Sell’s conjecture for non-autonomous dynamical systems

SELL’S CONJECTURE FOR NON-AUTONOMOUS DYNAMICALSYSTEMS

DAVID CHEBAN

Abstract. This paper is dedicated to the study of the G. Sell’s conjecturefor general non-autonomous dynamical systems. We give a positive answer forthis conjecture and we apply this result to different classes of non-autonomousevolution equations: Ordinary Differential Equations, Functional DifferentialEquations and Semi-linear Parabolic Equations.

1. Introduction

The aim of this paper is the study the problem of global asymptotic stability oftrivial solution for non-autonomous dynamical systems. We study this problem inthe framework of general non-autonomous dynamical systems (NDS).

Consider a differential equation

(1) x′ = f(t, x) (f ∈ C(R×W,Rn)),

where R := (−∞,+∞), Rn is a product space of n copies of R, W is an opensubset from Rn containing the origin (i.e., 0 ∈ W ), C(R ×W,Rn) is the space ofall continuous functions f : R ×W 7→ Rn equipped with compact open topology.This topology is defined, for example [5, 15], by the following distance

ρ(f, g) :=+∞∑

k=1

12k

ρk(f, g)1 + ρk(f, g)

,

where ρk(f, g) := max|f(t, x)−g(t, x)| : (t, x) ∈ [−k, k]×Wk, Wk is a family ofcompact subsets from W with the properties: Wk ⊂ Wk+1 for all k ∈ N,

⋃+∞k=1 Wk =

W , and | · | is a norm on Rn. Denote by (C(R ×W,Rn),R, σ) the shift dynamicalsystem [5, 15] on the space C(R × W,Rn) (dynamical system of translations orBebutov’s dynamical system), i.e., σ(τ, f) := fτ for all τ ∈ R and f ∈ C(R×W,Rn),where fτ (t, x) := f(t + τ, x) for all (t, x) ∈ R×W .

Below we will use the following conditions:

(A): for all (t0, x0) ∈ R+ × W the equation (1) admits a unique solutionx(t; t0, x0) with initial data (t0, x0) and defined on R+ := [0, +∞), i.e.,x(t0; t0, x0) = x0;

Date: December 27, 2011.1991 Mathematics Subject Classification. 34D23, 34D45, 37B25, 37B55, 39A11, 39C10, 39C55.Key words and phrases. Global attractor; non-autonomous dynamical system; asymptotic

stability.

1

2 DAVID CHEBAN

(B): the hand right side f is positively compact, if the set Σ+f := fτ : τ ∈ R+

is a relatively compact subset of C(R×W,Rn);(C): the equation

y′ = g(t, y), (g ∈ Ωf )is called a limiting equation for (1), where Ωf is the ω-limit set of f withrespect to the shift dynamical system (C(R×W,Rn),R, σ), i.e., Ωf := g :there exists a sequence τk → +∞ such that fτk

→ g as k →∞;(D): equation (1) (or its hand right side f) is regular , if for all p ∈ H+(f) the

equationy′ = p(t, y)

admits a unique solution ϕ(t, x0, p) defined on R+ with initial conditionϕ(0, x0, p) = x0 for all x0 ∈ W , where H+(f) := fτ : τ ∈ R+ and bybar is denoted the closure in the space C(R×W,Rn);

(E): equation (1) admits a null (trivial) solution, i.e., f(t, 0) = 0 for all t ∈ R+;(F): a function f satisfies to local (respectively, global) Lipschitz condition, if

there exists a function L : R+ 7→ R+ (respectively, a positive constant L)such that

|f(t, x1)− f(t, x2)| ≤ L(r)|x1 − x2|(respectively, |f(t, x1)− f(t, x2)| ≤ L|x1 − x2|) for all t ∈ R+ and x1, x2 ∈W with |x1|, |x2| ≤ r for all r > 0 (respectively, for all x1, x2 ∈ W ).

The trivial solution of equation (1) is said to be:

(i) uniformly stable, if for all positive number ε there exists a number δ = δ(ε)(δ ∈ (0, ε)) such that |x| < δ implies |ϕ(t, x, fτ )| < ε for all t, τ ∈ R+;

(ii) uniformly attracting, if there exists a positive number a

limt→+∞

|ϕ(t, x, fτ )| = 0

uniformly with respect to |x| ≤ a and τ ∈ R+;(iii) uniformly asymptotically stable, if it is uniformly stable and uniformly

attracting.

G. Sell’s conjecture ([15, Ch.VIII,p.134]). Let f ∈ C(R ×W,Rn) be a regularfunction and f be positively pre-compact. Assume that W contains the origin 0and f(t, 0) = 0 for all t ∈ R+. Assume further that there exists a positive numbera such that

limt→+∞

|ϕ(t, x, g)| = 0

takes place uniformly with respect to |x| ≤ a and g ∈ Ωf . Then the trivial solutionof (1) is uniformly asymptotically stable.

The positive solution of G. Sell’s conjecture was obtained by Z. Artstein [1] andBondi P. et al [2].

Remark 1.1. 1. Bondi P. et al [2] proved this conjecture under the additionalassumption that the function f is local Lipschitzian.

2. Artstein Z. [1] proved this statement without Lipschitzian condition. In realityhe proved a small more general statement. Namely, he supposed that only limitingequations for (1) are regular, but the function f is not obligatory regular.

SELL’S CONJECTURE FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS 3

3. It is well known (see, for example,[11]) that for a wide class of ordinary differentialequations (ODEs) the notions of uniform asymptotic stability and the notion ofstability in the sense of Duboshin (total stability or stability under the perturbation)are equivalent. There is a series of works (see, for example,[2],[8],[10],[12]-[14], [16]-[17] and the references therein), where the authors study the analog of G. Sell’sproblem for total stability.

In this paper we will formulate G. Sell’s conjecture for the abstract NDS. Wewill give a positive answer to this conjecture and we will apply this result to dif-ferent classes of evolution equations: infinite-dimensional differential equations,functional-differential equations and semi-linear parabolic equations .

The paper is organized as follows.

In Section 2, we collect some notions (global attractor, stability, asymptotic stabil-ity, uniform asymptotic stability, minimal set, point/compact dissipativity, recur-rence, shift dynamical systems, etc) and facts from the theory of dynamical systemswhich will be necessary in this paper.

Section 3 is devoted to the analysis of G. Sell’s conjecture. In this Section weformulate an analog of G. Sell’s conjecture for cocycles and general NDS.

In Sections 4 we establish the relation between different types of stability of NDS.We prove that from uniform attractiveness it follows uniform asymptotic stability(Theorem 4.1). It is proved that for asymptotically compact dynamical systemasymptotic stability and uniform asymptotic stability are equivalent (Theorem 4.3).The main results of this Section are Theorem 4.7 and Corollary 4.9 which contains positive answer to G. Sell’s conjecture for general NDS.

Finally, Section 5 contains some applications of our general results from Sections 2-4for Ordinary Differential Equations (Theorem 5.1), Functional-Differential Equa-tions (Theorem 5.5) and semi-linear parabolic equations (Theorem 5.7).

2. Compact Global Attractors of Dynamical Systems

Let X be a topological space, R (Z) be a group of real (integer) numbers, R+ (Z+)be a semi-group of the nonnegative real (integer) numbers, S be one of the twosets R or Z and T ⊆ S be one of the sub-semigroups R+ (respectively, Z+) or R(respectively, Z).

A triplet (X,T, π), where π : T × X → X is a continuous mapping satisfying thefollowing conditions:

(2) π(0, x) = x;

(3) π(s, π(t, x)) = π(s + t, x);

is called a dynamical system. If T = R (R+) or Z (Z+), then (X,T, π) is calleda group (semi-group) dynamical system. In the case, when T = R+ or R thedynamical system (X,T, π) is called a flow, but if T ⊆ Z, then (X,T, π) is called acascade (discrete flow).

Below X will be a complete metric space with the metric ρ.

4 DAVID CHEBAN

The function π(·, x) : T → X is called a motion passing through the point x atmoment t = 0 and the set Σx := π(T, x) is called a trajectory of this motion.

A nonempty set M ⊆ X is called positively invariant (negatively invariant, in-variant) with respect to dynamical system (X,T, π) or, simple, positively invariant(negatively invariant, invariant), if π(t,M) ⊆ M (M ⊆ π(t,M), π(t,M) = M) forevery t ∈ T.

A closed positively invariant set (respectively, invariant set), which does not containown closed positively invariant (respectively, invariant) subset, is called minimal.

Let M ⊆ X. The set

ω(M) :=⋂

t≥0

⋃

τ≥t

π(τ, M)

is called ω-limit for M .

The set W s(Λ), defined by equality

W s(Λ) := x ∈ X| limt→+∞

ρ(π(t, x), Λ) = 0

is called a stable manifold of the set Λ ⊆ X.

The set M is called:

- orbital stable, if for every ε > 0 there exists δ = δ(ε) > 0 such thatρ(x, M) < δ implies ρ(π(t, x),M) < ε for all t ≥ 0;

- attracting, if there exists γ > 0 such that B(M,γ) ⊂ W s(M), whereB(M, γ) := x ∈ X : ρ(x,M) < γ;

- asymptotic stable, if it is orbital stable and attracting;- global asymptotic stable, if it is asymptotic stable and W s(M) = X;- uniform attracting, if there exists γ > 0 such that

limt→+∞

supx∈B(M,γ)

ρ(π(t, x),M) = 0.

The system (X,T, π) is called:

− point dissipative if there exist a nonempty compact subset K ⊆ X suchthat for every x ∈ X

(4) limt→+∞

ρ(π(t, x),K) = 0;

− compactly dissipative if equality (4) takes place uniformly with respect tox on the compact subsets from X;

− locally dissipative if for any point p ∈ X there exist δp > 0 such thatequality (4) takes place uniformly with respect to x ∈ B(p, δp);

− bounded dissipative if equality (4) takes place uniformly with respect to xon every bounded subset from X;

− local completely continuous (compact) if for all point p ∈ X there are twopositive numbers δp and lp such that the set π(lp, B(p, δp)) is relativelycompact.


Let (X,T, π) be compactly dissipative and K be a compact set attracting everycompact subset from X. Let us set

(5) J = ω(K) :=⋂

t≥0

⋃

τ≥t

π(τ,K).

It can be shown [5, Ch.I] that the set J defined by equality (5) doesn’t dependson the choice of the attractor K, but is characterized only by the properties ofthe dynamical system (X,T, π) itself. The set J is called a Levinson center of thecompactly dissipative dynamical system (X,T, π).

Denote by

D+(M) :=⋂ε>0

⋃π(t, B(M, ε))|t ≥ 0,

J+(M) :=⋂ε>0

⋂

t≥0

⋃π(τ, B(M, ε))|τ ≥ t,

D+x := D+(x) and J+

x := J+(x).Lemma 2.1. Let (X,T, π) be a dynamical system and x ∈ X be a point withrelatively compact semi-trajectory Σ+

x := π(t, x) : t ≥ 0. Then the followingstatements hold:

(i) the dynamical system (X,T, π) induces on the H+(x) := Σ+

x a dynamicalsystem (H+(x),T+, π), where by bar is denoted the closure of Σ+

x in thespace X;

(ii) the dynamical system (H+(x),T+, π) is compactly dissipative;(iii) Levinson center JH+(x) of (H+(x),T+, π) coincides with ω-limit set ωx of

the point x.

Proof. The first and second statements are evident. Now we will establish theequality JH+(x) = ωx. It is clear that ωx ⊆ JH+(x) because ωx ⊆ H+(x) is acompact and invariant subset of (H+(x),T+, π). To finish the proof of Lemmait is sufficient to show that JH+(x) ⊆ ωx. Let y ∈ ωx, then there are sequencesxn ⊆ H+(x) and tn ⊆ T+ such that tn →∞ and π(tn, xn) → y as n → ∞.Logically two cases are possible:

(i) there exists a subsequence xnk ⊆ xn such that xnk

⊆ Σ+x . This

means that there is a sequence sk ⊆ T+ such that xnk= π(sk, x)) and,

consequently, we have

y = limk→∞

π(tnk, xnk

) = limk→∞

π(tnk, π(sk, x)) = lim

k→∞π(tnk

+ sk, x).

Since sk + tnk→∞ as k →∞, then y ∈ ωx.

(ii) there exists a subsequence xnk ⊆ xn such that xnk

⊆ ωx and,consequently, π(tnk

, xnk) ∈ ωx for all k ∈ N. Since the set ωx is invariant

and closed, then y also belongs to ωx.

Lemma is completely proved. ¤

6 DAVID CHEBAN

3. G. Sell’s conjecture for non-autonomous dynamical systems

Let T1 ⊆ T2 ⊆ T be two sub-semigroups of S and ((Y,T2, σ) be a dynamical systemon metric space Y . Recall that a triplet 〈W,ϕ, (Y, T2, σ)〉 (or shortly ϕ), where Wis a metric space and ϕ is a mapping from T1 × W × Y into W , is said to be acocycle over (Y,T2, σ) with the fiber W , if the following conditions are fulfilled:

(i) ϕ(0, u, y) = u for all u ∈ W and y ∈ Y ;(ii) ϕ(t + τ, u, y) = ϕ(t, ϕ(τ, u, y), σ(τ, y)) for all t, τ ∈ T1, u ∈ W and y ∈ Y ;(iii) the mapping ϕ : T1 ×W × Y 7→ W is continuous.

Example 3.1. (Bebutov’s dynamical system) Let X, W be two metric space.Denote by C(T × W,X) the space of all continuous mappings f : T × W 7→ Xequipped with the compact-open topology and σ be the mapping from T× C(T×W,X) into C(T × W,X) defined by the equality σ(τ, f) := fτ for all τ ∈ T andf ∈ C(T×W,X), where fτ is the τ -translation (shift) of f with respect to variablet, i.e., fτ (t, x) = f(t+ τ, x) for all (t, x) ∈ T×W . Then [5, Ch.I] the triplet (C(T×W,X),T, σ) is a dynamical system on C(T×W,X) which is called a shift dynamicalsystem (dynamical system of translations or Bebutov’s dynamical system).

A function f ∈ C(T × W,X) is said to be recurrent with respect to time t ∈ Tuniformly with respect to spacial variable x ∈ W on every compact subset fromW , if f ∈ C(T × W,X) is a recurrent point of the Bebutov’s dynamical system(C(T×W,X),T, σ).

Example 3.2. Consider differential equation (1) with regular second right handf ∈ C(R × W,Rn), where W ⊆ Rn. Denote by (H+(f),R+, σ) a semi-groupshift dynamical system on H+(f) induced by Bebutov’s dynamical system (C(R×W,Rn),R, σ), where H+(f) := fτ : τ ∈ R+. Let ϕ(t, u, g) a unique solution ofequation

y′ = g(t, y), (g ∈ H+(f)),then from the general properties of the solutions of non-autonomous equations itfollows that the following statements hold:

(i) ϕ(0, u, g) = u for all u ∈ W and g ∈ H+(f);(ii) ϕ(t + τ, u, g) = ϕ(t, ϕ(τ, u, g), gτ ) for all t, τ ∈ R+, u ∈ W and g ∈ H+(f);(iii) the mapping ϕ : R+ ×W ×H+(f) 7→ W is continuous.

From above it follows that the triplet 〈W,ϕ, (H+(f),R+, σ)〉 is a cocycle over(H+(f), R+, σ) with the fiber W ⊆ Rn. Thus, every non-autonomous equation (1)with regular f naturally generates a cocycle which plays a very important role inthe qualitative study of equation (1).

Suppose that W ⊆ E, where E is a Banach space with the norm | · |, 0 ∈ W (0is the null element of E) and the cocycle 〈W,ϕ, (Y,T2, σ)〉 admits a trivial (null)motion/solution, i.e., ϕ(t, 0, y) = 0 for all t ∈ T1 and y ∈ Y .

The trivial motion/solution of cocycle ϕ is said to be:

(i) uniformly stable, if for all positive number ε there exists a number δ = δ(ε)(δ ∈ (0, ε)) such that |u| < δ implies |ϕ(t, u, y)| < ε for all t ≥ 0 and y ∈ Y ;


(ii) uniformly attracting, if there exists a positive number a such that

(6) limt→+∞

|ϕ(t, u, y)| = 0

uniformly with respect to |u| ≤ a and y ∈ Y ;(iii) uniformly asymptotically stable, if it is uniformly stable and uniformly

attracting.

G.Sell’s conjecture for cocycle. Suppose that 〈W,ϕ, (Y, T2, σ)〉 is a cocycleunder (Y,T2, σ) with the fiber W and the following conditions are fulfilled:

(i) the cocycle ϕ admits a trivial motion/solution;(ii) the space Y is compact;(iii) there exists a positive constant a such that (6) takes place uniformly with

respect to |u| ≤ a and y ∈ JY , where JY is Levinson center (maximal com-pact invariant set) of compactly dissipative dynamical system (Y,T2, σ);

Recall [5] that a triplet 〈(X,T1, π), (Y,T2, σ), h〉 is said to be a NDS, where (X,T1, π)(respectively, (Y,T2, σ)) is a dynamical system on X (respectively, Y ) and h is anhomomorphism from (X,T1, π) onto (Y,T2, σ).

Below we will give some examples of nonautonomous dynamical systems which playa very important role in the study of nonautonomous differential equations.

Example 3.3. (NDS generated by cocyle.) Note that every cocycle 〈W, ϕ,(Y, T2, σ)〉 naturally generates a NDS. In fact. Let X := W × Y and (X,T1, π)be a skew-product dynamical system on X (i.e., π(t, x) := (ϕ(t, u, y), σ(t, y)) forall t ∈ T1 and x := (u, y) ∈ X), then the triplet 〈(X,T1, π), (Y,T2, σ), h〉, whereh := pr2 : X 7→ Y is the second projection (i.e., h(u, y) = y for all u ∈ W andy ∈ Y ), is a NDS.

Example 3.4. (NDS generated by tangent flow.) Let M be a compact dif-ferentiable manifold and (M,R, σ) be a differentiable flow on M . Denote by TMthe tangent bundle of M and by Dσ : R× TM 7→ TM the derivative flow on TM :if x := (u, y) ∈ TM , i.e., u ∈ TyM (TyM is the tangent space for M at the pointy ∈ M), then π(t, (u, y)) := (Dσt(y)u, σt(y)) (where σt(y) := σ(t, y) for all t ∈ Rand y ∈ Y ). The triplet 〈(TM,R, π), (M,R, σ), h〉, where h : TM 7→ M is thenatural projection (h(u, y) := y for all y ∈ Y and u ∈ TyM), is a nonautonomousdynamical system.

Example 3.5. (NDS generated by equation (1) with non-regular righthand side.) Let f ∈ C(R ×W,Rn) be a local Lipschitzian function and considera differential equation (1). Let Y = H(f) := fτ : τ ∈ R and by bar is denoted aclosure in the space C(R×W,Rn) and (Y,R, σ) be a shift dynamical system on Y .Denote by X the set of all points x := (u, g) ∈ W × Y such that the equation

(7) y′ = g(t, y)

admits a (unique) solution ϕ(t, u, g) defined on R+ with initial data ϕ(0, u, g) = u.Now we define a mapping π : R+ × X 7→ X by the following way: π(τ, (u, g)) :=(ϕ(τ, u, g), gτ ) for all τ ∈ R+ and (u, g) = x ∈ X. From the general properties ofsolutions of ODEs it follows that the triplet (X,R+, π) is a semi-group dynamicalsystem. Note that a triplet 〈(X,R+, π), (Y,R, σ), h〉, where h : X 7→ Y is the secondprojection, is a nonautonomous dynamical system generated by equation (1).

8 DAVID CHEBAN

Remark 3.6. 1. Note that the NDS in Example 3.4 (respectively, Example 3.5) isnot generated by a cocycle.

2. Equation (1) from Example 3.5, generally speaking, is not regular because wesuppose the uniqueness of Cauchy problem for (1) (the function f is local Lip-schitzian), but we do not suppose the existence on the all space W . As a conse-quence we obtain that the bundle space (X, h, Y ), figuring in Example 3.5, is nottrivial.

Let (X,h, Y ) be a vectorial bundle. Denote by θy the null element of the vectorialspace Xy := x ∈ X : h(x) = y and Θ := θy : y ∈ Y is the null section and of(X, h, Y ).

A vectorial bundle (X, h, Y is said to be local trivial with fiber F if for every pointy ∈ Y there exists a neighborhood U of the point y (U is an open subset of Ycontaining y) such that h−1(U) and U × F are homeomorphic, i.e., there exists anhomeomorphism α : h−1(U) 7→ U × F (trivialization).

Lemma 3.7. Let (X, h, Y ) be a vectorial bundle and Θ be its null section. Supposethat the following conditions hold:

(i) the space Y is compact;(ii) the vectorial bundle (X, h, Y ) is local trivial.

Then the trivial section Θ is compact.

Proof. Let Y be compact and y ∈ Y . Since (X, h, Y ) is trivial, then there exists aneighborhood Uy of the point y such that h−1(Uy) and Uy ×F are homeomorphic.Let θyk

be an arbitrary sequence from Θ. We will prove that from θyk can be

extracted a convergent subsequence. In fact, since the space Y is compact thenwithout loss of generality we may suppose that the sequence yk converges toy0 ∈ Y and, consequently, there exists a natural number k0 such that yk ∈ Uy0 forall k ≥ k0. Denote by α the homeomorphism (trivialization) between h−1(Uy0) andUy0 × F , then θyk

= α−1(yk, 0) (0 is the null element of the vectorial space F ) forall k ≥ k0 and, consequently, θyk

→ θy0 as k →∞. ¤

Consider a NDS 〈(X,T1, π), (Y,T2, σ), h〉 on the vectorial bundle (X, h, Y ). Every-where in this paper we suppose that the null section Θ of (X, h, Y ) is a positivelyinvariant set, i.e., π(t, θ) ∈ Θ for all θ ∈ Θ and t ≥ 0 (t ∈ T1).

The null (trivial) section Θ of NDS 〈(X,T1, π), (Y,T2, σ), h〉 is said to be:

(i) uniformly stable, if for every ε > 0 there exists a δ = δ(ε) > 0 such that|x| < δ implies |π(t, x)| < ε for all t ≥ 0 (t ∈ T1);

(ii) attracting, if there exists a number ν > 0 such that B(Θ, ν) ⊆ W s(Θ),where B(Θ, ν) := x ∈ X| |x| < ν;

(iii) uniform attracting, if there exists a number ν > 0 such that

limt→+∞

sup|π(t, x)| : |x| ≤ ν = 0;

(iv) asymptotically stable (respectively, uniformly asymptotically stable) if, Θis uniformly stable and attracting (respectively, uniformly attracting).


Let (Y,T2, σ) be a compactly dissipative dynamical system, JY its Levinson centerand 〈(X,T1, π), (Y,T2, σ), h〉 be a NDS. Denote by X := h−1(JY ) = x ∈ X :h(x) = y ∈ JY , then evidently the following statements are fulfilled:

(i) X is closed;(ii) π(t, X) ⊆ X for all t ∈ T1 and, consequently, on the set X is induced by

(X,T1, π) a dynamical system (X,T1, π));(iii) the triplet 〈(X,T1, π), (JY ,T2, σ), h〉 is a NDS.

G.Sell’s conjecture for NDS. Suppose that 〈(X,T1, π), (Y,T2, σ), h〉 is a NDSand the following conditions are fulfilled:

(i) the null/trivial section Θ of (X, h, Y ) is a positively invariant set;(ii) the space Y is compact;(iii) the null section Θ of the NDS 〈(X,T1, π), (JY ,T2, σ), h〉 is uniformly

attracting.

Then the trivial section Θ of NDS 〈(X,T1, π), (Y,T2, σ), h〉 is uniformly asymptot-ically stable.

One of the main goal of this paper is a positive answer to G.Sell’s conjecture forgeneral NDS (Corollary 4.9).

4. Relation between different types of stability for NDS

In this Section we establish the relation between different types of stability of NDS.In particular, it is proved that for asymptotically compact dynamical system as-ymptotic stability and uniform asymptotic stability are equivalent (Theorem 4.3).The main results of this Section are Theorem 4.7 and Corollary 4.9 which contains positive answer to G. Sell’s conjecture for general NDS.

Theorem 4.1. Let Y be compact. If the null section Θ of non-autonomous dynam-ical system 〈(X,T1, π), (Y,T2, σ), h〉 is uniformly attracting, then it is uniformlyasymptotically stable.

Proof. To prove this statement it is sufficient that under the condition of Theorem,the null section Θ is uniformly stable. Let r be a positive number such that

(8) limt→+∞

sup|x|≤r

|π(t, x)| = 0.

If we suppose that under the condition of Theorem the set Θ is not uniformly stable,then there are ε0 > 0, |xn| ≤ 1/n and tn ≥ n such that

(9) |π(tn, xn)| ≥ ε0

for all n ∈ N. On the other hand from (8) we have for any 0 < ε < ε0/2 a positivenumber L = L(ε) such that

(10) |π(t, x)| < ε

for all |x| ≤ r and t ≥ L(ε). For sufficiently large n ∈ N (n > maxL, r−1) wehave |xn| < r and, consequently, from (10) we obtain

(11) |π(tn, xn)| < ε < ε0/2.

10 DAVID CHEBAN

Inequalities (9) and (11) are contradictory. The obtained contradiction completesthe proof of Theorem. ¤

Remark 4.2. 1. Note that Theorem 4.1 remains true without assumption of thecompactness of Y .

2. By Theorem 4.1, if Θ is uniformly attracting, then it is uniformly asymptoticallystable.

3. It is evident that from uniform asymptotic stability of Θ it follows its asymptoticstability. The converse statement, generally speaking, is not true (see, for example,[5, Ch.I] Example 1.8).

Theorem 4.3. Suppose that the following conditions are fulfilled:

(i) the space Y is compact;(ii) the null section Θ is asymptotically stable;(iii) the dynamical system (X,T1, π) is asymptotically compact.

Then the null section Θ is uniformly asymptotically stable.

Proof. By Lemma 3.7 the trivial section Θ is a compact positively invariant set.Since the null section Θ is asymptotically stable, then there exists a positive numberδ such that B(Θ, δ) ⊂ W s(Θ). Let µ be a positive number such that |π(t, x)| ≤ δ forall x ∈ X with |x| ≤ µ and t ≥ 0. Consider the set M := π(t, x) : |x| ≤ µ, t ≥ 0.It is cleat that M is bounded and positively invariant. Since the dynamical system(X,T1, π) is asymptotically compact, then there exists a nonempty compact set Kfrom X such that

limt→+∞

β(π(t,M),K) = 0,

where β(A,B) := supa∈A

ρ(a,B) and ρ(a,B) := infb∈B

ρ(a, b). According to Lemma 1.3

[5, Ch.I] the set Ω(M) is compact, invariant and

(12) limt→+∞

β(π(t,M), Ω(M)) = 0,

where π(s,M) := π(s, x) : x ∈ M. It is clear that Ω(M) ⊆ M ⊆ W s(Θ).According to Theorem 1.37 [5, Ch.I] the set Ω(Θ) ⊆ Θ is a maximal compactinvariant set of dynamical system (X,T1, π) in W s(Θ) and, consequently,

(13) Ω(M) ⊆ Ω(Θ) ⊆ Θ.

From (12) and (13) it follows that

limt→+∞

sup|x|≤µ

|π(t, x)| = 0.

Indeed, if we suppose that it is not true, then there are ε0 > 0, xn with |xn| ≤ µand tn → +∞ such that

(14) |π(tn, xn)| ≥ ε0

for all n ∈ N. According to Lemma 1.3 [5, Ch.I] the sequence π(tn, xn) is rela-tively compact and, consequently, without loss of generality we can suppose thatπ(tn, xn) is convergent. We denote by x := lim

n→∞π(tn, xn), then from (14) we


obtain |x| ≥ ε0 > 0. On the other hand x ∈ Ω(M) ⊆ Θ and, consequently, |x| = 0.The obtained contradiction proves our statement. The theorem is proved. ¤

Directly from Theorem 4.3 it follows the following statement.

Corollary 4.4. Suppose that the following conditions are fulfilled:

(i) the space Y is compact;(ii) the null section Θ is globally asymptotically stable;(iii) the dynamical system (X,T1, π) is asymptotically compact.

Then the null section Θ is globally uniformly asymptotically stable.


(i) the space Y is compact;(ii) the null section Θ is asymptotically stable;(iii) one of the following two conditions hold:

a. the dynamical system (X,T1, π) is completely continuous;b. the fiber bundle (X, h, Y ) is finite-dimensional.

Then the null section Θ is uniformly asymptotically stable.

Proof. This statement it follows from Theorem 4.3. Indeed, condition a. (or b.)implies the asymptotically compactness of the dynamical system (X,T1, π) and nowit is sufficient to apply Theorem 4.3. ¤

From Theorem 4.1 and Theorem 4.3 we have.


(i) the space Y is compact;(ii) the null section Θ is asymptotically stable;(iii) the dynamical system (X,T1, π) is asymptotically compact.

Then the null section Θ is uniformly asymptotically stable if and only if it is auniform attracting set.

Theorem 4.7. Let 〈(X,T1, π), (Y,T2, σ), h〉 be a NDS. Suppose that the followingconditions are fulfilled:

(i) the space Y is compact;(ii) the dynamical system (X,T1, π) is locally compact;(iii) the trivial section Θ of (X, h, Y ) is positively invariant;(iv) the trivial section Θ of NDS 〈(X,T1, π), (JY ,T2, σ), h〉 is uniformly at-

tracting.

Then the trivial section Θ of non-autonomous dynamical system 〈(X,T1, π), (Y,T2, σ),h〉 is uniformly stable.

12 DAVID CHEBAN

Proof. Since Θ is uniformly attracting with respect to NDS 〈(X,T1, π), (JY ,T2, σ),h〉, then there exists a positive number α such that

(15) limt→+∞

sup|π(t, x)| : |x| ≤ α, x ∈ X = 0.

According to local compactness of the dynamical system (X,T1, π) for all y ∈Y there exists a positive number ly such that the set π(ly, B[θy, α]) is relativelycompact, where B[x, δ] := x ∈ X : |x| ≤ δ. By Lemma 3.7 the trivial section Θis compact, then from the covering B[θy, α] : y ∈ Y of Θ we can extract a finitesub-covering B[θyi

, α] : i = 1, 2, . . . ,m. Put l := maxlyi: i = 1, 2, . . . , m, then

the set π(l, B(Θ, α)) is relatively compact, where α ∈ (0, α) such that B(Θ, α) ⊂⋃mi=1 B[θyi

, α].

Now we will show that for arbitrary ε ∈ (0, α) there exists a number δ(ε) ∈ (0, ε)such that |x| < δ implies |π(t, x)| < ε for all t ≥ 0 (t ∈ T1). If we suppose thatit is not true, then there are ε0 ∈ (0, α), a sequence 0 < δn → 0 as n → +∞ andtn → +∞ such that

|xn| ≤ δn, |π(tn, xn| = ε0 (if the time T1 is continuous, i.e., R+ ⊆ T1)

or|π(tn + 1, xn)| > ε0 (if the time T1 is discrete, i.e., T1 ⊆ Z)

and|π(t, xn)| ≤ ε0 for all t ∈ [0, tn].

Denote by xn := π(tn, xn) and xn := π(l, xn). According to choose of the numberl, the sequence xn is relatively compact. Let γn be a continuous mapping definedon S with the values from X, defined by equality

γn(s) :=

π(s + l + tn, xn) = π(s, xn), s ≥ −l − tnxn, s ≤ −l − tn.

The sequence γn is relatively compact with respect to compact-open topology. Infact, to establish this fact it is sufficient to show that for all r > 0 the sequence ϕn,where ϕn(t) := γn(t) for all t ∈ [−r, r], satisfies the conditions of Arcela-Ascolytheorem. This means that there exists a compact Kr ⊂ X that ϕn([−r, r]) ⊆ Kr

for all n ∈ N and ϕn is equi-continuous on [−r, r]. Since tn → +∞ as n →∞, thenfor the given r > 0 there exists a number n0 ∈ N such that [−r, r] ⊂ [−r−l−tn,+∞)for all n ≥ n0. Since γn([−r, r]) ⊆ π(l, B[Θ, α]), then in the quality of Kr we cantake the set π(l, B[Θ, α])

⋃M , where M :=

⋃n0i=1 γi([−r, r]). Now we will show

that for arbitrary ε > 0 there exists a δ = δ(ε) > 0 such that |s1 − s2| < δ impliesρ(ϕn(s1), ϕn(s2)) < ε. If we suppose that it is not true, then there are ε0 > 0,0 < δn → 0 and si

n ∈ [−r, r] (i = 1, 2) such that

(16) ρ(ϕn(s1n), ϕn(s2

n)) ≥ ε0.

From (16) we have

(17) ρ(π(s1n + l + tn, xn), π(s2

n + l + tn, xn)) ≥ ε0

for all n ≥ n0. Without loss of generality we can suppose that the sequence xn isconvergent. Denote by x := lim

n→∞xn and si := lim

n→∞si

n (i = 1, 2). Since |s1n − s2

n| <δn, then s1 = s2. Passing into limit in (17) as n →∞ we obtain

0 = ρ(π(s1 + r, x), π(s2 + r, x) ≥ ε0.


The obtained contradiction proves our statement. Thus the sequence γn is rel-atively compact in the compact-open topology. Without loss of generality we cansuppose that the sequence γn converges. Denote by γ the limit of γn, thenaccording to Theorem 1.3.5 [6] γ is an entire trajectory of (X,T1, π) with γ(0) = x.On the other hand we have

(18) |γ(−l)| = limn→∞

|γn(−l)| = limn→∞

|π(tn, xn)| = ε0 > 0,

if the time T1 is continuous and

(19) |γ(−l)| = limn→∞

|γn(−l)| = limn→∞

|π(tn + 1, xn)| ≥ ε0 > 0,

if the time T1 is discrete. Thus we have γ(−l)| ≥ ε0 > 0, i.e., the entire trajectoryγ of (X,T1, π) is not trivial. It easy to see that under the conditions of Theorem

(20) |γ(s)| ≤ α

for all s ≤ −l. On the other hand from (15) it follows that for ε0 > 0 there existsa positive number L = L(ε0) such that

(21) |π(t, x)| < ε0/2

for all t ≥ L and |x| ≤ α. Let t0 ≥ L, then from (18)-(21) we obtain

0 < ε0 ≤ |γ(−l)| = |π(t0, γ(−t0 − l))| < ε0/2.

The obtained contradiction complete the proof of Theorem. ¤

Remark 4.8. 1. Note that the assumption of the compactness of Y plays a veryimportant role in the proof of Theorem 4.3 and Theorem 4.7. Probably these The-orems remain true if we replace this condition by the weaker one: the dynamicalsystem (Y,T2, σ) is compactly dissipative. But it is an open problem.

2. Theorem 4.7 remains true:

(i) if we replace the condition of uniform attraction of Θ by the following:there exists a positive number α such that for all compact subset K ⊆B[Θ, α] we have

limt→+∞

sup|π(t, x)| : x ∈ K = 0;

(ii) if we replace the condition of local compactness for (X,T1, π) by the fol-lowing: there are positive numbers α and l such that the set π(l, B(Θ, α))is relatively compact.

Corollary 4.9. Under the conditions of Theorem 4.7 the trivial section Θ of NDS〈(X,T1, π), (Y,T2, σ), h〉 is uniformly asymptotically stable.

Proof. To prove this statement it is sufficient to show that Θ is attracting. Accord-ing to Theorem 4.7 there are positive constants α and l such that the set πlB(Θ, α)is relatively compact. Let ε0 ∈ (0, α] and a := δ(ε0) > 0 (without loss of generalitywe can suppose that a ≤ minα, α, where α is a positive number figuring in Corol-lary 4.9) be a positive number from uniform stability of Θ, then |π(t, x)| ≤ a for allt ≥ 0 and |x ≥ a. Thus the positive semi-trajectory Σ+

x with |x| ≤ a is relatively

14 DAVID CHEBAN

compact. Now it is easy to see that limt→+∞

|π(t, x)| = 0. In fact. If we suppose that

it is not true, then there exist ε0 > 0, 0 < |x0| ≤ a and tn → +∞ such that

(22) |π(tn, x0)| ≥ ε0.

Since the sequence π(tn, x0) is relatively compact, then we can suppose that itconverges. Denote by x0 its limit. Passing into limit in (22) as n → +∞ we obtain

(23) |x0| ≥ ε0 > 0.

Note that x0 is an ω-limit point for x0. Taking into account that the ω-limit set ωx0

is a nonempty, compact invariant set, then there exists an entire motion γ ∈ Φx0(π)such that |γ(s)| ≤ a ≤ α for all s ∈ S and we will have

(24) ε0 ≤ |x0| = |π(t, γ(−t))| ≤ supx∈H(γ)

|π(t, x)

for all t ≥ 0, where H(γ) := γ(s) : ∈ S is a compact subset of ωx0 . Passinginto limit in (24) and taking into account Remark 4.8 we receive ε0 ≤ 0. The lastinequality contradicts to (23). The obtained contradiction proves our statement.

¤

Remark 4.10. 1. Note that Corollary 4.9 gives a positive answer to G.Sell’sconjecture for local-compact NDS.

2. Application this result (Corollary 4.9) to ODEs (classical G. Sell’s conjecture[15, Ch.VIII,p.134]) we give below (Theorem 5.1 and its proof).

5. Some applications

5.1. Ordinary differential equations. Let E be a Banach space with the norm| · |, W be an open subset of E and 0 ∈ W . Denote by C(S×W,E) the space of allcontinuous mappings f : S ×W 7→ E equipped with the compact open topology.On the space C(S×W,E) it is defined a shift dynamical system [5, ChI] (dynamicalsystem of translations or Bebutov’s dynamical system) (C(S×W,E), S, σ), where σis a mapping from S×C(S×W,E) onto C(S×W,E) defined as follow σ(τ, f) := fτ

for all (τ, f) ∈ S × C(S × W,E), where fτ is the τ -translation of f with respectto time t, i.e., fτ (t, x) := f(t + τ, x) for all (t, x) ∈ S ×W . Consider a differentialequation

(25) u′ = f(t, u),

where f ∈ C(R×W,E).

If the function f is regular, then the equation (25) naturally defines a cocycle〈W,ϕ, (H+(f),R+, σ)〉, where (H+(f),R+, σ) is a (semi-group) dynamical systemon H+(f) induced by Bebutov’s dynamical system.

Applying the general results from Sections 2-4 we will obtain a series of results forequation (25). Below we formulate some of them.

Denote by Ωf := g ∈ H+(f) : there exists a sequenceτn → +∞ such that g =lim

n→∞fτn the ω-limit set of f .


Theorem 5.1. Assume that the following conditions are fulfilled:

(i) the function f is regular;(ii) the set H+(f) is compact;(iii) f(t, 0) = 0 for all t ∈ R+;(iv) There exists a neighborhood U of the origin 0 and a positive number l such

that the set ϕ(l, U,H+(f)) is relatively compact;(v) there exists a positive number a such that

(26) limt→+∞

sup|v|≤a,g∈Ωf

|ϕ(t, v, g)| = 0.

Then the null solution of equation (25) is uniformly asymptotically stable.

Proof. Consider the dynamical system (H+(f),R+, σ). Since the space H+(f)is compact, then (H+(f),R+, σ) is compactly dissipative and its Levinson cen-ter (maximal compact invariant set) JH+(f) evidently coincides with ω-limit setΩf of f . Let Y := H+(f) and (Y,R+, σ) be the shift dynamical system onY . Denote by X := W × Y and (X,R+, π) the skew-product dynamical sys-tem generates by (Y,R+, σ) and cocycle ϕ, i.e., π(t, (v, g)) := (ϕ(t, v, g), σ(t, g)) forall t ∈ R+ and (v, g) ∈ X. Now consider a non-autonomous dynamical system〈(X,R+, π), (Y,R+, π), h〉 (h := pr2) associated by equation (25). It easy to verifythis NDS posses the following properties:

(i) by Lemma 2.1 the dynamical system (Y,R+, σ) is compactly dissipativeand its Levinson center JY coincides with Ωf ;

(ii) the null section Θ of 〈(X,R+, π), (Y,R+, π), h〉 coincides with 0 × Y ;(iii) Θ is a positively invariant subset of (X,R+, π);(iv) according to (26) the null section Θ of NDS 〈(X,R+, π), (JY ,R+, σ), h〉

is uniformly attracting because |π(t, x)| = |ϕ(t, v, g)| for all t ∈ R+ andx := (v, g) ∈ X.

Now to finish the proof it is sufficient to apply Corollary 4.9. ¤Remark 5.2. If the space E is finite-dimensional, then Theorem 5.1 coincides withthe result of Artstein Z. [1] and Bondi P. et all [2] because in this case the cocycleϕ associated by equation (25) is local compact.

5.2. Functional differential-equations. We will apply now the abstract theorydeveloped in the previous Sections to the analysis of a class of functional differentialequations.

5.2.1. Functional-differential equations (FDEs) with finite delay. Let us first recallsome notions and notations from [9]. Let r > 0, C([a, b],Rn) be the Banachspace of all continuous functions ϕ : [a, b] → Rn equipped with the sup–norm. If[a, b] = [−r, 0], then we set Cr := C([−r, 0],Rn). Let σ ∈ R, A ≥ 0 and u ∈C([σ − r, σ + A],Rn). We will define ut ∈ Cr for all t ∈ [σ, σ + A] by the equalityut(θ) := u(t + θ), −r ≤ θ ≤ 0. Consider a functional differential equation

(27) u = f(t, ut),

where f : R× Cr → Rn is continuous.

16 DAVID CHEBAN

Denote by C(R × C,Rn) the space of all continuous mappings f : R × C 7→ Rn

equipped with the compact open topology. On the space C(R × C,Rn) is defined(see, for example, [5, ChI]) a shift dynamical system (C(R × C,Rn),R, σ), whereσ(τ, f) := fτ for all f ∈ C(R× C,Rn) and τ ∈ R and fτ is τ -translation of f , i.e.,fτ (t, φ) := f(t + τ, φ) for all (t, φ) ∈ R× C.Let us set H+(f) := fs : s ∈ R+, where by bar we denote the closure in C(R ×C,Rn).

Along with the equation (27) let us consider the family of equations

(28) v = g(t, vt),

where g ∈ H+(f).

Below, in this subsection, we suppose that equation (27) is regular.

Remark 5.3. 1. Denote by ϕ(t, u, f) the solution of equation (27) defined on R+

(respectively, on R) with the initial condition ϕ(0, u, f) = u ∈ Cr, i.e., ϕ(s, u, f)= u(s) for all s ∈ [−r, 0]. By ϕ(t, u, f) we will denote below the trajectory ofequation (27), corresponding to the solution ϕ(t, u, f), i.e., the mapping from R+

(respectively, R) into Cr, defined by ϕ(t, u, f)(s) := ϕ(t + s, u, f) for all t ∈ R+

(respectively, t ∈ R) and s ∈ [−r, 0].

2. Due to item 1. of this remark, below we will use the notions of “solution” and“trajectory” for equation (27) as synonym concepts.

It is well known [3, 15] that the mapping ϕ : R+×Cr ×H+(f) 7→ Rn possesses thefollowing properties:

(i) ϕ(0, v, g) = u for all v ∈ Cr and g ∈ H+(f);(ii) ϕ(t + τ, v, g) = ϕ(t, ϕ(τ, v, g), σ(τ, g)) for all t, τ ∈ R+, v ∈ Cr and g ∈

H+(f);(iii) the mapping ϕ is continuous.

Thus, a triplet 〈Cr, ϕ, (H+(f),R+, σ)〉 is a cocycle which is associated to equation(27). Applying the results from Sections 2-4 we will obtain a series of results forfunctional differential equation (27). Below we formulate some of them.

Lemma 5.4. Suppose that the following conditions hold:

(i) the function f ∈ C(R×W, C) is regular;(ii) the set H+(f) is compact;(iii) the function f is completely continuous, i.e., the set f(R+×A) is bounded

for all bounded subset A ⊆ C.

Then the cocycle ϕ associated by (27) is completely continuous, i.e., for all boundedsubset A ⊆ W there exists a positive number l = l(A) such that the set ϕ(l, A,H+(f))is relatively compact in C.

Proof. This statement follows from the general properties of solutions of equation(27) (see, for example [9], Lemma 2.2.3 and Lemma 3.3.1) because the set H+(f)is compact. ¤



(i) the function f is regular;(ii) the set H+(f) is compact;(iii) f(t, 0) = 0 for all t ∈ R+;(iv) there exists a positive number a such that

(29) limt→+∞

sup|v|≤a,g∈Ωf

|ϕ(t, v, g)| = 0.


Proof. Consider the dynamical system (H+(f),R+, σ). Since the space H+(f) iscompact, then (H+(f),R+, σ) is compactly dissipative and by Lemma 2.1 its Levin-son center JH+(f) coincides with ω-limit set Ωf of f . Let Y := H+(f) and (Y,R+, σ)be the shift dynamical system on Y . Denote by X := C × Y and (X,R+, π) theskew-product dynamical system generates by (Y,R+, σ) and cocycle ϕ. Now con-sider a NDS 〈(X,R+, π), (Y,R+, π), h〉 (h := pr2) associated by equation (27). Iteasy to verify this NDS posses the following properties:

(i) the dynamical system (Y,R+, σ) is compact dissipative and its Levinsoncenter JY coincides with Ωf ;


is uniformly attracting because |π(t, x)| = |ϕ(t, v, g)| for all t ∈ R+ andx := (v, g) ∈ X;

(v) according to Lemma 5.4 the dynamical system (X,R+, π) is completelycontinuous.

Now to finish the proof it is sufficient to apply Corollary 4.9. ¤

5.3. Semi-linear parabolic equations. Let H be a separable Hilbert space withinner product 〈·, ·〉 and the norm | · |2 := 〈·, ·〉, and A be a self-adjoint operator withdomain D(A).

An operator is said (see, for example, [7]) to have a discrete spectrum if in the spaceH, there exists an ortho-normal basis ek of eigenvectors, such that 〈ek, ej〉 = δkj ,Aek = λkek (k, j = 1, 2, . . .) and 0 < λ1 ≤ λ2 ≤ . . . , λk ≤ . . ., and λk → +∞ ask → +∞.

One can define an operator f(A) for a wide class of functions f defined on thepositive semi-axis as follows:

D(f(A)) := h =∑∞

k=1 ckek ∈ H :∑∞

k=1 ck[f(λk)]2 < +∞,f(A)h :=

∑∞k=1 ckf(λk)ek, h ∈ D(f(A)).(30)

In particular, we can define operators Aα for all α ∈ R. For α = −β < 0 thisoperator is bounded. The space D(A−β) can be regarded as the completion of thespace H with respect to the norm | · |β := |A−β · |.The following statements hold [7]:

18 DAVID CHEBAN

(i) The space F−β := D(A−β) with β > 0 can be identified with the space offormal series

∑∞k=1 ckek such that

∞∑

k=1

ckλ−2βk < +∞;

(ii) For any β ∈ R, the operator Aβ can be defined on every space D(Aα) asa bounded operator mapping D(Aα) into D(Aα−β) such that

AβD(Aα) = D(Aα−β), Aβ1+β2 = Aβ1Aβ2 .

(iii) For all α ∈ R, the space F := D(Aα) is a separable Hilbert space with theinner product 〈·, ·〉α := 〈Aα·, Aα·〉 and the norm | · |α := |Aα · |.

(iv) The operator A with the domain F1+α is a positive operator with discretespectrum in each space Fα.

(v) The embedding of the space Fα into Fβ for α > β is continuous, i.e.,Fα ⊂ Fβ and there exists a positive constant C = C(α, β) such that| · |β ≤ C| · |α.

(vi) Fα is dense in Fβ for any α > β.(vii) Let α1 > α2, then the space Fα1 is compactly embedded into Fα2 , i.e.,

every sequence bounded in Fα1 is relatively compact in Fα2 .(viii) The resolvent Rλ(A) := (A−λI)−1, λ 6= λk is a compact operator in each

space Fα, where I is the identity operator.

According to (30) we can define an exponential operator e−tA, t ≥ 0, in the scalespaces Fα . Note some of its properties [7]:

a. For any α ∈ R and t > 0 the linear operator e−tA maps Fα into⋂

β≥0

Fβ

and|e−tAx|α ≤ e−λ1t|x|α

for all x ∈ Fα.b. e−t1Ae−t2A = e−(t1+t2)A for all t1, t2 ∈ R+;c.

|e−tAx− e−τAx|β → 0

as t → τ for every x ∈ Fβ and β ∈ R;d. For any β ∈ R the exponential operator e−tA defines a dissipative compact

dynamical system (Fβ), e−tA);e.

|Aαe−tAh| ≤[(

α−βt

)α−β + λα−β1

]e−tλ1 |Aβh|, α ≥ β

||Aαe−tA|| ≤ (αt

)αe−α, t > 0, α > 0.

Consider an evolutionary differential equation

(31) u′ + Au = F (t, u)

in the separable Hilbert space H, where A is a linear (generally speaking un-bounded) positive operator with discrete spectrum, and F is a nonlinear continuousmapping acting from R×Fθ into H, 0 ≤ θ < 1, possessing the property

(32) |F (t, u1)− F (t, u2)| ≤ L(r)|Aθ(u1 − u2)|


for all u1, u2 ∈ Bθ(0, r) := u ∈ Fθ : |u|θ ≤ r. Here L(r) denotes the Lipschitzconstant of F on the set Bθ(0, r).

A function u : [0, a) 7→ Fθ is said to be a weak solution (in Fθ) of equation (31)passing through the point x ∈ Fθ at moment t = 0 (notation ϕ(t, x, F )) if u ∈C([0, T ],Fθ) and satisfies the integral equation

u(t) = e−tAx +∫ t

0

e−(t−τ)AF (τ, u(τ))dτ

for all t ∈ [0, T ] and 0 < T < a.

In the book [7], it is proved that, under the conditions listed above, there exists aunique solution ϕ(t, x, F ) of equation (32) passing through the point x at momentt = 0, and it is defined on a maximal interval [0, a), where a is some positive numberdepending on (x, F ).

Denote by C(R× Fθ,H) the space of all continuous mappings equipped with thecompact open topology and by (C(R×Fθ,H),R, σ) the shift dynamical system onC(R×Fθ,H).

Denote by (H+(F ),R+, σ) a shift dynamical system on H+(F ) induced by (C(R×Fθ,H),R, σ). From general properties of solutions of evolution equation (31) andTheorem 5.1 [4] it follows that the triplet 〈Fθ, ϕ, (H+(F ),R+, σ)〉 is a cocycle over(H+(F ),R+, σ) with the fiber Fθ.

Applying results from Sections 2-4 we obtain a series of results for evolution equa-tion (31). Now we will formulate some of them.

Lemma 5.6. Under the conditions listed above, if the function F is regular and theset H+(F ) is compact, then the cocycle ϕ associated by equation (31) is completelycontinuous.

Proof. This statement can be proved with the slight modification of the proof ofLemma 5.3 [4]. ¤


(i) the function F is regular;(ii) the set H+(F ) is compact;(iii) F (t, 0) = 0 for all t ∈ R+;(iv) there exists a positive number a such that

(33) limt→+∞

sup|v|≤a,G∈ΩF

|ϕ(t, v, G)| = 0.


Proof. Consider the dynamical system (H+(F ),R+, σ). Since the space H+(F )is compact, then (H+(f),R+, σ) is compactly dissipative and its Levinson centerJH+(F ) coincides with ω-limit set ΩF of F . Let Y := H+(F ) and (Y,R+, σ) bethe shift dynamical system on Y . Denote by X := Fθ × Y and (X,R+, π) theskew-product dynamical system generates by (Y,R+, σ) and cocycle ϕ. Consider a

20 DAVID CHEBAN

non-autonomous dynamical system 〈(X,R+, π), (Y,R+, π), h〉 (h := pr2) associatedby equation (31). It easy to verify that this NDS posses the following properties:

(i) the dynamical system (Y,R+, σ) is compact dissipative and by Lemma 2.1its Levinson center JY coincides with ΩF ;


is uniformly attracting because |π(t, x)| = |ϕ(t, v, g)| for all t ∈ R+ andx := (v, g) ∈ X;

(v) by Lemma 5.6 the cocycle ϕ and, consequently, the skew-product dynam-ical system (X,R+, π) too, is completely continuous.

Now to finish the proof it is sufficient to apply Corollary 4.9. ¤

Acknowledgements. We would like to thank the anonymous referees for thehelpful suggestions which allowed us to improve the presentation of this article.

References

[1] Artstein Zvi, Uniform asymptotic stability via the limiting equations. Journal of DifferentialEquations, 27(2), 172-189, 1978.

[2] Bondi P., Moaurd V. and Visentin F., Limiting equations in the stability problem. NonlinearAnalisys, v.1, no.2, 123–128, 1977.

[3] Bronsteyn I. U., Extensions of Minimal Transformation Group, Noordhoff, 1979.[4] Caraballo Tomas and Cheban David, On the Structure of the Global Attractor for Infinite-

Dimensional Non-autonomous Dynamical Systems with Weak Convergence. Communica-tions in Pure and Applied Analysis, to appear (2012).

[5] Cheban D. N., Global Attractors of Non-Autonomous Dissipstive Dynamical Systems. In-terdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, xxiii+502pp.

[6] Cheban D. N., Global Attractors of Set-Valued Dynamical and Control Systems. Nova Sci-ence Publishers Inc, New York, 2010, xvii+269 pp.

[7] Chueshov, I. D., Vvedenie v teoriyu beskonechnomernykh dissipativnykh sistem. Univer-sitetskie Lektsii po Sovremennoi Matematike. AKTA, Kharkiv, 1999. 436 pp. (in Russian)[English translation: Introduction to the theory of infinite-dimensional dissipative systems.University Lectures in Contemporary Mathematics. AKTA, Kharkiv, 1999. 436 pp.]

[8] D’Anna A., Total stability properties for an AP equation by means of limiting equations.FEs, 27 (1984), 201–209.

[9] Hale J. K., Theory of Functional-Differential Equations. Springer-Verlag, New York-Heidelberg-Berlin, 1977. [Russian translation: Theory of Functional-Differential Equations.Mir, Moscow, 1984.]

[10] Hino Yoshiyuki and Murakami Satoru, Limiting equations and some stability properties forasymptotically almost periodic FDEs with infinite delay. The Tohoku Mathematical Journal,54 (2002), pp.239-257.

[11] Junji Kato, A.A. Martynyuk and A.A. Shestakov, Stability of Motion of NonautonomousSystems: Methods of Limiting Equations. Taylor & Francis Ltd,1996, 304 pp.

[12] Kato J. and Yoshizawa T., A relationship between uniform asymptotic stability and totalstability. Funkcialaj Ekvacioj, 12 (1969), 233–238.

[13] Kato J. and Yoshizawa T., Uniformly asymptotic stability and total stability. The TohokuMathematical Journal, 22 (1970), 254–269.

[14] Kato J. and Yoshizawa T., Stability under the perturbations by a class of functions. TheTohoku Mathematical Journal, 22 (1970), 254–269.

[15] Sell G. R., Topological Dynamics and Ordinary Differential Equations. Van Nostrand-Reinhold, London, 1971.


[16] Visentin F., Limiting equations and total stability. Applied Nonlinear Analysis, 1977,pp.721–726.

[17] Visentin F., Limiting equations in the problem of total stability. Ricerche Mat., 28 (1979),no.2, 323–331.

(D. Cheban) State University of Moldova, Department of Mathematics and Informatics,A. Mateevich Street 60, MD–2009 Chisinau, Moldova

E-mail address, D. Cheban: [email protected]

Documents

Sell’s conjecture for non-autonomous dynamical systems