bohr/levitan almost periodic and almost automorphic solutions of linear stochastic differential

36 Pages·2017·0.21 MB·English

BOHR/LEVITAN ALMOST PERIODIC AND ALMOST
AUTOMORPHIC SOLUTIONS OF LINEAR STOCHASTIC
DIFFERENTIAL EQUATIONS WITHOUT FAVARD’S
SEPARATION CONDITION.
DAVIDCHEBAN
Abstract. Weprovethatthelinearstochasticequationdx(t)=(A(t)x(t)+
f(t))dt+g(t)dW(t) with linear operator A(t) generating a continuous linear
cocycleφandBohr/Levitanalmostperiodicoralmostautomorphiccoefficients
(A(t);f(t);g(t)) admits a unique Bohr/Levitan almost periodic (respectively,
almost automorphic) solution in distribution sense if it has at least one pre-
compactsolutiononR+ andthelinearcocycleφisasymptoticallystable.
Dedicated to the memory of Professor V. V. Zhikov.
1. Introduction
This paper is dedicated to the study of linear stochastic differential equations with
Bohr/Levitan almost periodic and almost automorphic coefficients. This (cid:12)eld is
called Favard’s theory [20, 33], due to the fundamental contributions made by J.
Favard[15]. In1927,J.Favardpublishedhiscelebratedpaper,wherehestudiedthe
problem of existence of almost periodic solutions of equation in R of the following
form:
′
(1) x =A(t)x+f(t)
with the matrix A(t) and vector-function f(t) almost periodic in the sense of Bohr
(see, for example, [16, 20]).
Along with equation (1), consider the homogeneous equation
′
x =A(t)x
and the corresponding family of limiting equations
′
(2) x =B(t)x;
where B 2H(A); and H(A) denotes the hull of almost periodic matrix A(t) which
is composed by those functions B(t) obtained as uniform limits on R of the type
B(t):= lim A(t+t ); where ft g is some sequence in R.
n n
n!1
Date:October4,2017.
1991 Mathematics Subject Classi(cid:12)cation. 34C27,35B15,37B55,60H10,60H15.
Key words and phrases. Bohr/Levitan almost periodic solutions, almost automorphic solu-
tions,linearstochasticdifferentialequations,Favardtheory.
1
2 DAVIDCHEBAN
Theorem 1.1. (Favard’s theorem [15]) The linear differential equation (1) with
BohralmostperiodiccoefficientsadmitsatleastoneBohralmostperiodicsolutionif
it has a bounded solution, and each bounded solution φ(t) of every limiting equation
(2) (B 2H(A)) is separated from zero, i.e.
(3) inf jφ(t)j>0:
t2R
This result was generalized in(cid:12)nite-dimensional equation in the works of V. V.
Zhikov and B. M. Levitan [33] (see also B. M. Levitan and V. V. Zhikov [20,
ChVIII]).
Favard’s theorem for linear differential equation with Levitan almost periodic (re-
spectively, almost automorphic) coefficients was established by B. M. Levitan [19,
ChIV] (respectively, by Lin F. [21]).
For linear stochastic differential equation Favard’s theorem was established by Liu
Z. and Wang W. in [22].
In the work [2] it was proved that Favard’s theorem remains true if we replace
condition (3) by the following:
(4) inf jφ(t)j=0:
t!+1
In this paper we establish that Favard’s theorem remains true for linear stochastic
differential equations under the condition (4).
This paper is organized as follows.
InSection2wecollectsomewellknownfactsfromthetheoryofdynamicalsystems
(both autonomous and non-autonomous). Namely, the notions of almost periodic
(both in the Bohr and Levitan sense), almost automorphic and recurrent motions;
cocycle, skew-product dynamical system, and general non-autonomous dynami-
cal system, comparability of motions by character of recurrence in the sense of
Shcherbakov [26]-[28].
Section 3 is dedicated to the proof of classical Birkoff’s theorem (about existence
of compact minimal set) for non-autonomous dynamical systems.
InSection4westudytheproblemofstronglycomparabilityofmotionsbycharacter
of recurrence of semi-group non-autonomous dynamical systems. The main result
is contained in Theorem 4.11 and it generalizes the known Shcherbakov’s result
[26, 28].
Section 5 is dedicated to the shift dynamical systems and different classes of Pois-
son stable functions. In particular: quasi-periodic, Bohr almost periodic, almost
automorphic functions and many others.
In Section 6 we collect some results and constructions related with Linear (homo-
geneous and nonhomogeneous) Differential Systems. We also discusses here the
relationbetweentwode(cid:12)nitionsofhyperbolicity(exponentialdichotomy)forlinear
non-autonomous systems. The main result of this section (Theorem 6.13) estab-
lish the equivalence of two de(cid:12)nitions for (cid:12)nite-dimensional and for some classes of
in(cid:12)nite-dimensional systems.
BOHR/LEVITAN ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF ...3
Section 7 is dedicated to the study of Bohr/Levitan almost periodic and almost
automorphicsolutionsofLinearStochasticDifferentialEquations. Themainresults
(Theorems7.8, 7.13andCorollaries7.9, 7.14)showthatclassicalFavard’stheorem
remains true (under some conditions) for linear stochastic differential equations.
2. Cocycles, Skew-Product Dynamical Systems and Non-Autonomous
Dynamical Systems
Let X be a complete metric space, R (Z) be a group of real (integer) numbers, R
+
(Z ) be a semi-group of nonnegative real (integer) numbers, T be one of the two
+
setsRorZandS(cid:18)T(T (cid:18)S)beasub-semigroupoftheadditivegroupT,where
+
T :=ft2T: t(cid:21)0g.
+
Let (X;S;(cid:25)) be a dynamical system.
De(cid:12)nition 2.1. Let (X;T ;(cid:25)) and (Y;T ;(cid:27)) (T (cid:18) T (cid:18) T (cid:18) T) be two dy-
1 2 + 1 2
namical systems. A mapping h : X ! Y is called a homomorphism (isomor-
phism, respectively) of the dynamical system (X;T ;(cid:25)) onto (Y;T ;(cid:27)), if the map-
1 2
ping h is continuous (homeomorphic, respectively) and h((cid:25)(x;t)) = (cid:27)(h(x);t) (
t 2 T ; x 2 X). In this case the dynamical system (X;T ;(cid:25)) is an extension of
1 1
the dynamical system (Y;T ;(cid:27)) by the homomorphism h, but the dynamical system
2
(Y;T ;(cid:27))iscalledafactorofthedynamicalsystem(X;T ;(cid:25))bythehomomorphism
2 1
h. The dynamical system (Y;T ;(cid:27)) is called also a base of the extension (X;T ;(cid:25)).
2 1
De(cid:12)nition 2.2. A triplet ⟨(X;T ;(cid:25)); (Y;T ;(cid:27)); h⟩, where h is a homomorphism
1 2
from(X;T ;(cid:25))onto(Y;T ;(cid:27));iscalledanon-autonomousdynamicalsystem(NDS).
1 2
De(cid:12)nition 2.3. A triplet ⟨W;φ;(Y;T ;(cid:27))⟩ (or shortly φ), where (Y;T ;(cid:27)) is a dy-
2 2
namical system on Y, W is a complete metric space and φ is a continuous mapping
from T (cid:2)W (cid:2)Y to W, satisfying the following conditions:
1
a. φ(0;u;y)=u (u2W;y 2Y);
b. φ(t+(cid:28);u;y)=φ((cid:28);φ(t;u;y);(cid:27)(t;y)) (t;(cid:28) 2T ; u2W;y 2Y);
1
is called [24] a cocycle on (Y;T ;(cid:27)) with the (cid:12)ber W.
2
De(cid:12)nition 2.4. Let X := W (cid:2)Y and de(cid:12)ne a mapping (cid:25) : X (cid:2)T ! X as
1
following: (cid:25)((u;y);t) := (φ(t;u;y);(cid:27)(t;y)) (i.e., (cid:25) = (φ;(cid:27))). Then it is easy
to see that (X;T ;(cid:25)) is a dynamical system on X which is called a skew-product
1
dynamical system [24] and h = pr : X ! Y is a homomorphism from (X;T ;(cid:25))
2 1
onto (Y;T ;(cid:27)) and, consequently, ⟨(X;T ;(cid:25)); (Y;T ;(cid:27));h⟩ is a non-autonomous
2 1 2
dynamical system.
Thus,ifwehaveacocycle⟨W;φ;(Y;T ;(cid:27))⟩onthedynamicalsystem(Y;T ;(cid:27))with
2 2
the (cid:12)ber W, then it generates a non-autonomous dynamical system ⟨(X;T ;(cid:25));
1
(Y;T ;(cid:27));h⟩ (X :=W (cid:2)Y) called a non-autonomous dynamical system generated
2
by the cocycle ⟨W;φ;(Y;T ;(cid:27))⟩ on (Y;T ;(cid:27)).
2 2
Non-autonomous dynamical systems (cocycles) play a very important role in the
study of non-autonomous evolutionary differential equations. Under appropriate
assumptionseverynon-autonomousdifferentialequationgeneratesacocycle(anon-
autonomous dynamical system). Below we give some examples of theses.
4 DAVIDCHEBAN
Example 2.5. Let E be a real or complex Banach space and Y be a metric space.
Denote by C(Y (cid:2) E;E) the space of all continuous mappings f : Y (cid:2) E 7! E
endowed by compact-open topology. Consider the system of differential equations
{
u′ = F(y;u)
(5)
y′ = G(y);
where Y (cid:18)E;G2C(Y;E) and F 2C(Y (cid:2)E;E). Suppose that for the system (5)
the conditions of the existence, uniqueness, continuous dependence of initial data
and extendability on R are ful(cid:12)lled. Denote by (Y;R ;(cid:27)) a dynamical system
+ +
on Y generated by the second equation of the system (5) and by φ(t;u;y) { the
solution of equation
′
(6) u =F(yt;u) (yt:=(cid:27)(t;y))
passingthroughthepointu2E fort=0. Thenthemappingφ:R (cid:2)E(cid:2)Y !E
+
is continuous and satis(cid:12)es the conditions: φ(0;u;y) = u and φ(t + (cid:28);u;y) =
φ(t;φ((cid:28);u;y);yt) for all t;(cid:28) 2 R , u 2 E and y 2 Y and, consequently, the sys-
+
tem (5) generates a non-autonomous dynamical system ⟨(X;R ;(cid:25));(Y;R ;(cid:27));h⟩
+ +
(where X :=E(cid:2)Y, (cid:25) :=(φ;(cid:27)) and h:=pr :X !Y).
2
We will give some generalization of the system (5). Namely, let (Y;R ;(cid:27)) be a
+
dynamical system on the metric space Y. Consider the system
{
u′ = F(yt;u)
(7)
y 2 Y;
whereF 2C(Y(cid:2)E;E). Supposethatfortheequation(6)theconditionsoftheexis-
tence,uniquenessandextendabilityonR areful(cid:12)lled. Thesystem⟨(X;R ;(cid:25));(Y;
+ +
R ;(cid:27));h⟩, where X := E (cid:2)Y, (cid:25) := (φ;(cid:27)), φ((cid:1);u;y) is the solution of (6) and
+
h := pr : X ! Y is a non-autonomous dynamical system generated by the equa-
2
tion (7).
Example 2.6. Let us consider a differential equation
′
(8) u =f(t;u);
where f 2 C(R(cid:2)E;E). Along with equation (8) we consider its H-class [1],[20],
[24], [28], i.e., the family of equations
′
(9) v =g(t;v);
whereg 2H(f):=ff(cid:28) :(cid:28) 2Rg,f(cid:28)(t;u):=f(t+(cid:28);u)forall(t;u)2R(cid:2)E andby
bar we denote the closure in C(R(cid:2)E;E). We will suppose also that the function
f is regular, i.e. for every equation (9) the conditions of the existence, uniqueness
and extendability on R are ful(cid:12)lled. Denote by φ((cid:1);v;g) the solution of equation
+
(9) passing through the point v 2 E at the initial moment t = 0. Then there
is a correctly de(cid:12)ned mapping φ : R (cid:2)E (cid:2)H(f) ! E satisfying the following
+
conditions (see, for example, [1], [24]):
1) φ(0;v;g)=v for all v 2E and g 2H(f);
2) φ(t;φ((cid:28);v;g);g(cid:28))=φ(t+(cid:28);v;g)foreveryv 2E, g 2H(f)andt;(cid:28) 2R ;
+
3) the mapping φ:R (cid:2)E(cid:2)H(f)!E is continuous.
+
Denote by Y := H(f) and (Y;R ;(cid:27)) a dynamical system of translations (a semi-
+
group system) on Y, induced by the dynamical system of translations (C(R (cid:2)
BOHR/LEVITAN ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF ...5
E;E);R;(cid:27)). The triplet ⟨E;φ;(Y;R ;(cid:27))⟩ is a cocycle on (Y;R ;(cid:27)) with the (cid:12)ber
+ +
E. Thus,equation(8)generatesacocycle⟨E;φ;(Y;R ;(cid:27))⟩andanon-autonomous
+
dynamical system ⟨(X;R ;(cid:25)); (Y;R ;(cid:27));h⟩, where X := E (cid:2)Y, (cid:25) := (φ;(cid:27)) and
+ +
h:=pr :X !Y.
2
Remark 2.7. Let Y := H(f) and (Y;R;(cid:25)) be the shift dynamical system on Y.
The equation (8) (the family of equation (9)) may be written in the form (6), where
F : Y (cid:2)E 7! E is de(cid:12)ned by equality F(g;u) := g(0;u) for all g 2 H(f) = Y and
u 2 E; then F(gt;u)= g(t;u) (gt(s;u):= (cid:27)(t;g)(s;u)= g(t+s;u) for all t;s 2 R
and u2E).
2.1. Recurrent, Almost Periodic and Almost Automorphic Motions. Let
(X;S;(cid:25)) be a dynamical system.
De(cid:12)nition 2.8. A number (cid:28) 2S is called an ">0 shift of x (respectively, almost
period of x), if (cid:26)(x(cid:28);x)<" (respectively, (cid:26)(x((cid:28) +t);xt)<" for all t2S).
De(cid:12)nition2.9. Apointx2X iscalledalmostrecurrent(respectively,Bohralmost
periodic), if for any ">0 there exists a positive number l such that at any segment
of length l there is an " shift (respectively, almost period) of point x2X.
De(cid:12)nition 2.10. If the point x 2 X is almost recurrent and the set H(x) :=
fxt j t2Sg is compact, then x is called recurrent.
Denote by N :=fft g: ft g(cid:26)S such that f(cid:25)(t ;x)g!x as n!1g.
x n n n
De(cid:12)nition2.11. Apointx2X ofthedynamicalsystem(X;S;(cid:25))iscalledLevitan
almost periodic [20], if there exists a dynamical system (Y;S;(cid:27)) and a Bohr almost
periodic point y 2Y such that N (cid:18)N :
y x
De(cid:12)nition 2.12. A point x2X is called stable in the sense of Lagrange (st:L), if
its trajectory (cid:6) :=(cid:8)f(cid:25)(t;x) : t2Sg is relatively compact.
x
De(cid:12)nition 2.13. A point x 2 X is called almost automorphic in the dynamical
system (X;S;(cid:25)); if the following conditions hold:
(i) x is st.L;
(ii) the point x2X is Levitan almost periodic.
Lemma 2.14. [9] Let (X;S;(cid:25)) and (Y;S;(cid:27)) be two dynamical systems, x2X and
the following conditions be ful(cid:12)lled:
(i) a point y 2Y is Levitan almost periodic;
(ii) N (cid:18)N .
y x
Then the point x is Levitan almost periodic, too.
Corollary 2.15. Let x2X be a st.L point, y 2Y be an almost automorphic point
and N (cid:18)N . Then the point x is almost automorphic too.
y x
Proof. Lety beanalmostautomorphicpoint,thenbyLemma2.14thepointx2X
is Levitan almost periodic. Since x is st.L, then it is almost automorphic. (cid:3)
6 DAVIDCHEBAN
Remark 2.16. We note (see, for example, [20] and [28]) that if y 2Y is a station-
ary ((cid:28)-periodic, almost periodic, quasi periodic, recurrent) point of the dynamical
system (Y;T ;(cid:27)) and h : Y ! X is a homomorphism of the dynamical system
2
(Y;T ;(cid:27)) onto (X;T ;(cid:25)), then the point x = h(y) is a stationary ((cid:28)-periodic, al-
2 1
most periodic, quasi periodic, recurrent) point of the system (X;T ;(cid:25)).
1
De(cid:12)nition 2.17. A point x 2X is called [28, 30]
0
- pseudo recurrent if for any ">0, t 2T and p2(cid:6) there exist numbers
0 x0
L=L(";t )>0 and (cid:28) =(cid:28)(";t ;p)2[t ;t +L] such that (cid:28) 2T(p;"));
0 0 0 0
- pseudoperiodic(oruniformlyPoissonstable)ifforany">0, t 2Tthere
0
exists a number (cid:28) =(cid:28)(";t )>t such that (cid:28) 2T(p;")) for any p2(cid:6) .
0 0 x0
Remark 2.18. 1. Every pseudo periodic point is pseudo recurrent.
2. If x2X is pseudo recurrent, then
- it is Poisson stable;
- every point p2H(x) is pseudo recurrent;
- there exist pseudo recurrent points for which the set H(x ) is compact but
0
not minimal [26, ChV];
- there exist pseudo recurrent points which are not almost automorphic (re-
spectively, pseudo periodic) [26, ChV].
2.2. Comparability of Motions by the Character of Recurrence. In this
subsection following B. A. Shcherbakov [27, 28] (see also [3], [4, ChI]) we introduce
the notion of comparability of motions of dynamical system by the character of
their recurrence. While studying stable in the sense of Poisson motions this notion
plays the very important role (see, for example, [26, 28]).
Let (X;S;(cid:25)) and (Y;S;(cid:27)) be dynamical systems, x 2 X and y 2 Y. Denote by
(cid:6) := f(cid:25)(t;x) : t 2 Sg and M := fft g : such that f(cid:25)(t ;x)g converges as
x x n n
n!1g.
De(cid:12)nition 2.19. A point x 2X is called
0
a. comparable by the character of recurrence with y 2 Y if there exists a
0
continuous mapping h:(cid:6) 7!(cid:6) satisfying the condition
y0 x0
(10) h((cid:27)(t;y ))=(cid:25)(t;x ) for any t2R;
0 0
b. strongly comparable by the character of recurrence with y 2 Y if there
0
exists a continuous mapping h:H(y )7!H(x ) satisfying the condition
0 0
(11) h(y )=x and h((cid:27)(t;y))=(cid:25)(t;h(x)) for any y 2H(x ) and t2R;
0 0 0
c. uniformly comparable by the character of recurrence with y 2 Y if there
0
exists a uniformly continuous mapping h:(cid:6) 7!(cid:6) satisfying condition
y0 x0
(10).
Theorem2.20. Letx 2X beuniformlycomparablebythecharacterofrecurrence
0
with y 2 Y. If the spaces X and Y are complete, then x is strongly comparable
0 0
by the character of recurrence with y 2Y.
0
BOHR/LEVITAN ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF ...7
Proof. Let h :(cid:6) 7! (cid:6) be a uniformly continuous mapping satisfying condition
y0 x0
(10) and the spaces X and Y be complete. Then h admits a unique continuous
extension h:H(y )!H(x ). Now we will show that this map possesses property
0 0
(11). Tho this end we note that by condition h satis(cid:12)es equality (10). Let now
y 2H(y )andt2R,thenthereexistsasequenceft g(cid:26)Tsuchthat(cid:27)(t ;y )!y
0 n n 0
asn!1and,consequently,(cid:27)(t+t ;y )!(cid:27)(t;y). Sincethesequencef(cid:27)(t ;y )g
n 0 n 0
isconvergentandthemaph:(cid:6) 7!(cid:6) isuniformlycontinuous,satis(cid:12)es(10)and
y0 x0
the spaces X and Y are complete, then the sequence f(cid:25)(t ;x )g = fh((cid:27)(t ;y ))g
n 0 n 0
is also convergent. Denote by x:= lim (cid:25)(t ;x ). Then we have
n 0
n!1
h((cid:27)(t;y))= lim h((cid:27)(t +t;y ))= lim (cid:25)(t +t;x )=
n 0 n 0
n!1 n!1
lim (cid:25)(t;(cid:25)(t ;x ))=(cid:25)(t;x)=(cid:25)(t;h(y)):
n 0
n!1
Theorem is proved. (cid:3)
Corollary 2.21. The uniform comparability implies strong comparability (if the
phase spaces are complete) and strong comparability implies the (simple) compara-
bility.
Theorem 2.22. [3],[4, ChI] Let X and Y be two complete metric spaces, then the
following statement are equivalent:
(i) the point x is strongly comparable by the character of recurrence with
0
y 2Y;
0
(ii) M (cid:18)M .
y0 x0
Theorem 2.23. [27]IfthespacesX andY arecompleteandy isLagrangestable,
0
then the strong comparability implies uniform comparability and, consequently, they
are equivalent.
Remark 2.24. From Theorems 2:20 and 2:23 follows that the strong comparability
of the point x with y is equivalent to their uniform comparability if the point y
0 0 0
is st. L and the phase space X and Y are complete. In general case these notions
are apparently different (though we do not know the according example).
Theorem 2.25. [26, 28] Let x 2 X be uniformly comparable by the character
0
of recurrence with y 2 Y. If y 2 Y is pseudo recurrent (respectively, pseudo
0 0
periodic), then x is so.
0
Proof. If y is pseudo recurrent (respectively, pseudo periodic) then for any ">0,
0
t 2 T and p 2 (cid:6) there exist L = L(";t ) > 0 and (cid:28) = (cid:28)(";t ;p) 2 [t ;t +L]
0 x0 0 0 0 0
(respectively, there exists (cid:28) = (cid:28)(";t ) > t ) such that (cid:28) 2 T(p;") for any p 2 (cid:6) .
0 0 x0
Since x is uniformly comparable by the character of recurrence with y , then
0 0
there exists a uniformly continuous mapping h : (cid:6) 7! (cid:6) satisfying (10). Let
p 2 (cid:6) , q 2 h(cid:0)1(p), (cid:14) = (cid:14)(") > 0 be chosen fromy0 the uxn0iform continuity of h
x0
and L~(";t ) := L((cid:14)(");t ) > 0 and (cid:28)~(";t ;p) := (cid:28)((cid:14)(");t ;q) 2 [t ;t +L~(";t )]
0 0 0 0 0 0 0
(respectively, (cid:28)~(";t ) := (cid:28)((cid:14)(");t ) > t ), then (cid:28) 2 T(p;") because (cid:28) 2 T(q;(cid:14)) and
0 0 0
h(q)=p. Theorem is proved. (cid:3)
Theorem 2.26. Let ⟨(X;T;(cid:25));(Y;T;(cid:27));h⟩ be a nonautonomous dynamical system
and x 2X be a conditionally Lagrange stable point (i.e., the set (cid:6) is condition-
0 x0
ally precompact), then the following statement hold:
8 DAVIDCHEBAN
∩
(i) ifH(x ) X consistsasinglepointfx g,wherey :=h(x ),thenN (cid:18)
0 y0 0 0 0 y0
Nx0; ∩
(ii) if the set H(x ) X contains at most one point for any q 2H(y ), then
0 q 0
M (cid:18)M .
y0 x0
Pprreocoof.mLpaetctfatnndgf2(cid:25)N(ty0;,xth)gen=(cid:27)(cid:6)(tn;∩y0h)(cid:0)!1(fy(cid:27)0(ats;ny!)g)1,t.heSninfc(cid:25)e(t(cid:6)x;0xis)gcoisnadiptrioencoamlly-
n 0 x0 n 0 n 0
pactsequence. Thoshowthatft g2N itissufficienttoprovethatthesequence
n x0
f(cid:25)(t ;x )ghasatmostonelimitingpoint. Letp (i=1;2)betwolimitingpointsof
n 0 i
f(cid:25)(t ;x )g, then there are ft g (cid:18) ft g such that p := lim (cid:25)(t ;x ) (i = 1;2).
n 0 kni ∩n i n!1 kni 0
Since ft g 2 N , then p 2 H(x ) X = fx g (i = 1;2) and, consequently,
p =p =kni x . Thyu0s we havie N (cid:18)0N . y0 0
1 2 0 y0 x0
∩
Let now ft g2M , q 2H(y ) such that q = lim (cid:27)(t ;y ) and H(x ) X con-
n y0 0 n!1 n 0 0 q
tains at most one point. By the same arguments as above the sequence f(cid:25)(t ;x )g
n 0
is precompact. To show that the sequence f(cid:25)(t ;x )g converges we will use the
n 0
similar reasoning as above. Let p (i = 1;2) be two limiting points of f(cid:25)(t ;x )g,
i n 0
then there are ft g (cid:18) ft g such that p := lim (cid:25)(t ;x ) (i = 1;2). Since
kni n i ∩n!1 kni 0
(cid:27)(t ;y ) ! q as n ! 1, then p 2 H(x ) X (i = 1;2) and, consequently,
ki 0 i 0 q
p =np . Thus we have M (cid:18)M . Theorem is completely proved. (cid:3)
1 2 y0 x0
Theorem 2.27. Let ⟨(X;T;(cid:25));(Y;T;(cid:27));h⟩ be a nonautonomous dynamical system
and x 2 X be a conditionally Lagrange stable point. Suppose that the following
0
conditions are ful(cid:12)lled:
a. Y is m∩inimal;
b. H(x ) X consists a single point fx g, where y :=h(x );
0 y0 0 0 0
c. the set H(x ) is distal, i.e., inf(cid:26)((cid:25)(t;x );(cid:25)(t;x )) > 0 for any x ;x 2
0 1 2 1 2
t2T
H(x ) with x ̸=x and h(x )=h(x ).
0 1 2 1 2
Then M (cid:18)M .
y0 x0
∩
Proof. ByTheorem2.26toprovethisstatementitissufficienttoshowthatH(x ) X
0 q
contains at most one point for any q 2 H(y0). If∩we suppose that it is not true,
then there are points q 2 H(y ) and pi 2 H(x ) X such that p1 ̸= p2. Then
0 0 0 0 q0 0 0
by condition c. there exists a number (cid:11)=(cid:11)(p1;p2)>0 such that
0 0
(12) (cid:26)((cid:25)(t;p1);(cid:25)(t;p2)(cid:21)(cid:11)
0 0
for any t 2 T. Since Y is minimal, then H(q ) = H(y ) = Y and, consequently,
0 0
there exists a sequence ft g (cid:26) T such that (cid:27)(t ;q ) ! y as n ! 1. Consider
n n 0 0
the sequences f(cid:25)(t ;pi)g (i = 1;2). Since (cid:6) is conditionally precompact, then
n 0 x0
by the same arguments as in Theorem 2.26 the sequences f(cid:25)(tc ;pi)g (i = 1;2)
n 0
are precompact too. Without loss of generality we may suppose that ∩they are
convergent. Denote by xi := lim (cid:25)(t ;pi) (i = 1;2). Then xi 2 H(x ) X =
n!1 n 0 0 y0
fx g (i = 1;2) and, consequently, x1 = x2 = x . By the other hand according to
0 0
inequality (12) we have (cid:26)(x1;x2) (cid:21) (cid:11) > 0. The obtained contradict∩ion show that
our assumption is falls, i.e., under the conditions of Theorem H(x ) X contains
0 q
at most one point. Theorem is proved. (cid:3)
BOHR/LEVITAN ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF ...9
Remark2.28. NotethatTheorems2.26and2.27coincidewiththeresultsofB.A.
Shcherbakov [26, ChIII] (see also [28, ChIII]) when the point x is Lagrange stable.
0
3. Birkhoff’s theorem for non-autonomous dynamical systems (NDS)
Let X;Y be two complete metric spaces and
(13) ⟨(X;T ;(cid:25));(Y;T ;(cid:27));h)
1 2
be a non-autonomous dynamical system.
De(cid:12)nition 3.1. A subset M of X is said to be a minimal set of NDS (13) if it
possesses the following properties:
a. h(M)=Y;
b. M is positively invariant, i.e., (cid:25)(t;M)(cid:18)M for any t2T ;
1
c. M is a minimal subset of X possessing properties a. and b..
Remark 3.2. 1. In the case of autonomous dynamical systems (i.e., Y consists a
single point) the de(cid:12)nition above coincides with the usual notion of minimality.
2. If the NDS ⟨(X;T ;(cid:25));(Y;T ;(cid:27));h) is periodic (i.e., there exists a (cid:28)-periodic
1 2
point y0 2Y such that Y =f(cid:27)(t;y0): t2[0;(cid:28))g), then the nonempty com∩pact set
M (cid:26) X is a minimal set of NDS (13) if and only if the set M := X M is a
y0 y0
minimal set of the discrete (autonomous) dynamical system (X ;P) generated by
y0
positive powers of the map P :=(cid:25)((cid:28);(cid:1)):X !X .
y0 y0
Lemma 3.3. Let M (cid:26)X be a nonempty, closed and positively invariant subset of
NDS (13) such that h(M)=Y, then the following statements hold:
(i) if H(x) = M for any x 2 M, where H(x) := f(cid:25)(t;x): t2T g, then M
1
is a minimal set of NDS (13);
(ii) if
(a) T =T ;
1 2
(b) Y is a minimal set of autonomous dynamical system (Y;T ;(cid:27));
2
(c) M is a minimal subset of NDS (13) and it is conditionally compact,
then H(x)=M for any x2M.
Proof. Let M (cid:26) X be a nonempty, closed and positively invariant subset of NDS
(13)suchthath(M)=Y andH(x)=M foranyx2M. Wewillshowthatinthis
case M is a minimal set of NDS (13). If we suppose that it is not true, then there
exists a subset M~ (cid:26) M such that M~ is a nonempty, closed, positively invariant,
h(M~) = Y and M~ ̸= M. Let x 2 M~ (cid:26) M, then by condition of Lemma we have
M = H(x) (cid:18) M~ (cid:26) M and, consequently, M = M~. The obtained contradiction
proves our statement.
Suppose that M is a minimal subset of NDS (13) and it is conditionally compact.
We will establish that, then H(x) = M for any x 2 M. In fact. If it is not so,
then there exists a point x 2 M such that H(x ) (cid:26) M and H(x ) ̸= M. Since
0 0 0
h((cid:25)(t;x )) = (cid:27)(t;y ) (where y := h(x )) for any t 2 T and Y is minimal, then
0 0 0 0 1
for any y 2Y there exists a sequence ft g(cid:26)T such that (cid:27)(t ;y )!y as n!1.
n n 0
Since the set M is conditionally compact without loss of generality we can suppose
10 DAVIDCHEBAN
that the sequence f(cid:25)(t ;x )g converges. Denote by x := lim (cid:25)(t ;x ), then we
n 0 n 0
n!1
have h(x) = y. Since y 2 Y is an arbitrary point, then h(H(x )) = Y. Thus we
0
have a nonempty, positively invariant subset H(x )(cid:26)M such that h(H(x ))=Y
0 0
and H(x ) ̸= M. This fact contradicts to the minimality of M. The obtained
0
contradiction proves the second statement of Lemma. (cid:3)
Corollary 3.4. Let ⟨(X;T;(cid:25));(Y;T;(cid:27));h) be a non-autonomous dynamical system
and M (cid:26) X be a nonempty, conditionally compact and positively invariant set. If
the dynamical system (Y;T;(cid:27)) is minimal, then the subset M is a minimal subset
of NDS (13) if and only if H(x)=M for any x2M.
Theorem3.5. Supposethat⟨(X;T ;(cid:25));(Y;T ;(cid:27));h)isanon-autonomousdynam-
1 2
ical system and X is conditionally compact, then there exists a minimal subset M.
Proof. Denote by A(X) the family of all nonempty, positively invariant and condi-
tionally compact subsets A (cid:18) X. Note that A(X) ̸= ∅ because X 2 A(X). It is
clear that the family A(X) partially ordered with respect to inclusion (cid:18). Namely:
A (cid:20) A if and only if A (cid:18) A for all A ;A 2 A(X). If A (cid:18) A(X) is a linear
1 2 1 2 1 2
ordered subfamily of A(X), then the intersection M of subsets of the family A is
nonempty. In f∩act. For any y 2 Y the family of subsets Ay := fAy : A 2 Ag,
where A :=A X , is linear ordered and, consequently,
y y
∩
M = fA : A2Ag̸=∅
y y
because X is compact. Thus M is a closed, positively invariant set such that
y
h(M) = Y and, consequently, M 2 A(X). By Lemma of Zorn the family A(X)
contains at least one minimal element M. It is clear that M is a minimal set.
Theorem is proved. (cid:3)
If X is a compact metric space, then XX denote the collection of all maps from
X to itself, provided with the product topology, or, what is the same thing, the
topology of pointwise convergence. By Tikhonov theorem, XX is compact.
XX has a semigroup structure de(cid:12)ned by the composition of maps.
Let⟨(X;T ;(cid:25));(Y;T ;(cid:27));h⟩beanon-autonomousdynamicalsystemandy 2Y be
1 2
a Poisson stable point. Denote by
E+ :=f(cid:24)j 9ft g2N+1 such that (cid:25)tnj !(cid:24)g;
y n y Xy
where X := fx 2 Xj h(x) = yg and ! means the pointwise convergence and
y
N+1 :=fft g2N : t !+1 as n!1g.
x n x n
Lemma 3.6. [6, ChIX,XV] Let y 2 Y be a Poisson stable point, ⟨(X;T ;(cid:25));
1
(Y;T ;(cid:27));h⟩beanon-autonomousdynamicalsystemandX beaconditionallycom-
2
pact set. Then E+ is a nonempty compact sub-semigroup of the semigroup XXy:
y y
Lemma 3.7. [6, ChIX,XV] Let X be a conditionally compact metric space and
⟨(X;T ;(cid:25)); (Y;T ;(cid:27));h⟩ be a non-autonomous dynamical system. Suppose that the
1 2
following conditions are ful(cid:12)lled:
(i) The point y 2Y is Poisson stable;