Modified Random Mann Iterations Systems

ZHANG Xiao-cai, LV Ping, DU Hong-xia

(1. College of Science, Henan University Technology, Zhenzhou 450001, China;2. College of Science, Hangzhou Normal University, Hangzhou 310036, China;3. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China)

1 Introduction

Probabilistic functional analysis has come out as one of the momentous mathematical disciplines in view of its requirements in dealing with probabilistic models in applied problems. 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. An interesting aspect of nonlinear analysis is to randomize deterministic fixed point theorems of nonlinear mappings. In linear spaces there are two general iterations which have been successfully applied to fixed point problems of operators and also to obtaining solutions of operator equations.

These are Ishikawa iteration scheme[1]and Mann iteration scheme[2]. Random fixed point theory has received much attention since the publication of the survey article by Bharuch-Reid[3]in 1976, in which the stochastic version of some well-known fixed point theorems were proved. The machinery of random fixed point theory provides a convenient way of modeling many problems arising from economic theory, see for example [4] and references mentioned therein. Random methods have revolutionized the financial markets. We note some important recent works on random fixed points in[5-8]. In an attempt to construct iterations for finding fixed points of random operators defined on linear spaces, random Ishikawa iteration scheme was introduced in [9]. This iteration and also some other random iterations based on the same ideas have been applied for finding solutions of random operator equations and fixed points of random operators[9-12].

The purpose of the paper is to introduce a random fixed-point iteration that is a random Mann iteration scheme with randomφ-hemicontractive operator and to show the iteration will under certain conditions converge to a random fixed point of a random operator defined in the context of a separable Hilbert space.

2 Preliminary

Throughout this paper, (Ω,F) denotes a measurable space andHstands for a separable Hilbert space.Cis a nonempty subset ofH.

A functionf:Ω→Cis said to be measurable with respect to F, iff-1(B∩C)∈F for every Borel subsetBofH. Denote G the class of measurable functions with respect to F.

A functionT:Ω×C→Cis said to be a random operator, ifT(·,x) is measurable for everyx∈C.

A measurable functiong:Ω→Cis said to be a random fixed point of the random operatorT, ifT(ω,g(ω))=g(ω) for allω∈Ω.

MANN ITERATION SCHEME. (See[2]) LetVbe a linear space. IfT:V→Vis a mapping andx0∈Vis any element ofV, then {xn} is defined iteratively as following:

xn+1=(1-cn)xn+cnTxn,n=0,1,2,…,

where {cn} is a real sequence satisfying the following conditions,

Definition1[RANDOM MANN ITERATION SCHEME. (See [13])] LetT:Ω×C→CwhereCis a nonempty convex subset of a separable Hilbert spaceHbe a random operator. Then the sequence of functions {gn} is defined in the following,g0: Ω→Cbe an arbitrary measurable function,

forω∈Ω,n≥0,

gn+1(ω)=(1-cn)gn(ω)+cnT(ω,gn(ω)),

where 0

SinceCis convex it follows from the above construction thatgnis a mapping from Ω toCfor alln.

LetF(T):={g∈G:T(ω,g(ω))=g(ω), forallω∈Ω} be the class of the random fixed points of the random operatorT.

Our main object in this work, is to consider a modified random Mann iteration system described as following:

LetKbe a nonempty subset of a separable Hilbert spaceH, andT: Ω×K→Kbe a random operator. Define the new iteration system {gn}:

g0:Ω→Kbe an arbitraty mesuarable function,

(1)

and for anyω∈Ω,n≥1,

gn(ω)=cngn-1(ω)+(1-cn)T(ω,gn(ω)),

(2)

where the sequence {cn} taking values in (0,1) and satisfying some conditions.

In the following we define theφ-hemicontractive mapping.

Definition2[φ-HEMICONTRACTIVE] LetCbe a nonempty subset ofH. We call a mappingT:C→Cφ-hemicontractive ifF(T)≠? and there exists a functionφsuch that

‖T(x)-g‖2≤‖x-g‖2+φ(‖x-T(x)‖),?x∈C,g∈F(T),

(3)

where the functionφ:R+→R+satisfies the following conditions: for anyγ∈(0,1) and for anyα∈[γ,1-γ], there existsγα>0 such thatα2x2-φ(αx)≥γαx2, ?x≥0.

Remarks1If we takeφ(α2x)=α2xthen theφ-hemicontractive mapping induces the classical hemicontractive mapping.

Before the proof of Theorem 1, we firstly need some lemmas.

Lemma1[See[14]] Suppose that {ρn},{σn} are two sequences of nonnegative numbers such that for some real numberN0≥1,

ρn+1≤ρn+σn,?n≥N0.

Then, 1) If ∑σn<∞, limn→∞ρnexists;

2) If ∑σn<∞ and {ρn} has a subsequence converging to zero, limn→∞ρ=0.

Lemma2[See[1]] For allx,y∈Handλ∈[0,1], the following identity holds:

‖(1-λ)x+λy‖2=(1-λ)2‖x‖2+λ‖y‖2-λ(1-λ)‖x-y‖2.

3 Main results

After the preparations of the above section, we now in aposition to state our main results.

Theorem1LetKbe a compact convex subset of a separable Hilbert spaceH,T:Ω×K→Kbe a randomφ-hemicontractive mapping (i.e., for allω∈Ω,T(ω,·):K→Kisφ-hemicontractive). Assume {cn} a real sequence in [0,1] satisfying {cn}?[γ,1-γ] for someγ∈(0,1) and infn≥1γcn>0. For arbitrary measurable functiong0:Ω→K, the iteration system {gn} defined by (2) converges to a random fixed pointgofT.

ProofLetg∈F(T) be a random fixed point ofT. Applying the fact theTis randomφ-hemicontractive operator, we have for allω∈Ω,

‖T(ω,gn(ω))-g(ω)‖2≤‖gn(ω)-g(ω)‖2+φ(‖gn(ω)-T(ω,gn(ω))‖),

(4)

Hence, by lemma 2 and the definition of the iteration system (2),

‖gn(ω)-g(ω)‖2=‖cn(gn-1(ω)-g(ω))+(1-cn)(T(ω,gn(ω))-g(ω))‖2=

cn‖gn-1(ω)-g(ω)‖2+(1-cn)‖T(ω,gn(ω))-g(ω)‖2-

cn(1-cn)‖gn-1(ω)-T(ω,gn(ω))‖2,

(5)

and

‖gn(ω)-T(ω,gn(ω))‖2=‖cngn-1(ω)+(1-cn)T(ω,gn(ω))-T(ω,gn(ω))‖2=

(6)

Substituting (4) in (5), we get

‖gn(ω)-g(ω)‖2≤cn‖gn-1(ω)-g(ω)‖2+(1-cn)‖gn(ω)-g(ω)‖2+
(1-cn)(φ(‖gn(ω)-T(ω,gn(ω))‖)-cn‖gn-1(ω)-T(ω,gn(ω))‖2).

By (6) and the properties of functionφimply,

(1-cn)γcn‖gn-1(ω)-T(ω,gn(ω))‖2≤‖gn-1(ω)-g(ω)‖2-‖gn(ω)-g(ω)‖2.

Since {cn}?[γ,1-γ] for someγ∈(0,1), we haveγ≤1-cn.

Hence, for allω∈Ω,

for any elements ofF(T). Therefore,

‖g0(ω)-g(ω)‖2

This shows

By (6), we obtain

By compactness ofK, there is a subsequence {gnj} of {gn} which converges to a random fixed point ofT, denoteh. Therefore,

‖gn(ω)-h(ω)‖2≤‖gn-1(ω)-h(ω)‖2-δ2‖gn-1(ω)-T(ω,gn(ω))‖2.

Corollary1LetH,K,Tand {cn} be as in Theorem 1. LetPK:H→Kbe the projection operator ofHontoK. Then the sequence {gn(ω)} defined iteratively by,

gn(ω)=PK(cngn-1(ω)+(1-cn)T(ω,gn(ω))) for anyω∈Ω,n≥1,

converges to a random fixed point ofT.

ProofThe operatorPKis nonexpansive (see, e.g.,[15]).Kis a Chebyshev subset ofH, soPKis a single-valued map. Hence, for allω∈Ω, we have the following estimate:

‖gn(ω)-g(ω)‖2=‖PK(cngn-1(ω)+(1-cn)T(ω,gn(ω)))-PKg(ω)‖2≤

‖cn(gn-1(ω)-g(ω))+(1-cn)(T(ω,gn(ω))-g(ω))‖2=

cn‖gn-1(ω)-g(ω)‖2+(1-cn)‖T(ω,gn(ω))-g(ω)‖2-

cn(1-cn)‖T(ω,gn(ω))-gn(ω)‖2.

The setK∪T(K) is compact and so the sequence {‖gn(ω)-T(ω,gn(ω))‖} is bounded for allω∈Ω. The rest of the argument follows exactly as in the proof of Theorem 1 and the proof is completed.

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