Volume 2010, Article ID 723216,13pages doi:10.1155/2010/723216
Research Article
Random Periodic Point and
Fixed Point Results for Random Monotone Mappings in Ordered Polish Spaces
Xing-Hua Zhu and Jian-Zhong Xiao
College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Correspondence should be addressed to Jian-Zhong Xiao,[email protected] Received 5 November 2010; Accepted 24 December 2010
Academic Editor: M. de la Sen
Copyrightq2010 X.-H. Zhu and J.-Z. Xiao. 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.
The measurability of order continuous random mappings in ordered Polish spaces is studied.
Using order continuity, some random fixed point theorems and random periodic point theorems for increasing, decreasing, and mixed monotone random mappings are presented.
1. Introduction and Preliminaries
The study of random fixed points forms a central topic in probabilistic functional analysis.
It was initiated by ˇSpaˇcek1, Hanˇs2, and Wang3. Some random fixed point theorems play an important role in the theory of random differential and random integral equations see Bharucha-Reid4,5. Since the recent 30 years, many interesting random fixed point theorems and applications have been developed, for example, see Beg and Shahzad6,7, Beg and Abbas8, Chang9, Ding10, Fierro et al.11, Itoh12, Li and Duan13, O’Regan et al.14, Xiao and Tao15, Xu16, and Zhu and Xu17.
In 1976, Caristi18introduced a partial ordering in metric spaces by a function and proved the famous Caristi fixed point theorem, which is one of the most important results in nonlinear analysis. From then on, there appeared many papers concerning fixed point theory and abstract monotone iterative technique in ordered metric spaces or ordered Banach spaces.
In particular, some useful fixed point theorems for monotone mappings were proved by Zhang19, Guo and Lakshmikantham20, and Bhaskar and Lakshmikantham21under some weak assumptions.
In this paper, motivated by ideas in18–21, we study random version of fixed point theorems for increasing, decreasing, and mixed monotone random mappings in ordered Polish spaces. In Section 2, we introduce order continuous random mapping and discuss its measurability. A well-known result is generalizedseeRemark 2.4. In Sections3–5, we present some existence results of random periodic point and fixed point for increasing, decreasing and, mixed monotone random mappings, respectively.
We begin with some definitions that are essential for this work. LetX, dbe a metric space andBXbe a Borel algebra ofX, wheredis a metric function onX. IfXis separable and complete, thenX, dis called a Polish space. We denote byΩ,A, Pa complete probability measure spacebriefly, a measure space, whereΩ,Ais a measurable space,Ais a sigma algebra of subsets ofΩ, andPis a probability measure. The notation “a.e.” stands for “almost every.”
Definition 1.1see3,5,9,12. A mappingy:Ω → Xis said to be measurable if y−1G
ω∈Ω:yω∈G
∈ A 1.1
for each open subset G of X. A measurable mapping is also called a random variable.
A mappingT :Ω×X → Xis called a random mapping, if for each fixedx∈X, the mapping T·, x:Ω → Xis measurable. A random mapping is said to be continuous, if forω∈Ωa.e., the mappingTω,· :X → Xis continuous. A measurable mappingy : Ω → X is said to be a random fixed point of the random mappingT :Ω×X → X, ifTω, yω yω, for ω ∈Ωa.e. Let 2Xbe the family of all nonempty subsets ofX andF : Ω → 2X a set-valued mapping.Fis said to be measurable, if
F−1G {ω∈Ω:Fω∩G /∅} ∈ A 1.2
for each open subsetGofX. A mappingy :Ω → X is said to be a measurable selection of a measurable mappingF:Ω → 2X, ifyis measurable andyω∈Fωa.e.
We denote byFRTthe set of all random fixed points of a random mappingT. Ifkis a positive integer andu∈FRTk, thenuis a randomk-periodic points of a random mapping T. By Tnω, xwe denote thenth iterate Tω, Tω, T. . . , Tω, xof T, whereT0 I,I : Ω×X → Xis defined byIω, x x.
Lemma 1.2see3,22. LetX, dbe a Polish space andΩ,A, Pa measure space. LetT :Ω× X → Xbe a continuous random mapping. Ify:Ω → Xis measurable, thenTω, yω:Ω → X is measurable.
Lemma 1.3see3,4. LetX, dbe a Polish space andΩ,A, Pa measure space. If{ynω}is a sequence of measurable mappings inXand limn→ ∞ynω yω∈X a.e., theny :Ω → X is measurable.
Lemma 1.4cf.23. LetX, dbe a Polish space andΩ,A, Pa measure space. LetF:Ω → 2X be a set-valued mapping. Then,
1Fis measurable if and only if Graph F {ω, x:x∈Fω}isA × BXmeasurable;
2ifFis measurable andFωis closed a.e., then there exists a measurable selection ofF.
Lemma 1.5see18. LetX, dbe a metric space andφ:X → Êa functional. Then the relation onXdefined by
xy⇐⇒d x, y
≤φx−φ y
, x, y∈X, 1.3
is a partial ordering.
ByLemma 1.5, ifis the partial ordering induced byφ, thenxyimpliesφx≥φy.
IfX, dis a Polish space andis the partial ordering induced byφ, thenX, d, φis called an ordered Polish space. Ifx0, y0∈Xandx0 y0, thenx0, y0 {x∈X : x0xy0}is called an order interval inX.
Definition 1.6 cf. 19. Let X, d, φ be an ordered Polish space andΩ,A, P a measure space. LetT :Ω×X → Xis a random mapping.T is is said to be increasing if
xy⇒Tω, xT ω, y
, ∀ω∈Ωa.e.; 1.4
Tis said to be decreasing if
xy⇒Tω, xT ω, y
, ∀ω∈Ωa.e.; 1.5
a random mappingS:Ω×X×X → Xis said to be mixed monotone if x1y1, y2x2⇒Sω, x1, x2S
ω, y1, y2
, ∀ω∈Ωa.e. 1.6
It is evident that, ifS:Ω×X×X → Xis mixed monotone, thenS·,·, x:Ω×X → X is increasing andS·, x,·:Ω×X → Xis decreasing, for every fixedx∈X.
2. Measurability of Order Continuous Random Mappings
Definition 2.1. LetX, d, φbe an ordered Polish space andΩ,A, Pa measure space. Let T :Ω×X → Xbe a random mapping.Tis said to be order continuous if for every monotone sequence{xn},
xn −→x⇒Tω, xn−→Tω, x, ∀ω∈Ωa.e. 2.1
Tis is said to be order contractive if there existsαω∈0,1such that xy⇒d
Tω, x, T ω, y
≤αωd x, y
, ∀ω∈Ωa.e. 2.2
It is evident that continuity implies order continuity. IfT :Ω×X → X is order con- tractive, thenTis order continuous. A mixed monotone random mappingS:Ω×X×X → X is said to be order continuous if and only if for monotone sequences{xn}and{yn},
xn−→x, yn−→y⇒T
ω, xn, yn
−→T ω, x, y
, ∀ω∈Ωa.e. 2.3
Example 2.2. LetΩ 1,2andX Ê2. Letφ:X → ÊandT:Ω×X → Xbe defined by
φx1, x2 −x1 x2, Tω,x1, x2
⎧⎨
⎩
0,0, ifx1x2≤0;
ω, ω, ifx1x2>0. 2.4
It is easy to check thatTis order continuous, butTis not continuous at0,0.
Now we prove the following theorem which plays an important role in the sequel.
Theorem 2.3. LetX, d, φbe an ordered Polish space andΩ,A, Pa measure space, whereφ is continuous. Let T : Ω×X → X be an order continuous random mapping. Ify : Ω → X is measurable, thenTω, yω:Ω → Xis measurable.
Proof. LetEω {x∈X :x yω},Hω {x∈X :yω x}, andQεω {x ∈X : dx, yω≤ε}, whereε >0. Clearly,Eω,Hω, andQεωare all nonempty subsets ofX for allω ∈Ω. Sincedis continuous,Qεωis closed for allω ∈ Ω. Let{xn}∞n1 ⊂ Eωand xn → x0n → ∞. Then, fromxnyω, we have
d
xn, yω
≤φxn−φ
yω
. 2.5
Sinceφis continuous, we havedx0, yω≤φx0−φyω, that is,x0yω. This shows thatx0 ∈Eω, and soEωis closed for allω∈ Ω. Similarly,Hωis closed for allω ∈Ω.
We claim that
E, H, Qε:Ω−→2X are all measurable. 2.6
In fact, ifBx{y∈X :xy}, thenBxis a closed subset ofX. LetGbe an open subset ofX, WX\G, andE−1W {ω∈Ω:Eω⊂W}. Then, we have
E−1W
ω∈Ω:xyω, x∈W
x∈W
ω∈Ω:xyω
x∈W
y−1Bx y−1
x∈W
Bx
.
2.7
Since y is measurable and
x∈WBx is closed, E−1W is measurable. From E−1G Ω\ E−1W, we see that E−1Gis measurable. Hence, E is measurable. Similarly, H is meas- urable. Now we prove thatQε is measurable. Since d is continuous andy is measurable, dx, yω:Ω×X → Êis measurable. Note that
GraphQε{ω, x:x∈Qεω}
ω, x:d
x, yω
≤ε
2.8 isA × BXmeasurable. UsingLemma 1.41, we obtain thatQεis measurable. Therefore,2.6 holds. Let F1ω {x ∈ X : x yω, dx, yω ≤ 1}. Then,F1ω Eω∩Q1ω is
nonempty and closed for allω∈Ω. By2.6,F1is measurable. ByLemma 1.42, we can take y1ω∈F1ω, wherey1:Ω → Xis measurable. Forn2,3, . . ., let
Fnω
x∈X:yn−1ωxyω, d
x, yω
≤ 1 n
. 2.9
Then, Fnω is nonempty and closed for all ω ∈ Ω. Whenyn−1 is measurable, from2.6, we obtain thatFn is measurable. UsingLemma 1.42, we can takeynω ∈ Fnω, where yn : Ω → X is measurable. By induction, there exists a measurable sequence{ynω}such that
y1ωy2ω · · · ynω · · · yω, ynω−→yωn−→ ∞, ∀ω∈Ω.
2.10 SetY ∞
n1{ynω:ω∈Ω} ∪ {yω:ω∈Ω}. ThenY is a Polish subspace ofX. SinceT : Ω×X → Xis order continuous,T:Ω×Y → Xis continuous. By2.10, we have
T
ω, ynω
−→T
ω, yω
n−→ ∞, ∀ω∈Ωa.e. 2.11
By Lemma 1.2, Tω, ynω is measurable for all n. Thus, from 2.11 and Lemma 1.3 it follows thatTω, yωis measurable. This completes the Proof.
Remark 2.4. Theorem 2.3is a generalization ofLemma 1.2.
3. Random Periodic Points and Fixed Points for Increasing Random Mappings
Theorem 3.1. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0 → Xbe an order continuous and increasing random mapping withx0Tkω, x0andTkω, y0y0
forω ∈ Ωa.e., wherek is a positive integer. Then there exist a minimum randomk-periodic point uωand a maximum randomk-periodic pointvωinx0, y0such thatuωzωvωa.e., for allz∈FRTk.
Proof. Without loss of generality, we may assume thatΩ0 ⊂ Ω,PΩ0 1,Tω,·is order continuous for allω ∈ Ω0, andx0 Tkω, x0,Tkω, y0 y0 for all ω ∈ Ω0. Letω ∈ Ω0, STk,xnω Snω, x0, andynω Snω, y0. Sincex0Sω, x0,Sω, y0y0, andT is increasing, we have
x0x1ω · · · xnω · · · ynω · · · y1ωy0. 3.1
Then, it follows from3.1that
φx0≥φx1ω≥ · · · ≥φxnω≥ · · · ≥φ ynω
≥ · · · ≥φ y1ω
≥φ y0
. 3.2
From3.2we see that{φxnω}and{φynω}are two convergent sequences of numbers.
For everyε >0 there exists a positive integerNsuch that
dxnω, xmω≤φxnω−φxmω< ε, ∀m > n > N;
d
ymω, ynω
≤φ ymω
−φ ynω
< ε, ∀m > n > N. 3.3 This shows that{xnω}and{ynω}are two Cauchy sequences inX. The completeness of Ximplies that{xnω}and{ynω}are all convergent. Defineuωandvωby
uω
⎧⎨
⎩
nlim→ ∞xnω, ifω∈Ω0,
x0, ifω∈Ω\Ω0; vω
⎧⎨
⎩
nlim→ ∞ynω, ifω∈Ω0, y0, ifω∈Ω\Ω0.
3.4
SinceT is order continuous,Sis order continuous. Then, we have Sω, uω lim
n→ ∞Sω, xnω lim
n→ ∞xn 1ω uω, ∀ω∈Ω0; Sω, vω lim
n→ ∞S
ω, ynω lim
n→ ∞yn 1ω vω, ∀ω∈Ω0.
3.5
Note thatPΩ\Ω0 0. ByTheorem 2.3,xnωandynωare all measurable. ByLemma 1.3, uωand vωare all measurable. Therefore, from3.5we see thatuωand vωare all random fixed points ofS, that is,u, v∈FRS FRTk. Sinceφis continuous, we have, for ω∈Ωa.e.,
dx0, uω lim
n→ ∞dx0, xnω≤ lim
n→ ∞
φx0−φxnω
φx0−φuω;
d
vω, y0
lim
n→ ∞d
ynω, y0
≤ lim
n→ ∞
φ ynω
−φ y0
φvω−φ y0
;
duω, vω lim
n→ ∞d
xnω, ynω
≤ lim
n→ ∞
φxnω−φ
ynω
φuω−φvω.
3.6 This shows thatx0 uω vω y0 a.e. Ifz ∈ FRTk FRS, then we havexnω zωynωa.e., for alln. Thus, forω∈Ωa.e.,
duω, zω lim
n→ ∞dxnω, zω≤ lim
n→ ∞
φxnω−φzω
φuω−φzω;
dzω, vω lim
n→ ∞d
zω, ynω
≤ lim
n→ ∞
φzω−φ
ynω
φzω−φvω.
3.7
This shows thatuωzωvωa.e., which is the desired conclusion.
Corollary 3.2. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0 → Xbe an order continuous and increasing random mapping withx0Tω, x0andTω, y0y0for ω∈Ωa.e.. Then there exist a minimum random fixed pointuωand a maximum random fixed point vωinx0, y0such thatuωzωvωa.e., for allz∈FRT.
Proof. It is obtained by takingk1 inTheorem 3.1.
Corollary 3.3. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0 → Xbe a increasing random mapping withx0 Tkω, x0andTkω, y0 y0 forω ∈Ωa.e., where kis a positive integer. IfT is an order contraction mapping, then there exists a unique random fixed pointuωinx0, y0.
Proof. From order contraction ofTit follows thatTis order continuous. ByTheorem 3.1, there exist a minimum randomk-periodic pointuωand a maximum randomk-periodic point vωinx0, y0. SinceTis an order contraction mapping, forω∈Ωa.e., we have
duω, vω d
Tkω, uω, Tkω, vω
≤αωkduω, vω, 3.8
whereαω∈0,1. This shows thatuω vωa.e., namely, there is a uniqueu∈FRTk. LetTω, uω zω. Then we havezω∈x0, y0a.e. and
zω T
ω, Tkω, uω
Tk 1ω, uω Tkω, zω, 3.9
that is,z ∈ FRTk. Hence, we haveu z. This shows thatu ∈ FRT. If y ∈ FRTand yω∈x0, y0a.e., theny ∈FRTk, and soy u, that is, there is a uniqueu∈FRT. This completes the proof.
4. Random Periodic Points and Fixed Points for Decreasing Random Mappings
Theorem 4.1. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0 → Xbe an order continuous and decreasing random mapping withx0Tω, y0andTω, x0y0for ω∈Ωa.e. Then there exists a random 2-periodic pointuinx0, y0such thatTω, uω∈x0, y0 a.e.
Proof. Without loss of generality, we may assume thatΩ0 ⊂ Ω,PΩ0 1,Tω,·is order continuous for all ω ∈ Ω0 and x0 Tω, y0,Tω, x0 y0 for allω ∈ Ω0. Letω ∈ Ω0, xnω Tω, yn−1ω, andynω Tω, xn−1ω,n 1,2, . . .. SinceT is decreasing, we have
x0x1ω · · · xnω · · · ynω · · · y1ωy0. 4.1
Then, from4.1it follows that
φx0≥φx1ω≥ · · · ≥φxnω≥ · · · ≥φ ynω
≥ · · · ≥φ y1ω
≥φ y0
. 4.2
From4.2we see that{φxnω}and{φynω}are two convergent sequences of numbers.
For everyε >0 there exists a positive integerNsuch that
dxnω, xmω≤φxnω−φxmω< ε, ∀m > n > N;
d
ymω, ynω
≤φ ymω
−φ ynω
< ε, ∀m > n > N. 4.3 This shows that{xnω}and{ynω}are two Cauchy sequences inX. By the completeness of Xwe see that{xnω}and{ynω}are all convergent. Defineuωandvωby3.4. Since Tis order continuous, we have
Tω, uω lim
n→ ∞Tω, xn−1ω lim
n→ ∞ynω vω, ∀ω∈Ω0; Tω, vω lim
n→ ∞T
ω, yn−1ω lim
n→ ∞xnω uω, ∀ω∈Ω0.
4.4
By the continuity ofφ, we have, forω∈Ω0, dx0, uω lim
n→ ∞dx0, xnω≤ lim
n→ ∞
φx0−φxnω
φx0−φuω;
d
vω, y0
lim
n→ ∞d
ynω, y0
≤ lim
n→ ∞
φ ynω
−φ y0
φvω−φ y0
. 4.5
SincePΩ\Ω0 0, we haveuω, vω∈x0, y0a.e.. ByTheorem 2.3,xnωandynωare all measurable. ByLemma 1.3,uωandvωare all measurable. Therefore, from4.4we have
T2ω, uω Tω, vω uω, ∀ω∈Ωa.e. 4.6
This shows thatu∈FRT2, which is the desired conclusion.
Corollary 4.2. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0 → Xbe a decreasing random mapping withx0 Tω, y0andTω, x0y0forω∈Ωa.e. IfT is an order contraction mapping, then there exists a unique random fixed pointuωinx0, y0.
Proof. SinceTis an order contraction mapping,T is order continuous. ByTheorem 4.1, there exists a random 2-periodic pointuinx0, y0such thatTω, uω vω ∈x0, y0a.e. We claim thatuω vωa.e. In fact that, from4.1we haveuω vωa.e. Ifuω/vω a.e., then there existsαω∈0,1such that
duω, vω dTω, vω, Tω, uω≤αωdvω, uω
< dvω, uω, ∀ω∈Ωa.e., 4.7
which is a contradiction. Hence,u ∈ FRT. Ify ∈ FRTandyω ∈ x0, y0a.e., then we have
xnωyωynω, ∀ω∈Ω a.e., 4.8
where{xnω}and{ynω}are the iterations in the proof ofTheorem 4.1. It is easy to check thatuωyωvω, for allω∈Ωa.e. Butuv, and so we haveyu. This completes the proof.
Theorem 4.3. LetX, d, φbe an ordered Polish space, whereφis continuous andφXis bounded.
LetT :Ω×x0, y0 → Xbe an order continuous and decreasing random mapping withy0Tω, x0 andTω, y0x0forω∈Ωa.e. Then there exists a random 2-periodic pointuinX.
Proof. Without loss of generality, we may assume thatΩ0 ⊂ Ω,PΩ0 1,Tω,·is order continuous for allω ∈ Ω0, andy0 Tω, x0,Tω, y0 x0 for allω ∈ Ω0. Let ω ∈ Ω0, xnω Tω, yn−1ω, andynω Tω, xn−1ω,n 1,2, . . .. SinceT is decreasing, we have
· · · xnω · · · x1ωx0y0y1ω · · · ynω · · · 4.9
Then, it follows from4.9that
· · · ≥φxnω≥ · · · ≥φx1ω≥φx0≥φ y0
≥φ y1ω
≥ · · · ≥φ ynω
≥ · · · 4.10 This shows that{φxnω}and{φynω}are two convergent sequences of numbers by the boundedness ofφX. For everyε >0 there exists a positive integerNsuch that
dxnω, xmω≤φxnω−φxmω< ε, ∀n > m > N. 4.11 This shows that {xnω} is a Cauchy sequence in X. The completeness of X implies that {xnω}is convergent. Similarly,{ynω}is convergent. Defineuωandvωby3.4. Since Tis order continuous, we have
Tω, uω lim
n→ ∞Tω, xn−1ω lim
n→ ∞ynω vω, ∀ω∈Ω0; Tω, vω lim
n→ ∞T
ω, yn−1ω lim
n→ ∞xnω uω, ∀ω∈Ω0. 4.12 SincePΩ\Ω0 0, by Theorem 2.3,xnωand ynω are all measurable; byLemma 1.3, uωandvωare all measurable. Therefore, from4.12we have
T2ω, uω Tω, vω uω, ∀ω∈Ωa.e. 4.13 This shows thatu∈FRT2, which is the desired conclusion.
5. Coupled Random Periodic Point and Fixed Point Theorems
Theorem 5.1. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0× x0, y0 → Xbe an order continuous and mixed monotone random mapping withx0 Tkω, x0, y0 and Tkω, y0, x0 y0 forω ∈ Ω a.e., wherek is a positive integer. Then there exists a coupled
randomk-periodic pointu, vsuch thatTkω, uω, vω uω,Tkω, vω, uω vω, and uω, vω ⊂ x0, y0 a.e. If u1, v1 is a coupled random k-periodic point such that u1ω, v1ω⊂x0, y0a.e., thenu1ω, v1ω⊂uω, vωa.e.
Proof. Without loss of generality, we may assume thatΩ0 ⊂ Ω,PΩ0 1,Tω,·,·is order continuous for allω∈Ω0andx0Tkω, x0, y0,Tkω, y0, x0y0for allω∈Ω0. Letω∈Ω0, STk,xnω Snω, xn−1ω, yn−1ω, andynω Snω, yn−1ω, xn−1ω,n1,2, . . ..
SinceT is a mixed monotone mapping, we have x0 x1ω S
ω, x0, y0
S
ω, y0, y0
S
ω, y0, x0
y1ωy0. 5.1
By induction, we have
x0x1ω · · · xnω · · · ynω · · · y1ωy0. 5.2
Thus, from5.2it follows that
φx0≥φx1ω≥ · · · ≥φxnω≥ · · · ≥φ ynω
≥ · · · ≥φ y1ω
≥φ y0
. 5.3
This shows that{φxnω}and {φynω} are two convergent sequences of numbers. In a similar way to the proof ofTheorem 3.1, we can check that{xnω}and{ynω}are two Cauchy sequences in X. The completeness of X implies that {xnω} and {ynω} are all convergent. Defineuω and vω by3.4. Sinceφ is continuous, it is easy to prove that xnω uω vω ynωfor alln. SinceT is order continuous,Sis order continuous.
Then, we have
Sω, uω, vω lim
n→ ∞S
ω, xnω, ynω lim
n→ ∞xn 1ω uω, ∀ω∈Ω0; Sω, vω, uω lim
n→ ∞S
ω, ynω, xnω lim
n→ ∞yn 1ω vω, ∀ω∈Ω0. 5.4 Note thatPΩ\Ω0 0. ByTheorem 2.3,xnωandynωare all measurable. ByLemma 1.3, uω and vω are all measurable. Therefore, from 5.4 we see that u, v is a coupled random fixed point ofS, that is, it is a coupled randomk-periodic point ofT. If u1, v1is a coupled randomk-periodic point such thatu1ω, v1ω ⊂ x0, y0a.e., then, by mixed monotonicity of T, we have x1ω Tω, x0, y0 Tω, u1ω, v1ω u1ω a.e. and v1ω Tω, v1ω, u1ωTω, y0, x0 y1ωa.e. Then, by induction, we have
xnωu1ωa.e., v1ωynωa.e., ∀n. 5.5
SinceT is order continuous, we haveu1ω, v1ω ⊂ uω, vωa.e.. This completes the proof.
Corollary 5.2. LetX, d, φbe an ordered Polish space, whereφis continuous. LetT :Ω×x0, y0× x0, y0 → X be an order continuous and mixed monotone random mapping withx0Tω, x0, y0 andTω, y0, x0y0forω∈Ωa.e. Then there exists a coupled random fixed pointu, vsuch that
Tω, uω, vω uω,Tω, vω, uω vωand uω, vω ⊂ x0, y0a.e. Ifu1, v1 is also a coupled random fixed point such thatu1ω, v1ω⊂x0, y0a.e., thenu1ω, v1ω⊂ uω, vωa.e.
Proof. It is obtained by takingk1 inTheorem 5.1.
Theorem 5.3. LetX, d, φbe an ordered Polish space, whereφis continuous andφXis bounded.
LetT :Ω×x0, y0×x0, y0 → Xbe an order continuous and mixed monotone random mapping with Tω, x0, y0 x0 and y0 Tω, y0, x0forω ∈ Ω a.e., wherex0/y0. Then there exists a coupled random fixed pointuω, vωsuch thatTω, uω, vω uω,Tω, vω, uω vω, andx0, y0 ⊂ uω, vωa.e. Ifu1, v1is also a coupled random fixed point such that x0, y0⊂u1ω, v1ωa.e., thenuω, vω⊂u1ω, v1ωa.e.
Proof. Without loss of generality, we may assume thatΩ0 ⊂ Ω,PΩ0 1,Tω,·,·is order continuous for allω∈Ω0andTω, x0, y0x0,y0 Tω, y0, x0for allω ∈Ω0. Letω∈Ω0, xnω Tω, xn−1ω, yn−1ω, andynω Tω, yn−1ω, xn−1ω,n1,2, . . .. Then,
x1ω T
ω, x0, y0
x0y0 T
ω, y0, x0
y1ω. 5.6
SinceTis a mixed monotone mapping, we havex2ω Tω, x1ω, y1ωTω, x0, y0
x1ω, andy1ω Tω, y0, x0Tω, y1ω, x1ω y2ω. By induction, we have
· · · xnω · · · x1ωx0y0y1ω · · · ynω · · · 5.7
Thus, from5.7it follows that
· · · ≥φxnω≥ · · · ≥φx1ω≥φx0≥φ y0
≥φ y1ω
≥ · · · ≥φ ynω
≥ · · · 5.8 This shows that{φxnω}and{φynω}are two convergent sequences of numbers by the boundedness ofφX. In a similar way to the proof ofTheorem 4.3, we can check that{xnω}
and{ynω}are two Cauchy sequences inX. The completeness ofXimplies that{xnω}and {ynω}are all convergent. Defineuωandvωby3.4. Sinceφis continuous, it is easy to prove thatuω xnω x0andy0 ynω vωfor alln. SinceT is order continuous, we have
Tω, uω, vω lim
n→ ∞T
ω, xnω, ynω lim
n→ ∞xn 1ω uω, ∀ω∈Ω0; Tω, vω, uω lim
n→ ∞T
ω, ynω, xnω lim
n→ ∞yn 1ω vω, ∀ω∈Ω0. 5.9 Note thatPΩ\Ω0 0. ByTheorem 2.3,xnωandynωare all measurable. ByLemma 1.3, uωandvωare all measurable. Therefore, from5.9we see thatu, vis a coupled random fixed point ofT. Ifu1, v1is a coupled random point ofTwithx0, y0⊂u1ω, v1ωa.e., then, by mixed monotonicity ofT, we haveu1ω Tω, u1ω, v1ωTω, x0, y0 x1ω a.e., andv1ω Tω, v1ω, u1ω Tω, y0, x0 y1ωa.e., namely,x1ω, y1ω ⊂ u1ω, v1ωa.e. By induction, we have
u1ωxnωa.e., ynωv1ωa.e., ∀n. 5.10
SinceT is order continuous, we haveuω, vω ⊂ u1ω, v1ωa.e. This completes the proof.
Acknowledgments
The authors are grateful to the referees for their suggestions to improve the legibility of the paper. This work is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions Grant no. 10KJB110006and by the Natural Science Foundation of Nanjing University of Information Science and Technology of China20080286.
References
1 A. ˇSpaˇcek, “Zuf¨allige Gleichungen,” Czechoslovak Mathematical Journal, vol. 5, pp. 462–466, 1955.
2 O. Hanˇs, “Random fixed point theorems,” in Transactions of the 1st Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, pp. 105–125, Publishing House of the Czechoslovak Academy of Sciences, Prague, Czech Republic, 1957.
3 Z. K. Wang, “An introduction to random functional analysis,” Advances in Mathematics, vol. 5, no. 1, pp. 45–71, 1962Chinese.
4 A. T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, NY, USA, 1972, Mathematics in Science and Engineering, Vol. 9.
5 A. T. Bharucha-Reid, “Fixed point theorems in probabilistic analysis,” Bulletin of the American Mathematical Society, vol. 82, no. 5, pp. 641–657, 1976.
6 I. Beg and N. Shahzad, “Random fixed points of random multivalued operators on Polish spaces,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 7, pp. 835–847, 1993.
7 I. Beg and N. Shahzad, “On random approximation and coincidence point theorems for multivalued operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 6, pp. 1035–1041, 1996.
8 I. Beg and M. Abbas, “Random fixed point theorems for Caristi type random operators,” Journal of Applied Mathematics & Computing, vol. 25, no. 1-2, pp. 425–434, 2007.
9 S.-S. Chang, “Some random fixed point theorems for continuous random operators,” Pacific Journal of Mathematics, vol. 105, no. 1, pp. 21–31, 1983.
10 X. P. Ding, “Solutions for a system of random operator equations and some applications,” Scientia Sinica. Series A, vol. 30, no. 8, pp. 785–795, 1987.
11 R. Fierro, C. Mart´ınez, and C. H. Morales, “Fixed point theorems for random lower semi-continuous mappings,” Fixed Point Theory and Applications, vol. 2009, Article ID 584178, 7 pages, 2009.
12 S. Itoh, “A random fixed point theorem for a multivalued contraction mapping,” Pacific Journal of Mathematics, vol. 68, no. 1, pp. 85–90, 1977.
13 G. Li and H. Duan, “On random fixed point theorems of random monotone operators,” Applied Mathematics Letters, vol. 18, no. 9, pp. 1019–1026, 2005.
14 D. O’Regan, N. Shahzad, and R. P. Agarwal, “Random fixed point theory in spaces with two metrics,”
Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 2, pp. 171–176, 2003.
15 J. Z. Xiao and Y. Tao, “Property of surjection for monotonic random operator and its application,”
Acta Mathematica Sinica (Chinese series), vol. 51, no. 2, pp. 391–400, 2008.
16 H.-K. Xu, “Random fixed point theorems for nonlinear uniformly Lipschitzian mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 7, pp. 1301–1311, 1996.
17 C.-x. Zhu and Z.-b. Xu, “Random ambiguous point of randomkω-set-contractive operator,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 2–6, 2007.
18 J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of the American Mathematical Society, vol. 215, pp. 241–251, 1976.
19 X. Zhang, “Fixed point theorems of monotone mappings and coupled fixed point theorems of mixed monotone mappings in ordered metric spaces,” Acta Mathematica Sinica. Chinese Series, vol. 44, no. 4, pp. 641–646, 2001.
20 D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987.
21 T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006.
22 R. Kannan and H. Salehi, “Random nonlinear equations and monotonic nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 57, no. 1, pp. 234–256, 1977.
23 J.-P. Aubin and H. Frankowska, Set-Valued Analysis, vol. 2 of Systems & Control: Foundations &
Applications, Birkh¨auser, Boston, Mass, USA, 1990.
