periodic solutions for a class of nonlinear hyperbolic equations

Nonhnear Anolysrs. Theory, Merhods & Applicafiom, Vol. 15, No. 8, pp. 787-l%, 1990. 0362-546X/90 53.CO+ .@I

Printed in Great Britain. Q 1990 Pergamon Press plc

PERIODIC SOLUTIONS FOR A CLASS OF NONLINEAR HYPERBOLIC EQUATIONS

ANDRZEJ NOWAKOWSKI and ANDRZEJ ROGOWSKI Institute of Mathematics, tbdi University, Banacha 22, 90-238 t6di, Poland

(Received 9 August 1989; received for publication 8 March 1990)

Key words and phrases: Periodic solutions, abstract hyperbolic equations, duality theory, calculus of variations.

INTRODUCTION AND STATEMENT OF THE MAIN RESULT

IN THIS article we shall study the nonlinear abstract problem:

$ ay/(t, x’(t)) + a$@, x(t)) 3 0 (a.e.) almost everywhere in R (O.la)

x(t + T) = x(t), (0. lb)

where T is a given positive number, w, 4: R x X -+ R are T-periodic in f, X is a real separable reflexive Banach space; au/, 84 are the subdifferentials of convex, lower semicontinuous (1.s.c.) functions I&, s), 4(t, s).

For (O.la) we shall consider the functional

4x(*)) = ;-4(&x(t)) + M,x’(t)))dt + I(x(O),x(T)) s 0

,(u, @ = Mu) + Ma

(0.2)

(0.3)

Mu) = I 0 if u = 0,

,204

0 if u = 0, =

+oo if 2.4 # 0, [

+oo if u # 0,

on the space A(X) of absolutely continuous functions x: [0, T] + X. Note that equation (0. la) is the generalized Euler-Lagrange equation for functional (0.2) and further, that the integrand in it is a concave-convex function. The last fact makes functional (0.2) indefinite, i.e. it is, in general, unbounded from below and from above. To avoid such a situation we use some growth assumptions on 4 and w, and then J is bounded from below in A(X). However, J is still not lower semicontinuous in any reasonable topology of A(X).

We obtain two kinds of results. The first is the duality theory for (0.2) and (O.la). It is a modification of duality of Toland [ll, 121 for two points boundary problem (see also Auchmuty [ 11). The second concerns of a minimization of Jon A(X). Both allow us to charac- terize solutions for (0.1).

We would like to stress that our duality principle (see theorem 0.1 below) differs from that of [12, 1, 5, 7, 81.

Let 1.1 denote the norm of X, X* is dual to X with the norm ( - I* and ( -, *> is the duality between X and X*.

181

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ayl u! ualza ‘Mau aq 01 suIaas (1.0) uralqoJd JO suognlos ayl JO uogez!Jai3eJey3 aAoqe ayL

*(X)v u! iDedtuo3 lCppaM lClle!iuanbas s! (~'o)%u!1I~syas (X)v 3 (.)x

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CL’01 tow [73 (.)A

‘(( .)&a[ = (( .)d)Qf dns y.q =

Hx)v 3 c-)x :((-)x)flJu! = cc-)xh-

pue ‘r(la+vadsaJ ‘(*X)v‘(X)v 01 %uoIaq ‘(.)tZ‘(.)g ‘[L ‘01

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.r=;+W O (9’0) ‘L! ‘01 3 I ‘1 < ul

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I

4 (41

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IXSMOOO~ ‘V PUi? IXSMOXVMON ‘V 88L

Periodic solutions 189

1. AUXILIARY RESULTS

Let L’(0, T; X) and L-(0, T, X) denote the usual Banach spaces of functions from [0, T] to X, with the norms ]Iu(-)]\~~ = {~]u(t>] dt and ]]u(-)IIL Q - ess sup] u(t)1 . Let B”(X) be the linear -

OStST

space X @ L”(0, T, X) with the norm ]](c, u(*))]], = maxllcl, ]]u(-)]]~Q). It is known [3] that, for each absolutely continuous function x(a) E A(X), the derivative x’(t) exists almost everywhere in [0, T] and A(X) can be identified with X @ L’(0, T; X) normed by ]]x(-)]],~, = Ix(O)/ + ]]x’(-)]]~l. Similarly, we define L’(0, T; X*), L-(0, T; X*), B”(X*), A(X*). The dual A*(X*) of A(X*) will be identified with B”(X) under the pairing

(P(.),(C, U('))> = (P(O),d + 1

T

W(t), v(t)) dt. (1.1) 0

The Fenchel conjugate of a function g: X + [- 00, + 001 is the function g*: X* -, [ - 00, + m] defined by g*(x*) = sup((x*, x) - g(x) : x E X); it is lower semicontinuous and convex.

For a concave-convex function G: X x X + R, c?G(~,p) is the set of all (u, u) E X* x X*, such that

G(% P) 2 G(Z, P) + (u, P - la> for all p E X*,

G&P) I G&P) + (u, z - i;) for all 2 E X*.

To indicate of what topology the space X is just considered we shall write X,, X, for the sequentially weak topology and the strong one, respectively.

Let L: [0, T] x X x X + I? be IL x lb-measurable and let L(t, x, u) be lower semicontinuous in (x, u) (in X,,, x X,,,) and convex in u. The Hamiltonian associated with L is the function H: [0, T] x X x X* + I? defined by H(t, x,p) = sup((p, u) - L(t, x, u) : u E X). We say that L satisfies the growth condition if there exists a function f: [0, T] x X* + R, f (t, p) = E(t)lpl”, + F(t), E(a), F(s) E L’(0, T; R), l/n + l/m = 1, satisfying

H(t, x, PI 5 f ct, PJ for all t E [0, T], x E X, p E X*.

To get that J attains its minimum, we use the following theorem which is a particular case of results in [6].

THEOREM 1.1. Let L and H be as above and let L satisfy the growth condition. If I is defined as in (0.3) then the functional

JL(X(.)) =

s

T

Ut,x(t), x'(t)) dt + I(tiO),x(t)) 0

attains its minimum in A(X). The sets lx(*) E A(X) : JL(x(*)) I c), c E R, are sequentially weakly compact in A(X).

THEOREM 1.2 [9]. Let g: [0, T] x X + R be 11 x lb-measurable and let g(t, *), t E [0, T], be lower semicontinuous and convex. Assume that x(s) --t jrg(t, x(t)) dt is finite on L”(O, T; B), B=(x~X:Ix]~j]foracertainj>O.Thent + sup(g(t, x): 1x1 I jl), j, < j is summable in IO, Tl.

790 A. NOWAKOWSKI and A. ROGOWSKI

THEOREM 1.3 [IO]. Let g be the same as in theorem 1.2 and assume that there is a( *) E L”(0, T; X*) such that jrg*(t,p(t)) dt < +oo. Then the functionals u(m) -, jrg(r, u(t)) dt, p(s) + jrg*(t, p(t)) dt are convex and lower semicontinuous in L’(0, T; X), L-(0, T; X*), respectively, and they are in duality with respect to the pairing (u( -),p( -)> = jF(~(t),~(t)) dt.

A measurable function w: R + X will be called T-periodic if w(t + T) = w(t) for almost all t E R. The set of all such functions will be denoted by P(X). Let A’(X) stand for the subspace of P(X) consisting of all functions which are absolutely continuous on each finite subinterval of R.

2. DUALITY RESULTS

To conclude that a minimizer for J may also be identified on [0, T] with a solution of (0. l), we apply a modification of the duality theory of Toland.

In what follows we assume that hypotheses (a) and (b) of theorem 0.1 hold. We define, for each x(a) E A(X), the perturbation of J as

J&r, g(a)) = -(,(x(O) + a) - MG’-)) + I

r($(C x(t) + g(t)) - v(r, x’(t))) dr (2.1) 0

for (a, g( *)) in B”(X). Of course, J,(O) = -J(x( e)). By theorem 1.3 and the definition of Ii, we see that, for each x( *) E A(X), J,( *, g(m)) is upper

semicontinuous and concave in X for all g( *) and J,(a, *) is *-weakly lower semicontinuous and convex in L-(0, T, X) for all a. That is the reason for which we cannot use the results from [ 121 or [I] directly.

For x( *) in A(X) and p( *) in A(X*), we define

(g(r),p’(t)> dt - s

=(M, x(r) + g(t)) - r&t, x’(r))) dr + M(T)) 0

+ $(a, P(O)) + I&40) + all. (2.2)

for p(a) E A(X*).

(2.3) +

1

7 T

W,x'Q)) dt +

s

4*@, p'(t)) dt. 0 0

Hence

sup X(~)EACY)

- J,#(-~(3 = ;~px((c, -p(T)) - MC))

T 7

+ sup (s "(.)EL' 0

(n(t), P@)> dt - I'

W, v(t)) df 0 I

j

T

- 9*V, -p’(t)) dt

0

T T

= ry*K p(l)) dt - d*V, -p'(t))dt = -JD(P(.)) (2.4) 0 0

Again by theorem 1.3, a direct calculation gives

J,#(P(*)) = -WThp(T)) + MT1) + s ;x'(t),~Wd~ 0


Further, for x( *) E A(X) and (a, g(s)) E B”(X), we define

s ;-x(t), u(t)> dt

0

T - d*U, v(t)) dt - W-V)) -

0 s T

W, x’(t)) dt 0 1

It is easy to see that J:‘(O) = J,(O) for all x( *) E A(X). We can also compute, using the minmax theorem [3] to the function $(p(.)) (note that (x’(e) EL’ : jr((x’(t),p(f)) + y(t, x’(t))) dt I A), A E R, are weakly compact-see[lO]), (2.3) and (2.4), that

sup J,##(O) = sup sup inf -X(-P(*)) XC.) EACY) xc.) EA(X) p’(.) l L‘(O,T;X*) p(O) EX*

= sup inf P’C.1 ELI p(0) EX’

- JD(P(.))

=- inf sup p’(.) ELI P(O) sx*

JD(P( - 1). (2.5)

THEOREM 2.1. inf J@(m)) = inf sup JD(P( - 1). XC.) EAW) P’(.) ELI p(0) ex*

Proof, From the above notes and (2.5) we obtain

- x(.)%(x) J(x(*)) = x(.;2cJ) J,(O) = Sup J:“(O) = - inf sup JD(P( * )I xc.1 EACY) P’(.) ELI P(O) EX*

and the proof is completed. This theorem shows the main difference between our duality principle and those of [12, 11.

THEOREM 2.2. Let ,?(e) E A(X) be a minimizer for (0.2) and let the set U,(O) be nonempty. Then there exists (-j?(O), -jr’(*)) E H,(O) wherep’(.) E L’(0, T; X*), p(t) = p(O) + jbp’(.s) ds, such that p( *) E A(X*) and J&J(*)) = inf sup J&( *)). Furthermore,

P’(.) EL’ p(0) EX*

J,(O) + J,(-p( *)) = 0, (2.6)

JdP(.)) - J&Pa(*)) = 0. (2.7)

Proof. First, we note that the form of J and the assumptions on it imply that X(O) = x(T) = 0. Further, in view of theorem 2.1, for the first assertion, it will suffice to prove that J(N*N 2 SUP J&X.)) = JDhXs)) where d(f) = ~(0) + jbP’(s)d.s and (-NO, -BY*)) E

P(O) ex*

tJJ,(O). Let J@(e)) = inf XC.1 EACY)

J(x(-)) = i, i E R. By the definition of dJ, for the concave-

convex function J,, we have, for each (-p(O), -p’(q)) E aJ,(O) (it is clear by (0.3) that any


point of X* can be such a p(O)),

-11(a) 5 (-P(O), a> for all a E X, p(O) E X*,

j

T T

i

T

4(f,xCt) + g(t)) dt 2 +(t,X(t)) + W), -P'(t)> dt 0 0 0

for all g( *) E L-(0, T; X).

From (2.9) we get, for all g(e) E L”(0, T; X),

J,(O, g(e)) 2 -i + .r

;g(r), -F(t)> dt 0

and further, for p(t) = p(O) + Judy ds, p(O) E X*,

(2.8)

(2.9)

(g(t), -P’(t)> dt - J,(O, g(a)) I i.

Hence -J&p(*)) = sup - J,“(-p(m)) L -i, i.e. JD(p(*)) I i for all p(0) E X*, thus XC.) EACY)

sup JD(@( e)) 5 i. Further, by the assumption on I,V and the definition of w*, we notice P(O) ex* that ~*(t,@(t)) + sup tq(t, z) L j,l(fi(t)l, for t in [0, T] and B, = (z E X: IzI 5 ji), j, c j.

Z-EB,

By theorem 1.2, the function r(t) = sup I,Y(~, z) is summable in [0, T]. Observing that

;;$;;)I* dt 2 T(~P(O)~, - j;kW)l* df;:B’ we easily obtain that jl v*(t,p(t)) dt -+ +a0 as * + +a.

Thus, in virtue of the convexity and lower semicontinuity of p(O) -, jl ~*(t,@(t)) dt, we conclude that there exists y(O) in X* such that j,‘v*(t, p(t)) = min jr w*(t,@(t)) dt where

P(O) E x* P(t) = p(0) + jhp’(s) d.s, i.e. sup J&j(.)) = J&Y(*)). This proves the first assertion of the

P(O) E x* theorem. Since J,(O) = -i, and J&p(*)) = i, therefore

J,(O) + J&P(.)) = 0.

As we have proved that (-a(O), -p’(m)) E N,(O),

J,(O) + J&D(*)) = 0.

Applying (2.10) to (2.1 l), we obtain (2.7). The proof is completed.

(2.10)

(2.11)

By A,(X) we mean the subspace of A(X) consisting of all functions x(e) such that x(0) = x(T) = 0.

COROLLARY 2.1. Let _Y( *) E A,(X) minimize J over ,4(X). Then there exists p(a) E ,4(X*), p(t) = p(O) + Jim’ ds, such that p(t) E at&, Y(t)) and -p’(t) E a&t, x(t)) for almost all t in [0, T], and

J&7(*)) = inf sup JD(P(-)) = JM- 1). p’(.) EL* p(0) l x*

(2.12)

Proof. By theorem 1.3, (0.4) and (2.8), we conclude (see [8, proposition 1.5.21) that the set dJ,(O) is nonempty. Therefore, we can apply theorem 2.2 and find some p(m) in A(X*)


satisfying, along with x(e) E A,(X), (2.6), (2.7) and (2.12). Rewriting (2.6) explicitly, we get

s

T

(ddt,x(t)) + +*(t, -F(t)) - <x(t), -~‘(t)>)dt = 0.

0

However, by the definition of +*, the integrand above is nonnegative, and so, 4(?, X(t)) + r#~*(t, -p’(i)) - (f(t), -p’(t)> = 0 for almost all t in [0, T]. This means that -p’(t) E a+([, X(t)) for almost all t in [0, T]. In the same way we obtain from (2.7) that P(t) E aw(t, Y(t)). The proof is completed.

Remark 2.1. In view of the T-periodicity of the functions r#~(* , x), w(*, x), x E X, we notice that each minimizer x(m), of (0.2) can be identified with some function nr(‘) E AT(X) restricted to [0, T]. This means that if we consider the functional

3

(i+ UT (-4(t,tit)) + w(t,x’(t))) dt + UCjT),tiO’ + l)T)),

j = 0, +I, +2 , -me, instead of (O.2), then XT(‘) restricted to DT, (j + l)T] is a minimizer for it in the space of absolutely continuous functions x: DT, (_j + l)T] --* X. We further observe that, for the same reason, the function p’(e) E L’(0, T, X*) from the assertion of theorem 1.3 can be identified with some function &( *) E P(X*) restricted to [0, T]. Define p,(t) = p(O) + l;,fl;(.r) d.9, r E DT, (_j + l)T), j = 0, +l, +2, . . . . Of course, aT( *) E P(X*). From corollary 2.1 we infer that

PT(t) E &dt, %(I)) a.e. in R,

-iii(t) E a+(ts XT(f)) a.e. in R.

3. EXISTENCE OF A MINIMUM FOR J

In this section we prove that Jattains its minimum in A(X), and that the set of minimizers for J is sequentially weakly compact in A(X). We assume the hypotheses of theorem 0.1 to be satisfied.

We begin with the following lemma.

LEMMA 3.1. Let S, = [x(e) E A(X) : J(x( s)) 5 b), b > 0. For sufficiently large b, S, are nonempty and bounded in the norm ]I - IIc (Ilx(-)llc = sup(Ix(t)I : t E [0, T])). Moreover, J is bounded below.

Proof. The first assertion follows from the assumptions on 4 and ly. Indeed, we fix b > 0. Let x(e) E A(X) and J(x(*)) I b. Then x(m) E A,(X) and, by (0.5), (0.6), we have

5

T

b L J(x(.)) 1 l(t)jx’(t)l” dt + jTe(t)dt - 5’ k(t)~x(t)~” dt - jTd(r)dr. (3.1)

As x(e) E A,(X), we hive the estimate lx(;)i_ 5 P’~~lx’(s)l” ds, and this,

b + s

‘(d(t) - e(t)) dr _ 0

> [;(l(f) - ~:k(s)s”‘“dr)Ix’(t)l’dr

Tlx’(t)lm dt 5: (Q/T”‘“)lx(t)l” 0

(3.2)

for all t in [0, T]. This also proves the second part of lemma 3.1.

794 A. NOWAKOWSKI and A. R~GOWSKI

Take b E R such that S, # 0. Put E = (b + jl(d(t) - e(t)) dt)(,m”‘/Q)l’m and D = (x E X; 1x1 5 E]. By (3.2) each x(m) E S, satisfies x(t) E D, t E [0, T]. Define

&9x> = 40% xl ifxED, _-43

otherwise,

L(t, x, u) = -&t, x) + ~(t, V) in [0, T] x X x X. We easily see that L(t, x, *) is convex, L is It x [B-measurable and satisfies the growth condition from Section 1 with

f(GP) = l n(ml(t))“-l Ipj: - e(t) + k(t)E”’ + d(t).

We assume that E, D defined above are the same as in (c) of theorem 0.1. Therefore L(t, -, -) is lower semicontinuous in X, x X,.

THEOREM 3.1. J attains its minimum in A(X) and the set of minimizers is sequentially weakly compact in A(X).

Proof. Denote 1(x(.)) = jcL(t, x(t), x’(t)) dt + /(x(O), x(T)). Then J(x(*)) = &x(e)) if J@(e)) I b, and thus

inf(J(x(*)) :x(e) E A(X)] = inf(&x(.)) : x(a) E A(X)).

By theorem 1.1, Jattains its minimum in A(X) for some A?(*). Of course, .J(X( *)) 5 b. Thus X( *) is also a minimizer for J. The same theorem implies that the set of all those minimizers is sequentially weakly compact in A(X).

4. THE PROOF OF THEOREM 0.1

It is an obvious consequence of the above results. Indeed, the last assertion of theorem 0.1 follows from theorem 3.1. This theorem, corollary 2.1 and remark 2.1 imply the first assertion. The second one follows directly from corollary 2.1 and remark 2.1.

5. AN EXAMPLE OF EQUATION (0.1)

Let SJ be a bounded and open subset of R” with a sufficiently smooth boundary r. We set P=RxCJandE’=Rxr.Letj,:RxR + (- 00, +oo) be bounded, k-measurable and T- periodic with respect to the first variable, continuously differentiable and convex with respect to the second variable and such that j, (t, r) L e(t), t E [0, T], r E R, e(e) is as in (0.5). We put

WI(f, z) = s j,tt, z(u)) du R

for t in R and z E L’(a). Let X = l+$,1p2(CJ) (W01*2(sZ) is the Sobolev space of functions from r?(a) with compact support in a for which the distributional derivative belong to L’(n)) with the norm 1~1’ = I;= 1 jn(zU,(u))’ du + jn(z(u))2 du. vl(t, .) is convex and lower semicontinuous in X (see e.g. [2]), t E [0, T]; v/l is also IL x [B-measurable in R x X. We set ~(t, z) = tlz12 + ~,(t, z) for t E R, z E X. Then (0.5) is satisfied with m = 2 and r(t) =i. Moreover, z( .) + jrvl(t, z(t)) dt is finite on L-(0, T;B), B defined in theorem 0.1.

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‘,5: ml 0 = (n ‘z)x

'd 3 (n ‘I) ‘(n '1)X = (n ‘J + I)X

‘0 = (I ‘n ‘(n '1)x)y + (n ‘3)x@ ‘j)% +

~qqord aql 01 uoynlos t? ale lCaq$ put? ‘h ‘$ mo JOJ (~‘0) AJS~ES put? ‘rCja+padsa~ ‘(*x)v‘(x)v Ol %uo\aq (*)d‘(.& [_z“ol 01 uo!l3~wa.I aSOqM (.)=B‘(.)'_x lyxa a.Iay1

:I’() uxa~oay~ liq ‘aJoJa.IaqJ_ '#1 = (2‘$) v?ql qms *_y u! luaurala UE s! *2 alayM ‘Ala,ypadsal ‘(x‘J)~~

+ x(l)V ‘(2‘3)'&? + *2 01 wba pm lUawala-a@!fS ale cpe ‘A@ slas aql ~eql awln3pi?~ aM

'Z = UI ql!M 9 JOJ pagS!lES ale 1'0 UIaloayl JO suogdumsse IIF? uaqL '5 + x = + )a?

'mu03 s! (.‘I)0 + (.‘r)x pwx x 8 us alqemwaw

-gi X ll s! 3 ‘Jaql.W 'x u! snonuguo3 LIyEaM dIlepuanbas sr ‘[J ‘01 3 1 ‘(. ‘j)f~ snql lDoduro3 s! (C&7 3 (f_l),~~“~ WppaquI! aq$ sv -(n ‘I) qsea 103 M 01 vadsal ql!M xamo3 s! (I VI ‘MM)8

+ p(” ‘I)Oa 1w.p qsns ST sp (I ‘n ‘s)y ij = (I ‘n ‘~)a leqi aumssr! ahi ‘laAoa_IoK .(y !J ‘@7 01 zuoraq [_L ‘o] 01 suwvisa~ r!aqi pue ypoVad;l am (.)p ‘( .)y ‘($)p + ,Ix/(~)y+ bq papunoq s!

0 lJ

npsp(Vn W = (X‘I)t) (W ss

uoy3un~ aqi pue a!po!-rad-J SI (I ‘n ‘s)y +- J ‘a~qamseaw s! (J ‘n ‘s)y + (I ‘n) ‘snonupuos s! (I ‘n ‘s)y + s vzql qsns 21 + H x 0 x y::yuo~wm~papunoqe'a~e~_

'_y II! snonuyuo~ lCIqt?aM dllepuanbas so ‘[J‘o] 3 J ‘(.‘j)_y

a~~JaJaq1 ((~1 *%a aas) ixdmo3 s! ‘[J‘o] 3 I ‘(1)~ ams *,Ix\(~)y+ 5 (x‘J)~ pm ‘X II! xaAuo3

‘snonuguo:, pue I u! aIqemsoauI s! (x ‘x(~)y)Q = (x ‘I)X ieqi aas ah *(gyj) sagsgos ‘[J so]

UI alqal%aw! (.)y pm ‘u x x 3 (n ‘I) ‘(1)~ 5: (n ‘r)Oo 5: 0 ‘u ‘e** ‘2 6 1 = ! ‘(~)a s (n ‘I)% 5 0 lEq1 qms put? I u! ypoyad-J ‘7~ x x II! suopmy a[qe_uwaauI ale (n ‘I)% ‘(n'j)!a aJaqM

-(n)x(n ‘j)Oo + ($ (n s,)!o) F I$‘- = (N(x(t)V)

S6L

se Wol 3 I ‘(a),~,_~ + W,.,"~ :(W aui3aa

suognlos J!pO!Jad


7. EKELAND I. & LASRY I. M., Duality in nonconvex variational problems, in Advances in Hamiltonian Systems (Edited by J. P. AUBIN, A. BENSOUSSAN & I. EKELAND). BirkGser, Boston (1983).

8. EKELAND I. & TEMAM R., Convex Analysis and Va,rarional Problems. North-Holland, Amsterdam (1976). 9. NOWAKOWSKI A., Note on convex integral functionals, Demons?. Math. (to appear).

IO. ROCKAFELLAR R. T., Convex integral functionals and duality, in Conrriburions to Nonlinear Funcfional Analysis (Edited by F. H. ZARANTONELLO). Academic Press, New York (1971).

Il. TOLAND J. F., A duality principle for non-convex optimization and the calculus of variations, Archs Ration. Mech. Analysis 71, 41-61 (1979).

12. TOLAND J. F., Duality in nonconvex optimization, J. math. Analysis Applic. 66, 399-415 (1978).

periodic solutions for a class of nonlinear hyperbolic equations

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