A continuation theorem for periodic boundary value problems with ...

NoDEA 2 (1995) 133 163

1021-9722/95/020133-31 $ 1.50+0.20 @ 1995 Birkh~.user Verlag, Basel

A continuation theorem for periodic boundary value problems with oscillatory nonlinearities * Anna CAPIETTO Dipartimento di Matematiea, Universit/~ di Torino Via Carlo Alberto 10, 1-10123 Torino, Italy Jean MAWHIN Institut de Mathgmatique Pure et Appl., Universit@ Catholique de Louvain Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium Fabio ZANOLIN Dipartimento di Matematica e Informatica, Universith di Udine Via Zanon 6, 1-33100 Udine, Italy

Abstract We prove a continuation theorem for the solvability of the coincidence equation Lx = N x in normed spaces. Applications are given to the periodic boundary value problem for second order ordinary differential equations. Dealing, in particular, with the periodically forced Duffing equation

(~)

x" + g(x) = p(t) = p(t + T),

we show that our main theorem can be applied to the case in which g ( x ) / x crosses an arbitrary number of eigenvalues for Ix[ large. A typical result in this direction is the following (see Corollary 4.2): Assume g odd with l i m ~ + ~ g(x) = + ~ and, for G'(x) = g(x), suppose that ] ~ X/-G~I

bounded for x and y positive and large, implies that Ix - Yl is bounded. Then (~) has at least one T-periodic solution provided that lim i n f ~ + ~ 2 G ( x ) / x 2 < l i m s u p x ~ + ~ 2 G ( x ) / x 2. The technical condition on V ~ generalizes various growth restrictions on g previously considered in the literature and, in general, it is always satisfied for a function g having order of growth at infinity like Ix]=, with c~ > 1. (See Proposition 3.1 and Remark 3.1 for more precise informations about the technical condition on V~.) *The research of A. Capietto and F. Zanolin has been supported by M.U.R.S.T. 60~o, University of Udine.

134

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Introduction

Let r : X --+ X be a completely continuous operator defined on a real Banach space X. We denote by I the identity in X and by B(xo, R) the open ball of center the point x0 and radius R > 0. The Leray-Schauder continuation principle, a very efficient tool in proving the existence of fixed points for ~5, can be described, in one of its variants, as follows. First, the operator ~ is embedded into a one-parameter family of completely continuous operators of the form ~* = ~*(z, A) : X x [0, 1] --+ X, where ~5*(x, 1) = r while ~*(z, 0) := qSo(X), is suitably "simple", in the sense that, for instance, the set E0 of the fixed points of q)0 is bounded and, moreover, the

degree condition X0 := I d e g ( I - - ~ 0 , B ( 0 , R),0)l r

for E0 c B ( 0 , R),

is satisfied. Then, according to the homotopic invariance of the topological degree, we can conclude that equation x = ~(x) has at least one solution if the homotopy I - ~* is admissible, that is, provided that we find an open bounded set ~, with D E0, such that no solution (x, A) of

. =

a),

a

[0,1),

is such that x 6 cqt~ (see [17]). Denote by E C X • [0, 1] the set of solutions (x, A) of x = ~*(x, A) and let EA := { x : (x, A) E E} be the section of E at A. In the applications of the above method to nonlinear differential equations, one usually takes for ~* a convex homotopy between 9 and a / / n e a r completely continuous operator s>0, with r not having 1 as an eigenvalue. In this manner, the degree condition is easily satisfied with Xo = 1. As for,the admissibility condition, the most standard attempts consist into the search of a priori bounds for the solutions of the homotopic equation, that is, in looking for the existence of a constant R > 0 such that U kE[0,1)E ~ C B(0,/~). There are, however, various examples, arising in a natural manner from the theory of boundary value problems for ordinary and partial differential equations, where no a priori bounds can be obtained. In this case, working with the LeraySchauder continuation principle, one has to find in a very careful manner a suitable open and bounded set f~ (with a D E0) such that Of~ N (Oxe[0,1)Ea) = {a. In a recent article [2] we proposed an approach to find the set f~ in a somehow "indirect" manner. Namely, we consider a continuous functional r/: X x [0, 1] --+ ~r~ which is proper on the solution set E. Then, setting r]_ := inf rl(x, 0) and rl+ := sup

~7(x,0)

xEE0

we prove the existence of fixed points for 9 if there are constants

xEEo

c_ < r/_ and c+ > rl+ which are never achieved by ~ along the solutions of the homotopie equation, i.e., c-E ~ rl(E). A simple proof of this result can be obtained, arguing by contradiction, as follows. If ~2 = ~*(., 1) has no fixed points, then, a combination of a theorem of Leray-Schauder [17, ThdorSme Fondamental] with a

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topological lemma of Kuratowski-Whyburn ensures the existence of an unbounded closed connected set C C E such that C A (E0 • {0}) ~ 0 (see [13]). Then, using the properness of ~1~, we see that ~(C) is an unbounded set of ~ , actually, an unbounded interval (since C is connected and U is continuous). Moreover, U(C) N [~-, 7+] ~ 0 (as C N (E0 • {0}) ~ 0) and therefore at least one of the numbers c_,c+ must belong to U(C) and thus a contradiction with the initial hypothesis is found. The result in [2] is proved in the more general framework of "coincidence degree" theory (see [20]). In [2], [3] this continuation approach was applied to the existence of periodic solutions x(.) to some ordinary and functional differential equations with superlinear nonlinearities (typical cases of lack of a priori bounds for the solutions). In these examples the functional 77is equal to ~1 • number of the zeros of the solutions in the interval of periodicity, or, equivalently, to the number of rotations around the origin of the vector (x(t),x~(t)) in the phase - plane during a period. The properness of ~Ir comes from the fact that (in the superlinear case), larger solutions have faster oscillations. The numbers c• are easily found taking into account that ~ has only discrete values. Analogous results were independently obtained by Furi and Pera in [14], [15] for the study of forced oscillations on even dimensional spheres. There are, however, several (other) problems for which the functional U (numbet of rotations) introduced in [2], [3] is not proper. In this situation, we need to find a second (possibly proper) functional ~ : X • [0, 1] --~ ~ , in order to show that ~(E) does not contain some specific values (like cj: above). On the other hand, now we can take advantage of the fact that ~(x, ~) must be constant on a connected set (recall that in our example ~ takes integer values). Hence, if we argue as in the above proof, we can obtain a contradiction assuming that ~(x, A) does not take some values along the solutions (x,)~) E E for which U(x, A) = k (k a fixed integer). This modified continuation theorem (with two functionals) is the content of Theorem 2.1 below; it is given (like in [2], [3]) within the setting of coincidence degree. In the applications to nonlinear boundary value problems, we shall choose r ~) = m a x x ( t ) , so that, in order to apply Theorem 2.1, we have to find estimates on the maxima only for those solutions which have a prescribed number of zeros (i.e. ~(x, ~) = k). Clearly, all these results reduce to the classical LeraySchauder principle whenever a priori bounds for all the solutions are available, or to Corollary 2 in [2] if there are a priori bounds for the solutions with a fixed rotation number ~. Our main result for the second order equation x " + F(t, x, z') = 0, (Theorem 2.2) is applicable to some problems which cannot be handled via the above quoted theorems. In order to describe more concrete applications of Theorem 2.2, let us consider the solvability of the periodic boundary value problem for the periodically

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perturbed second order (Duffing) equation

x" + g(x) = p(t, x, x').

(1.1)

We assume that g : Z~ --+ ~ is continuous and satisfies lim g(x)sign(x) = +oc,

(91)

while p = p(t, x, y) :/R x ~ 2 --+ f t , is continuous, T-periodic (T > 0) in the first variable and globally bounded on its domain, that is jPloo := sup tp(t,x,y)] < +oo. R3

We are interested in "nonresonance", that is the existence of at least one solution of (1.1), satisfying x(O) - x ( T ) = x'(O) - x'(T) = O, (1.2) for any (bounded) forcing term p. This problem has been widely investigated in the literature, starting with the classical papers of Dolph [9] (in the setting of Hammerstein equations and Dirichlet problems) and Loud [18] and Leach [16] for (1.1) - (1.2) (see the references in [19]). In [19] the existence of T-periodic solutions to (1.1) was achieved under a condition of asymptotic non-interference with the spectrum specT(--X" ) of the linear differential operator x ~-~ - x " subject to the T-periodic boundary condition (1.2). Namely, in [19] the solvability of (1.1) - (1.2) was proved by the hypothesis

w2J 2 < g. := liminf g ( x ) / x 0, or in an interval where x is negative and m i n x = c < 0. In order to achieve this goal, we need a technical condition, which reads as follows: (G1)

Vel > 0, 3e2 > 0 : A B > 0 & I ~

- ~ l

< el ~

IB

-

-

AI < e2.

T h e corresponding technical details are the content of the Appendix. W i t h all these tools, we can prove: Assume (91), (Gt) and (1.7). Then (1.1) has at least one T-periodic solution. (See Corollary 4.1). T h e nonresonance condition (1.7) is expressed by means of the asymptotics for the t i m e - m a p s 7-g (or Tg), but it is satisfied under more direct assumptions concerning the ratios g(x)/x and 2G(x)/x 2. W i t h this respect, it is sufficient to

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recall some estimates due to Opial [22] (see also [11], [7]). Indeed, we have: lim g(x)/x = L+ ==~ chmooTg(c) = 7c/ V / - ~

x--4.•

(1.8)

and

l i m i n f 2 G ( x ) / x 2 K~t = x - ~ •

limsupTg(c) >_ ~ r / y / - ~ ,

--

liminfTg(e) < 7 c / V / ~ . c--~•

(1.10)

--

Hence we have: Assume g odd and satisfying (gl)~ ( G 1 ) a n d suppose that

liminf2G(x)/x21imsup2G(x)/x

] #{~2j2},

yjCz~+.

(1.11)

Then (1.1) has at least one T-periodic solution. (See Corollary 4.2.) The meaning of (1.11) is that either the limit l i m x ~ + ~ 2G(x)/x 2 does not exists, or, if it exists, then it does not belong to specT(--x" ) A Kt +. It seems worth mentioning that (1.11) is sharp. Indeed, in [7, w an example can be found of an odd function g such that l i m x ~ + ~ 2G(x)/x 2 = w2k 2 (for some k E 2~ +) and (1.1) does not possess T-periodic solutions for some p : p(t). Note that the function g in the example in [7] satisfies (G1) and is not linear, since the limit limx_++~ g(x)/x does not exist. We also remark that (1.11) covers various situations of superquadratic growth at infinity for the potential. Indeed, (1.11) is satisfied if lira sup~__++~ 2G(x) / x 2 = +c~. Further corollaries can be obtained from the main result for equation (1.1), using (1.8), (1.9) and (1.10). Asymmetric nonlinearities and the interaction with the Fu~ik spectrum [12], [4], can be considered as well (see Section 4 for more details). Finally, as far as (G1) is concerned, we just note that our condition is strictly more general than (1.6) (see Proposition 3.1, Remark 3.1 and Example 5.1). All the results can be written for p satisfying the Caratheodory assumptions [20], and, if p p(t), we can assume p E Lloc too. The following basic list of notations is used throughout the paper. For A = (reals), or A = 2g (integers), we denote by A + and A+, respectively, the sets of positive and non-negative numbers. Thus, in particular, 2~ + = tV is the set of natural numbers (0 excluded). For a measurable function u, we denote by lulp, with 1 _< p < ~ , its LP-norm. In case of T-periodic functions, the norm is referred to an interval of length T. In particular, for u : [0, T] --+ /R of class C 1, we set IIu]l~ : : lul~ + lu'l~ 9 Other notations will be introduced later when necessary. =

2

The main result

Let X and Z be real Banach spaces with norms I" Ix = I1 It and ]. Iz, respectively. Let L : dotaL C X --+ Z be a linear Fredholm mapping of index zero and let N : X • [0, 1] -+ Z be an L-completely continuous operator, according to the

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terminology

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in [20]. In what follows, we denote by :: {(x,l) 9 d o m L x [0,I]: Lx = N(x;A)}

the (possibly empty) set of solutions of equation Lx = N(x;A),

x 9 domL

and by E~, with I 9 [0, 1], the section of E at l, i.e. E~ := {x 9 d o m L : (x,A) 9 E}. We assume: (ii) E0 is bounded Then, define

in X.

Xo := I D L ( L - N(';O),X)I = I D L ( L - - N(.;O),~)[, where ~ D E0 is any open bounded subset of X and DL denotes the coincidence degree (cf. [20]). Observe that from the excision property ([20, page 15]) it follows that X0 is well defined; moreover, E0 # 0 if X0 # 0. Accordingly, we assume

(i2)

xo

r

o.

We introduce now two continuous functionals

x

x

[0,11

and suppose that (i3) ~b is proper on E; (i4) 3R > 0,3d _> 0 such that for each (x,A) 9 E with Ilxll >_ m it follows that

Then we have: T h e o r e m 2.1 A s s u m e (il), (i2), (i3) and (Q). Suppose that f o r each k 9 2g there is a sequence {c(k)}~, with lim ~ + ~ ~~(k) = +oc, and there is art index n k, such that

(i5)

~b(x,A)

r (k)n,

V(x,A) 9

n>n~.

Then, equation Lx = N(x,1),

x 9 domL,

(2.1)

has at least one solution.

Assume, by contradiction, that (2.1) has no solution. Then, according to [13], there exists a closed unbounded connected set C, with C C E, such that C r3 (Eo • {0}) r 0 (see also [2, proof of Lemma 1]). Let R0 _> R be a fixed radius such that B(0, R0) D Eo. Consider Proof.

:Do := E r~ (B[0, Ro] x [0, 1]) D C A (B[0, Ro] x [0, 1]),

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a compact set (by the local compactness of E), so that the following constants are defined: ao := max{~/ (x, ),) : (x, A) 9 :Do}, K :-- max{lr/(x,A)l :

(z,A) 9 :Do}.

Consider now only the sequences {c(nk)}n, with k 9 IK := 2g n I - K , K]. For any k 9 IK, we can find an index nk, with nk > n~, such t h a t a0 < c (-k) In order - -

n k

"

to simplify the notation, we set c k := c

. Choose now a constant b0, with b0 > # m a x { m a x { e ~ : k 9 IK},d}. In this manner, we have t h a t ao < c k < bo,Vk 9 IK. Finally, as a last step, we use the properness of ~1~ and find a radius ~1 > ~:~0 such t h a t Ilxll < R~, V(x,a) 9 E N r B y the definition of R1, it follows t h a t Ir which in t u r n implies t h a t

A)I > bo, V(x, A) 9 E \ ( B (0, R1) x [0, 1]),

(by (i4) and since R1 > R0 _> R). After these preliminary choices of the constants a0, c #k (with k 9 IK), b0 and R1, we proceed as follows.

B y W h y b u r n lemma (see [24]), there exists a subcontinuum Cl of C joining

OB(O, Ro) x [0, 1] with OB(O,~t:~1) X [0, 1]; more precisely, we have t h a t ,4:= c~ n (oB(0, Ro) x [0,11) # 0, u := Cl n

(OB(O,n l ) • [0, 11) r 0,

Ro ~ Ilxll ~ R1,

V(x,~) 9 C~.

Now, by the property (i4) for r] and the choice R0 >_ R, it follows t h a t

Indeed, r] is continuous and takes only discrete values outside B(O, Ro) x [0, 1], so t h a t r] is constant outside t h a t set. In particular, r~(x, A) = k, V(x, A) E A. On the other hand, ,4 C :Do, so t h a t I~(x,A)l _< ~;, for all (x,A) E ,4. In conclusion, - K < k < K, i.e. k E IK. Consider now the set ~(C1). It is a compact (since C1 is compact) connected subset of ~ , i.e. a closed b o u n d e d interval. Thus, we set ~(C1) = [a, fl]. We have: a = i n f , ( C 1 ) _< inf ~(`4) _< sup ~(`4) _< sup ~(:D0) = do, fl = supS(C1) _> s u p S ( B ) _> infg)(B) > b0. Hence, [a0, b0] C ~(C1) and we can conclude t h a t there is (~, A) c C1 such t h a t g?(2, A) = c~. On the other hand, we also have t h a t r](2, X) = ~, and we contradict (i5). T h e proof is complete. []

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We consider now the second order b o u n d a r y value p r o b l e m

x" + F(t, x, x ' ) = 0,

(2.2)

x(O) - x ( T ) = x'(O) - x ' ( T ) = 0,

(2.3)

where F : ~ x ff~2 --~ ff~ is a continuous function which is T-periodic in the first variable ( T > 0). We e m b e d p r o b l e m (2.2) - (2.3) into a o n e - p a r a m e t e r family of problems of the form x"+f(t,x,x';;~) =0, A E [0,1], (2.2);,

x(O) - x(T) = x'(O) - x'(T) = 0,

(2.3)

with f = f(t,x,y;)~) : ~ x /R 2 x [0,1] --* ~ continuous and T-periodic in t and such t h a t f ( t , x , y ; 1) = F ( t , x , y ) and f ( t , x , y ; O ) := fo(x,y). Note t h a t any solution x : [0, T] -~ ~ of (2.2)~ - (2.3) is actually the restriction on [0, T] of a Tperiodic solution of class C 1 defined over ~t. This r e m a r k will be used t h r o u g h o u t in the sequel, assuming, when convenient, t h a t x(.) is defined in ~ (by periodicity). Finally define, for any given (T-periodic) function x(.), n(x) := c a r d ( Z z ) , where Zx := x -1 (0) A [0, T). Then, the following result holds. Theorem

2.2 Let d > 0 be such that

f(t,x,O;;~)x>O,

VtE[0,T],

AE[0,1]

and

Ixl>d,

(2.4)

and suppose that (Jo) 3R0 > 0 : []xllo~ 1.

It is i m m e d i a t e to check t h a t ~ is continuous. Moreover, by (2.7), rl(x; A) -- n(x; A) 2

for every x 9 Xx \ B(0, R*).

(2.9)

Hence,

x ~ ~ \ B(o, R*) ~

n(x; ~) ~ z L .

(2.1o)

Note t h a t if Zx = 0 for some x E E~ then [x(t)l > 0 for all t. Let t. E [0,T) be such t h a t lx(t,)l = mint~[0,T] I~(t)l. T h e n x'(t,) = 0 and xs'(t,)-x(t,) >_O. F r o m (2.2)~ we obtain f(t.,x(t.),O; A)z(t.) < 0, so t h a t (2.4) implies Ix(t,)l < d. Hence, mint~[O,T]{lx(t)] + Ix'(t)l} < ]x(t.)l + [x'(t.)] < d and (2.5) imply t h a t

]]x]l~ rd and Ilxllo~ _> R*, then Zx r 0. In this manner, (2.10) can be read a s

9 x~, \ B(0, R*) ~

~(~; ~) e ~ + .

(2.11)

Secondly, we define the functional ~b : X x [0, 1] --+ ~ as follows:

~(x; a) := max ~(~). tE[0,TI

it is i m m e d i a t e to check t h a t tb is continuous. Now, we prove t h a t !b is proper on E, i.e. for every f > 0 the set ~ := {(x,A) 9 E : I m a • _< M }

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is compact in X x [0, 1]. Indeed, for any fixed (x, ~) E A d , let { be such that x ( ~ = maxtc[0,T] x(t). Then, x'(t-) = 0 and I (t)l _< M. By assumption (jl), there exists K > 0 such that M >_ Ix(t-)l + [xl(t~l > min {Ix(t)l + ]x'(t)[} ~ te[0,T]

I[x]l~ < K.

This fact ensures, by Ascoli theorem, that the set 3,t is compact in X • [0, 1] (recall also that E is locally compact and Ad C E). Thus, we have checked that e l f is proper and (i3) is proved. Now, since we have already proved (i4) for the functional ~ with an arbitrary R _> R*, in order to complete the proof of (i4) for r we proceed as follows. Assume by contradiction that there exists (2, ~) E E \ (B(0, R) x [0, 1]) such that 2(t*) = maxte[0,r ] 2(t) < - d (with t* E [0, T) ). Since 2'(t*) = 0 and 2"(t*) 2'I(t *) = - f ( t * , 2(t*), 0; A), a contradiction with (2.4). Hence, (i4) is completely proved with R _> R* and d as in (2.4). Finally, we observe that (J2) is the same as (i5) (by the choice of the functional and (2.11)). Thus, we can apply Theorem 2.1 and the proof is complete. []

3

Applications

An application of Theorem 2.2 can be given for the Duffing equation

x" + g(x) =

x, x'),

(3.1)

with the associated periodic boundary conditions x(O) - x(T)

= x'(O) - x'(T)

= O,

(3.2)

where g : Z~ -~ ~ is a continuous function and p = p(t, x, y) : ~ x ~ 2 --~ 1R is continuous, T-periodic (T > 0) in the first variable and globally bounded on its domain, that is ]PI~ :-- sup Ip(t,x,y)l < +oo. R3

Throughout this section we assume (91)

lira

g(x)sign(x) = + ~ ,

and define the potential

a(x) :=

Ji

g(s) ds.

By (gl) it follows that limlx]~+~ G(x) = +c~, G is bounded from below in H~, and so we can take a constant L0 > 0 such that 2 G ( x ) + L 0 > 1,

VxC/R.

(3.3)

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As a further step, we consider a continuous function r : ~ -~ z~ such t h a t I~(x)l_~ 1,

Vx~,

x~(x)

Iwlo~,

-

Vlx I _~ do.

Now, setting f ( t , x, y; A) : = g(x) - e(t, x, y; A), we have t h a t (2.4) and (j0) are satisfied for d = R0 = do. As for the "elastic condition" (jl) we observe t h a t it has been already checked in L e m m a 1 of [8] and therefore it does not need to be proved here. T h e n T h e o r e m 2.2 can be rephrased as follows. Theorem

3.1 S u p p o s e t h a t f o r each k E 2g + there is a sequence {c~)}~ w i t h

c (k ) --+ + o o as n - ~ +oo, and an i n d e x n k* s u c h that

max x(t) tE[O,T]

# c n(k) ,

n

>

n k*

(3.4)

f o r every T - p e r i o d i c s o l u t i o n x o f

9 "+g(x)

:e(t,x,,';A),

Ae

[o,1],

(3.1)~

with n(x) = 2k. Then problem (3.1) - (3.2) has at least one solution. In order to apply T h e o r e m 3.1, we suppose from now on t h a t u : ~ --+ ~ is a T-periodic solution of (3,1)a (for some A E [0, 1]), such t h a t m a x u(t) = u(t*) = M > do ~ [O,T]

and n(x)=2k,

with k c 2 g +.

We shall now describe the qualitative behaviour of the solution u(.) when M is sufficiently large and k is fixed. This p r o g r a m will be developed by means of some lemmata.

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L e m m a 3.1 There is R1 > do such that M>_R~

~ u , ( t ) 2 + 2 G ( u ( t ) ) > 1,

Vt~.

Proof. We start by introducing, as an auxiliary function, the continuously differentiable and T-periodic map re(t) = u'(t) 2 + 2G(u(t)) + Lo, with L0 defined in (3.3). We have:

tm'(t)l

=

12u"(t)u'(t) + 2g(u(t))u'(t)l [2e(t,u(t),u'(t);A)l. ]u'(t)l _< 2w(t)]u'(t)l _< w(t)(1 + u'(t) 2) 0,

for t~ < t < ti+l,

i odd,

u'(ti) < 0,

i odd,

u(t) < 0

i even,

for ti < t < ti+l,

i even.

Denote by si E (ti,ti+l) the first point such t h a t ur(si) = O, and set so = s2k - T . We want to prove t h a t u'(t) = 0 if and only if t -= si (modT), for some i = 1, 2 , . . . , 2k. To this end, we introduce the auxiliary function l(t) = V/u'(t) ~ + 2G(u(t)). Note t h a t by L e m m a 3.1 it turns out t h a t 1 is globally defined in ~ as a T-periodic and C 1 function, provided t h a t M > R1. Lemma

3.2 There is E > 0 such that

M > Rl ~

ll(s) - l(t)l do,

w:

~'(s) = 0.

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Let us take now M _> R2 and s such t h a t u~(s) = 0. B y L e m m a 3.3 we h a v e lu(s)l > do and therefore, by the assumptions on g, =

+

0 u'(s) = O,u(s) < 0. Hence, we i m m e d i a t e l y obtain t h a t if t - s~ ( m o d T ) , for some i = 1 , 2 , . . . ,2k, and thus the ensure t h a t

u'(t) >0,

tE (si-l,si)

foriodd

u'(t) < O,

t e (si-a,si)

f o r i even.

In particular, for any i odd, si is the unique m a x i m u m point of u in [ti,ti+l], while for i even si is the unique m i n i m u m point of u in [ti, ti+l]. In order to introduce our m a i n result, as a last step, we assume condition (G1)

Vcl > 0, 3c2 > 0 : A B > 0 & Iv/G(B) - X ~ - ~ [

< Cl ~

Ij~

-

-

AI < c2.

Observe t h a t in order to fulfill (G1) it is sufficient to take IAI, [B[ large enough. Proposition

3.1 Suppose that

(G2)

lim sup G(x) Izl-~+o~ ~ < +co.

Then, (G1) holds. Proof. It is sufficient to observe t h a t for IAI and IBI sufficiently large and having tile same sign, we can write IB - A I = 2 I x / ~ - ~l~/g(~), for a suitable ~ between A and B. From this the result easily follows. [] 3.1 Condition (G2) can also be found in [23], and it turns out to be satisfied under very general assumptions on g. For example, (G2) holds if 9 grows like ]x]%ignx, for x --* +co, with a > 1, and, in particular, if,

Remark

0 < liminf g(x) _< l i m s u p g(x) x---+=t=oc X

x--~•

X

< +co.

In this latter case, we generalize the Lipschitz and growth restrictions on g considered in [5], [6], [7]. Other simple cases in which (G2) is fulfilled are, for instance: (a) when g is nondscreasing in a neighbourhood of - c o and in a neighbourhood of +co and l i m i n f g ( x ) / x > 0, x---+Q-oo

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or (b) when Iv/G) -1 is (globally) lipschitzian in a neighbourhood o f - o c and in a neighbourhood of +oc, or, further, (c) if there is a constant 0 > 1 such that

xg(x) >_oa(x) > o,

for Ixl

large.

(In this case, observe that Ig(z)l has order of growth at + ~ greater than or equal to 0 >_ 1.) A t t h i s p o i n t , recalling t h e definition of t h e t i m e - m a p for g at t h e level

M > o, rg(M ) = 2 f s M ) ( 2 ( G ( M ) - G(s))) -1/2 ds, we c a n state: Lemma

3 . 4 Assume (gl) and (G1). Then, there is Ra >_ R2 such that

M > R3 - - z cr(M)T~(M) - o(

) < ~ < cr(M)5(M )+o(

),

where o(s) --* 0 + as s --+ 0 +, and a(s) ~ I as s --+ +oc. Proof. R e c a l l t h a t u is a T - p e r i o d i c solution of (3.1)~ such t h a t m a x u = M having 2k zeros in [0, T ) / T a k e at first M > R2 a n d a p p l y L e m m a 3.2, so t h a t for all t E - E < ~/(u'(t)) 2 + 2a(u(t)) < Vh-d(M) + E and

V/2C(h(M)) - E < V/(u'(t)) 2 + 2G(u(t)) < X/2G(h(M)) + E. B y (gl) a n d h ( M ) --+ - e c for M --+ + e c , we can find r4 > R2 such t h a t for a n y M > r4 t h e r e a r e u n i q u e l y d e t e r m i n e d c o n s t a n t s do < A -- A ( M ) < B = B ( M ) a n d D = D ( M ) < C = C ( M ) < - d o such t h a t C = h(A), D = h(B) a n d =

2E

=

B y (G1) a n d o b s e r v i n g t h a t

A,B>0 and X/2G(D)-~=4E,

C,D 0 such t h a t

B-A_ fs~ ~/(A)ds -

f~ w ( s ) d s >_ (si - t)'y(A) - IWll,

(3.9)

w ( s ) d s > (t - s~)7(A) - I w l ~ .

(3.10)

while if t 9 [si,/3i]

-u'(t) >

7(A)ds i

i

Integration of (3.9) over [ai, si] yields 1

L > u(si) - u(ai) >_ ~(si - a i ) 2 y ( A ) - IWllT, so t h a t

(~

-

~)~


0 and we obtain

v/2(O(A) --b(~(~)))dt > ~ - t~ >

~/2(O(B) - a(~(t)))

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Hence,

/0

V/2(G(A)

-

< 0

If/3i < t _< Q+i, t h e n u'(t)

~0"A

du ,J2(G(A)

G(u))

Q >

/0

-V/2(G(B)

G(u))"

-

a n d similar c o m p u t a t i o n s yield

fO A

G(u))

-

> o~i

> Q+I -/3~ >

du ,v/2(G(B)

-

G(u))'

so that , recalling (3.11), we can conclude

2

F

~ / 2 ( G ( A ) - G(u))

> ti+l -- ti > 2

+0

_> 2 fo~ d 2 ( c ( ~ du - - c(~))

/o

d2(c(B) - c(~))

-

o(~ ),

using Lemma 5.3. If now i is even, we have only to repeat the same changes and obtain

2

~ / 2 ( G ( C ) - G(u)) + o(

-o(~)

) _~ Q+I - Q _> 2

argument

with obvious

~ / 2 ( G ( D ) - G(u))

S u m m i n g up all t h e a b o v e r e l a t i o n s for i = 1 , 2 , . . . , 2 k a n d recalling G(C) = C ( A ) , G ( D ) = C ( B ) , C = h(A), D = h ( B ) , a n d t h e definition of ~g, we o b t a i n

2v1

1v1

so that

~ ( B ) - o(~-) < ~ < ~(A) + o(

).

Finally, using L e m m a 5.4 we have

~(M)~(M) -

o(

) < ~ < ~(M)~g(M) + o(~--)

a n d t h e result is proved. [] Now, we are in p o s i t i o n to state: Theorem

3 . 2 Assume (91), (G1) and suppose that there is a sequence M~ ~ +cxD

such that lim

~-~+~

T

- -

~ (M.~)

- p C [O,+ce],

with

pC ~+. Then, problem (3.1) - (3.2)

has at least one solution.

Vol. 2, 1995

A continuation theorem for periodic boundary value problems

153

Proof. We apply T h e o r e m 3.1 with c(~k) = Mn for each n and k. Assume by contradiction t h a t there is k E 2~ + and there is a subsequence Mnj of M~ such t h a t for each j there is a T-periodic solution xj(.) of (3.1)~ for A = Aj E [0, 1] with m a x xj(t) = M~j

and

tC[O,T]

n ( x j ) = 2k.

Vj = 1 , 2 , . . . .

Taking j such t h a t M~j > R3, with R3 coming from L e m m a 3.4, we have

o-(M~j)Tg(M~j) - O( M--~j ) < =-Tk-< (7(M~j)Tg(M~j) + O( M--~j ). Dividing by T > 0 and passing to the limit as j -~ + o c we have p1 _ contradiction. []

4

Corollaries of Theorem

~1 ' &

3.2 a n d r e m a r k s

We consider some applications of T h e o r e m 3.2 to the solvability of the p r o b l e m f x " + 9(x) = p(t, x, x') = p(t + T, x, x') [ x(.), T - periodic.

(P)

with 9 : H~ ~ ~ , p : ~ x ZT~2 --~ Z~, continuous and sup ]p(t, x, y)] < +oc, R3

lim g(x)sign(x) = +ec Ixl~+or

(91) and (al)

VC 1 > 0, ~C 2 > 0:

AB > 0 & I ~

- ~ l

< cl ~

IB - AI < e2.

Following [6] we say t h a t the p r o b l e m (P) is asymptotically resonant if there is k E 2g + such t h a t for rg(c) = 2 fl~(~) (2(G(c) - G(s))) -1/2 ds, T

limOOrg(c ) z ~ . C~-b Let us now define ~-* := limsupTg(C),

T. := liminfTg(C)

and consider the interval [T., T*] C [0, +oc]. W i t h the above notation, T h e o r e m 3.2 can be rephrased as follows.

Corollary 4.1 Assume (gl)

and

(G1) and suppose that problem (P) is not asymp-

totically resonant, i.e. [r.,T*]ci{T},

Then, problem (P) has at least one solution.

V k E 2 g +.

(4.1)

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and Fabio Zanolin

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Corollary 4.1 extends T h e o r e m 1 in [6], T h e o r e m 2.2 in [7] and the main result in [23], as far as the existence of at least one solution is concerned. Indeed, condition (4.1) generalizing the corresponding a s s u m p t i o n in [6] has been already assumed in [5] and [23] together with more restrictive conditions on the function 9 (see R e m a r k 3.1). T h e evaluation of ~-,, ~-, is particularly simple if (g2)

g

is odd .

In this case, %(c) = 2Tg(c) := 4 f o (2(G(c) T* = 2T*, where T, = lim inf

G(s))) -1/2 ds,

and so ~-, = 2T, and

x/2 f~ du Jo ,/a(c) -

T* = lira sup

V/~ f~

du

Following [22], [11], [7] and (1.9), (1.10), it is easy to see t h a t IT,, T*] D

,

,

where G , := lim inf

2G(x)

x--++-oc

X2

'

G* :=

lira sup -2G(x) x---+ + o e

X2

and we use the convenction 1 / + cc = 0, 1/0 + = +oo. Set also, 9* := liminf x~+~

9(z----~), g, X

:= l i m s u p x--*+oo

g(x~) X

and recall that, according to the generalized de L' HSpitai's rule, g. < G . < G* < g*. Finally, denote by A~ = (2_~)2 k r 2g +, the k-th positive eigenvalue of the differential o p e r a t o r x ~-* - x " in the space of T-periodic functions. Now, we have: Corollary

4.2 Assume (gl), (g2) and (G1). Then, problem (P) has at least one

solution provided that [G.,G* l # {~},

Vk r ~ + .

(4.2)

To c o m p a r e this result with previous theorems, we r e m a r k t h a t the condition assumed in [19] was [9,,g*] N {/kk : k e , ~ + } = [~ and observe t h a t [9,,g*] D [G,, G*], so that, the result in [19] implies [G,, G*] N {A~: k E 2~'+} = 0 and thus (4.2) follows. On the other hand, in [7], one can find conditions like (4.2), but for g globally lipsehitzian in ff~, an a s s u m p t i o n which is not required here. Finally, to a p p l y the t h e o r e m in [23], one should replace (G~) with the more restrictive a s s u m p t i o n (G2). Note also t h a t (4.2) is always satisfied if G* = + o c , with no condition on G , . As a further application of Corollary 4.1, we have the following:

Vol. 2, 1995

A continuation theorem for periodic boundary value problems

C o r o l l a r y 4.3 Assume (gO and

(G1) and suppose

155

that

T, > T.

(4.3)

Then, problem (P) has at least one solution. Corollary 4.3 can be compared with previous results of Opial [21] and [11]. Actually, under condition (gl) it was proved in [21] that if lim inf Tg (c) + lim inf Tg (c) > T C - - + - - OG

(4.4)

C--+~-O0

problem (P) is solvable. Subsequently, in [11] Opial's assumption was improved to

liminf Tg(c) + limsup Tg(c) > T. C - - + - - C>O

(4.5)

C_+ ~ ~

Clearly, (4.3) contains all the above conditions as particular cases. In our situation, however, the extra hypothesis (G1) has to be required. A comparison among all these results in the odd case shows that (under (92)) condition (4.4) reads as

T while (4.5) becomes T* + T, > T, and finally (4.3) gives T

T* > Y'

(4.6)

Note that (4.6) is satisfied whenever

lim of: (x) x--~+oo m Y -


- ~ '~

=

T~(z)

-

~

~o y

du

{G(z)

-

c(~)

du .v/O(z)

- a(~)'

where, from the assumption of L e m m a 5.4 and (G1), y E [K1 - z, z], where K1 depends on K via condition (G1). Hence, recalling again L e m m a 5.3 we can write Tg(y ) _> Tg(z)

- o(1).

(5.5)

By (5.4) and (5.5) we can conclude t h a t lim z~-~o

Tg(y) - cr(z)Tg(z)

= 0,

(5.6)

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Anna Capietto, Jean Mawhin and Fabio Zanolin

NoDEA

with or(z) conveniently defined from ~0(z). Prom (5.6) the proof can be completed for the function 7-9, arguing by symmetry. [] For the reader's convenience and in order to make the article selfcontained, we report here (following [7]) also a proof of the estimates (1.9) and (1.10) which play a crucial r61e in the corollaries of Section 4. L e m m a 5.5 Assume (gl ), then

lim inf 2G (x ) / x 2 < K• ~

limsupTg(c) _> 7 r / X / / ~ ,

x~•

c~•

l i m s u p 2 G ( x ) / x 2 >_ K• ~

liminf Tg(c) _< 7r/~rK-~.

x~•

c~•

Proof. We discuss only the cases at +co. Suppose at first that l i m i n f x ~ + ~ 2 G ( x ) / x 2 _ O. Then, assume K C f t + and take an arbitrary constant L > K. We define the function p(x) := 89 2 C(x) = ~x 1 2 ( L - 2G(x)x -2) and observe that ~(0) = 0 and limsupx~+ ~ ~(x) = +cx~, so that there is an increasing sequence {Cn},~, with cn --+ + ~ as n -~ +cxD, such that L 2 ~(x) < ~(Cn), for all x e [0, e~). Hence, a(cn) - G ( x ) < 7(c~ ' ~ ) , fo~ all 9 e [0, c~). From the definition of Tg(x), now we find Tg(e~) > 2 f o ~ (L(c~ - u2)) -1/2 du = ~/v/-L, and therefore lim s u p ~ + ~ Tg (c) > 7r/v/L. Thus the first of the above inferences is proved by letting L "~ K. Assume now that l i m s u p ~ + ~ 2 G ( x ) / x 2 >_ K E [0, +cxD1. Here the only nontrivial case is for K > 0. Taking an arbitrary constant L E (0, K), we consider the function defined above and observe that ~(0) = 0 and l i m i n f ~ + ~ ~(x) = - ~ , so that there is an increasing sequence {d~}~, with d~ --~ +co as n --+ + ~ , such that ~(x) > ~(d~), L 2 for all x e [0, d~). Hence, G(d~) - G(x) > y(d~ - x2), for all x e [0, d~) and we find

Tg(d~) < 2 f : ~ (L(d~ - u 2 ) ) -1/~ du = ~/xffL, and therefore lira i n f ~ + ~ Tg(c ) < 7r/V~. Thus the second inference is proved by letting L / z K. [] At the end of the paragraph, we observe that condition (G1) is invariant with respect to bounded perturbations of the function G(x). Namely, if g satisfies (gl) and (G1), and we take any continuous function h : ~ --~/R such that lim

(9(x) + h(x))sign(x) = +cx~,

with bounded primitive H(x) := f o h(s) ds, then the function G(x) := G(x) § H(x) satisfies (G1) as well. (The straightforward proof is omitted). Now, using this remark, we can provide an example showing that (G1) is more general than Qian's condition (G2)

lim sup G(x)

E x a m p l e 5.1 Take g(x) = x + h(x), where h : Kt --* ~ is odd, continuous and satisfies

the following conditions: 3K > 0 : I

h(s) dsl < K,

Vx ~ ~ ,

(5.7)

Vol. 2, 1995

A continuation theorem for periodic boundary value problems

3an E ~ + , V n E tVV, with lima~ = + o c :

h(x) > v ~ - x,

h(an) = ~ -

Vx > o.

an,

163

(5.8)

(5.9)

B y the above remark and since the function x ~ x satisfies (gl) and (G1), the same conditions are fulfilled by g(x) as well (use (5.9) and (5.7)). On the other hand, (5.8) implies that

lira G ( a n ) / g ( a ~ ) 2

n~+cc

1 lim

= 2 n~+~

2 + 2H(a~) an --+cx~. an

This shows that (G2) does not hold. []

Using similar arguments, a wide class of functions satisfying (Gi) but not (G2) can be found. Received December 14, 1993