable Banach space X and T : Ω × E → E be a random operator. Let x0 : Ω → E be any measurable function. Then a random iteration scheme is defined by the following rule.: u(ω) ∈ E, x1 (ω) ∈ E and (4)

xn+1 (ω) = bn u(ω) + (1 − bn ) T (ω, xn (ω)), n ≥ 0, ω ∈ Ω,

where {bn } is a sequence in [0, 1]. Now we use this random iteration procedure to obtain some convergence results and as well as stability results for a very general class of random operators over a separable Banach space. Theorem 3.1. Let (E, k.k) be a separable Banach space, T : Ω × E → E be a continuous random operator with a random fixed point x∗ (ω) ∈ E satisfying kx∗ (ω) − T (ω, y)k ≤ a(ω) kx∗ (ω) − y(ω)k ∞

for some 0 ≤ a(ω) < 1 and for each y(ω) ∈ E. If

∑ bn = ∞, then the sequence {xn(ω)}∞n=0

n=1

of function defined by random Halpern iteration scheme (4) converges strongly to x∗ (ω) almost surely.

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Proof. Let A = {ω ∈ Ω : 0 ≤ a(ω) < 1} and Cx∗ ,y = {ω ∈ Ω : kx∗ (ω) − T (ω, y)k ≤ a(ω) kx∗ (ω) − y(ω)k} . Let S be a countable dense subset of E and take s ∈ S. We next show that \

\

(Cx∗ ,y ∩ A) =

x∗ ,y∈E

(Cx∗ ,s ∩ A) .

x∗ ,s∈S

Let ω ∈

\

(Cx∗ ,s ∩ A) .

x∗ ,s∈S

Now kx∗ (ω) − T (ω, y)k ≤ kx∗ (ω) − T (ω, s)k + kT (ω, s) − T (ω, y)k ≤ a(ω) kx∗ (ω) − s(ω)k + kT (ω, s) − T (ω, y)k ≤ a(ω) [kx∗ (ω) − y(ω)k + ky(ω) − s(ω)k] + kT (ω, s) − T (ω, y)k .

(5)

Choose ε > 0. So we can find a δ (y) > 0 such that kT (ω, s) − T (ω, y)k < ε, whenever ks(ω) − y(ω)k < δ . From (6), we get kx∗ (ω) − T (ω, y)k ≤ a(ω) kx∗ (ω) − y(ω)k . Consequently, we have ω∈

\

(Cx∗ ,y ∩ A)

x∗ ,y∈E

and hence \

\

(Cx∗ ,s ∩ A) ⊂

(Cx∗ ,y ∩ A) .

x∗ ,y∈E

x∗ ,s∈S

Also it is obvious that \

(Cx∗ ,y ∩ A) ⊂

x∗ ,y∈E

\

(Cx∗ ,s ∩ A)

x∗ ,s∈S

and so \ x∗ ,s∈S

(Cx∗ ,s ∩ A) =

\ x∗ ,y∈E

(Cx∗ ,y ∩ A) .


Let N =

\

9

(Cx∗ ,y ∩ A) .

x∗ ,y∈E

Then µ (N) = 1. Now take ω ∈ N and n ≥ 1. Then using (4) and (5) we get kxn+1 (ω) − x∗ (ω)k = kbn u(ω) + (1 − bn )T (ω, xn (ω)) − x∗ (ω)k = kbn u(ω) + (1 − bn )T (ω, xn (ω)) − (1 − bn )x∗ (ω) − bn x∗ (ω)k ≤ (1 − bn ) kx∗ (ω) − T (ω, xn (ω))k + bn kx∗ (ω) − u(ω)k ≤ (1 − bn )a(ω) kx∗ (ω) − xn (ω)k + bn kx∗ (ω) − u(ω)k ≤ (1 − bn ) kx∗ (ω) − xn (ω)k + bn kx∗ (ω) − u(ω)k , since a(ω) < 1 = (1 − bn ) kxn (ω) − x∗ (ω)k + rn , where rn = bn kx∗ (ω) − u(ω)k Now using Lemma 2.1 we get lim kxn (ω) − x∗ (ω)k = 0. Hence {xn (ω)}∞ n=0 as defined by rann→∞

dom Halpern iteration scheme (4) converges strongly to x∗ (ω) almost surely. Example 3.1. Let Ω = R and E = [0, 1]. Define T : Ω × E → E by T (ω, x) = 2x . So the random operator T (ω, x) is continuous and has the random fixed point 0. Next choose a(ω) such that 1 2

< a(ω) < 1, then it is clear that the inequality (5) satisfies. Also take u(ω) = 1 and bn = ∞

(n ≥ 1) where b0 = 0 and so

∑ bn = ∞.

1 n

Then the sequence {xn (ω)} of function defined by

n=1

random Halpern iteration scheme (4) converges strongly to 0 almost surely. Similar convergence result for random Xu-Mann iteration can be derived for such a general class of random operator over a separable Banach space. Corollary 3.2. Let (E, k.k) be a separable Banach space, T : Ω × E → E be a continuous random operator with a random fixed point x∗ (ω) ∈ E satisfying kx∗ (ω) − T (ω, y)k ≤ a(ω) kx∗ (ω) − y(ω)k for some 0 ≤ a(ω) < 1 and for each y(ω) ∈ E. If ∞

∑ bn = ∞, cn = o(bn)

n=1

10


and sup kun − x∗ (ω)k = M(> 0). n≥0

Then the sequence {xn (ω)}∞ n=0 of function defined by random Xu-Mann iteration scheme (2) converges strongly to x∗ (ω) almost surely. Proof. Proceeding in the same fashion as in the proof of Theorem 3.1 and using (7) and (2) we get kxn+1 (ω) − x∗ (ω)k = kan xn (ω) + bn T (ω, xn (ω)) + cn un − x∗ (ω)k = kan xn (ω) + bn T (ω, xn (ω)) + cn un − (an + bn + cn )x∗ (ω)k ≤ an kxn (ω) − x∗ (ω)k + bn kT (ω, xn (ω)) − x∗ (ω)k +cn kun − x∗ (ω)k = (1 − bn − cn ) kxn (ω) − x∗ (ω)k +bn kT (ω, xn (ω)) − x∗ (ω)k + cn kun − x∗ (ω)k ≤ (1 − bn − cn ) kxn (ω) − x∗ (ω)k +bn a(ω) kxn (ω) − x∗ (ω)k + cn kun − x∗ (ω)k ≤ (1 − bn + a(ω)bn ) kxn (ω) − x∗ (ω)k +cn kun − x∗ (ω)k ≤ [1 − (1 − a(ω))bn ] kxn (ω) − x∗ (ω)k + Mcn . Then by Lemma 2.1 we get lim kxn (ω) − x∗ (ω)k = 0. Hence {xn (ω)}∞ n=0 as defined by random n→∞

Xu-Mann iteration scheme (2) converges strongly to x∗ (ω) almost surely.

4. Stability results Theorem 4.1. Let (E, k.k) be a separable Banach space, T : Ω × E → E be a continuous random operator with a random fixed point x∗ (ω) ∈ E satisfying kx∗ (ω) − T (ω, y)k ≤ a(ω) kx∗ (ω) − y(ω)k


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for some 0 ≤ a(ω) < 1 and for each y(ω) ∈ E. Let {xn (ω)}∞ n=0 be a random Xu-Mann iteration scheme defined by (2) converging strongly to x∗ (ω) almost surely. If

∞

∑ bn = ∞, cn = o(bn) and

n=1

sup kun − x∗ (ω)k = M(> 0). Then the the random Xu-Mann iteration scheme (2) is T -stable n≥0

almost surely. Proof. Let {yn (ω)}∞ n=0 be any sequence of random variables in E and ε n (ω) = kyn+1 (ω) − an yn (ω) − bn T (ω, yn (ω)) − cn un k . Now following same approach as in the proof of Theorem 3.1 and then using (2) and (8) we get kyn+1 (ω) − x∗ (ω)k = kyn+1 (ω) − an yn (ω) − bn T (ω, yn (ω)) − cn un k + kan yn (ω) + bn T (ω, yn (ω)) + cn un − x∗ (ω)k = ε n (ω) + kan yn (ω) + bn T (ω, yn (ω)) + cn un − (an + bn + cn )x∗ (ω)k ≤ ε n (ω) + (1 − bn − cn ) kyn (ω) − x∗ (ω)k +bn kT (ω, yn (ω)) − x∗ (ω)k + cn kun − x∗ (ω)k ≤ ε n (ω) + (1 − bn − cn ) kyn (ω) − x∗ (ω)k +bn a(ω) kyn (ω) − x∗ (ω)k + cn kun − x∗ (ω)k ≤ ε n (ω) + (1 − bn + a(ω)bn ) kyn (ω) − x∗ (ω)k +cn kun − x∗ (ω)k ≤ ε n (ω) + [1 − (1 − a(ω))bn ] kyn (ω) − x∗ (ω)k + Mcn

(6)

Now if ε n (ω) → 0 as n → ∞, then by Lemma 2.1 we get lim kyn (ω) − x∗ (ω)k = 0, which n→∞

shows that the random Xu-Mann iteration scheme (2) is T -stable almost surely. Example 4.1.

Let Ω = R and E = [0, 1]. Define T : Ω × E → E by T (ω, x) = 2x . Then the

inequality (8) satisfies. Take un = 1, an = ∞

an + bn + cn = 1 and

n−1 n ,

bn = n1 , cn = 0 for all n and a0 = b0 = 0 so that

∑ bn = ∞. Then random Xu-Mann iteration scheme

n=1

xn+1 (ω) = an xn (ω) + bn T (ω, xn (ω)) + cn un = 0, for all n

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and hence is T -stable almost surely. Theorem 4.2. Let (E, k.k) be a separable Banach space, T : Ω × E → E be a continuous random operator with a random fixed point x∗ (ω) ∈ E satisfying kx∗ (ω) − T (ω, y)k ≤ a(ω) kx∗ (ω) − y(ω)k for some 0 ≤ a(ω) < 1 and for each y(ω) ∈ E. Let {xn (ω)}∞ n=0 be a random Halpern iteration scheme defined by (4) converging strongly to x∗ (ω) almost surely. Then the random Halpern iteration scheme (4) is T -stable almost surely. Proof. First we proceed in the similar fashion as in the proof of Theorem 3.1 and then let {yn (ω)}∞ n=0 be any sequence of random variables in E. Also let ε n (ω) = kyn+1 (ω) − bn u(ω) − (1 − bn )T (ω, yn (ω))k . Then using (4) and (10) we get kyn+1 (ω) − x∗ (ω)k ≤ kyn+1 (ω) − bn u(ω) − (1 − bn )T (ω, yn (ω))k + kbn u(ω) + (1 − bn )T (ω, yn (ω)) − x∗ (ω)k = ε n (ω) + k(1 − bn )T (ω, yn (ω)) − (1 − bn )x∗ (ω) + bn u(ω) − bn x∗ (ω)k ≤ ε n (ω) + (1 − bn ) kx∗ (ω) − T (ω, yn (ω))k + bn kx∗ (ω) − u(ω)k ≤ ε n (ω) + (1 − bn )a(ω) kx∗ (ω) − yn (ω)k + bn kx∗ (ω) − u(ω)k ≤ ε n (ω) + (1 − bn ) kx∗ (ω) − yn (ω)k + bn kx∗ (ω) − u(ω)k , since a(ω) < 1 = ε n (ω) + (1 − bn ) kx∗ (ω) − yn (ω)k + rn , where rn = bn kx∗ (ω) − u(ω)k Now if ε n (ω) → 0 as n → ∞, then by Lemma 2.1 we get lim kyn (ω) − x∗ (ω)k = 0, which n→∞

shows that the random Halpern iteration scheme (4) is T -stable almost surely.


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Remark 4.1. In both the random iteration algorithms, random Mann iteration scheme is a special case. So the results obtained here are the generalization of random Mann iteration scheme. Conflict of Interests The authors declare that there is no conflict of interests.

R EFERENCES [1] I. Beg, Random fixed points of random operators satisfying semicontractivity conditions, Math. Japan. 46 (1997), 151-155. [2] I. Beg, Approximation of random fixed points in normed spaces, Nonlinear Anal. 51 (2002), 1363-1372. [3] I. Beg, Minimal displacement of random variables under Lipschitz random maps, Topol. Methods Nonlinear Anal. 19 (2002), 391-397. [4] I. Beg, M. Abbas, Iterative procedures for solutions of random operator equations in Banach spaces J. Math. Anal. Appl. 315 (2006), 181-201. [5] I. Beg, M. Saha, A. Ganguly, D. Dey, Random fixed point of Greguˇs mapping and its application to nonlinear stochastic integral equations, Kuwait J. Science 41 (2014) 1-14. [6] I. Beg, N. Shahzad, Random approximations and random fixed point theorems, J. Appl. Math. Stochastic Anal. 7 (1994), 145-150. [7] A.T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976), 641657. [8] A. O. Bosede and B. E. Rhoades, Stability of Picard and Mann iteration for a general class of functions, J. Adv. Math. Stud.3 (2010), 23-25. [9] B. S. Choudhury, Random Mann iteration scheme, Appl. Math. Lett. 16 (2003), 93-96. [10] B. S. Choudhury, Convergence of a random iteration scheme to a random fixed point, J. Appl. Math. Stochastic Anal. 8 (1995), 139-142. [11] B. S. Choudhury, M. Ray, Convergence of an iteration leading to a solution of a random operator equation, J. Appl. Math. Stochastic Anal. 12 (1999), 161-168. ´ c, A generalization of Banach’s contraction principle , Proc. Amer. Math. Soc. 45 (1974), 727-730. [12] Lj. B. Ciri´ [13] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957-961. [14] O. Hanˇs, Reduzierende zuf¨allige transformationen, Czechoslovak Math. J. 7 (1957), 154-158. [15] O. Hanˇs, Random operator equations, Proceedings of 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, University of California Press, California , part I, (1961), 185-202.

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[16] C. O. Imoru, M. O. Olatinwo, On the stability of Picard and Mann iteration processes, Carpathian J. Math. 19 (2003), 155-160. [17] S. Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), 261-273. [18] M.C. Joshi, R.K. Bose, Some topics in non linear functional analysis, Wiley Eastern Ltd. (1984). [19] L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114-125. [20] Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 256 (2001), 1-7. [21] W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (1953), 506-510. [22] A. Mukherjee, Transformation aleatoives separables theorem all point fixed aleatoire, C. R. Acad. Sci. Paris Ser. A - B 263 (1966), 393-395. [23] M.O. Osilike, Stability results for fixed point iteration procedures, J. Nigerian Math. Soc. 14/15 (1995/96), 17-28. [24] N.S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 507-514. [25] B. E. Rhoades, Iteration to obtain random solutions and fixed points of operators in uniformly convex Banach spaces, Soochow J. Math. 27 (2001), 401-404. [26] E. Rothe, Zu¨r Theorie der topologische ordnung und der vektorfelder in Banachschen Raumen, Composito Math. 5 (1937), 177-197. [27] M. Saha, D. Dey, Some Random fixed point theorems for (θ , L)-weak contractions, Hacettepe J. Math. Stat. 41 (2012), 795-812. [28] M. Saha, L. Debnath, Random fixed point of mappings over a Hilbert space with a probability measure, Adv. Stud. Contemp. Math. 1 (2007), 79-84. [29] M. Saha, On some random fixed point of mappings over a Banach space with a probability measure, Proc. Nat. Acad. Sci. India 76 (2006), 219-224. [30] M. Saha and A. Ganguly, Random fixed point theorem on Ciric type contractive mapping and its consequence, Fixed Point Theory Appl. 2012 (2012), Article ID 209. ´ c [31] M. Saha, D. Dey, Investigation on convergence of a random iteration scheme to a random fixed point of Ciri´ quasi-contractive operator, Italian J. Pure. Appl. Math. (accepted and to appear). [32] V. M. Sehgal, C. Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc. 90 (1984), 425-429. ˇ cek, Zuf¨allige Gleichungen, Czechoslovak Math. J. 5 (1955), 462-466. [33] A. Spaˇ


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[34] Y. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equation, J. Math. Anal. Appl. 224 (1998), 91- 101. [35] H.K. Xu, Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990), 103-123. [36] H.K. Xu, I. Beg, Measurability of fixed point sets of multivalued random operators, J. Math. Anal. Appl. 225 (1998), 62-72. [37] S. Zhang, X. Wang, M. Liu, J. Zhu, Almost sure T-stability and convergence for random iterative algorithms, Appl. Math. Mech. -Engl. Ed. 32 (2011), 805-810.