Random fixed point theorems for asymptotic 1 set contraction operators

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PII. S0161171202006105 http://ijmms.hindawi.com © Hindawi Publishing Corp.

RANDOM FIXED POINT THEOREMS FOR ASYMPTOTIC

1 _{-SET CONTRACTION OPERATORS}

P. VIJAYARAJU

Received 3 November 2000

Random fixed point theorems for condensing 1-set contraction selfmaps are known. But no random fixed point theorem for more general asymptotic 1-set contraction selfmaps is yet available. The purpose of this paper is to prove random fixed point theorems for such maps.

2000 Mathematics Subject Classification: 47H10, 47H09, 60H25.

1. Introduction. Random fixed point theorems are stochastic versions of (classi-cal or deterministic) fixed point theorems and are required for the theory of random equations. The study of random fixed point theorems for contraction maps in sepa-rable complete metric spaces was initiated by Spacek [8] and Hans [3]. In sepasepa-rable Banach spaces, Itoh [4] proved a random fixed point theorem for condensing self-maps. This Itoh’s result was extended by Xu [11] to condensing non-selfmapT, by assuming an additional condition thatT satisfies either weakly inward (see [11]) or the Leray-Schauder condition (see [11]). Lin [5] proved a random fixed point theorem for 1-set contraction selfmapT, by assuming thatI−T is demiclosed at zero, where

Idenotes the identity map.

Rao [6] obtained a probabilitic version of Krasnoselskii’s theorem which states that a sumT+Sof a contraction random operatorT :Ω×K→X, and a compact random operator S:Ω×K→X with T (ω,x)+S(ω,y)∈Kfor all x,y∈K,ω∈Ω, has a random fixed point, whereKis a closed convex subset of a separable Banach spaceX

and(Ω,Σ)is a measurable space. Some results on random fixed points of Krasnoselskii type were obtained by Bharucha-Reid [1,2].

Itoh [4] extended Rao’s result to a sumT+S of a nonexpansive random operator

T :Ω×K→X and a completely continuous random operator S :Ω×K→X with

T (ω,x)+S(ω,x)∈Kfor eachx∈K,ω∈Ω, whereKis a weakly compact convex sub-set of a separable uniformly convex Banach spaceX. This Itoh’s result was extended by Lin [5] to a sum T+S of a locally almost nonexpansive (LANE) (cf. [5]) random operatorT :Ω×K→Kand a completely continuous random operatorS:Ω×K→K, whereKis a nonempty closed convex bounded subset of a separable uniformly convex Banach spaceX.

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random operators includes the classes of condensing, nonexpansive, and LANE ran-dom operators. In this paper, we prove ranran-dom fixed point theorems of Krasnoselskii type for sum of an asymptotic 1-set contraction and a compact (completely contin-uous) map from which we will be able to deduce random fixed point theorems for asymptotic 1-set contraction in the setting of separable Banach spaces, and we also prove random fixed point theorems of Krasnoselskii type for sum of a 1-set contrac-tion and a compact (completely continuous) map.

2. Preliminaries. Throughout this paper,(Ω,Σ)denotes a measurable space with Σ, a sigma algebra of subsets ofΩ. Let 2x be the family of all subsets of a Banach space X. A mapping F :Ω→2x is called measurable if for each open subsetA of

X, F−1_(A)_∈_Σ_{. A mapping}_φ_:_Ω_→_X _{is called a measurable selector of a} measur-able mappingF:Ω→2x, ifφis measurable and for any ω∈Ω, φ(ω)∈F(ω). Let

K be a nonempty subset of X. A map T :Ω×K →X is called a random operator if for each fixedx∈K, the map T (·,x):Ω→X is measurable. A measurable map

φ:Ω→Kis a random fixed point of a random operatorT ifT (ω,φ(ω))=φ(ω)for eachω∈Ω. A random operatorT:Ω×K→Kis called continuous (k-set contraction, condensing, 1-set contraction, asymptotic 1-set contraction, asymptotically regular, uniformly asymptotically regular (UAR), nonexpansive, locally almost nonexpansive, asymptotically nonexpansive, compact, completely continuous, weakly inward, demi-closed, etc.) if the mapTω:K→K, defined byTω(x)=T (ω,x)for allx∈K, is so, for

eachω∈Ω.

LetAbe a nonempty bounded subset ofX. The Kuratowski’s measure of noncom-pactness ofAis defined as the numberα(A)=inf{ε >0 :Acan be covered by a finite number of sets of diameter≤ε}. LetKbe a nonempty subset ofXand letTbe a con-tinuous mapping ofKintoK. If there existsk, 0≤k <1, such that for each nonempty bounded subsetAofK, we haveα(T (A))≤kα(A), thenTis called ak-set contraction. Ifk=1, thenT is called a 1-set contraction. If for every nonempty bounded subset

AofKwithα(A) >0, we haveα(T (A)) < α(A), thenT is called a condensing map. It is clear that ak-set contraction map is a condensing map and a condensing map is a 1-set contraction map. If there is a sequence{kn}of real numbers withkn≥1,

kn≥kn+1, andkn→1 asn→ ∞such thatα(Tn(A))≤knα(A)forn≥1 and for every

nonempty bounded subsetAofK, then T is called an asymptotic 1-set contraction (see [9]). As shown in [9, Theorem 2.2 and Remark 2.1] the sum of an asymptotic 1-set contraction and a compact map is an asymptotic 1-set contraction.

IfT maps weakly convergent sequences into strongly convergent sequences, then

T is called a completely continuous or strongly continuous map. IfT is continuous andT (K)is precompact, thenT is called a compact map. If for any sequence{xn}

inK, the conditions{xn}converge weakly toxinKandT (xn)converges toyinX,

which imply thatT (x)=y, thenT is said to be demiclosed aty.

A map T :K→X is said to be nonexpansive ifT (x)−T (y) ≤ x−yfor all

x,y∈K. A selfmapT of Kis said to be asymptotically nonexpansive if there is a sequence{kn}of real numbers withkn≥1,kn≥kn+1, andkn→1 asn→ ∞such that

Tn_(x)₋_Tn_(y)_≤_k_n_x₋_y_{for all}_x,y_∈_K_{and for each integer}_n_≥_{1. The class}

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RANDOM FIXED POINT THEOREMS FOR ASYMPTOTIC... 409

maps. A selfmapT ofK is called an asymptotically regular map if for any x∈K, Tn_(x)₋_Tn+1_(x)_→_{0 as}_n_{→ ∞}_{. If}_T_and_S_map_K_into_X_{, then}_T_{is called a uniformly} asymptotically regular (UAR) map with respect toS[10], if for eachε >0 there exists an integerNsuch thatTn_(x)₋_Tn−1_(x)₊_S(x)_{< ε}_{for all}_n_≥_N_{and for all}_x_∈_K_{. If}

TandSare random operators ofΩ×KintoX, then we say thatT is UAR with respect toSifTωis UAR with respect toSωfor eachω∈Ωin the above classical definition.

The mapTis UAR ifTis UAR with respect to the zero operator. It is clear that uniform asymptotic regularity is stronger than asymptotic regularity.

3. Main results. We will need the following theorem to prove random fixed point theorems of Krasnoselskii type.

Theorem _3.1_{(see [5])}. _LetK _{be a nonempty weakly compact convex subset of a}

separable Banach spaceX. IfT :Ω×K→Kis a continuous1-set contraction random operator such thatI−Tω is demiclosed for eachω∈Ω, thenT has a random ﬁxed

point.

Theorem_3.2. _LetK_{be a nonempty weakly compact convex subset of a separable}

Banach spaceX,T:Ω×K→Kan asymptotic1-set contraction random operator. Let S:Ω×K→Kbe a compact random operator. Assume further thatTis UAR with respect toS such thatI−(T+S)is demiclosed at zero,Tn

ω(K)+Sω(K)⊆Kforn≥1and for

eachω∈Ω. ThenT+Shas a random ﬁxed point.

Proof. Letvbe a fixed element ofK. For eachn, define a mapT_n:Ω×K→Kby

Tn(ω,x)=anv+1−anTωn(x)+Sω(x) ∀x∈K, ω∈Ω, (3.1)

where{an}is a sequence of real numbers in(0,1)with 1−1/kn≤an≤1−1/kn−1 and{kn}is as in the definition of asymptotic 1-set contraction.

LetMbe a bounded subset ofK. Then

αTn(ω,M)≤anα{v}+1−anαTωn(M)+Sω(M)

≤1−anknα(M) sinceT is asymptotic 1-set contraction,Sis compact

≤α(M).

(3.2)

Therefore eachTn is a 1-set contraction random operator. By Theorem 3.1, Tn has

a random fixed pointφn, that is, there is a measurable mapφn:Ω→K such that

Tn(ω,φn(ω))=φn(ω)for eachω∈Ω.

Let be the family of all nonempty weakly compact subsets ofK.

Let w−cl(A)denote the weak closure ofA. For eachn, define a mapFn:Ω→ by

Fn(ω)=w−cl{φi(ω):i≥n}.

Define a mapF:Ω→ byF(ω)=∞n=1Fn(ω).

SinceKis weakly compact andXis separable, it follows that the weak topology on

Kis a metric topology. SinceKandFn(ω)are weakly compact,Fis measurable with

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For eachx∗_∈_X∗_{, dual of}_X_,_x∗_(φ(_·₎₎_{is measurable as a numerically-valued} func-tion inΩ. SinceXis separable,φis measurable [1, pages 14–16].

Now we prove thatφis a random fixed point ofT+S.

Fixing anyω∈Ω, sinceφ(ω)∈F(ω), there is a subsequence{φm(x)}of{φn(x)}

that converges weakly toφ(ω).

Sinceφm(ω)=Tm(ω,φm(ω))=amv+(1−am)[Tωm(φm(ω))+Sω(φm(ω))], it

follows that

φm(ω)−Tωmφm(ω)−Sωφm(ω)

=amv−Tωm

φm(ω)−Sωφm(ω)→0 asm → ∞.

(3.3)

SinceTωis UAR with respect toSωfor eachω∈Ω,

Tm

ωφm(ω)−Tωm−1φm(ω)+Sωφm(ω) →0 asm→ ∞. (3.4)

Thereforeφm(ω)−Tωm−1(φm(ω))→0 asm→ ∞.

Now,

_φ_m_(ω)₋_T_ω

φm(ω)−Sωφm(ω)

=φm(ω)−Tωm

φm(ω)−Sωφm(ω)+Tωm

φm(ω)

+Sωφm(ω)−Tωφm(ω)−Sωφm(ω)

≤φm(ω)−Tωmφm(ω)−Sωφm(ω)+Tωmφm(ω)−Tωφm(ω)

≤φm(ω)−Tωm

φm(w)−Sωφm(ω)+k1T_ωm−1

φm(ω)−φm(w)

→0 asm → ∞.

(3.5)

SinceI−(Tω+Sω)is demiclosed at zero for eachω∈Ω,[I−(Tω+Sω)](φ(ω))=0

and thereforeφ(ω)=T (ω,φ(ω))+S(ω,φ(ω)).

Corollary3.3. LetKbe a nonempty closed convex bounded subset of a separable

reﬂexive Banach spaceX,T:Ω×K→Kan asymptotic1-set contraction random oper-ator. LetS:Ω×K→Kbe a completely continuous random operator. Assume further thatT is UAR with respect toS such thatI−(T+S) is demiclosed at zero and that Tn

ω(K)+Sω(K)⊆K for n≥1and for each ω∈Ω. ThenT+S has a random ﬁxed

point.

Proof. SinceKis a closed convex bounded subset of a reflexive Banach spaceX,

Kis weakly compact and hence every sequence{xn}inK has a weakly convergent

subsequence{xm}. SinceS is completely continuous,S(ω,xm)→S(ω,x)asm→ ∞.

HenceSis compact. All the conditions ofTheorem 3.2are satisfied. The result follows fromTheorem 3.2.

Remark_3.4. _IfS=_{0 in}_{Theorem 3.2}_and_{Corollary 3.3, then random fixed point}

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RANDOM FIXED POINT THEOREMS FOR ASYMPTOTIC... 411

Theorem3.5. LetKandXbe as inCorollary 3.3,T:Ω×K→Kan asymptotically

nonexpansive, UAR random operator such thatI−Tis demiclosed at zero. ThenT has a random ﬁxed point.

Corollary3.6. LetKbe a nonempty closed convex bounded subset of a separable

uniformly convex Banach spaceX,T:Ω×K→Kan asymptotically nonexpansive, UAR random operator. ThenT has a random ﬁxed point.

Proof. SinceXis uniformly convex andT_ω:K→Kis asymptotically nonexpan-sive, it follows as in Xu [12] thatI−Tω is demiclosed at zero for eachω∈Ω. The

remaining part of proof follows fromTheorem 3.5.

InTheorem 3.2andCorollary 3.3ifT is a 1-set contraction random operator, then uniform asymptotic regularity ofT with respect toS is not required.

Theorem 3.7. LetK be a weakly compact convex subset of a separable Banach

spaceX. LetT :Ω×K→Xbe a1-set contraction random operator andS:Ω×K→X a compact random operator. Suppose that for anyω∈Ω,Tω(x)+Sω(y)∈Kfor all

x,y∈KandI−(T+S)is demiclosed at zero. ThenT+Shas a random ﬁxed point.

Proof. _{Motivated by Itoh [4, Theorem 3.5], choose an element} v _inK_{and a}

se-quence{an}of real numbers such that 0< an<1 andan→0 asn→ ∞. For eachn,

define a mapTn:Ω×K→Kby

Tn(ω,x)=anv+1−anTω(x)+Sω(x) ∀x∈K, ω∈Ω, (3.6)

thenTnis a(1−an)-set contraction random operator. The remaining part of proof of

this theorem is almost identical to [4, Proof of Theorem 3.5].

Corollary_3.8. _LetK_{be a nonempty closed convex bounded subset of a separable}

uniformly convex Banach spaceX. LetT be as inTheorem 3.7such thatI−T is demi-closed at zero. LetS:Ω×K→Xbe a completely continuous random operator. Suppose that for anyω∈Ω,Tω(x)+Sω(y)∈Kfor allx,y∈K. ThenT+Shas a random ﬁxed

point.

The proof of this corollary follows from the proof ofTheorem 3.7.

Acknowledgment. _{The author would like to thank Professor T. R. Dhanapalan}

for constant encouragement and helpful discussions in the preparation of this paper.

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