Fixed Point Theorems for Random Lower Semi-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

1Instituto de Matemáticas, Pontificia Universidad Católica de Valpara´ıso, Cerro Barón, Valpara´ıso, Chile

2Laboratorio de An´alisis Estoc´astico CIMFAV, Universidad de Valpara´ıso, Casilla 5030, Valpara´ıso, Chile

3Department 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 2^Xresp., CXthe family of all nonempty resp., nonempty and closed subsets ofX. A mapping T : S → 2^Xis said to satisfyconditionPif, for every closed ballBofSwith radiusr ≥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 → 2^X satisfiesconditionPif for each ω ∈ Ω, the mappingTω,· : S → 2^X satisfiesconditionP. 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 → CXa continuous random operator satisfying condition (A). ThenT has a deterministic fixed point if and only ifThas a random fixed point.

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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 → 2^X, 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 → 2^X, then we say thatFis lowerupper 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 → 2^Yis also called a random operator if, for eachx∈X,F·, x:Ω → 2^Y is measurable. A measurable mapping ξ : Ω → Y is called a measurable selection of the operatorF : Ω → 2^Y ifξω ∈Fωfor eachω∈Ω. A measurable mappingξ:Ω → Xis called a random fixed point of the random operatorF :Ω×X → X orF : Ω×X → 2^Xif for everyω ∈ Ω, ξω Fω, ξω 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 theFis demiclosed aty0 ∈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 is hemicompact if 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 → 2^X 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 → 2^CbyFω {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

, 3.2

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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 → 2^X 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 to Theorem 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, y

d y, zu

< α, 3.6

and consequently,du, Tω, u< α. Therefore,Ais open, and proof is complete.

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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 → 2^Rbe defined by

Tx

⎧⎨

⎩

1, x /0

2,3, x0. 3.7

Thenhx |x−1|forx /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 → 2^Xbe a lower semi-continuous random operator. Suppose, for eachω∈Ω, thatI−Tω,·is demiclosed at 0 and 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 of T, there exists a convergent subsequence{xnk} of {xn}, so that xnk → x ∈ B0. Hence

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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 such thatTω, x x. This means that the setFω, defined inTheorem 3.1, is nonempty.

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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 a random contraction if there exists a mappingk:Ω → 0,1such that

d

Tω, x, T ω, y

≤kωd x, 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 kωfor eachω ∈ Ω. ThenT has a unique random fixed point.

Proof. For eachω∈Ω, the mappingT²has a unique fixed point,ξω, 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 contraction operator with constantkωfor eachω∈Ω. ThenT has a unique random fixed point.

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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 thatTⁿis 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 thatT²is a contraction with constantkωfor eachω∈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 ólica de Valpara´ıso under grant 124.719/2009. In addition, the first author was supported by Laboratory of Stochastic Analysis PBCT-ACT 13.

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