Forced systems with almost periodic and quasiperiodic forcing term

Nonlinear Analysis, Theory,Methods&Applications, Vol. 32, No. 6, pp. 727-739, 1998 © 1998Elsevier Science Ltd. All rights reserved Printed in Great Britain 0362-546X/98 $19.00+ 0.00

Pergamon PII: S0362-546X(97)00513-0 FORCED

SYSTEMS WITH QUASIPERIODIC

ALMOST PERIODIC FORCING TERM

AND

CARLO CARMINATI D i p a r t i m e n t o di M a t e m a t i c a , Universith di Pisa, via B u o n a r r o t i 2, 56127 Pisa, Italy

(Received 13 September 1996; received for publication 14 May 1997) Key words and phrases: F o r c e d systems, a l m o s t periodic, q u a s i p e r i o d i c forcing term.

1. I N T R O D U C T I O N

In this paper we shall study the bounded solutions of a system of differential equations of the type g-

VV(x) = h(t)

(1)

where the forcing term h(t) is bounded. If in addition h(t) is also almost periodic (or quasiperiodic) we shall be interested in finding almost periodic (or quasiperiodic) solutions to (1). Our work is motivated by two papers of Berger and Chen ([1, 2]) where they find almost periodic and quasiperiodic solutions of equation (l) in the case the potential is of the form 1

V(x) = -~ A x " x + U(x)

where A is a symmetric positive definite matrix and U ~ CZ(~N; ~e) is a convex function satisfying some growth hypothesis. Namely in the scalar case [1] they assume lim

IU(x)[

(2)

_ +oo

while in the case of systems [2] the assumption on U is 3M>O:M<

Ixil

vi=

<-[Yil

1. . . . . N ~

U(x)<_ U(y).

(3)

In our work neither condition (2) nor (3) are assumed; in fact what is needed to get precise estimates for the solution is just the position of the minimum of the potential. Moreover, by virtue of the estimates we have found we are able to prove a "local" version of the results of Berger and Chen, assuming that U is convex only near the minimum of V. THEOREM 1. Assume that the potential V e Cl(f~o; ~) has a local minimum at Xo e f2o and for all x e Bp(X o) = Ix e ~ N : IX -- XOI <-- P} the potential is of the form

1

V(x) = -~ A x . x 727

+ U(x)

(4)

728

C. C A R M I N A T I

where A is a symmetric matrix such that

A x " x >- alxl 2

(5)

and U(x) ~ Cx(O: R) is a convex function defined on some open set f ] D Bp(xo). Assume also that h(t) is a bounded continuous function and s u p [h(t)[ _< a p . t~lR

Under this assumption the system (1) admits one bounded solution u(t) such that

1

s u p lu(t) - x01 <-- - s u p Ih(t)l tel~

(6)

(7 t e R

in fact u(t) is the only solution of (1) satisfying (6). If we assume also that the forcing h(t) is almost periodic (or quasiperiodic) we can prove that the solution u(t) is almost periodic (quasiperiodic) as well.

The result of [1] has been generalized in [3] for elliptic equations using the method of super and sub solutions introduced in [4]. Obviously such a method is not fit for systems. For sake of completeness let us recall that the problem of almost periodic and quasiperiodic solutions o f e q u a t i o n (1) has been studied in some cases when V(x) is concave; in this situation the dynamics of the system is much more complicated even in the scalar case; for some results in various directions see [5-8]. This paper is structured as follows: in Section 2 we prove Theorem 1 in the case p = oo (that is the setting of [2] without assumption (3)), in Section 3 we prove Theorem 1 in the general case using a truncation argument and, finally, in Section 4 we give a variational characterization of the solution found in Section 2.

2. N O T A T I O N S

AND PRELIMINARY

RESULTS

Let us begin recalling some definitions and results concerning the class of almost periodic functions (see [9] for more details). A subset T C E is called relatively dense if there exists some / > 0 such that

T D [ a , a + I] ~ (~ ¥a ~ ff~. A continuous function f : R ~ [~N is called almost periodic if for any e > 0 the set

T(f'e) := Ir e ~ l s u p l f ( t + r) - f(t)] < is relatively dense. The set T(f, e) is called the set of e-periods of the function f . One remarkable feature of an almost periodic function is that one can define the mean value that is the following limit exists

9E(f(t)) = lim 1 f r f(t) dt. T~+oo ~ ~-T

Forced systems

729

In fact, since the product of an almost periodic function by an almost periodic scalar function is almost periodic, for all ;t e R one can define

a(f, 2) := ~)g(f(t) e -ixt) moreover it can be proved that the set of the Fourier coefficients of f namely e x p ( f ) := [)~ e ~ [ a ( f , A) # O] is at most countable. We shall call m o d ( f ) the additive group generated by the set exp(f). An almost periodic function f is called quasiperiodic with fundamental frequencies A1 . . . . . Ak if (i) A, . . . . . A~ are rationally independent; (ii) e x p ( f ) is contained in the additive group generated by A 1. . . . . Ak; (iii) there is no set which satisfies (i) and (ii) and has fewer elements. In the following BC(~; [Rs ) denotes the Banach space of bounded continuous functions endowed with the norm of uniform convergence ]Ix[[ = sup~[x(t)[ and A P k ( ~ ; ~U) denotes the almost periodic functions of class C g whose first k derivatives are almost periodic. Let us consider the following forced system:

2 - A x - VU(x) = h(t)

(7)

where

(i) h ~ BC([R; ~N); (ii) A is a symmetric positive definite matrix, and Ax.

x >_ a l x l 2

( a > 0);

(iii) U e CI(~N; ~) and there exists R such that VU(x) • x >_ 0

v Ix[ > R.

To prove the almost periodicity of the solution we shall "strengthen" (i) and (iii), namely we shall require (i') h e AP(R; [~N); (iii') U ~ CI([RN; ~) is convex. THEOREM 2. If (i), (ii) and (iii) hold then system (7) has one bounded solution u(t) defined on the whole real line. THEOREM 3. If (i), (ii) and (iii') hold then system (7) has a solution u(t) and this solution is unique; in fact. the function A A: BC(R; [Rm) ~ BC(R; ~ s )

h(t) ~ A(h) = u(t) that maps the forcing term in the unique solution of (7) is Lipschitz continuous with Lipschitz constant l / a .

730

C. C A R M I N A T I

In particular, if V(x) = 1 / 2 A x . x following estimate holds:

+ U(x) attains its minimum at the point Xo the

]lh!l

sup l u ( t ) - x0[ -< te~

(7

THEOREM 4. If (i'), (ii) and (iii') hold then the solution u(t) of (7) belongs to Ap2([R; []~N) and mod(u) = mod(h). In particular if h(t) is quasiperiodic, u(t) is quasiperiodic with the same fundamental frequencies. P r o o f o f T h e o r e m 2. It is easy to see that U is bounded from below and it attains its minimum, therefore up to a translation we will choose U positive and min U = 0. Let T > 0 be fixed; let E = H I ( [ - T, T]; R N) with its usual norm; we define

Jr(x) =

S*'

~ ([2[ 2 + A x " x ) d t +

-T

f*

U(x) dt +

-T

l*

h • x dt.

-T

Standard calculations show that: • Jr e C~(E; ~); • J r is bounded from below and coercive; • J r is weakly lower semi-continuous. Therefore there exist some v e E such that J r ( v ) = rain J r ; such a v e C2([- T, T]; ~N) is a classical solution of (7) satisfying the Neumann boundary conditions v(-T)

= v ( T ) = O.

We shall now find, with some maximum principle techniques, some hounds in uniform norm for this solution. Let

llxllr= M=HhI[

sup [x(t)l;

te[-T,T]

and

CI=max[R;M

1.

CLAIM.

Itvllr-< c,. To see this let ~o(t) =

[v(t)l z.

We see that ~b(t) = 2v • b so that v satisfies ~ ( - T ) = ~(T) = 0 and (b(t) = 2[t)(t)[ 2 + 2v(t) • 6(t) = 2lv(t)l z +

2Av(t). v(t) + 2 V U ( v ( t ) ) " v(t) + 2h(t) • v(t).

(8)

Forced systems

73 ]

N o w , suppose by contradiction that I = It e [ - T , T]:~0(t) > C~I ;~ • ; by (ii) and (iii) it is readily seen that v t e I, (b(t) >__ 2a¢(t) + 2h(t) • v(t) _> 2ere(t) - 2Mlv(t)l >_ 2x/~0(t)(crx/q~(t) - M). Since a~/~0(t) - M > 0 v t e / w e have that some to e I, then it is clear that [to, T] C ~b(t) > 0 which is a contradiction with (8). proving our claim. Since v(t) satisfies (7) we have that f(t)

II II

~b(t) > 0 v t ~ I. Supose now that (b(to) -> 0 for I and since ~b is strictly increasing on I we get Similarly we get the contradiction if ~b(to) <- 0 is b o u n d e d as well, namely:

~ s u p ( l A x + VU(x)l) + M = C 2. x<_C

N o w we recall the following l e m m a . ? LEMMA 5. If X ~ BCZ(I; ~U) then

Ilxill --- max 2 ~ l l x , lll/211,~lI1J2;

41Ix, Ill ~ )

i:

1 . . . . . N.

So we have proved there exist some a priori b o u n d s independent o f T >_ 1 for this solution, namely IlvllT-< c , ,

Ilvll~_< c 3 = (2x/-Nx/C, C2 + 4C,).

Let us define u e C~(~; ~N) in the following way: v(-T)

ur(t) = ~ v(t)

if t _< - T , if t e [ - T , T],

!

\ v(T)

if t >_ T.

Clearly Ilu~ll = Ilvllrand Ilurll -- Ilvllr. s o , if we choose a sequence T, --* oo, Un = ur, are equicontinuous and equibounded, by the Ascoli-Arzelh t h e o r e m together with a diagonal process we get that there is a subsequence u,~ ~ u

u n i f o r m l y on c o m p a c t subsets o f ~.

Using elementary standard techniques one also gets that • /g e C 2 ( ~ ; []~N); • u satisfies (7) on the whole real line;

• u, ft, ii ~ BC(R;

~N).

t T h e proof of this lemma can be easily carried over using the same arguments as in L e m m a 3 of [1].

732

C. C A R M I N A T I

P r o o f o f Theorem 3. Since Uis convex and A is strictly positive definite we immediately see that V(x) = ½ A X . x + U(x) satisfies

lim V(x) = + oo Ixl~oo and it admits a unique point of minimum x0; we may assume without loss of generality that Xo = 0 and VU(0) = 0. By the convexity of U we get also that (VU(x) - VU(y)) • (x - y) ~ 0

Taking y theorem bounded and g(t),

v x , y ~. [~N.

= 0 in the formula above we see immediately that condition (iii) of the previous is satisfied (with R = 0); thus we deduce that equation (7) has a t least one solution. If x(t) and y(t) are two bounded solutions of (7) with forcing term h(t) respectively, then letting w(t) = x(t) - y(t)

~u(t) = [w(t)[ z,

we see that (dropping the variable t) O? = 2[w[ z + 2 A w .

w + 2(VU(x) - VU(y)) • (x - y) + w . (h - g)

_> 2trq/ - [[h - g[[ x/~, so, by arguing by contradiction, if ~f~/o) > 1/tr[[h -g[[, then q/is a strict convex function defined on a neighborhood of t o, and by repeating the arguments of the p r o o f of Theorem 2 we get that w(t) is unbounded, which is absurd. In particular, taking f = g we get q/(t) - 0, namely the solution of (7) is unique. So we have proved also the Lipschitz continuous dependence from the forcing because we have shown that ifx(t) and y(t) are two bounded solutions of (7) with forcing term h(t) and g(t), respectively, then 1 IIx - yll -< - I I h t7

- gll.

Finally, since the function x(t) - 0 is the solution of the problem without forcing by the last formula we get immediately the following estimate for the solution u(t)

Ilull = Jlhll ~7

P r o o f o f Theorem 4. If u(t) is the solution with forcing term h(t) then u~(t) = u(t + r) is the solution with forcing term h~(t) = h(t + r), so the previous result implies 1

Ilu - ull -- -Irh a

- hit.

This assures that u(t) is almost periodic by virtue of Theorem 4.5 of [9], which in fact tells us that u(t) is almost periodic and mod(u) C rood(h).

Forced systems 3. P R O O F

733

OF THEOREM

1

Now we shall use the results of Section 2 to prove Theorem 1. Let the hypothesis of Theorem 1 be satisfied, and let us assume for sake of simplicity that x o = 0 and U(0) = 0. Let us fix p ' > p such that Bp,(0) is still contained in f2 the domain of definition of U, and define O: R N ~ R in the following way O(x)

=

sup

[lz(x) = U(z) + V U ( z ) • ( x - z ) ] .

Izl <_p'

It is immediate to check that O(x) < +Qo q x ~ A N, 0 is convex (and hence continuous) and O(x) = U(x) v x ~ B;,(O) since U is convex on Bo,(0). Unfortunately 0 might not be in CI(RN; R) (even if U ~ C =) so we shall consider the sequence of Un ~ C=(RN; P,) defined by Un

=

O*¢pn

where {0n is a sequence of mollifiers, namely ~Pn > 0,

supp ~0n C B(0, 1/n),

t

i~ [RN

~on(y ) dy = 1

vn.

It is easy to check that each Un is convex and [In ~ O in CI(Bp,(0); R). Finally let V.(x) = ~1 A x . x + Un(x)

1

('(x) = ~ A x " x + U(x)

and xn the minimum of V,. If n > 2 / p then xn ~ Bp(0), moreover it is easy to see that xn converges and lim xn ~ N Ix[ I?(x) < el = 10]. n

~:>0

By Theorem 3 for all n ~ N there exists some u n ~ C2(~; R N) such that an - VV~(un) = h(t)

and

Ilhll

sup [Un(t) - xn[ <- - t~

O"

(9)

So it is clear that ]1Un][are equibounded and repeating the arguments used in Section 2 also ]]anl] and []un[] are equibounded; so we get that (up to a subsequence) un -~ u in c t ( I ; EN) for all compact interval I e R, and by the usual bootstrap argument u e C2(~; R N) is the solution of a - if(u) = h(t) such that Ilull-< Ilhll/a. But since we have assumed [[hll-< ~rp and ¥ x e Bp(0) then u is a bounded solution to the original equation a - A u - VU(u) = h(t).

17(x)= V(x)

(10)

By Theorem 4 it is obvious that if the forcing h e A P ( R ; R ~) then the solution u e A p 2 ( R ; RN). The theorem is thus completely proved.

734

C. CARMINATI

Remark. T h e results we shall o b t a i n in Section 4 will show t h a t u(t) is a local m i n i m u m on AP1(~; R N) for the f u n c t i o n ~ ( x ) = Ogt(L(t, x, .~)) with L(t, x, x) = 1 / 2 1 x l z - V(x) + h(t). Let us see a simple c o n s e q u e n c e o f this t h e o r e m . A s s u m e n o w o n l y that V(x) • C2(f~0; ~) a n d 3Xo • ~ o , a0 > 0 such that ( V l ) vV(xo) = 0, ( v 2 ) D2V(xo)[V, v] >_ 2aolVl 2 v v • []~N. Let us define a f u n c t i o n

a(r) = min D2V(x)[~, (] Ixl < r I~l = 1

a n d let M = sup

ra(r).

r>0

T h e f o l l o w i n g result holds. THEOREM 6. F o r all

h • BC(~;

~N) such 2

has a b o u n d e d s o l u t i o n u(t). M o r e o v e r if h • AP(~; ~,U) then

-

t h a t Ilhll < M the e q u a t i o n VV(x)

h(t)

=

(11)

u • Ap2(R; [~U) a n d m o d ( u ) = rood(h).

Let us p o i n t o u t that this is m o r e t h a n a p e r t u r b a t i o n result since we have an explicit e s t i m a t e on the m a x i m u m n o r m o f the forcing a l l o w e d a n d this e s t i m a t e d e p e n d s o n l y on the g e o m e t r y o f the p o t e n t i a l .

Proof of Theorem 6. F o r the sake o f simplicity we shall here t r e a t the case Xo = 0. Ilhll = m < M ; since the f u n c t i o n ra(r) is c o n t i n u o u s there is s o m e fi > 0 such that

Let

m
Example. Let us c o n s i d e r the e q u a t i o n x" - sinx = in this case the f u n c t i o n

h(t)

(12)

a(r) d e f i n e d a b o v e is a(r)= I~°s(r)

rO<-r<-~r/2> ~r/2

a n d so M = a cos c~ where a is the u n i q u e s o l u t i o n in ]0, rr/2[ o f the e q u a t i o n c~ = cotg c~. So b y T h e o r e m 6 we have t h a t for all h e A P such that IIhlt -< c~ cos ~, e q u a t i o n (12) has an a l m o s t p e r i o d i c solution.

Forced systems 4. R A T E O F C O N V E R G E N C E

735

AND RELATED

RESULTS

In this section we shall give some more precise information on the way the functions UT converge to the solution u; this will be useful to give a variational characterization of the solution found in Section 2; in fact there are some works (see [10] and references therein) which have tried to attack the problem of the existence of almost periodic solutions using a variational approach. Let us consider equation (7) again

5?- A x -

V U = h(t)

and let the assumptions (i), (ii) and (iii') of Theorem 3 hold and for sake of simplicity assume also VU(0) = 0; let or(t) be the solution of (7) satisfying the Neumann boundary condition on the interval [ - T , T]. Let us recall that or(t) is the minimum of the functional Jr, moreover if h ~ 0 then Jr(or) < 0 =

J(O).

Since the norm of both vr and u are bounded by the constant [Ihl[/tr we get that sup

Ju(t) -

vr(t)l

-< 1/21,h,_1__2~

t~[-T,T]

(13)

(7"

Let

o(t)

=

lu(t)- OT(t)l z.

Carrying out the same calculations as in Section 2 it is easy to see that ~0 satisfies the differential inequality ~b(t) _> 2tr~0(t), moreover by (13) 0 <__~0(t) _<

41!hll 2

v t ~ I - T , TI.

0 2

Let ~(t) be the solution of the system I 0~(t) = 2trq/(t) 411hll 2

q/(T) = q / ( - T )

-

0" 2

and c~(t) --- ~o(t) - ~,(t). Since c~(+T) _< 0;

&(t) _> 2trc~(t)

we have that c~(t) _< 0 v t ~ [ - T , T] that is ~0(t) _< q/(t). On the other hand we have an explicit expression for ~,, namely: 4llhl[ 2 cosh'~-~ t qJ(t) - tr 2 c o s h x / ~ T "

736

C. CARMINATI

So we have established the following useful estimate 4[Ihll2 cosh x/2-at lu ( t ) - vr(t)[ 2-< " ~o cosh~T

v t e [-T, T].

(14)

In particular it is now clear that the functions u r defined in Section 2 converge uniformly on compact subsets of ~ to the function u(t) as t ~ +co. From now on we shall be interested in the "convergence in mean"; let (t) dt dee I

g(t) dt;

and, if g e AP°(~,; ~N) ~lg(g) def = lim

1 ( r g(t) dt.

T~+oo ~

,~-T

By equation (14) we get ~r -r

lu(t)

- vr(t)[ 2 dt <

~r j-r

1 (4~)

~u(t) dt =

2-T

sinh,J2-a T cosh x/2-a T

1 4[[hl[z T tr 2

(15)

on the other hand, by Holder's inequality we have lu(t) - Vr(t)l dt <_ -T

[u(t) - vr(t)l 2 dt -T

and so

l

r -T

lu(t) - Vr(t)[ dt <_

2[[h[[ ~/T 17

-T

(16)

We are now ready to prove the following. LEMMA 7. If VU(x) is locally Lipschitz continuousT and h(t) is almost periodic, then 10 def im T~+oo

where

l

L(t, u(t), u(t)) dt = lim -T

L(t, vr(t ), vr(t)) dt

(17)

T~+oo J - T

1 1712 + ~1A x .

L(t, x, y) = ~

x + U(x) + h(t) • x.

Proof. First let us remark that the limit on the left-hand side of (17) exists since the integrand is almost periodic in t, so it will be enough to prove that lim

T~+oo

-T

[L(t, u(t), ie(t)) - L(t, vr(t), vr(t))] dt = O.

t This hypothesis is in fact not necessary, but it simplifies the calculations.

Forced systems

737

If x e H ~ ( [ - T , T]; [RN) satisfies (7) then, integrating by parts we have

12(t)x(t)lr_r

L(t, x(t), k(O) dt = ~

+

-T

Loft, x(t)) dt .]-T

where 1

1

Lo(t, x) = U(x) - ~ VU(x) • x + ~ h(t) • x. Moreover since V'(x) is locally Lipschitz continuous it is immediate to check that there is some K > 0 such that

ILo(t, X l ) - Lo(t, x2)l <--KIxl - x21

Vt~;

IIhll

VXl,X2 : Ixil < _ -a-

i = 1,2

Therefore we get

l

L(t, u(t), i~(t)) dt -

L(t, Vr(t), br(t))

-T

-T

<-

lu(t)u(t)Jr_r +

d,

[Lo(t, u(t)) - Lo(t, vr(t)] -T

<-

+

lu(t) - Vr(t)l dt r - + : 0 -T

So our claim is proved.

Remark. In fact, under the assumption that VU is locally Lipschitz, using the estimate (14) and Lemma 5 one can prove lim T~ +~

Izi(t) -

vr(t)J a dt = 0.

(18)

-T

We omit the p r o o f of this statement because it is not essential for what follows. Now we are able to state the main result of this section. THEOREM 8.

1o =

L(t, u(t), u(t)) dt =

lim T--+ +oo ,J - T

min

lim

L(t, x(t), 2(0) dt

(19)

x e A p I ( [ R ; [RN ) ~, T ~ +oo ,J - T

In fact u(t) is the unique minimum of the class of functions AP~(~; R N) and 1o < O.

Proof. Since Vx ~ ApI([R; R N) L(t, x(t), 2(0) dt >_ ,)-T

t

L(t, Vr(t), Or(t)) dt -T

VT>O

738

C. C A R M I N A T !

passing to the limit as T ~ +oo the previous lemma proves that

L(t, x(t), x(t)) dt >_ lim

lim T~+oo

,)-T

L(t, vr(t), vr(t)) dt = lo.

T~+oo

J-T

Now, since by the previous lemma for x = u the equality holds, we have proved the existence of the minimum. Now we show that this minimum is unique. Let us consider the functional

r(x) = f r [L( t, x, Jc) - L( t, VT, VT)] dt - ~l ( i r - r [JC-- b r [ 2 d t + a I r -7"

'X-- Vrl2dt) -T

defined on H i ( [ - T , T] : RN). It is easy to check that O r is convex, OT(VT) = 0 and O~'(VT) = 0; it therefore is clear that VT is a minimum for • r and that

Or(X) >- OT(VT)

¥ X ~ H I ( [ - T , T] : [RN)

which implies

I

r L ( t , x , x ) dt -T

T

L(t, vr, vr) dt

J-T

> -

12 - br[ 2 dt +

-- 2

- vr[ 2 dt

-T

v x e H~([ - T, T];

(20)

[~N)

-T

So, if we fix x ~ A P ~ ( ~ ; ~N) we have

L(t, x, 2) dt -T

=

t l(l

t

L(t, u, iO dt -T

[L(t, x, 2) - L(t, Vr, vr)l dt +

-T

>2

[Jc - vr[Edt + -T

a[x

--

l

[L(t, vr, vr) - L(t, u, zi)] dt -T

UT[2 dt

)

-

J-T

[L(t, vr, VT) - L(t, u,

.

-T

Passing to the limit as T -0 oo, we note that by equation (17) the second term of the last expression tends to zero. On the other hand by equation (16) it is easy to see that lira T~+oo

f

-T

Ix(t)

-

vr(t)l 2 d t =

lira T~+oo

f

Ix(t)

- u(t)l 2 dt.

-T

Therefore we get 17

9g(L(t, x, Jc)) - lo >- ~ 9g(lx - ulZ), which implies

9g(L(t, x, Yc)) > 1o

vx ~ ApI(R;

~ N ) : X ~;~

U.

In particular, if we take x --- 0 in this last expression we see immediately that lo < 0 if h ~ 0, and the p r o o f is complete.

Forced systems

739

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