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Periodic solutions of variational systems of ordinary differential equations
JOURNAL
OF DIFFERENTIAL
EQUATIONS
28, 354-368
(1978)
Periodic Solutions of Variational Systems Ordinary Differential Equations DAVID
University
C.
CLARK
of Puerto Rico, Mayagiiez,
Received
November
9, 1976;
of
Puerto Rico 00708
revised
May
18, 1977
This paper is concerned with existence of periodic solutions variational systems of ordinary differential equations of the form
for certain
Here aij = aii is T-periodic, gi is odd in x and T-periodic in t, and there exists V(t, X) such that gi = aV’/axi . We seek solutions of (0.1) which are T-periodic. These occur in pairs &-x(t). The system (0.1) is formally the set of Euler equations of the functional 4(x)
=
I’
(t 0
.f
aijxj’3cj’
-
V(t,
x)j
dt.
r.i=l
Our existence theorems are based on two abstract theorems of critical point theory, one due to the author [2] and the other a combination of results due to Ambrosetti and Rabinowitz [ 11. The primary motivation for this paper was to extend and complement the results of [6]. It is to be noted that our methods apply equally well to boundary value problems for systems of ordinary differential equations and even to boundary value problems for partial differential equations. Indeed, in [l] the authors develop new results in critical point theory, similar to those used in this paper, and apply them to a Dirichlet problem for a partial differential equation and to integral equations. Theorem 3.9 below in analogous to Theorem 3.13 of [l], but its hypothesis (A6) is more general than the direct analog of the corresponding hypothesis (p&. Berger [3, 51 has studied existence of periodic solutions for systems of the form x” + grad U(X) = 0, where U(0) = 0 and U is even in [3] and is convex in [5]. The results are local to x = 0 in [3] and generalize a theorem of Lyapunov; the result in [5] is global. There is a more recent paper by Lazer [4] which refines some of the results of [3]. 354 0022-0396/78/0283-0354$02.00/O Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
PERIODIC SOLUTIONS OF DIFFERENTIAL
EQUATIONS
355
Section 1 contains background material on critical point theory for even real valued functions on a Banach space. Section 2 deals with background material on quadratic forms arising from systems of linear differential equations. Section 3 sets forth the hypothesis to be satisfied by aij and gi , and the main existence theorems for periodic solutions are developed. There are several different cases considered, corresponding to different behaviors of #J(X) in the neighborhood of 0 and in the neighborhood of infinity. Section 4 gives analogous existence theorems for odd and even solutions, and briefly indicates the changes necessary in the arguments.
1. CRITICAL
POINT
THEORY
Let E denote a real Banach space, Z(E) the class of closed sets A C E\(O) which are symmetric with respect to the origin, and let r(A) denote the genus of A, the least integer K such that there exists an odd continuous map of A into R”\(O). If no such integer exists we set r(A) = co, and we set y( D) = 0. The proof of (l)-(3) in the following lemma can be found in [7], and that of (4) in [l]. LEMMA
(1)
1.1.
Let A, B E Z(E).
If there exists an odd continuous map of A into B, then y(A) < y(B).
(2) Y A C 4 ~(4 < r(B). (3) metric,
If there exists an odd homeomorphism of A onto the boundary of a symbounded, open neighborhood of 0 in Rk, then y(A) = k.
(4) Let M be a k dimensional subspace of E, and N a complementary subspace. Ify(A)>k,thenAnN# QI. Let I’, = (A E Z(E) [ A compact, tions #: E + R, let c,(4)
= inf{max #(A) 1A E T’,,}.
If m < n, T, 1 I’, , hence c,(#) We introduce
y(A) > m}. For even, continuous
func-
(1.1)
< c,($).
the assumptions:
(El) There exists an Edimensional subspace M of E and p > 0 such that Z,!J< 0 on S, n M, where S, = {x E E / 11x 11= p]. (E2) There exists aj dimensional subspace il? of E with a complementary subspace N such that # is bounded below on N. LEMMA 1.2. Let 4: E - R be even, continuous, satisfy (El), (E2), with l > j, and let $(O) = 0. Then -CD < c,(#) < Oforj < m < 1.
356
DAVID C. CLARK
Proof. Let M and p be as in (El). By (3) of Lemma 1.1, y(S, n M) = 1. Since max $(S, n M) < 0, c,(z/) -C 0 f or m < 1. Let Ic;i and N be as in (E2) andolbealowerboundfor$onN.Form>jandAeI’,,AnN#@, by (4) of Lemma 1.1, hence c,(#) 2 01. Assuming also that I/Jpossessesa continuous Frechet derivative, we introduce the conditions (B) I VW < (C) contains
For every a > 0, there exist b, r > 0 such that // #‘(x)/l > b whenever a and II x II > r. If {+} C E is bounded, 1#(+)I is bounded, and #‘(xk) -+ 0, then {xJ a convergent subsequence.
Let K, = (x E E j 1+4(x)= c, Z/J’(X)= O}. By combining Lemma 1.2 with Theorem 8 of [2], we have THEOREM 1.3. Let #: E -+ R be even, of class Cl, satisfy (B), (C), (El), (E2), with 1 > j, and let 4(O) = 0. Then if j < m < 1, Kcm is compact and non-empty, and if -con--m+l. Hence # has at least 1 - jpairs fx of non-zero critical points.
It is to be noted that the critical levels obtained are negative and that we must have I > j. Ambrosetti and Robinowitz [l] obtained existence theorems for critical points which are complementary to Theorem 1.3, in some sense. Theorem 1.4 is essentially a combination of Theorems 2.19 and 2.23 of [l], and we have mainly followed their ideas in the proof. Let B, = {x E E j II x 11< p} and S, = {x E E I /j x )I = p}, and denote B, , S, by B, S, respectively. We assume (E3) There exists a j dimensional subspace il?i of E and a number Y > 0 such that $ < 0 on S, n iii? (E4) There exists an 1’ dimensional subspace M’ of E, a complementary subspace N’, and p, 01> 0 such that tc,> 01on S, n IV. Furthermore, let A* = {h E C(E, E) / h is an odd homeomorphism of E onto E with h(B) C BP u (cr-‘([O, CO))}, A, = {K C E I K is compact, symmetric, Y(K n h(S)) > m for all h E A*}, b,(4) = inf{max I/J(K) I K E A,}. THEOREM 1.4. Let $: E -+ R be even, of class Cl, satisfy (B), (C), (E3), (E4), where I’ < j and p < T, and let #(O) = 0. For m < j, A, # ,@, so thut b,(#) If b,=b,=b, where l’
PERIODIC SOLUTIONS OF DIFFERENTIAL
351
EQUATIONS
n < j, then y(Kb) >, n - m + 1. Hence qbhas at least j - 1’pairs -&x of non-zero critical points. Proof. Let a and r be as in (E3), let &I be an m dimensional subspace of &!I, and let K, = aT n a. We shall show that K, E A,, . From (E3) it follows that if heA* then h(B)nS,nti=
o.
Consequently, the component of h(B) n a, considered as an open subset of A?l, that contains the origin must be contained in K,. . Since the boundary of this component, relative to i@, is contained in K, n h(S), it follows by (2) and (3) that y(K, n h(S)) 3 m; hence K, E A, . Since A, r) Am+r , b, < b,+i for wz < j. To show that b,,,, 3 01, let K E A,+, and define h E A* by h(x) = px. Then y(K n S,) > I’ + 1, hence by (4) of Lemma 1.1, K n S, n N’ # B’, where N’ is as in (E4), so that max #(K n S,) 3 01and therefore b,,+i 3 oi. If 4 is an odd homeomorphism of E onto E such that 4(x) = x if 4(x) < 0, and such that @l(#-l([O, CD))) C #-l([O, co)), then 4-l 0 h E A* whenever h E A*. Therefore, if K E A, and h E A*, then YMK)
hence +(&) The proof assertion of of Theorem
n h(S))
= 44W
n 4-l
0 W)))
= Y(K n 4-l
0 h(S))
2 m;
C A, . of the assertion regarding y(Kb) is the same as for the corresponding Theorem 2.19 of [l], and follows standard ideas. The last assertion 1.4 is an immediate consequence of the preceding assertion.
2. QUADRATIC FORMS Let H denote the Hilbert space consisting of functions X: R ---f Rn whose components xi are T-periodic, absolutely continuous, and have square integrable derivatives on [0, T], together with the inner product
(x,Y) = J^,’f (xi’yi’ + w,) dt. i=l
We denote the associated norm by // 11. We shall use certain quadratic forms defined on H of the general form
(aij(t) xi/xi’ - &(t) xixj) dt,
XEH,
(2.2)
358
DAVID
C. CLARK
where aij = aji is T-periodic, of class Cl, and positive definite, and bij = bij is T-periodic and continuous. Also, for x, y E H we let
B(x,Y)
= lo’
i (a&> i,j=l
xi%’
- b(t)
X~YA dt.
(2.3)
We say that K is of index p (modified index p’) if p( p’) is the maximum dimension of subspaces on which K is negative definite (non-positive). It is clear that K is continuous and possessesa Frechet derivative given by
(K’(X), Y)= pcx, Y).
(2.4)
We say that K is regular if 11K’(X)// > C/j x 11,where C > 0. We let L2 denote the Hilbert space of functions x: R -+ IIn, whose components xi are T-periodic and square integrable on [0, T], with the inner product
and corresponding norm I/ 11s. Formally integrating by parts in (2.3), /3(x, y) = -(9x, Y)~ , where n (9~)~ = C ((aijxi’)’ + bijxj). j=l
Using integral equations in a standard way, it can be shown that there exist eigenvalues hi and corresponding eigenfunctions ui E Ca n H, satisfying L?ui + h&i = 0,
(2.6)
A, < A, < A, < ... + 03 (Ui, Ui)o =
(2.7) (2.8)
Sij
and such that (ui} is complete in L2. Moreover, there hold the Parseval relations, forx,yEH, /3(x, Y) = f bidi i=l ci = (x, ui)O )
9
di = (y, ui),, .
(2.9)
Through the remainder of this section and in subsequent sections we shall use C, C, , C, ,... to denote positive constants which do not depend on the particular element of H at hand. Now we let A4 = span(d j hi < 0}, M’ = span{d 1hi & 0}, N’ = M’I
= +ii{u”
1Xi > 0).
(2.10) (2.11) (2.12)
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
EQUATIONS
359
The index (modified index) of K is finite and equal to dim M is negatiwe definite on M andpositive dejnite on N’. Hence the index (modiJied index) of K is equal to the number of negative (non positive) eigenvalues hi . K is regular if and only if no & equals zero. 2.1.
LEMMA
(dim M’).
Proof.
K
dim M is finite, in view of (2.7). Since K(X) = @(x, x), K(X)
=
4 i
/\iCi2,
ci
=
(x,
u$
)
(2.13)
i=l
by (2.9), and so K is negative definite on M and non positive on M’. It is clear that K is not negative definite (non positive) on any subspace of dimension larger than dim M (dim M’). Hence index K = dim M and modified ‘index K = dim M’. To show that K is positive definite on IV (with respect to the norm of H), we first note that
using (2.2). Let
Ml = span(ui I 0 < hi < k}, M, = span{ui 1Xi >, k), Clearly N’ = Ml 0 M2, and by (2.13), if where y E Ml , z E M2 . By the finite dimensionality of Ml , K(Y) >, C, 11y lj2. It remains to show that K(z) 2 C I] z l12.By (2.13),
where
k > 2C2 + 2C,Cs/C,.
x E N’,
K(X)
=
K(y)
+
K(z),
K(x) 2 (k/2) 11X 11:-
(2.16)
Combining (2.16) and (2.15) with x set equal to z, and substituting into (2.14) with x set equal to z, we obtain that K(z) >, C II x II2 with C > 0. Hence K is positive definite on IV. If some Xi = 0, by (2.4) and (2.9), K’(Ui) = 0. If no A, = 0, then H = M @N and, writing x = y + z, y E M, 2 E N’, we have
(K’(X),aY + b> = 2‘43’) + 33’44, from which it follows, using the definiteness on M and IV’, that 11K’(x)~] > C II x jj, so that K is regular. Finally, let us consider the autonomous case, where the aij and b,, are constant. Here the index of K can be calculated and the regularity checked directly in
360
DAVID
C. CLARK
terms of the matrices A = (aij) and B = (&). In fact, using the positive definiteness of A, it is well known that there exists an n x n matrix M such that MTAM
= id,
MTBM
= D = diag(d,).
(2.17)
Letting x(t) = My(t), we have K(X) = i(y)
= & joT f
(y;” - diyi2) dt.
(2.18)
i=l
Clearly index K = index d. For the corresponding bilinear form P(Y, 4 = jo= il
(~r’zi’ - diyid
dt.
(2.19)
it is clear that W(Y)? 4 = ac Y, 4
(2.20)
We introduce the set ei ,
(sinj27rtlT) ei ,
(cosj27rt/T) ei ,
(2.21)
1 < i < n, j 3 1, where {e,} is the usual basis in R”. It is clear that the set in (2.21) is complete in H and is orthogonal with respect to p. Therefore, the index (modified index) of r7is equal to the number of elements of (2.21) on which E is negative (non-positive). Also, it is easily checked, using (2.20), that z is regular if and only if I? is nonzero on all the elements of (2.21); and since the norm 11M-lx Ilo is equivalent to /I x /Is regularity of K implies regularity of K. After a short calculation we arrive at the following result. LEMMA 2.2. Suppose that (aij), (bij) are constant matrices, and dejine (di) as in (2.17). Then the index (mod$ed index) of K is equal to the numbs of di > 0 (di > 0) plus twice the number of pairs (i, j) such that di > j2(2r/T)2, (di > j2(2?r/ T)2), with j >, 1. K is regular if and only if there doesnot hold di = j2(2rr/ T)2 for any 1 < i < n, j 3 0.
3. THE FUNCTIONAL 4, MAIN RESULTS The results of the previous two sections are applied to $ defined on H. Several special cases are considered. We begin with some hypotheses common to all cases. Suppose that aij(t), gi(t, u) are defined for all t E R, u E R”, aij = aji is of class Cr and T-periodic, gi is continuous and T-periodic, and that (Al)
(aij(t)) is positive definite for all t,
(-44 (A3)
gdt, -4 = -g&, 4, There exists V(t, U) such that gi = aP’/& .
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
EQUATIONS
361
Thus a,j satisfies the same conditions as in Section 2. We choose V SO that V(t, 0) = 0, and consequently $(O) = 0. It is clear that (0.2) makes sense for x E H, that 4 is even, bounded on bounded sets in H, and possessesa continuous Frechet derivative, bounded on bounded sets, and satisfying (3.1) By standard arguments in the calculus of variations it can be shown that d’(x) = 0 precisely for those T-periodic functions x which are solutions of (0.1). Since (b is even, critical points, and hence solutions, occur in pairs &:x. LEMMA
3.1. If (Al), (A3) are satisjied, 4 satisjes (C).
Proof. Let {&} be bounded and $‘(&) --f 0. Since H is separable, {x”} contains a weakly convergent subsequence, which we again denote by {xk}, with weak limit f, say. It is immediate that (6(X”) -f(x),
Xk - a) + 0.
(3.2)
On the other hand, by (3.1),
(c)‘(x”) - f(f),Xk - 2) + j' 2 (g&l x") - g,(t, f))(Xf" - iQ dt. 0 ix1 Weak convergence in H implies xk --f x uniformly, hence the second term on the right in (3.3) approaches zero. With this fact, (Al), (3.2), (3.3) imply xk + x strongly in H, which verifies (C). Consider now the hypothesis as /uI+O,
(-44)
where bij = bji is continuous and T-periodic. It follows from (A3) and (A4) that v(t, *) = 4 jJ bij(t) %lcj + O(l u I’), i.j=l
as IuI-+O.
(3.4)
362
DAVID C. CLARK
aij and bii in (A4) satisfy the same conditions as in Section 2. We define K as in (2.2). It follows from (0.2), (2.2), and (3.4) that $Nx) = “(4 + 41 x 112),
as jj x jl 3 0.
(3.5)
The following lemma is an immediate consequence of Lemma 2.1 and (3.5). LEMMA 3.2. If (Al)-(A4) are s&s-ed, 4 satisjes (El) ((E4)) with 1 (I’) equal to the index p (modified index p’) Of K.
We shall put forth various conditions for gi to satisfy at infinity. First we consider a case that is asymptotically linear in the sense of Krasnoselskii [S]. We assume that
where cii = tit is continuous and T-periodic, and where the quadratic form z(x) = 3 Jb7$,
(uijx;x;
- cijx,x,) dt
is regular. It is easily seen that if (A5) is satisfied, then
+Yx) = e4 + 4 x II>> IIx II.-+ =4
(3.6)
4(x> = $4
(3.7)
+ 41 * 112),
II A”II + cJ3.
LEMMA 3.3. If (Al)-(A3), (A5) are satisfied, then (B), (E2), (E3) are satis$ed, with j equal to the index q of K.
Proof. Equation (3.6) and the hypothesis that 2 is regular imply that ]j 4’(x)ll > const > 0 whenever Jjx (/ is sufficiently large, hence (B) is satisfied. Let j% be the q dimensional subspace in Lemma 2.1 on which 17is negative definite. Since i is regular, there is no Ai equal to zero, hence the subspace N of Lemma 2.1 is complementary to a. Also z is positive definite on N’. Hence, and in view of (3.7) and the fact that # is bounded on bounded sets, # is bounded below on N, thus verifying (E2). It is evident from (3.7) that (E3) is satisfied. In the preceding discussion and Lemmas 3.1-3.3 all the hypotheses of Theorem 1.3 (1.4) have been verified for 4 whenp > q (q > p’). In view of the equivalence between critical points and periodic solutions, we have THEOREM 3.4. Let (Al)-(A5) be satisjed. Let p (p’) be the index (modz$ed index) of K and q the index of 2. If p > q (q > p’) there are at least p - q(q - p’) pairs f x(t) of non-triwiul T-periodic solutions of (0.1).
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
Next we study a case where d(x) sional subspace. We assume
EQUATIONS
363
--co as I/ x II--+ co, in every Jinite dimen-
646) for some C > 0 andfor
j u 1suficiently
large.
(A7)
V(t, u)/l u I2 -+ 00, as j u I -
(A8)
g,(t, u) = O(l u I) as I u I + 0, uniformly
co. in t.
By some elementary calculus of variations, we have the following lemma. LEMMA 3.5. If x E H and min / x(t)\ < 4 max 1x(t)\, then /I x’ (I0 > C (1x (I0, where C > 0 depends only on T. LEMMA 3.6. Proof.
If (Al), (A3), (A6) are satisfied, (b satisjies (B).
By (0.2) and (3.1),
(f(x), x) =
2464 + s,’ (2W
x) -
f xi&, i=l
x)) dt.
(3.8)
By a standard imbedding result, max I x(t)] < C, I/ x 11;hence from (A6), dt<--C,lj~[l-‘~=Vdt+C~.
(3.9)
0
Supposethat IIx‘ II0< G IIx II0, where C, is chosen so this inequality implies that min 1x(t)1 > -&max I x(t)l, on the basis of Lemma 3.5. Then min I x(t)/ > Cs 11x II and from (A7), s”’ V, 4 dt > CoIIx II’,
(3.10)
for Ij x (/ sufficiently large. If I $(x)1 < a for some given a, then from (3.8)-(3.10) it follows that 11C’(x)11> const > 0 for II x II sufficiently large. Suppose now the other case, that 11x’ IJo> C, 11x Ilo, with ) +(x)1 < a. From (0.2) and (Al), = V(t, x) dt > C, II x’ 11:-
I0
a,
(3.11)
hence from (3.8), (3.9), (3.1 l), again II@(x)11> const > 0 for II x 11sufficiently large. This complete the verification of (B).
364
DAVID C. CLARK
LEMMA 3.7.
(A7) are satisfied, 4 satisfies (E3)for
1f(A3),
Proof. It is clear that on any finite dimensional as j/ x Ij --f 03, from which the result follows. LEMMA 3.8. If (Al), such that 4 satis-es (E4).
every value of j.
subspace J?!, $(x) -
--co
(A3), (A8) are satis$ed, given p > 0, there exists 1’
Proof. For a given p > 0, j x(t)1 < C, for x EB, and for all t. By (AS), g,(t, ZJ) < 2Ca ] u 1 and hence V(t, u) < Ca 1u 12, for 1u I < C, . Using (A3),
w
3 cs IIx’ II: - G’ IIx II:
(3.12)
for x E BP . It is standard to show that the expression on the right hand side of (3.12) is positive definite on the L, orthogonal complement in H of some finite dimensional subspace, and (E4) then follows. In Lemmas 3.1, 3.6, 3.7, 3.8 we have verified the hypotheses of Theorem 1.4, where j and hence j - I’ may be taken arbitrarily large, hence the theorem: (A6)-(A8) be satis$ed. There exist an in.nite THEOREM 3.9. Let (AI)-(A3), number of pairs *x(t) of T-periodic solutions of (0.1). If 4(x) + DC)as 11x /I -+ co, then (E3) and (B) hold, so that Theorem 1.3 will apply, given certain conditions at the origin. However, in applications it is perhaps likely that I’(u) > 0 for u # 0, and then 4(x) < 0 for x = const # 0. In Section 4, we get around this by considering the restriction of 4 to odd functions and seeking odd solutions of (0.1). For the present, we consider a situation where V(t, u) > 0 for u # 0 and 4(x) -+ co as 11x jl -+ 00, when x is of mean zero. Let H, denote the subspace of constant functions x(t) and H’ the subspace of functions x(t) of mean zero, that is, such that fi x(t) dt = 0. Then H, is of dimension n, and H = H, @ H’. We assume
649)
gl (gtk 4 - g& v))(ui - vi>
G i
C&)(% - v&f+ - Vj>+ Cl I U-VII,
(3.13)
i.i=l
where cij(t) is such that if x E H’, T
s0 (MO)
gl
12
c
(aii(t) xi’xj’ - ~ijxixj) dt > C’, ((x 11’.
(3.14)
i,i=l
(g&,
4
-
gi(t,
VW
-
vi)
2
G
I u -
v
I -
c,
.
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
365
EQUATIONS
(A9) are satisfied, #Jsatisfies (E2) with j = n.
LEMMA
3.10.
If(Al)-(A3),
Proof.
V in (A3) may be represented
as (3.15)
By (A2),g,(t,
Equations
0) = 0, hence from (3.13) and (3.15)
(0.2) (3.14), (3.16) imply K4
Since H’ is complementary
to HO , (E2) follows, with j = n.
If (Al)-(A3),
(A9), (AIO) are satisjied, 4 satisfies (B).
LEMMA
3.11.
Proof.
From (3.1) and (A9),
(#J’(x)-f(Y),X--y)
XEH’.
2 c2 II x II2 - c, I/ x II,
3c211~-Y112-~lll~-Yll,
x-YEH’.
(3.17)
For a given x E H, consider #( y) = +(x + y) for y E H’. It is a standard exercise in the calculus of variations to show that # attains a minimum, say at y = 9. We denote x + 9 by R. Since #‘( 9) = 0, ($‘(a), y) = 0 for ally E H’, hence (f(i),
(3.18)
x - a) = 0.
From (3.17), (3.18), and the Schwarz inequality, II 4WII
>
Cl
II x
-
2 II -
c2
-
(3.19)
We define x = (l/T) li x(t) dt. Then x - x E H’. Setting u = x and v = 0 in (AlO), and using (3.15), w h ic h is still valid, we have V(t, 2) >, C’s \ x 1 - C, . Since +(a) < $(a) we then have w Applying
< -G
II XII + ccl.
(3.20)
the mean value theorem, and using (3.18),
C(x) - $(a) = 0 (- ,tl
atj(xi’ - fi’)(xi’
- 4’) dt
- 0)2) - gi(t, a))@, - ~2~)dt, (3.21)
366
DAVID C. CLARK
where 0 < 0 < 1. Denoting
the latter integral
.l > qc,
in (3.21) by 1, and using (AlO),
II x - .GII -
(3.22)
C,).
Now we assume that ) +(zc)I < a, as in the hypotheses of(B). (3.22) and the boundedness of aij ,
Then from (3.20)-
IIXII < C, IIx - f II2- Cl, IIx - 2 II + %a +
Cl,
.
(3.23)
From (3.17) we have
IIC’(4ll 3 - II$wl
+ c, IIx - ff II - G *
(3.24)
If II x - 2 /) 3 const > C,/C, , then by (3.19), II r)‘(x)11 > const > 0.
(3.25)
In the contrary case, (3.23) implies that 11x II is bounded; then II +‘@)I] is bounded, and (3.24) implies that (3.25) holds when II x II is suffiicently large. Thus, we have verified (B) in all cases. Using Lemmas 3.1, 3.2, 3.10, 3.11 in applying Theorem 1.3, we have THEOREM 3.12. Let (Al)-(A4), (A9), (AlO) be satisfied, and let the index p of K be greater than n. Then there exist at least p - n pairs *x(t) of non-trivial T-periodic solutions of (0.1).
4. ODD AND EVEN SOLUTIONS The methods we have used may be modified to yield T-periodic solutions of (0.1) which are odd or even in t. An additional hypothesis is introduced below, and the changes necessary in the arguments are briefly indicated. Analogs of the existence theorems of Section 3 are stated. The only result that is conspicuously different is Theorem 4.5 for odd solutions, and this is still partly analogous to Theorem 3.12. Let H, C H (H, C H) denote the subspace of odd (even) functions, and that gd&+,) the restriction of 4 to H,,(H,). W e assume, in addition to (Al)-(A3), (Al 1) gi(-t,
u) = gi(t, u), aii(-t)
= aij(t).
Let x E H, be a critical point of 4s . Then 0 = (&‘(x), y) = (F(X), y) for all y E H, . Since the integral over [0, T] of a T-periodic odd function is zero, by (3.1) we see that (V(x), y) = 0 for all y E H, . Hence f(x) = 0, and, as before, x is solution of (0.1). Conversely, an odd solution of (0.1) is obviously a critical point of +e . The analogous result for +, also holds. Hence, LEMMA 4.1. If (Al)-(A3), (Al 1) hold, the set of T-periodic odd (even) solutions of (0.1) coincides with the set of criticalpoints of&,(&J.
PERIODIC SOLUTIONS OF DIFFERENTIAL
EQUATIONS
367
In the discussion of quadratic forms we replace K in (2.2) by its restriction (Ke) to H,, (H,), and assume, in addition to the previous hypotheses on aij, bij , that aij , bij are even functions of t. We define the index of K,,(K,) as the maximum dimension of subspaces of Ho (H,) on which K,, (KJ is negative definite, and the modified index similarly. Relations analogous to (2.6)-(2.9) hold, where the eigenfunctions are now odd or even. These relations are derived as before by use of integral equations. Since $4 maps C2 odd (even) functions into odd (even) functions, the corresponding integral operators act in Ho n L2 (H, n Lg) and one can apply the theory of symmetric, completely continuous operators as before. The analogs of Lemma 2.1 and Corollary 2.2 are derived in the same way as previously. If (A4), (A$ and (Al 1) are satisfied, aij , bij , and cii are even functions, hence the discussion of the preceding paragraph applied to Kg, Kg, K, , K, , defined analogously to K, k of Section 3. We arrive at the following analog of Theorem 3.4. K,,
THEOREM 4.2. Let (Al)-(A5), (All) be satisjed. Let p ( p’) be the index (modified index) of q, and q the index of i. . If p > q (q > p’), there are at least p - q (q - p’) pairs *x(t) of non-trim& T-periodic, odd solutions of (0.1).
The corresponding statement regarding even solutions of (0.1) is also valid. The modification of the proof for Theorem 3.9 is completely straightforward and yields the following result. THEOREM 4.3. Let (Al)-(A3), (A6)-(A8), (All) be satisjed. Then there exist an injkite number of pairs -& x(t) of T-periodic odd (even) solutions ?/ (0.1).
The situation with the final theorem of Section 3 is slightly different. First we consider even solutions. Let H’, be the subspace of H, consisting of the functions of mean zero. We assume (A9)e
Same as (A9), but with H’, replacing H’.
THEOREM
index of K, T-periodic,
,
4.4. Let (Al)-(A4), (A9J, (AlO), (All) be satisJied, let p be the and p > n. Then there exist at least p - n pairs &x(t) of non-trivial, ezen solutions of (0.1).
For odd solutions, we assume (AS),,
Same as (A9), but with H,, replacing H’.
THEOREM 4.5. Let (Al)-(A4), (AS),, , (All) be sutis$ed, and let p be the index of IQ, . Then there exist at least p pairs +x(t) of non-trivial, T-periodic, odd soZutions of (0.1).
368
DAVID C. CLARK
Proof. Theorem 1.3 is applied to $,, . (A9), implies that $0(x) + co as 11x // -+ co, so that (B) is satisfied and (E2) is satisfied, with j = 0. The other hypotheses are shown to satisfied, using the analogs of Lemma 3.1 and 3.2. Then Lemma 4.1 is applied.
ACKNOWLEDGMENT The author would
like to thank the referee for his helpful
criticisms
and suggestions.
REFERENCES 1. A. AMBROSETTI AND P. H. BABINOWITZ, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. 2. D. C. CLARK, A variant of the Lustemik-Schnirelman theory, Indiana Utz;niv.,Math. J. 22 (1972), 65-74. systems, J. Math. 3. M. S. BERGER, On periodic solutions of second order Hamiltonian Anal. Appl. 29 (1970), 512-522. 4. A. C. LAZER, Topological degree and symmetric families of periodic solutions of nondissipative second-order systems, J. Differential Equations 19 (1975). 62-69. 5. M. S. BERGER, Periodic solutions of second order dynamical systems and isoperimetric variational problems, Amer. J Math. 93 (1971), l-10. systems of ordinary 6. D. C. CLARK, On periodic solutions of autonomous Hamiltonian differential equations, Proc. Amer. Math. Sot. 39 (1973), 579-584. 7. C. V. COFFMAN, A maximum principle for a class of nonlinear integral equations, 1. Analyse Math. 22 (1969), 391-419. “Topological Methods in the Theory of Nonlinear Integral 8. M. A. KRASNOSELSKI, Macmillan, New York, 1964. Equations,”
OF DIFFERENTIAL
EQUATIONS
28, 354-368
(1978)
Periodic Solutions of Variational Systems Ordinary Differential Equations DAVID
University
C.
CLARK
of Puerto Rico, Mayagiiez,
Received
November
9, 1976;
of
Puerto Rico 00708
revised
May
18, 1977
This paper is concerned with existence of periodic solutions variational systems of ordinary differential equations of the form
for certain
Here aij = aii is T-periodic, gi is odd in x and T-periodic in t, and there exists V(t, X) such that gi = aV’/axi . We seek solutions of (0.1) which are T-periodic. These occur in pairs &-x(t). The system (0.1) is formally the set of Euler equations of the functional 4(x)
=
I’
(t 0
.f
aijxj’3cj’
-
V(t,
x)j
dt.
r.i=l
Our existence theorems are based on two abstract theorems of critical point theory, one due to the author [2] and the other a combination of results due to Ambrosetti and Rabinowitz [ 11. The primary motivation for this paper was to extend and complement the results of [6]. It is to be noted that our methods apply equally well to boundary value problems for systems of ordinary differential equations and even to boundary value problems for partial differential equations. Indeed, in [l] the authors develop new results in critical point theory, similar to those used in this paper, and apply them to a Dirichlet problem for a partial differential equation and to integral equations. Theorem 3.9 below in analogous to Theorem 3.13 of [l], but its hypothesis (A6) is more general than the direct analog of the corresponding hypothesis (p&. Berger [3, 51 has studied existence of periodic solutions for systems of the form x” + grad U(X) = 0, where U(0) = 0 and U is even in [3] and is convex in [5]. The results are local to x = 0 in [3] and generalize a theorem of Lyapunov; the result in [5] is global. There is a more recent paper by Lazer [4] which refines some of the results of [3]. 354 0022-0396/78/0283-0354$02.00/O Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
PERIODIC SOLUTIONS OF DIFFERENTIAL
EQUATIONS
355
Section 1 contains background material on critical point theory for even real valued functions on a Banach space. Section 2 deals with background material on quadratic forms arising from systems of linear differential equations. Section 3 sets forth the hypothesis to be satisfied by aij and gi , and the main existence theorems for periodic solutions are developed. There are several different cases considered, corresponding to different behaviors of #J(X) in the neighborhood of 0 and in the neighborhood of infinity. Section 4 gives analogous existence theorems for odd and even solutions, and briefly indicates the changes necessary in the arguments.
1. CRITICAL
POINT
THEORY
Let E denote a real Banach space, Z(E) the class of closed sets A C E\(O) which are symmetric with respect to the origin, and let r(A) denote the genus of A, the least integer K such that there exists an odd continuous map of A into R”\(O). If no such integer exists we set r(A) = co, and we set y( D) = 0. The proof of (l)-(3) in the following lemma can be found in [7], and that of (4) in [l]. LEMMA
(1)
1.1.
Let A, B E Z(E).
If there exists an odd continuous map of A into B, then y(A) < y(B).
(2) Y A C 4 ~(4 < r(B). (3) metric,
If there exists an odd homeomorphism of A onto the boundary of a symbounded, open neighborhood of 0 in Rk, then y(A) = k.
(4) Let M be a k dimensional subspace of E, and N a complementary subspace. Ify(A)>k,thenAnN# QI. Let I’, = (A E Z(E) [ A compact, tions #: E + R, let c,(4)
= inf{max #(A) 1A E T’,,}.
If m < n, T, 1 I’, , hence c,(#) We introduce
y(A) > m}. For even, continuous
func-
(1.1)
< c,($).
the assumptions:
(El) There exists an Edimensional subspace M of E and p > 0 such that Z,!J< 0 on S, n M, where S, = {x E E / 11x 11= p]. (E2) There exists aj dimensional subspace il? of E with a complementary subspace N such that # is bounded below on N. LEMMA 1.2. Let 4: E - R be even, continuous, satisfy (El), (E2), with l > j, and let $(O) = 0. Then -CD < c,(#) < Oforj < m < 1.
356
DAVID C. CLARK
Proof. Let M and p be as in (El). By (3) of Lemma 1.1, y(S, n M) = 1. Since max $(S, n M) < 0, c,(z/) -C 0 f or m < 1. Let Ic;i and N be as in (E2) andolbealowerboundfor$onN.Form>jandAeI’,,AnN#@, by (4) of Lemma 1.1, hence c,(#) 2 01. Assuming also that I/Jpossessesa continuous Frechet derivative, we introduce the conditions (B) I VW < (C) contains
For every a > 0, there exist b, r > 0 such that // #‘(x)/l > b whenever a and II x II > r. If {+} C E is bounded, 1#(+)I is bounded, and #‘(xk) -+ 0, then {xJ a convergent subsequence.
Let K, = (x E E j 1+4(x)= c, Z/J’(X)= O}. By combining Lemma 1.2 with Theorem 8 of [2], we have THEOREM 1.3. Let #: E -+ R be even, of class Cl, satisfy (B), (C), (El), (E2), with 1 > j, and let 4(O) = 0. Then if j < m < 1, Kcm is compact and non-empty, and if -co
It is to be noted that the critical levels obtained are negative and that we must have I > j. Ambrosetti and Robinowitz [l] obtained existence theorems for critical points which are complementary to Theorem 1.3, in some sense. Theorem 1.4 is essentially a combination of Theorems 2.19 and 2.23 of [l], and we have mainly followed their ideas in the proof. Let B, = {x E E j II x 11< p} and S, = {x E E I /j x )I = p}, and denote B, , S, by B, S, respectively. We assume (E3) There exists a j dimensional subspace il?i of E and a number Y > 0 such that $ < 0 on S, n iii? (E4) There exists an 1’ dimensional subspace M’ of E, a complementary subspace N’, and p, 01> 0 such that tc,> 01on S, n IV. Furthermore, let A* = {h E C(E, E) / h is an odd homeomorphism of E onto E with h(B) C BP u (cr-‘([O, CO))}, A, = {K C E I K is compact, symmetric, Y(K n h(S)) > m for all h E A*}, b,(4) = inf{max I/J(K) I K E A,}. THEOREM 1.4. Let $: E -+ R be even, of class Cl, satisfy (B), (C), (E3), (E4), where I’ < j and p < T, and let #(O) = 0. For m < j, A, # ,@, so thut b,(#) If b,=b,=b, where l’
PERIODIC SOLUTIONS OF DIFFERENTIAL
351
EQUATIONS
n < j, then y(Kb) >, n - m + 1. Hence qbhas at least j - 1’pairs -&x of non-zero critical points. Proof. Let a and r be as in (E3), let &I be an m dimensional subspace of &!I, and let K, = aT n a. We shall show that K, E A,, . From (E3) it follows that if heA* then h(B)nS,nti=
o.
Consequently, the component of h(B) n a, considered as an open subset of A?l, that contains the origin must be contained in K,. . Since the boundary of this component, relative to i@, is contained in K, n h(S), it follows by (2) and (3) that y(K, n h(S)) 3 m; hence K, E A, . Since A, r) Am+r , b, < b,+i for wz < j. To show that b,,,, 3 01, let K E A,+, and define h E A* by h(x) = px. Then y(K n S,) > I’ + 1, hence by (4) of Lemma 1.1, K n S, n N’ # B’, where N’ is as in (E4), so that max #(K n S,) 3 01and therefore b,,+i 3 oi. If 4 is an odd homeomorphism of E onto E such that 4(x) = x if 4(x) < 0, and such that @l(#-l([O, CD))) C #-l([O, co)), then 4-l 0 h E A* whenever h E A*. Therefore, if K E A, and h E A*, then YMK)
hence +(&) The proof assertion of of Theorem
n h(S))
= 44W
n 4-l
0 W)))
= Y(K n 4-l
0 h(S))
2 m;
C A, . of the assertion regarding y(Kb) is the same as for the corresponding Theorem 2.19 of [l], and follows standard ideas. The last assertion 1.4 is an immediate consequence of the preceding assertion.
2. QUADRATIC FORMS Let H denote the Hilbert space consisting of functions X: R ---f Rn whose components xi are T-periodic, absolutely continuous, and have square integrable derivatives on [0, T], together with the inner product
(x,Y) = J^,’f (xi’yi’ + w,) dt. i=l
We denote the associated norm by // 11. We shall use certain quadratic forms defined on H of the general form
(aij(t) xi/xi’ - &(t) xixj) dt,
XEH,
(2.2)
358
DAVID
C. CLARK
where aij = aji is T-periodic, of class Cl, and positive definite, and bij = bij is T-periodic and continuous. Also, for x, y E H we let
B(x,Y)
= lo’
i (a&> i,j=l
xi%’
- b(t)
X~YA dt.
(2.3)
We say that K is of index p (modified index p’) if p( p’) is the maximum dimension of subspaces on which K is negative definite (non-positive). It is clear that K is continuous and possessesa Frechet derivative given by
(K’(X), Y)= pcx, Y).
(2.4)
We say that K is regular if 11K’(X)// > C/j x 11,where C > 0. We let L2 denote the Hilbert space of functions x: R -+ IIn, whose components xi are T-periodic and square integrable on [0, T], with the inner product
and corresponding norm I/ 11s. Formally integrating by parts in (2.3), /3(x, y) = -(9x, Y)~ , where n (9~)~ = C ((aijxi’)’ + bijxj). j=l
Using integral equations in a standard way, it can be shown that there exist eigenvalues hi and corresponding eigenfunctions ui E Ca n H, satisfying L?ui + h&i = 0,
(2.6)
A, < A, < A, < ... + 03 (Ui, Ui)o =
(2.7) (2.8)
Sij
and such that (ui} is complete in L2. Moreover, there hold the Parseval relations, forx,yEH, /3(x, Y) = f bidi i=l ci = (x, ui)O )
9
di = (y, ui),, .
(2.9)
Through the remainder of this section and in subsequent sections we shall use C, C, , C, ,... to denote positive constants which do not depend on the particular element of H at hand. Now we let A4 = span(d j hi < 0}, M’ = span{d 1hi & 0}, N’ = M’I
= +ii{u”
1Xi > 0).
(2.10) (2.11) (2.12)
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
EQUATIONS
359
The index (modified index) of K is finite and equal to dim M is negatiwe definite on M andpositive dejnite on N’. Hence the index (modiJied index) of K is equal to the number of negative (non positive) eigenvalues hi . K is regular if and only if no & equals zero. 2.1.
LEMMA
(dim M’).
Proof.
K
dim M is finite, in view of (2.7). Since K(X) = @(x, x), K(X)
=
4 i
/\iCi2,
ci
=
(x,
u$
)
(2.13)
i=l
by (2.9), and so K is negative definite on M and non positive on M’. It is clear that K is not negative definite (non positive) on any subspace of dimension larger than dim M (dim M’). Hence index K = dim M and modified ‘index K = dim M’. To show that K is positive definite on IV (with respect to the norm of H), we first note that
using (2.2). Let
Ml = span(ui I 0 < hi < k}, M, = span{ui 1Xi >, k), Clearly N’ = Ml 0 M2, and by (2.13), if where y E Ml , z E M2 . By the finite dimensionality of Ml , K(Y) >, C, 11y lj2. It remains to show that K(z) 2 C I] z l12.By (2.13),
where
k > 2C2 + 2C,Cs/C,.
x E N’,
K(X)
=
K(y)
+
K(z),
K(x) 2 (k/2) 11X 11:-
(2.16)
Combining (2.16) and (2.15) with x set equal to z, and substituting into (2.14) with x set equal to z, we obtain that K(z) >, C II x II2 with C > 0. Hence K is positive definite on IV. If some Xi = 0, by (2.4) and (2.9), K’(Ui) = 0. If no A, = 0, then H = M @N and, writing x = y + z, y E M, 2 E N’, we have
(K’(X),aY + b> = 2‘43’) + 33’44, from which it follows, using the definiteness on M and IV’, that 11K’(x)~] > C II x jj, so that K is regular. Finally, let us consider the autonomous case, where the aij and b,, are constant. Here the index of K can be calculated and the regularity checked directly in
360
DAVID
C. CLARK
terms of the matrices A = (aij) and B = (&). In fact, using the positive definiteness of A, it is well known that there exists an n x n matrix M such that MTAM
= id,
MTBM
= D = diag(d,).
(2.17)
Letting x(t) = My(t), we have K(X) = i(y)
= & joT f
(y;” - diyi2) dt.
(2.18)
i=l
Clearly index K = index d. For the corresponding bilinear form P(Y, 4 = jo= il
(~r’zi’ - diyid
dt.
(2.19)
it is clear that W(Y)? 4 = ac Y, 4
(2.20)
We introduce the set ei ,
(sinj27rtlT) ei ,
(cosj27rt/T) ei ,
(2.21)
1 < i < n, j 3 1, where {e,} is the usual basis in R”. It is clear that the set in (2.21) is complete in H and is orthogonal with respect to p. Therefore, the index (modified index) of r7is equal to the number of elements of (2.21) on which E is negative (non-positive). Also, it is easily checked, using (2.20), that z is regular if and only if I? is nonzero on all the elements of (2.21); and since the norm 11M-lx Ilo is equivalent to /I x /Is regularity of K implies regularity of K. After a short calculation we arrive at the following result. LEMMA 2.2. Suppose that (aij), (bij) are constant matrices, and dejine (di) as in (2.17). Then the index (mod$ed index) of K is equal to the numbs of di > 0 (di > 0) plus twice the number of pairs (i, j) such that di > j2(2r/T)2, (di > j2(2?r/ T)2), with j >, 1. K is regular if and only if there doesnot hold di = j2(2rr/ T)2 for any 1 < i < n, j 3 0.
3. THE FUNCTIONAL 4, MAIN RESULTS The results of the previous two sections are applied to $ defined on H. Several special cases are considered. We begin with some hypotheses common to all cases. Suppose that aij(t), gi(t, u) are defined for all t E R, u E R”, aij = aji is of class Cr and T-periodic, gi is continuous and T-periodic, and that (Al)
(aij(t)) is positive definite for all t,
(-44 (A3)
gdt, -4 = -g&, 4, There exists V(t, U) such that gi = aP’/& .
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
EQUATIONS
361
Thus a,j satisfies the same conditions as in Section 2. We choose V SO that V(t, 0) = 0, and consequently $(O) = 0. It is clear that (0.2) makes sense for x E H, that 4 is even, bounded on bounded sets in H, and possessesa continuous Frechet derivative, bounded on bounded sets, and satisfying (3.1) By standard arguments in the calculus of variations it can be shown that d’(x) = 0 precisely for those T-periodic functions x which are solutions of (0.1). Since (b is even, critical points, and hence solutions, occur in pairs &:x. LEMMA
3.1. If (Al), (A3) are satisjied, 4 satisjes (C).
Proof. Let {&} be bounded and $‘(&) --f 0. Since H is separable, {x”} contains a weakly convergent subsequence, which we again denote by {xk}, with weak limit f, say. It is immediate that (6(X”) -f(x),
Xk - a) + 0.
(3.2)
On the other hand, by (3.1),
(c)‘(x”) - f(f),Xk - 2) + j' 2 (g&l x") - g,(t, f))(Xf" - iQ dt. 0 ix1 Weak convergence in H implies xk --f x uniformly, hence the second term on the right in (3.3) approaches zero. With this fact, (Al), (3.2), (3.3) imply xk + x strongly in H, which verifies (C). Consider now the hypothesis as /uI+O,
(-44)
where bij = bji is continuous and T-periodic. It follows from (A3) and (A4) that v(t, *) = 4 jJ bij(t) %lcj + O(l u I’), i.j=l
as IuI-+O.
(3.4)
362
DAVID C. CLARK
aij and bii in (A4) satisfy the same conditions as in Section 2. We define K as in (2.2). It follows from (0.2), (2.2), and (3.4) that $Nx) = “(4 + 41 x 112),
as jj x jl 3 0.
(3.5)
The following lemma is an immediate consequence of Lemma 2.1 and (3.5). LEMMA 3.2. If (Al)-(A4) are s&s-ed, 4 satisjes (El) ((E4)) with 1 (I’) equal to the index p (modified index p’) Of K.
We shall put forth various conditions for gi to satisfy at infinity. First we consider a case that is asymptotically linear in the sense of Krasnoselskii [S]. We assume that
where cii = tit is continuous and T-periodic, and where the quadratic form z(x) = 3 Jb7$,
(uijx;x;
- cijx,x,) dt
is regular. It is easily seen that if (A5) is satisfied, then
+Yx) = e4 + 4 x II>> IIx II.-+ =4
(3.6)
4(x> = $4
(3.7)
+ 41 * 112),
II A”II + cJ3.
LEMMA 3.3. If (Al)-(A3), (A5) are satisfied, then (B), (E2), (E3) are satis$ed, with j equal to the index q of K.
Proof. Equation (3.6) and the hypothesis that 2 is regular imply that ]j 4’(x)ll > const > 0 whenever Jjx (/ is sufficiently large, hence (B) is satisfied. Let j% be the q dimensional subspace in Lemma 2.1 on which 17is negative definite. Since i is regular, there is no Ai equal to zero, hence the subspace N of Lemma 2.1 is complementary to a. Also z is positive definite on N’. Hence, and in view of (3.7) and the fact that # is bounded on bounded sets, # is bounded below on N, thus verifying (E2). It is evident from (3.7) that (E3) is satisfied. In the preceding discussion and Lemmas 3.1-3.3 all the hypotheses of Theorem 1.3 (1.4) have been verified for 4 whenp > q (q > p’). In view of the equivalence between critical points and periodic solutions, we have THEOREM 3.4. Let (Al)-(A5) be satisjed. Let p (p’) be the index (modz$ed index) of K and q the index of 2. If p > q (q > p’) there are at least p - q(q - p’) pairs f x(t) of non-triwiul T-periodic solutions of (0.1).
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
Next we study a case where d(x) sional subspace. We assume
EQUATIONS
363
--co as I/ x II--+ co, in every Jinite dimen-
646) for some C > 0 andfor
j u 1suficiently
large.
(A7)
V(t, u)/l u I2 -+ 00, as j u I -
(A8)
g,(t, u) = O(l u I) as I u I + 0, uniformly
co. in t.
By some elementary calculus of variations, we have the following lemma. LEMMA 3.5. If x E H and min / x(t)\ < 4 max 1x(t)\, then /I x’ (I0 > C (1x (I0, where C > 0 depends only on T. LEMMA 3.6. Proof.
If (Al), (A3), (A6) are satisfied, (b satisjies (B).
By (0.2) and (3.1),
(f(x), x) =
2464 + s,’ (2W
x) -
f xi&, i=l
x)) dt.
(3.8)
By a standard imbedding result, max I x(t)] < C, I/ x 11;hence from (A6), dt<--C,lj~[l-‘~=Vdt+C~.
(3.9)
0
Supposethat IIx‘ II0< G IIx II0, where C, is chosen so this inequality implies that min 1x(t)1 > -&max I x(t)l, on the basis of Lemma 3.5. Then min I x(t)/ > Cs 11x II and from (A7), s”’ V, 4 dt > CoIIx II’,
(3.10)
for Ij x (/ sufficiently large. If I $(x)1 < a for some given a, then from (3.8)-(3.10) it follows that 11C’(x)11> const > 0 for II x II sufficiently large. Suppose now the other case, that 11x’ IJo> C, 11x Ilo, with ) +(x)1 < a. From (0.2) and (Al), = V(t, x) dt > C, II x’ 11:-
I0
a,
(3.11)
hence from (3.8), (3.9), (3.1 l), again II@(x)11> const > 0 for II x 11sufficiently large. This complete the verification of (B).
364
DAVID C. CLARK
LEMMA 3.7.
(A7) are satisfied, 4 satisfies (E3)for
1f(A3),
Proof. It is clear that on any finite dimensional as j/ x Ij --f 03, from which the result follows. LEMMA 3.8. If (Al), such that 4 satis-es (E4).
every value of j.
subspace J?!, $(x) -
--co
(A3), (A8) are satis$ed, given p > 0, there exists 1’
Proof. For a given p > 0, j x(t)1 < C, for x EB, and for all t. By (AS), g,(t, ZJ) < 2Ca ] u 1 and hence V(t, u) < Ca 1u 12, for 1u I < C, . Using (A3),
w
3 cs IIx’ II: - G’ IIx II:
(3.12)
for x E BP . It is standard to show that the expression on the right hand side of (3.12) is positive definite on the L, orthogonal complement in H of some finite dimensional subspace, and (E4) then follows. In Lemmas 3.1, 3.6, 3.7, 3.8 we have verified the hypotheses of Theorem 1.4, where j and hence j - I’ may be taken arbitrarily large, hence the theorem: (A6)-(A8) be satis$ed. There exist an in.nite THEOREM 3.9. Let (AI)-(A3), number of pairs *x(t) of T-periodic solutions of (0.1). If 4(x) + DC)as 11x /I -+ co, then (E3) and (B) hold, so that Theorem 1.3 will apply, given certain conditions at the origin. However, in applications it is perhaps likely that I’(u) > 0 for u # 0, and then 4(x) < 0 for x = const # 0. In Section 4, we get around this by considering the restriction of 4 to odd functions and seeking odd solutions of (0.1). For the present, we consider a situation where V(t, u) > 0 for u # 0 and 4(x) -+ co as 11x jl -+ 00, when x is of mean zero. Let H, denote the subspace of constant functions x(t) and H’ the subspace of functions x(t) of mean zero, that is, such that fi x(t) dt = 0. Then H, is of dimension n, and H = H, @ H’. We assume
649)
gl (gtk 4 - g& v))(ui - vi>
G i
C&)(% - v&f+ - Vj>+ Cl I U-VII,
(3.13)
i.i=l
where cij(t) is such that if x E H’, T
s0 (MO)
gl
12
c
(aii(t) xi’xj’ - ~ijxixj) dt > C’, ((x 11’.
(3.14)
i,i=l
(g&,
4
-
gi(t,
VW
-
vi)
2
G
I u -
v
I -
c,
.
PERIODIC
SOLUTIONS
OF DIFFERENTIAL
365
EQUATIONS
(A9) are satisfied, #Jsatisfies (E2) with j = n.
LEMMA
3.10.
If(Al)-(A3),
Proof.
V in (A3) may be represented
as (3.15)
By (A2),g,(t,
Equations
0) = 0, hence from (3.13) and (3.15)
(0.2) (3.14), (3.16) imply K4
Since H’ is complementary
to HO , (E2) follows, with j = n.
If (Al)-(A3),
(A9), (AIO) are satisjied, 4 satisfies (B).
LEMMA
3.11.
Proof.
From (3.1) and (A9),
(#J’(x)-f(Y),X--y)
XEH’.
2 c2 II x II2 - c, I/ x II,
3c211~-Y112-~lll~-Yll,
x-YEH’.
(3.17)
For a given x E H, consider #( y) = +(x + y) for y E H’. It is a standard exercise in the calculus of variations to show that # attains a minimum, say at y = 9. We denote x + 9 by R. Since #‘( 9) = 0, ($‘(a), y) = 0 for ally E H’, hence (f(i),
(3.18)
x - a) = 0.
From (3.17), (3.18), and the Schwarz inequality, II 4WII
>
Cl
II x
-
2 II -
c2
-
(3.19)
We define x = (l/T) li x(t) dt. Then x - x E H’. Setting u = x and v = 0 in (AlO), and using (3.15), w h ic h is still valid, we have V(t, 2) >, C’s \ x 1 - C, . Since +(a) < $(a) we then have w Applying
< -G
II XII + ccl.
(3.20)
the mean value theorem, and using (3.18),
C(x) - $(a) = 0 (- ,tl
atj(xi’ - fi’)(xi’
- 4’) dt
- 0)2) - gi(t, a))@, - ~2~)dt, (3.21)
366
DAVID C. CLARK
where 0 < 0 < 1. Denoting
the latter integral
.l > qc,
in (3.21) by 1, and using (AlO),
II x - .GII -
(3.22)
C,).
Now we assume that ) +(zc)I < a, as in the hypotheses of(B). (3.22) and the boundedness of aij ,
Then from (3.20)-
IIXII < C, IIx - f II2- Cl, IIx - 2 II + %a +
Cl,
.
(3.23)
From (3.17) we have
IIC’(4ll 3 - II$wl
+ c, IIx - ff II - G *
(3.24)
If II x - 2 /) 3 const > C,/C, , then by (3.19), II r)‘(x)11 > const > 0.
(3.25)
In the contrary case, (3.23) implies that 11x II is bounded; then II +‘@)I] is bounded, and (3.24) implies that (3.25) holds when II x II is suffiicently large. Thus, we have verified (B) in all cases. Using Lemmas 3.1, 3.2, 3.10, 3.11 in applying Theorem 1.3, we have THEOREM 3.12. Let (Al)-(A4), (A9), (AlO) be satisfied, and let the index p of K be greater than n. Then there exist at least p - n pairs *x(t) of non-trivial T-periodic solutions of (0.1).
4. ODD AND EVEN SOLUTIONS The methods we have used may be modified to yield T-periodic solutions of (0.1) which are odd or even in t. An additional hypothesis is introduced below, and the changes necessary in the arguments are briefly indicated. Analogs of the existence theorems of Section 3 are stated. The only result that is conspicuously different is Theorem 4.5 for odd solutions, and this is still partly analogous to Theorem 3.12. Let H, C H (H, C H) denote the subspace of odd (even) functions, and that gd&+,) the restriction of 4 to H,,(H,). W e assume, in addition to (Al)-(A3), (Al 1) gi(-t,
u) = gi(t, u), aii(-t)
= aij(t).
Let x E H, be a critical point of 4s . Then 0 = (&‘(x), y) = (F(X), y) for all y E H, . Since the integral over [0, T] of a T-periodic odd function is zero, by (3.1) we see that (V(x), y) = 0 for all y E H, . Hence f(x) = 0, and, as before, x is solution of (0.1). Conversely, an odd solution of (0.1) is obviously a critical point of +e . The analogous result for +, also holds. Hence, LEMMA 4.1. If (Al)-(A3), (Al 1) hold, the set of T-periodic odd (even) solutions of (0.1) coincides with the set of criticalpoints of&,(&J.
PERIODIC SOLUTIONS OF DIFFERENTIAL
EQUATIONS
367
In the discussion of quadratic forms we replace K in (2.2) by its restriction (Ke) to H,, (H,), and assume, in addition to the previous hypotheses on aij, bij , that aij , bij are even functions of t. We define the index of K,,(K,) as the maximum dimension of subspaces of Ho (H,) on which K,, (KJ is negative definite, and the modified index similarly. Relations analogous to (2.6)-(2.9) hold, where the eigenfunctions are now odd or even. These relations are derived as before by use of integral equations. Since $4 maps C2 odd (even) functions into odd (even) functions, the corresponding integral operators act in Ho n L2 (H, n Lg) and one can apply the theory of symmetric, completely continuous operators as before. The analogs of Lemma 2.1 and Corollary 2.2 are derived in the same way as previously. If (A4), (A$ and (Al 1) are satisfied, aij , bij , and cii are even functions, hence the discussion of the preceding paragraph applied to Kg, Kg, K, , K, , defined analogously to K, k of Section 3. We arrive at the following analog of Theorem 3.4. K,,
THEOREM 4.2. Let (Al)-(A5), (All) be satisjed. Let p ( p’) be the index (modified index) of q, and q the index of i. . If p > q (q > p’), there are at least p - q (q - p’) pairs *x(t) of non-trim& T-periodic, odd solutions of (0.1).
The corresponding statement regarding even solutions of (0.1) is also valid. The modification of the proof for Theorem 3.9 is completely straightforward and yields the following result. THEOREM 4.3. Let (Al)-(A3), (A6)-(A8), (All) be satisjed. Then there exist an injkite number of pairs -& x(t) of T-periodic odd (even) solutions ?/ (0.1).
The situation with the final theorem of Section 3 is slightly different. First we consider even solutions. Let H’, be the subspace of H, consisting of the functions of mean zero. We assume (A9)e
Same as (A9), but with H’, replacing H’.
THEOREM
index of K, T-periodic,
,
4.4. Let (Al)-(A4), (A9J, (AlO), (All) be satisJied, let p be the and p > n. Then there exist at least p - n pairs &x(t) of non-trivial, ezen solutions of (0.1).
For odd solutions, we assume (AS),,
Same as (A9), but with H,, replacing H’.
THEOREM 4.5. Let (Al)-(A4), (AS),, , (All) be sutis$ed, and let p be the index of IQ, . Then there exist at least p pairs +x(t) of non-trivial, T-periodic, odd soZutions of (0.1).
368
DAVID C. CLARK
Proof. Theorem 1.3 is applied to $,, . (A9), implies that $0(x) + co as 11x // -+ co, so that (B) is satisfied and (E2) is satisfied, with j = 0. The other hypotheses are shown to satisfied, using the analogs of Lemma 3.1 and 3.2. Then Lemma 4.1 is applied.
ACKNOWLEDGMENT The author would
like to thank the referee for his helpful
criticisms
and suggestions.
REFERENCES 1. A. AMBROSETTI AND P. H. BABINOWITZ, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. 2. D. C. CLARK, A variant of the Lustemik-Schnirelman theory, Indiana Utz;niv.,Math. J. 22 (1972), 65-74. systems, J. Math. 3. M. S. BERGER, On periodic solutions of second order Hamiltonian Anal. Appl. 29 (1970), 512-522. 4. A. C. LAZER, Topological degree and symmetric families of periodic solutions of nondissipative second-order systems, J. Differential Equations 19 (1975). 62-69. 5. M. S. BERGER, Periodic solutions of second order dynamical systems and isoperimetric variational problems, Amer. J Math. 93 (1971), l-10. systems of ordinary 6. D. C. CLARK, On periodic solutions of autonomous Hamiltonian differential equations, Proc. Amer. Math. Sot. 39 (1973), 579-584. 7. C. V. COFFMAN, A maximum principle for a class of nonlinear integral equations, 1. Analyse Math. 22 (1969), 391-419. “Topological Methods in the Theory of Nonlinear Integral 8. M. A. KRASNOSELSKI, Macmillan, New York, 1964. Equations,”