Applied Mathematics and Mechanics (English Edition), 2007, 28(11):1527–1534c©Editorial Committee of Appl. Math. Mech., ISSN 0253-4827

Random attractors for asymptotically upper semicompact multivaluerandom semiflows ∗

LI Ting (��)

(Department of Mathematics, Suzhou University, Suzhou 215006, Jiangsu Province, P. R. China)

(Communicated by LI Ji-bin)

Abstract The present paper studied the dynamics of some multivalued random semi-flow. The corresponding concept of random attractor for this case was introduced to studyasymptotic behavior. The existence of random attractor of multivalued random semiflowwas proved under the assumption of pullback asymptotically upper semicompact, andthis random attractor is random compact and invariant. Furthermore, if the system hasergodicity, then this random attractor is the limit set of a deterministic bounded set.

Key words random attractor, asymptotically upper semicompact, absorbing set

Chinese Library Classification O175.12000 Mathematics Subject Classification 37D35, 34D20Digital Object Identifier(DOI) 10.1007/s10483-007-1112-z

Introduction

The understanding of the asymptotic behavior of semiflow is one of the most importantproblems of modern mathematical physics. One way to treat this problem for a system havingsome dissipativity properties is to analyze the existence and structure of its global attractor,which in the autonomous case, is an invariant compact set that attracts all the trajectories of thesystem, uniformly on bounded sets. However, non-autonomous systems and random systems arealso of great importance and interest as they appear in many applications to natural sciences.

The attempts to extend the notion of global attractor to the non-autonomous and randomcases led to the concept of the so-called theory of pullback (or cocycle) attractors, which hasbeen developed for both the non-autonomous and random dynamical systems (see Crauel etal.[1], Kloeden and Schmalfuss[2], Langa and Schmalfuss[3], Schmalfuss[4]), and has shown to bevery useful in the understanding of the dynamics of non-autonomous and random dynamicalsystems.

However, some difficulties appear when we have to work without uniqueness of solutions inthe system or when the model is better described by, for instance, a differential inclusion. Inthese cases, it has been shown that the theory of multivalued flows makes suitable the treatmentof the asymptotic behavior of these differential equations and inclusions. A new and differentdifficulty appears when a random term is added to the deterministic equation, a white noisefor instance, so that the corresponding stochastic partial differential equation must be treatedin a different way.

To prove the existence of the attractors in nonautonomous and random multivalued semiflowcases, the simplest, and therefore the strongest assumption is the existence of compact absorb-ing set[5,6]. However, this kind of compactness does not hold in general. Instead we often havesome kind of asymptotic compactness. In Refs. [7, 8], the authors considered nonautonomoussingle-valued and multivalued semiflow with pullback asymptotically compact respectively and

∗ Received Jul. 18, 2006; Revised Sep. 5, 2007Project supported by the National Natural Science Foundation of China (No. 10571130)Corresponding author LI Ting, E-mail: liting@suda.edu.cn

1528 LI Ting

proved the existence of a pullback attractor. In Ref. [6], authors also considered nonautonomousmultivalued semiflow with pullback asymptotically upper semi-compact and proved the exis-tence of a pullback attractor. However the global attractor in their paper is neither compactnor invariant.

In this paper, we study the asymptotic behavior of some multivalued random semiflow byintroducing previously the corresponding concept of random attractor for this case. We provethat the existence of random attractor of multivalued random semiflow under the assumptionof pullback asymptotically upper semicompact, and this random attractor is random compactand invariant. Furthermore, if we have ergodicity, then this random attractor is the limit setof a deterministic bounded set.

The paper is divided into two parts. In Section 1, we generalize the concepts of randomdynamical systems in Ref. [1] to the case of multivalued functions. A multivalued randomdynamical system will be a multivalued measurable map satisfying the cocycle property. Inthis framework, we can introduce the concepts of invariance, absorbtion and attraction, whichlead us to a general result of existence of random attractors for multivalued random semiflows.In Section 2, we give the proof of the main theorem.

1 Random attractors for multivalued random semiflow

Let X be a complete metric space with the metric d and the Borel σ-algebra F(X). Let(Ω,F , P ) be a probability space and θt : Ω → Ω be a measure preserving group of transformationin Ω such that the map (t, ω) �→ θtω is measurable and satisfies

i) θ0ω = ω for all ω ∈ Ω;ii) θt(θτω) = θt+τω for all ω ∈ Ω, t, τ ∈ R.

The parameter t takes values in R endowed with the Borel σ-algebra B(R).Let P (X) be the set of all nonempty subsets of X . Denote

B(X) = {A ∈ P (X) : A is bounded},C(X) = {A ∈ P (X) : A is closed},K(X) = {A ∈ P (X) : A is compact}.

We denote by dist(C1, C2) the Hausdorff semi-distance between C1 and C2 defined as

dist(C1, C2) = supx∈C1

infy∈C2

d(x, y), for C1, C2 ⊂ X.

Definition 1.1 A set valued map G : R+×Ω×X → P (X) is called a multivalued randomsemiflow if it satisfies

(i) G(0, ω) = Id on X ;(ii) G(t + s, ω)x = G(t, θsω)G(s, ω)x (cocycle property) for all t, s ∈ R+, x ∈ X, ω ∈ Ω.

A random set is a set valued map C : Ω → 2X ( 2X is the set of all subsets of X), such thatthe map ω �→ d(x, C(ω)) is measurable for all x. We say a random set C is forward invariant if

G(t, ω)C(ω) ⊂ C(θtω)

for P almost ω and all t ∈ R+ and it is invariant if

G(t, ω)C(ω) = C(θtω)

for P almost ω and all t ∈ R+. A random bounded set C = {C(ω)} is a random set suchthat for P a.e. ω ∈ Ω, C(ω) ∈ B(X). Let B̂(X) denote the class of all random boundedsets. A random compact set C = {C(ω)} is a random set such that C(ω) is a compact set forP a.e. ω ∈ Ω.

Random attractors for multivalue random semiflows 1529

Definition 1.2 A multivalued random semiflow G is said to be upper semi-continuous iffor all t ∈ R+ and P a.e. ω ∈ Ω it follows that given x ∈ X and a neighbourhood O of G(t, ω)x,there exists δ > 0 such that if d(x, y) < δ then

G(t, ω)y ⊂ O.

On the other hand, G is called lower semi-continuous if for all t ∈ R+ and P a.e. ω ∈ Ω, givenxn → x (n → ∞) and y ∈ G(t, ω)x, there exist yn ∈ G(t, ω)xn such that yn → y. It is said tobe continuous if it is upper and lower semi-continuous.

For random bounded set D ∈ B̂(X), ω ∈ Ω and t ∈ R+, we set

γt(ω,D) =⋃

s≥t

G(s, θ−sω)D(θ−sω).

Definition 1.3 A multivalued random semiflow G is called pullback asymptotically uppersemi-compact if for any random bounded set D, and P a.e. ω ∈ Ω, there exists t0 ≥ 0 such thatγt0(ω,D) ∈ B(X), any sequence ξn ∈ G(sn, θ−snω)D(θ−snω), where sn → +∞, is pre-compact.

Definition 1.4 A random set {A(ω)} is said to attract a deterministic bounded set D if

lims→∞ dist(G(s, θ−sω)D, A(ω)) = 0

for P almost surely.Definition 1.5 A random set B = {B(ω)} is said to be absorbing with respect to B(X)

if for any deterministic bounded set D ∈ B(X), and P a.e. ω ∈ Ω, there exists t0(ω, D) ≥ 0such that

G(t, θ−tω)D ⊂ B(ω) for all t ≥ t0(ω, D).

Definition 1.6 The family {A(ω)} is said to be a random attractor of the multivaluedrandom semiflow G if

(i) A = {A(ω)} is a random compact set;(ii) A = {A(ω)} attracts all deterministic bounded sets D ∈ B(X);(iii) It is invariant.Define the notation of limit set Λ(D, ω) = ΛD(ω) of a deterministic bounded set D as

ΛD(ω) =⋂

T≥0

⋃

t≥T

G(t, θ−tω)D

and the notation of limit set Ω(D, ω) = ΩD(ω) of a random bounded set D = {D(ω)} as

ΩD(ω) =⋂

T≥0

⋃

t≥T

G(t, θ−tω)D(θ−tω).

Theorem Suppose that multivalued random semiflow G is continuous and pullback asymp-totically upper semi-compact and there exists a random bounded set B = {B(ω)} ∈ B̂(X) whichis absorbing with respect to B(X). Then A = {A(ω)} is a random attractor of the multivaluedrandom semiflow G, where A(ω) =

⋃D∈B(X)

ΛD(ω).

2 Proof of Theorem

In order to prove Theorem, we first prove the following results.Lemma 2.1 y ∈ ΛD(ω) if and only if there exist sequences tn → +∞, xn ∈ D and

yn ∈ G(tn, θ−tnω)xn such that limn→+∞ yn = y.

1530 LI Ting

Proof Suppose y ∈ ΛD(ω), then for all n ∈ N, y ∈ ⋃t≥n G(t, θ−tω)D, and then there

exists yn ∈ ⋃t≥n G(t, θ−tω)D, which satisfies |y − yn| < 1

n . Thus there exist sequences tn ≥ nand xn ∈ D such that yn ∈ G(tn, θ−tnω)xn and lim

n→+∞ yn = y.

On the other hand, if there exist sequences tn → +∞, xn ∈ D and yn ∈ G(tn, θ−tnω)xn

such that limn→+∞ yn = y, then for ∀T ∈ R+, if tn ≥ T , we have

yn ∈ G(tn, θ−tnω)xn ⊂⋃

t≥T

G(t, θ−tω)D.

This implies that

y ∈⋃

t≥T

G(t, θ−tω)D.

Thus y ∈ ΛD(ω).Lemma 2.2 If G is pullback asymptotically upper semi-compact, upper semi-continuous

and there exists a random bounded set B = {B(ω)} which is pullback absorbing with respect toB(X), then for D ∈ B(X) and P a.e. ω ∈ Ω, the set ΛD(ω) is nonempty, compact, and

limt→+∞dist(G(t, θ−tω)D, ΛD(ω)) = 0. (1)

Furthermore, if G is lower semi-continuous, then {ΛD(ω)} is invariant.Proof Let us fix D ∈ B(X). Since random bounded set B = {B(ω)} is pullback absorbing

with respect to B(X), for P almost all ω ∈ Ω, there exists t0 ≥ 0 such that

G(t, θ−tω)D ⊂ B(ω) for all t ≥ t0,

and then γt0(ω, D) ∈ B(X). Thus if we consider a sequence tn → +∞ and a sequencexn ∈ G(tn, θ−tnω)D, then from the pullback asymptotic upper semicompact, we can extract aconvergent subsequence which converges to y. By its construction, y belongs to ΛD(ω), andthus this set is nonempty P almost surely.

We know that ΛD(ω) is closed, and consequently in order to prove its compactness it isenough to see that for any given sequence {yn} ∈ ΛD(ω), we can extract a convergent subse-quence. First, observe that as yn ∈ ΛD(ω), there exist tn ≥ n and xn ∈ G(tn, θ−tnω)D suchthat

d(xn, yn) ≤ 1n

.

Since G is the pullback asymptotic upper semicompact, there exists a subsequence xnkof xn,

which converges to y, and then we have ynk→ y (k → ∞).

If Eq. (1) is not true, then there exists ε > 0, a sequence tn → +∞, and a sequencexn ∈ G(tn, θ−tnω)D, such that

d(xn, y) ≥ ε for all y ∈ ΛD(ω).

For the sequence xn, there exists a subsequence xnk→ y ∈ ΛD(ω). This gives a contradiction.

Next we prove that ΛD(ω) is invariant. Let (t, ω) ∈ R+ ×Ω be fixed. We want to show thatif y ∈ ΛD(ω), then G(t, ω)y ∈ ΛD(θtω). Thus we have

G(t, ω)ΛD(ω) ⊂ ΛD(θtω).

In fact, if y ∈ ΛD(ω), then there exist a sequence tn → +∞, a sequence xn ∈ D and yn ∈G(tn, θ−tnω, xn) such that

yn → y.

Random attractors for multivalue random semiflows 1531

The lower semi-continuity of G implies that ∀z ∈ G(t, ω)y, there exists a sequence zm ∈G(t, ω)ym such that zm → z (m → ∞). Hence

zm ∈ G(t, ω)ym ⊂ G(t + tm, θ−(t+tm)(θtω), xm) ⊂ G(t + tm, θ−(t+tm)(θtω))D

with t + tm → +∞ and xm ∈ D. Thus z ∈ ΛD(θtω).Now we prove ΛD(θtω) ⊂ G(t, ω)ΛD(ω). For y ∈ ΛD(θtω) there exist a sequence tn → ∞

and a sequence xn ∈ D and yn ∈ G(tn, θ−tnω)xn such that

yn → y.

If tn ≥ t, we haveG(tn, θ−tnω)xn = G(t, ω)G(tn − t, θ−(tn−t)ω)xn.

As G is pullback upper semi-compact, tn − t → +∞ and xn ∈ D, there exists a subsequenceznk

∈ G(tnk− t, θ−(tnk

−t)ω)xnkwhich converges to z ∈ ΛD(ω), and we have ynk

∈ G(t, ω)znk.

The upper semi-continuity of G ensures that

y ∈ G(t, ω)z ⊂ G(t, ω)ΛD(ω).

Lemma 2.3 Suppose that B = {B(ω)} is a random bounded set, then {ΩB(ω)} is nonemptyand compact.

Proof B = {B(ω)} is a random bounded set. Thus if we consider a sequence tn → +∞ anda sequence xn ∈ G(tn, θ−tnω)B(θ−tnω), then from the pullback asymptotic upper semicompact,we can extract a convergent subsequence which converges to y. By its construction, y belongsto ΩB(ω), and thus this set is nonempty.

We know that ΩB(ω) is closed, and consequently in order to prove its compactness it isenough to see that for any given sequence {yn} ∈ ΩB(ω), we can extract a convergent subse-quence. First, observe that as yn ∈ ΩB(ω), there exist tn ≥ n and xn ∈ G(tn, θ−tnω)B(θ−tnω)such that

d(xn, yn) ≤ 1n

.

Since G is the pullback asymptotic upper semicompact, there exists a subsequence xnkof xn,

which converges to y, and then we have ynk→ y (k → ∞).

Proposition 2.1 If B is a pullback absorbing family, then

ΛD(ω) ⊂ ΩB(ω) for all D ∈ B(X),

for P almost surely.Proof Let us fix D ∈ B(X), and y ∈ ΛD(ω). Then there exist a sequence tn → +∞, a

sequence xn ∈ D and yn ∈ G(tn, θ−tnω)xn such that

limn→∞ yn = y.

Since B = {B(ω)} is pullback absorbing, for each k ∈ N, there exists tnk∈ {tn} such that

tnk≥ k and

G(tnk− k, θ−(tnk

−k)(θ−kω))D ⊂ B(θ−kω).

xnk∈ D implies that

G(tnk− k, θ−(tnk

−k)(θ−kω))xnk⊂ B(θ−kω).

1532 LI Ting

Sinceynk

∈ G(tnk, θ−tnk

ω)xnk= G(k, θ−kω)G(tnk

− k, θ−(tnk−k)(θ−kω))xnk

,

we can take zk ∈ B(θ−kω) such that ynk∈ G(k, θ−kω)zk.

limk→+∞

ynk= y

implies that y ∈ ΩB(ω).The proof of Theorem Fixing a point x0 ∈ X , let Dn = {x|d(x, x0) ≤ n}, n ∈ N. For

any deterministic bounded set D, there exists n such that D ⊂ Dn. By Proposition 2.1, it hasfor a fixed n,

ΛDn(ω) ⊂ ΩB(ω)

for P almost surely. Hence ⋃

n∈N

ΛDn(ω) ⊂ ΩB(ω)

for P almost surely. The compactness of ΩB(ω) implies that

⋃

n∈N

ΛDn(ω) ⊂ ΩB(ω)

for P almost surely and A(ω) =⋃

n∈N ΛDn(ω) is a compact set. On the other hand,

⋃

n∈N

ΛDn(ω) =⋃

D∈B(X)

ΛD(ω)

for P almost surely and A = {A(ω)} is a random compact set.Next we prove that the random compact set is invariant. By Lemma 2.2, for any bounded

set, {ΛD(ω)} is invariant and we have G(t, ω)ΛDn(ω) = ΛDn(θtω). Thus ∀x ∈ A(ω), thereexists a sequence {xn} ⊂ ⋃

n∈N ΛDn(ω) such that limn→∞ xn = x, then lower semicontinuity

implies that for ∀y ∈ G(t, ω)x there exists yn ∈ G(t, ω)xn such that limn→∞ yn = y. Since

yn ∈ G(t, ω)xn ⊂ G(t, ω)⋃

n∈N

ΛDn(ω).

Hencey ∈

⋃

n∈N

G(t, ω)ΛDn(ω) =⋃

n∈N

ΛDn(θtω) = A(θtω).

This givesG(t, ω)A(ω) ⊂ A(θtω).

On the other hand,

A(θtω) =⋃

n∈N

ΛDn(θtω) =⋃

n∈N

G(t, ω)ΛDn(ω) = G(t, ω)⋃

n∈N

ΛDn(ω).

Given y ∈ G(t, ω)⋃

n∈N

ΛDn(ω), there exists a sequence {xk} ⊂ ⋃n∈N

ΛDn(ω) such that G(t, ω)xk →y (k → ∞). The compactness of

⋃n∈N

ΛDn(ω) implies that we can extract a subsequence xklsuch

Random attractors for multivalue random semiflows 1533

that liml→∞

xkl= x ∈ ⋃

n∈N

ΛDn(ω). Since G is upper semicontinuous, liml→∞

dist(G(t, ω)xkl, G(t, ω)x) =

0. This implies that y ∈ G(t, ω)x ⊂ G(t, ω)⋃

n∈N

ΛDn(ω). Hence

A(θtω) ⊂ G(t, ω)⋃

n∈N

ΛDn(ω) = G(t, ω)A(ω).

Now we prove that random compact set A attracts any deterministic bounded set. ByLemma 2.2, we have for any deterministic bounded set D,

limt→+∞dist(G(t, θ−tω)D, ΛD(ω)) = 0,

and ΛD(ω) ⊂ A(ω). Therefore, we have

limt→+∞dist(G(t, θ−tω)D, A(ω)) = 0.

This gives the proof of Theorem.Remark 1 Under the conditions of Theorem, if we further suppose that random bounded

set {B(ω)} is uniformly bounded, that is say, there exists a deterministic bounded set U suchthat B(ω) ⊂ U for all ω ∈ Ω, then

A(ω) = ΩB(ω)

for P almost all ω. This can be easily obtained from Proposition 2.1 and the fact of ΩB(ω) ⊂ΛU (ω).

Proposition 2.2 Under the conditions of Theorem, if we further suppose that θt isergodic, then there exists a deterministic bounded set U such that

A(ω) = ΛU (ω)

for P almost all ω.Proof The idea of proof of Proposition comes from Crauel in Ref. [9]. For the completeness,

we include the proof here.We set

R(ω) = inf{r ∈ R|A(ω) ⊂ B(x0, r)},where x0 is any fixed point of X . The function R(ω) is measurable. Hence there is R0 > 0 suchthat

P ({ω|R(ω) ≤ R0}) > 0.

Let U = B(x0, R0). Ergodicity of θt implies that for almost all ω there exists a sequencetn → ∞ such that

R(θ−tnω) ≤ R0.

Let x ∈ A(ω) and tn be the sequence defined above. By the invariant property of A, for eachn ∈ N there exists xn ∈ A(θ−tnω) such that

x ∈ G(tn, θ−tnω)xn.

Since A(θ−tnω) ⊂ B(x0, R0), by the definition of ΛU (ω), we have x ∈ ΛU (ω). Hence

A(ω) ⊂ ΛU (ω).

From the definition of A(ω), it hasΛU (ω) ⊂ A(ω).

1534 LI Ting

ThusA(ω) = ΛU (ω).

In this paper we prove that the existence of random attractor of multivalued random semi-flow under the assumption of pullback asymptotically upper semicompact, and this randomattractor is random compact and invariant. Furthermore, if we have ergodicity, then this ran-dom attractor is the limit set of a deterministic bounded set. We will consider the applicationof results obtained as above in the forthcoming paper.

Acknowledgements Author would like to thank Prof. CAO Yong-luo for his suggestion aboutProposition 2.2.

References

[1] Crauel H, Debussche A, Flandoli F. Random attractors[J]. Journal of Dynamics and DifferentialEquation, 1995, 9(2):307–341.

[2] Kloeden P E, Schmalfuss B. Asymptotic behaviour of nonautonomous difference inclusions[J].Systems Control Letter, 1998, 33(4):275–280.

[3] Langa J A, Schmalfuss B. Finite dimensionality of attractors for nonautonomous dynamical sys-tems given by partial differential equations[J]. Stochastics and Dynamics, 2004, 4(3):385–404.

[4] Schmalfuss B. Attractors for nonautonomous dynamical systems[C]. In: Fiedler B, Greger K, andSprekels J (eds). Proc Equadi 99. Berlin: World Scientific, 2000, 684–689.

[5] Caraballo T, Langa J, Valero J. Global attractors for multivalued random dynamical systems[J].Nonlinear Analysis, 2002, 48(6):805–829.

[6] Caraballo T, Langa J, Melnik V, Valero J. Pullback attractors of nonautonomous and stochasticmultivalued dynamical systems[J]. Set-Valued Analysis, 2003, 11(3):153–201.

[7] Caraballo T, Lukaszewicz G, Real J. Pullback attractors for asymptotically compact nonau-tonomous dynamical systems[J]. Nonlinear Analysis, 2006, 64(3):484–498.

[8] Caraballo T, Marin-Rubio P, Valero J. Autonomous and non-autonomous attractors for differentialequations with delays[J]. Journal of Differential Equations, 2005, 208(1):9–41.

[9] Crauel H. Global random attractors are uniquely determined by attracting deterministic compactsets[J]. Annali di Matematica Pura ed Applicata, 1999, 176(4):57–72.