Fixed Point Theorems for Random Lowersemi-continuous Mappings

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Volume 2009, Article ID 584178,7pages doi:10.1155/2009/584178

Research Article

Fixed Point Theorems for Random Lower

Semi-Continuous Mappings

Ra ´ul Fierro,

1, 2

_{Carlos Mart´ınez,}

1

_{and Claudio H. Morales}

3

1_{Instituto de Matemáticas, Pontificia Universidad Católica de Valpara´}_ı_{so, Cerro Barón, Valpara´}_ı_{so, Chile} 2_{Laboratorio de Análisis Estocástico CIMFAV, Universidad de Valpara´}_ı_{so, Casilla 5030, Valpara´}_ı_{so, Chile} 3_{Department of Mathematics, University of Alabama in Huntsville, Huntsville, AL 35899, USA}

Correspondence should be addressed to Claudio H. Morales,[email protected]

Received 31 January 2009; Accepted 1 July 2009

Recommended by Naseer Shahzad

We prove a general principle in Random Fixed Point Theory by introducing a condition named

Pwhich was inspired by some of Petryshyn’s work, and then we apply our result to prove some random fixed points theorems, including generalizations of some Bharucha-Reid theorems.

Copyrightq2009 Ra ´ul Fierro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetX, dbe a metric space andSa closed and nonempty subset ofX. Denote by 2X_resp.,

CXthe family of all nonempty resp., nonempty and closed subsets ofX. A mapping

T : S → 2X_{is said to satisfy}_condition_P_{if, for every closed ball}_B_of_S_{with radius}_r _≥₀

and any sequence {xn} in S for which dxn, B → 0 and dxn, Txn → 0 asn → ∞,

there existsx0 ∈ Bsuch thatx0 ∈ Tx0wheredx, B inf{dx, y : y ∈ B}. IfΩis any

nonempty set, we say that the operatorT : Ω×S → 2X _satisfies_condition_P_{if for each}

ω ∈ Ω, the mappingTω,· : S → 2X _satisfies_condition_P_{. We should observe that this}

latter condition is related to a condition that was originally introduced by Petryshyn1for

single-valued operators, in order to prove existence of fixed points. However, in our case, the condition is used to prove the measurability of a certain operator. On the other hand, in the

year 2001, Shahzadcf.2using an idea of Itohcf.3, see also4, proved that under a

somewhat more restrictive condition, named conditionA, the following result.

Theorem S. LetSbe a nonempty separable complete subset of a metric spaceX andT :Ω×C →

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We shall show that the above result is still valid if the operatorT is only lower

semi-continuous. In addition, the assumption that each value Tx is closed has been relaxed

without an extra assumption. Furthermore we state a new condition which generalizes

conditionAand allow us to generalize several known results, such as, Bharucha-Reid5,

Theorem 7, Dom´ınguez Benavides et al.6, Theorem 3.1and Shahzad2, Theorem 2.1.

2. Preliminaries

LetΩ,Abe a measurable space and letX, dbe a metric space. A mappingF :Ω → 2X_,

is said to be measurable ifF−1G {ω ∈ Ω : Fω∩G /φ}is measurable for each open

subsetGofX. This type of measurability is usually called weaklycf.7, but since this is

the only type of measurability we use in this paper, we omit the term “weakly”. Notice that ifXis separable and if, for each closed subsetCofX, the setF−1Cis measurable, thenFis measurable.

LetCbe a nonempty subset ofXandF:C → 2X_{, then we say that}_F_{is lower}_upper

semi-continuous ifF−1Ais openclosedfor all openclosedsubsetsAofX. We say that

Fis continuous ifFis lower and upper semi-continuous.

A mappingF:Ω×X → Y is called a random operator if, for eachx∈X, the mapping

F·, x:Ω → Yis measurable. Similarly a multivalued mappingF :Ω×X → 2Y_{is also called}

a random operator if, for eachx∈X,F·, x:Ω → 2Y _{is measurable. A measurable mapping}

ξ : Ω → Y is called a measurable selection of the operatorF : Ω → 2Y _if_ξ_ω _∈_F_ω_for

eachω∈Ω. A measurable mappingξ:Ω → Xis called a random fixed point of the random

operatorF :Ω×X → X orF : Ω×X → 2X_{if for every}_ω _∈ _Ω_{, ξ}_ω _F_{ω, ξ}_ω _or

ξω∈Fω, ξω. For the sake of clarity, we mention thatFω, ξω Fω,·ξω.

LetCbe a closed subset of the Banach spaceX, and suppose thatFis a mapping from

Cinto the topological vector spaceY. We say theFisdemiclosedaty0 ∈Y if, for any sequences {xn}inCand{yn}inY withyn ∈ Fxn,{xn}converges weakly tox0and{yn}converges

strongly toy0, then it is the case thatx0 ∈ C andy0 ∈ Fx0. On the other hand, we say

thatF ishemicompactif each sequence {xn}inChas a convergent subsequence, whenever

dxn, Fxn → 0 asn → ∞.

3. Main Results

Theorem 3.1. LetCbe a closed separable subset of a complete metric spaceX, and letT:Ω×C → 2X

be measurable inωand enjoyconditionP. Suppose, for eachω∈Ω, thathω, x dx, Tω, x is upper semi-continuous and the set

Fω:{x∈C:x∈Tω, x}/φ. 3.1

ThenT has a random fixed point.

Proof. LetZ{zn}be a countable dense subset ofC. DefineF :Ω → 2CbyFω {x∈C:

x∈Tω, x}. Firstly, we show thatFis measurable. To this end, letB0be an arbitrary closed

ball ofC, and set

LB0 ∞

k1

z∈Zk

ω∈Ω:dz, Tω, z< 1 k

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whereZk Bk∩ Zand Bk {x ∈ C : dx, B0 < 1/k}. We claim thatF−1B0 LB0.

To see this, letω ∈ F−1B0. Then there existsx ∈ B0 such thatx ∈ Tω, x. Sincehω,·is

upper semi-continuous, for eachk∈N, there existsznk ∈Zksuch thatdznk, Tω, znk<1/k.

Thereforeω ∈LB0. On the other hand, ifω ∈LB0, then there exists a subsequence{znk}

of{zn}such that

dznk, B0<

1

k, dznk, Tω, znk<

1

k 3.3

for all k ∈ N. This means that dznk, B0 → 0 and dznk, Tω, znk → 0 as n → ∞.

Consequently, by conditionP, there existsx0 ∈ B0 such that x0 ∈ Tω, x0. Henceω ∈

F−1B0. Then we conclude thatF−1B0 LB0, and thusF−1B0is measurable. To complete

the proof, letGbe an arbitrary open subset ofC. Then by the separability ofC,

G

∞

n1

Bn where eachBn is a closed ball ofC. 3.4

SinceF−1G ∞_n₁F−1Bn, we conclude thatFis measurable. Additionally, we show that

Fωis closed for eachω ∈ Ω. To see this, letxn ∈ Fωsuch thatxn → x ∈ C. Then, let

B0{x}be a degenerated ball centered atxand radiusr 0, and sincedxn, Tω, xn 0,

conditionPimplies that x ∈ Tω, x. Hencex ∈ Fω and thus by the Kuratowski and

Ryll-Nardzewski Theorem8,F has a measurable selectionξ : Ω → C such thatξω ∈

Tω, ξωfor eachω∈Ω.

As a consequence ofTheorem 3.1, we derive a new result for a lower semi-continuous

random operator.

Theorem 3.2. LetCbe a closed separable subset of a complete metric spaceX, and letT:Ω×C → 2X

be a lower semi-continuous random operator, which enjoysconditionP. Suppose, for eachω∈Ω, that the set

Fω:{x∈C:x∈Tω, x}/φ. 3.5

Proof. Due toTheorem 3.1, it is enough to show thathω,· is upper semi-continuous. To

see this, we will prove that A {x ∈ C : dx, Tω, x < α}is open inC forα > 0. Let

a∈Aand select α−da, Tω, a. Then there existsy ∈Tω, aso thatda, y< /3

da, Tω, a. SinceTω,·is lower semi-continuous, there exists a positive numberr < /3 such thatTω, u∩By;/3_/∅ for all u ∈ Ba;r. Hence, we may choosezu ∈ Tω, u∩

By;/3for which,

du, zu≤du, a d

a, ydy, zu

< α, 3.6

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We observe that if the mappinghx dx, Txis upper semi-continuous, then not

necessarily the mappingTis lower semi-continuous. Consider the following example.

LetT :R → 2Rbe defined by

Tx

⎧ ⎨ ⎩

1, x /0

2,3, x0.

3.7

Thenhx |x−1|for_{x /}0 whileh0 2, which is upper semi-continuous. On the other

hand,T is not lower semi-continuous.

Now, we derive several consequences ofTheorem 3.2. We first obtain an extension of

one of the main results of6.

Theorem 3.3. LetCbe a weakly compact separable subset of a Banach spaceX, and letT :Ω×C →

2X_{be a lower semi-continuous random operator. Suppose, for each}_ω_∈_Ω_{, that}_I₋_T_ω,_·_{is demiclosed}

at0and the set

Fω:{x∈C:x∈Tω, x}/φ. 3.8

Proof. In order to applyTheorem 3.2, we just need to prove that T enjoysconditionP. To

this end, letωbe fixed inΩ. Suppose thatB0 is a closed ball ofCwith radiusr ≥ 0 where

{xn}is a sequence inCsuch thatdxn, B0 → 0 anddxn, Tω, xn → 0 asn → ∞. SinceC

is separable, the weak topology onCis metrizable, and thus there exists a weakly convergent

subsequence{xnk}of{xn}, so thatxnk → xweakly, whiledxnk, Tω, xnk → 0 ask → ∞.

Consequently, for eachk∈N, there existszk∈Tω, xnksuch that

xnk−zk −→0 ask−→ ∞. 3.9

Hence, the demiclosedness ofI −Tω,·implies thatx ∈ Tω, x, and thusTω,· enjoys

conditionP.

Before we give an extension of the main result of4, we observe thatconditionPis

basically applied to those closed balls directly used to prove the measurability of the mapping

F, as will be seen in the proof of the next result.

Theorem 3.4. LetCbe a closed separable subset of a complete metric spaceX, and letT :Ω×C →

CXbe a continuous hemicompact random operator. If, for eachω∈Ω, the set

Fω:{x∈C:x∈Tω, x}/φ, 3.10

thenT has a random fixed point.

Proof. Due toTheorem 3.2, it would be enough to show thatTω,·enjoysconditionPfor

everyω ∈ Ω. To see this, letB0 be a closed ball ofC, and let{xn}be a sequence inCsuch

that dxn, B0 → 0 and dxn, Tω, xn → 0 as n → ∞. Then by the hemicompactness

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dxnk, Tω, xnk → 0 ask → ∞. This means that, for eachk∈N, there existszk∈Tω, xnk

such that

dxnk, zk−→0 as k−→ ∞. 3.11

Consequently,zk → x. On the other hand, sinceT is upper semi-continuous atx, for every

>0 there existk0∈Nsuch that

Tω, xnk⊂BTω, x; for allk≥k0. 3.12

Hence,x∈BTω, x;. Sinceis arbitrary andTω, xis closed, we derive thatx∈Tω, x, and thusTsatisfiesconditionP.

Corollary 3.5. LetC be a locally compact separable subset of a complete metric space X, and let

T :Ω×C → CXbe a continuous random operator. Suppose, for eachω∈Ω, that the set

Fω:{x∈C:x∈Tω, x}/φ. 3.13

Proof. LetGbe an arbitrary open subset ofC, and letx∈G. SinceCis locally compact, there

exists a compact ball B centered at xsuch that B ⊂ G. Now, we prove thatconditionP

holds with respect toB. To see this, letω ∈ Ω, and let{xn} be a sequence inX such that

dxn, B → 0 anddxn, Tω, xn → 0 asn → ∞. Then there exists a sequence{yn}inBso

thatdxn, yn → 0 as n → ∞. SinceBis compact, there exists a convergent subsequence

{ynk} of {yn} such that ynk → x, and consequently xnk → x with x ∈ B as well as

dxnk, Tω, xnk → 0 ask → ∞. SinceTis upper semi-continuous, we derive, as in the proof

ofTheorem 3.4, thatx∈Tx. In addition, sinceT is lower semi-continuous, we may follow

the proof ofTheorem 3.1, to conclude thatF−1Bis measurable. Hence, the separability ofC

implies that we can select countably many compact ballsBicentered at corresponding points

xi∈Gsuch that

F−1G

i∈N

F−1Bi. 3.14

Therefore,Fis measurable.

Next, we get a stochastic version of Schauder’s Theorem, which is also an extension of

a Theorem of Bharucha-Reidsee5, Theorem 10. We also observe that our proof is much

easier and quite short.

Corollary 3.6. LetCbe a compact and convex subset of a Fr´echet spaceX, and letT :Ω×C → C

be a continuous random operator. ThenThas a random fixed point.

Proof. As we know, every Fr´echet space is a complete metric space, and sinceCis compact,

Citself is a complete separable metric space. In addition, for eachω ∈Ω, there existsx∈C

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SinceCis compact, any sequence inCcontains a convergent subsequence, which means that

T is trivially a hemicompact operator. Consequently, byTheorem 3.4,T has a random fixed

point.

Before obtaining an extension of Bharucha-Reid 5, Theorem 3.7, we define a

contraction mapping for metric spaces. LetX be a metric space, and letΩbe a measurable

space. A random operatorT :Ω×X → X is said to be arandom contractionif there exists a

mappingk:Ω → 0,1such that

dTω, x, Tω, y ≤kωdx, y for allx, y∈X. 3.15

Theorem 3.7. LetXbe a complete separable metric space, and letT :Ω×X → X be a continuous

random operator such thatT2 _{is a contraction with constant} _k_ω_{for each}_ω _∈ _Ω_{. Then}_T _{has a}

unique random fixed point.

Proof. For eachω∈Ω, the mappingT2_{has a unique fixed point,}_ξ_ω_{, which is also the unique}

fixed point ofT. It remains to show that the mappingξ:Ω → Xdefined byTω, ξω ξω

is measurable. To see this, letf0:Ω → Xbe an arbitrary measurable function. Then, we claim

thatTω, f0ωis measurable. To this end, letZ {zn}be a countable dense set ofX. Let

ω∈Ωand letk∈N. Define

hk:Ω−→X byhkω zm, 3.16

wheremis the smallest natural number for whichdzm, f0ω<1/k. Sincef0is measurable,

so are the sets Em {ω ∈ Ω : dzm, f0ω < 1/k}, which, as a matter of fact, conform

a disjoint covering of Ω. Consequently, {hk} is a sequence of measurable functions that

converges pointwise to f0. On the other hand, the range of eachhk is a subset of Z, and

hence constant on each setEm. Since the mappingTis measurable inω, then, for eachk∈N,

Tω, hkωis also measurable. Therefore the continuity ofT on the second variable implies

that

Tω, hkω−→T

ω, f0ω

ask−→ ∞, 3.17

for eachω∈Ω. HenceTω, f0ωis measurable. Define the sequence

fnω T

ω, fn−1ω, n∈N. 3.18

Then{fn}is a sequence of measurable functions. Sincefnω Tnω, f0ω, the fact thatT2

is a contraction implies thatfnω → ξω. Therefore, the mappingξis measurable, which

completes the proof.

As a direct consequence ofTheorem 3.7, we derive the extension mentioned earlier

where the spaceXis more general, and the randomness on the mappingkhas been removed.

Corollary 3.8. LetX be a complete separable metric space, and letT : Ω×X → X be a random

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Next, one can derive a corollary of the proof ofTheorem 3.7, which is a theorem of Hans9.

Corollary 3.9. LetXbe a complete separable metric space, and letT :Ω×X → Xbe a continuous

random operator. Suppose, for eachω∈Ω, that there existsn∈Nsuch thatTn_{is a contraction with}

constantkω. ThenThas a unique random fixed point.

Proof. As in the proof of the theorem, the mappingThas a unique fixed point for eachω∈Ω. The rest of the proof follows the proof of the theorem with the appropriate changes of the

second power ofT by thenth power ofT.

Notice thatTheorem 3.7holds for single-valued operators. The following question is

formulated for multivalued operators taking closed and bounded values inX.

Open Question

Suppose that X is a complete separable metric space, and let T : Ω×X → CBX be a

continuous random operator such thatT2_{is a contraction with constant}_k_ω_{for each}_ω_∈_Ω_.

Then doesThave a unique random fixed point?

Acknowledgments

This work was partially supported by Direcci ón de Investigaci ón e Innovaci ón de la Pontificia

Universidad Cat ´olica de Valpara´ıso under grant 124.719/2009. In addition, the first author

was supported by Laboratory of Stochastic Analysis PBCT-ACT 13.

References

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3 S. Itoh, “Random fixed-point theorems with an application to random diﬀerential equations in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 261–273, 1979.

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