A note on nonlinear oscillations at resonance

Acta Mathematica Siniea, New Series

1987, Voi 3, No. 4, pp. 351--361 tt A Note on Nonlinear Oscillations at Resonance

Pierpaolo Omari

Dipartimento di Scienze Matematiche

Uaiversi~ di Trieste

34127-Trieste

Italy

Fabio Zanolin

Dipartimento di Svienze Matematiche

UniversitZt di Trieste

34127-Trieste

Italy

Received February 9, 1987

Abotrac*t. Existence of 2n-periodic solutions to the equation ~ + g(x)=p(t) is proved, under sharp

nonresonance conditions on the interaction of g(x)/x with two consecutive eigenvalues m 2 and (m + 1)2:

touch with the eigenvalues is allowed.

Introduction and the Main Result

We are concerned with the solvability of the periodic boundary value problem

(1) ~ + g(x) = p(t) , (~ = dx/dt)

(2) x(0) - a(2~) = ~(0) - ~(2~) = 0,

where g: R - - ) R is continuous and p: [0, 2 g ] - * R is (Lebesgue) integrable.

Solutions to (1)-(2) are intended in the Carathbodory sense and will be referred to as 2 g -

periodic.

We investigate the existence of solutions to (1)-(2) assuming that g verifies,for some integer

m > 1, the condition

(3) m z <= g(x)/x <= (m + 1) z, for [xl ~ d > 0.

This kind of study was initiated by Loud [16] in 1967 and, since then on, it has received

attention from several authors like Lazer and Leach [13], I.e.ach [14], Fabry and Franchetti [10],

Cesari [5], Mawhin [17], [18], Reissig [23], Ding [81 [9], Omari and Zanolin [19], Iannacci and

Nkashama [12], Deng [7].

Since condition (3) is not sufficient, by itself, to ensure the existence of 2rt-periodic solutions

to (1), for a general peLt(O, 2:t), as trivial linear counterexamples show, some additional

restrictions have been put on the range of g(x)/x (and even on g'(x)), or on the function p. Here, in

352 P ie rpao lo O m a r i & F a b i o Zano l in

order to control the closeness of g(x)/x to the constants m 2 and (m + 1) 2, we introduce two functions

a• R - * R, with possibly lim a• = 0, such that

(4) m 2 + ~-(x) <= g(x)/x ~_ (m + 1) 2 - ~+(x), for I~1 >-- d.

Moreover, we define, for each integer k >-0, the spaces

P t : = fp~Ll(O, 2n): fi 'p(s)cos(ks)ds ffi fi'p(s)~in(~)as = 0 }.

Then the following result holds true.

Theorem. The periodic BVP (1)-(2) admits at least a solution i f (4) holds and one of the fol lowing conditions is ful f i l led

(i) p ~ P . ( 3 v .+l and ltm ;~(~) +o0,

(ii) p e Pm x2~-W'" = ltm " "-';~i=+(x)= + oo, a n d lim Is t - + oo ~ + z o

(iii) P~Pm+I and lim Ixla-(x)---- lira xZct+(x)= + m, lst-- + o0 Ixl-- + oo

(iv) p is arbitrary and lim Ixla• + oo. lai~ + oo

Remark 1. We point out that the condition

lira Ix l~+(x)= + ~ (resp. lira I x l ~ - ( x ) = + oo), 1 ~ + ~ M ~ + r

cannot be relaxed, if Pr (resp. pCPm). Indeed, a particular counterexample (already proposed in [8]) is provided by the differential equation

(5) ~ + ~(x)= ~(,),

with g(x): = (m + 1)2x - arctg x and p(t): = 4 cos((m + 1)t). Obviously, ~ satisfies (4), with a-(x): = 2m (a constant function) and rt+(x): = x -1 arctg x, for x r 0, ~+(0): = 1. Clearly,

lim xZct-(x)-- + oo, but lim Ixla+(x)--1t/2 < +oo.

Moreover, p ~ P~,, but p r Pm+ l- Therefore, condition (4)-(ii) of our theorem is fulfilled, except for

the requirement lim Ixla+(x) -- + oo, yet equation (5) has no 27t-periodic solution (see [8, Sect. Ixl~ 4-0o

III], or [13, Th. 2.11). Rema-k 2. We obvserve that our theorem constitutes a refinement of some results we proved

in [19, p. 147] and [20, p. 2921, where conditions (4)-(ii), (4)--(iii) and (4)-(iv) were permitted but (4)-(i) was not. However, we stress the fact that the statements in [191, [201, by themselves, are improvements both of the results in [171, [181, where (generalizing the existence theorem of Loud [16]) (4) with a+: positive constants is assumed, and of the results which can be deduced from [3],

A Note on Nonl inear Oscillations at Resonance 353

[2], [6], [12], where (on the lines of Lazer and Leach's paper [13]) it is essentially supposed that

(4) holds with lim [xl0t-(x) = + oo and 0t + a positive constant (resp. ~ - a positive constant and Ixl~+oo

lim [ x [ ~ + ( x ) = + co). Actually, in [3], [23, [6], [183, [123, g is assumed also to depend on the t -

variable, g: = g(t, x) and some kinds of non-uniform conditions (with respect to t) are considered.

However, they reduce to the ones just explained when g is independent of t. Furthermore, in [3],

[2], [63, [12], to control the interaction of g(x)/x with m z, the" so--called Landesman-Lazer

condition is assumed [131 yet it becomes the previous hypothesis on ct-, if one is concerned with

the solvability of (1)-(2) for every pr 2rt) (see also the final section).

As regards the paper [8], the author gives therein a rather complete treatment of problem

(1)-(2), under the condition

(6) m2<=g'(x)<(m+l) 2, for all x ~ R .

He proves that (6) is sufficient for the existence of 27r-periodic solutions to (1), i f p ~ P,, A Pro+ 1. Otherwise, he assumes, besides (6), condition (4)-(iv). It is clear that our result improves this one,

whenp r Pm N P,,+ 1, and is independent of it in the opposite case. Observe that anyhow we do not

even suppose the existence of g'. Finally, we remark that the recent result [7], when applied to the scalar equation (1),

essentially corresponds to our theorem, under condition (4)-(iv). Namely, in [7], the author on the

one side assumes a monotonicity condition on (m + 1)2x - g(x) we do not need, but on the other

t side he replaces the hypothesis lim Ixl~-(x) = + oo, with the weaker one lim Ixl- ' (g(u) I . ~ + Qo Ixl~ + Qo 0

-- rn2u)du = + oo.

Remark 3. As concerns the cases in which (3) is violated, we refer to [11], when lim g(x)/x

= + c~, to [223, when g(x)/x 5 0, for Ixl large, and to [21], when 0 _-< g(x)/x <-_ 1, for Ixl large. In the last two cases, the results in [22], [21] work as well for the (more general) Li6nard equation

(7) + Ax) + g(x) = p(t),

withf . 'R ~ R being an arbitrary continuous function. More precisely, in [21], we proved that the

periodic BVP (7)-(2) has at least one solution, for each p ~LI(0, 2rr), if there are two functions

such that

(8)

a + : R ~ R , with lim ~+(x) = 0, Ixl~ + oo

oC(x) <= g(x)/x =< 1 - ct+(x), for Ix[ > d > 0,

with lim Ixt0~-(x)= =lAin txi0t+(=)= + oo and x0~+(x) bounded for x _-> 0. Note that the

condition on 0c + allows g(x)/x to cross the eigenvalue 1 on the whole positive semiaxis. Finally, we

wish to observe that, combining the approach of this paper with some devices of [21], a previous

statement of Reissig [24], for problem (7)-(2), could be refined on the lines of our theorem, by

assuming (8) with 0t-: = 0 (a constant function) and lim x20t+(x)= + oo, if p ~Po 0 P1. Ixl--, + oo

354 Pierpaolo Omari & Fabio Zanolin

Proof of the Theorem. In what follows, we shall use the classical Banach spaces C~ 27t) and

i f (0 , 21t) (1 ~ p < ~ ) and the Sobolev spaces W2'P(0, 27t)(1 ~ p < oo), whose respective norms

will be indicated by I'lc o, I'1, P, I-IFr2.J" (see e.g. [1]). The L2-bilinear pairing will be denoted by (., %.

According to Remark 2, we prove the theorem only when (4)-(i) is assumed; for the other

cases we refer to [19] and [20J. Moreover, we replace condition (4)-(i) with the more general one:

there are two functions f l - : R ~ R + and fl+: R + ~ R + (with R + being the set of nonnegative

reals) such that

(9)

for Ixl >- d'( ~ d), with

(lO)

and (ll)

m2x 2 + fl-(x) <__ xg(x) <= (m + 1)2x 2 - fl+(Jxl),

lim fl +(~) = + ~3

hm ~-2/~+(~) = 0.

There is no restriction as to supposing also if+ to be non-decreasing functions. Let us fix a constant 0 < v < 2m + 1 and consider the equation (equivalent to (1))

(12) ~ + (m 2 + v)x = p(t) - g(x) + (m 2 + v)x.

Observe that, for each ueL l (0 , 2~), there exists exactly one function x = K u e Wa'I(0, 27t)

verifying (2) and

~i + (m 2 + v)x = u, for a.e. t z [0, 21t].

Moreover, define the Nemytzkii operator

N: C~ 2~) ~ L'(0, 2~), Nx : = p(') - d x ) + (m" + ,)x.

Then it is easily checked that the Hammerstein operator

KN: C~ 2~) ~ CO(0, 21t)

is completely continuous and its (possible) fixed points are the solutions of (12)-(2) in Wa'I(0, 2~).

Therefore, for solving (12)-(2), that is (1)-(2), we can apply the Leray-Schauder continuation

theorem [15] to the operator equation

x = KNx, x e C~ 2~).

Accordingly, we are looking for a constant R > 0 such that

(13) Ixlco < R,

for every function x satisfying, for some ),6 [0, lJ ,

x = ).KNx, x~C~ 2~:),

or, equivalently, (2) and (14~) ~. + (~2 + v)x + ;.(d~) - ( m2 + ,)~) = ~p, x~ ~ . ' ( o , 2~).

Setting, for x e R and ), ~ [0, 1],

h(x, ),) : = ),(dx) - m2x) + (1 - ).)vx,

A Note on Nonlinear Oscillations at Resonance 355

equation (14z) then reads

(15~) ~ + ,n 'x + h(x, 2) = 4p.

The function h verifies the following condition, which is a consequenc of (9),

(16) ~-(x) < xh(x, .I) <= (2m + 1)x 2 - fl+(Ixl),

for Ixl _>-d' and 2~[0,'~1]. For the same x and 4, we derive, in particular, that

(17) sign(x)h(x, ~.) = Ih(x, 4)1 =< (2m + 1)lxl.

Furthermore, multiplying (16) by x-lh(x, 2) = Ixl-'lh(x, 4)1, for Ixl > d' and 2~[0, 1], we get

(18) xh(x, 2)> (2m + 1)-~}h(x, 2)12 + fl+(Ix})l(2m + 1)xl-~lh(x, 2)1.

Hence, we find that, for every, x ~ R and 4~[0, 1.1,

(19) xh(x, 4) > (2m + 1)-~lh(x, ).)l 2 + 2fl+(2m + 1)-Xlh(x, 4)1) - C~,

with C l > 0 being a suitable constant. lndeeed, if Ixl > d' and 4~ [0, 1] are such that

lh(x, 4)1 < ~(2m + l)lxl,

we obtain, for such x and 2,

(20) xh(x, ;'.) > 2(2m + 1)-Xlh(x, 4)12

> (2m + 1)-rib(x, 2)1 + 2fl+((2m + 1)-rib(x, 2)1)- C2,

since from (11) we have, for every ~ e R +,

/?+((2m + 1)- t~) < 2(2m + 1)- t~2 + 2C2,

with C2 > 0 being a suitable constant. Yet, if Ixl > d' and 4 e l 0 , 1] are such that

Ih(x, ~-)1 > ~(2m + 1)lx[,

we derive, from (18),

1 xh(x, 4) > (2,n + 1)- %(x, 4)t 5 + ~§

Hence, using 8 + non-decreasing and {17), we get, for such x and 2,

(21) xh(x, ~.) > (2,n + 1)-llh(x, 4)12 + ~ + ( ( 2 m + 1)- ' lh(x, 4)1).

From (20) and (21), it follows

(22) xh(x, ,t) > (2m + 1)- '[h(x, 2)12 +2/~+((2m + 1)-rib(x, 4)1) - C2,

for Ix] -_> d' and 4 ~ [0, 1"1. Finally, using the continuity of h: R x [0, 1"] ~ R and the boundedness

356 Pierpaolo Cmari & Fabio Zanolin

from below of fl+, relation (19) is obtained, for a suitable choice of C 1 > C2( > 0). Moreover, since fl+ is non-negative and (10) and (11) hold, it is possible to find a concave function 7+: R + --* R +, with 7+(0)= (0), such that

.

7+(~) =< fl+((2m + 1)-~) - M +, for every ~ ' R +, with M + > 0, (23)

(24)

and (25)

lim 7+(~)= + a3

lim ~-~7+(~) = 0.

Therefore, owing to (19) and (23), the function h verifies

(26) (v,h(v, 2)) 2 > (2m + 1)-'lh(v, 2)1~2 + I~+(Ih(v, A)I)IL~ -2r:(C, + M+),

for every v~ W2'I(0, 27r) and ).~[0, 11. Now, we are in a position to prove the existence of the a-priori bound claimed in (13). Let

x~ W/2'1(0, 2r 0 be a possible solution to (15~}-(2), for some 2~]0, 1[. Correspondingly, set

~(.) : = h(4-), ~)

and let ~ ~Pk be the Fourier expansion of r 2u), that is, k=O

I: ~o : = (2~) -~ ~0(~)a~

and, for k > 1,

f2~ f2;T ,p~(t) : = (1/~)( ~(~)cos(k~)a~)cos(kt) + 0 / ~ ) ( ,p(~)~inI,~)a~)sinIt, t), 0 0

for every t ~ [0, 2rr-]. From the assumption p c Pm and from equation (15~), we easily check that ~p,, = 0.

Moreover, if we define

, : = (m2_ k=O kc=m

then (I)E ~ '2 (0 , 2r 0 and it satisfies (2) and ~ + mz~ --- q~. An easy computation also shows

= 2 - 1 2 - ( 2 m + (9, *)2 ( , : - k ) 1 l,2 >= k=O k#m

(27)

and (28)

and

I~lc ~ < ~1~1:

where 0:--- ~ ~Pk , ~ : = Z (rn2--k2)-ltPk k=O k=O

k#m,m+l kC:m, m+l

,m2_k2,2) k~m, m+l


Let us multiply both sides of equation (15~) by q~(') and integrate between 0 and 2=. Integration by parts (with the use of the boundary conditions (2)), the hypothesis p e P . , N P.,+I and the continuity of the L2-bilinear pairing give

(x + r~x, Ch = (~, q')z. = (x, h(x, 2))2 ,

(h(:, ~), r (~o, "h I~ (~ 1,2)-,,,_ = = - , ~ , L 2 k ~ O

kC:m

k = O

k ,~ m,m + l

Hence, by (28), HSlder inequality and 2 < 1, we find

(29) (~.hO,, ,~-)h-< - ~ (~-k~)-t{o,,,{[~ + ~-~Lt{'~I o k=O C

k#m

<= ~.(m z kZ) - , z - - [e.JL~ + "=~{LtI~iL2"

k=O

k~m

Now, compare (29) with

(x,h(x, 2))z _>_ (2m + 1)- ' ~ {(Pkl~z- 2n(C, + M+), k = O k ~ m

which follows from (26), by putting v = x and using }7+(]r 1 ~ 0. By easy calculations, w~ derive that

(4(., + 1)(2,. + 1))-'(2.~ + z)l~lh =< ~ ((2~ + a)-' + (m 2 - k2) -')l~0k{~, L k = O

k # m , m + l

< K[PILII~IL: + 2~(C1 + M+).

So that, there exists a constant C3 > 0 (independent of x and 2) such that

(30 ) {~{L2 <= C 3 .

Next, compare (29) with (26), where we put again t ' = x, By means of (30), we get

(31) 7+([q)(s)l)ds = {?+([(pl)lLt < K[plL,I@[L2 + 2~(C, + M +}

<= s:~oiL,C3 + 2n~Ct + M +) : = C,.

Hence, as :~+ is subadditive (being concave and non-negative), and, by (25), verifies 7+(~) _< {z + C5, for every ~ e R +, with C 5 > 0 being a suitable constant, we have

;: f: ~,+({~",~.,(s)i)es = ~,+Oo,.-~,(s)+ r ~,){~s

j:- f:- =< ~+({~0)0~s + ,,,+O~,(s)Oas

=< ;,+(lo,(~){)d~ + {~,(~){~ ,+ 2~C,,

358 Pierpaolo Cmari & Fabio l~nolin

and then, by (30) and (31),

c , .

Moreover, as ?+ verifies the condition

(32) ~+(~:) _~ #V+(~), for every ~ E a + and #~[0, 1], (a consequence of ),+ being concave and y+(0)= 0), we obtain

f: C 6 ~_ ?+(l~p.+t(s)l)ds = ?+((1/V/-~)}~p.+ tit 2lsin((m + 1)s + O)l)ds

f" _~ y+((1/,r 2) �9 Isin((m + 1)s + 0)lds = 4y+((1/,,/~)lcpm+ tla2). 0

Therefore, from (24), it follows that there exists a constant C7 > 0 (independent of x and ,l) such that

(33) Iq'.,+llL= -< C7.

Finally, using (30) and (33), we conclude that there is a constant C a > 0 (independent of x and 4) such that

(34) Ih(x, 2)1L2 = IrplL2 < C3 + C7: = Ca. Then, from equation (15~), we have, for 2 < 1,

Is + m2Xlat =< 2[h(x, 2)lrt + 2[Pit t < ~/2-~Cs + [Plat : = C9.

Next, denote by ~ x~ the Fourier expansion of xE IV2'1(0, 270, with kffiO

f" Xo:= (2=)-t x(s)ds, and for k ~ 1, 0

x~(t) : = (1/70(f2o'.X(s)cos(ks)ds)cos(kt)+(1/x) ( f i~x(s )sin(ks)ds)sin(kt) for t~[0, 21t]. An easy computaion shows that

f with X : = (270 - 1 ~m-"

Hence, we derive that (35) and, therefore,

Xl:~ + mZXlL1 > Ix -- x,.Ico ,

+ 4 ,ffi,~ Im 2 - k= l - ' } . k~ta

Ix - x.,IcO < xC9 : = Clo ,

(36) (x, h(x, 2))2 = ( = - =.,, h(x, 2))= ~ I x - =,.Icolh(x, 2)1 L, _~ C, oC9:= c,, .

Moreover, since 1~- is non-negative and (10) and (11) hold, it is possible to find a function 7 R +, with 7-(~)= 0, for every ~ < 0, 7- being concave for ~ _-> 0, such that

(37) 7-(~) < / ~ - ( ~ ) - M - , for every ~ R +, with M - => 0,

- : R

A Note on Nonl inear Oscillations at Resonance 359

and

(38) lim ~ - ( ~ ) = + oo.

Then, form (37), (16), (34) and (35), we get, for some constant C12 > O,

(39) ~,-(4s))ds - C~2 < B-(~s) )ds - Ct2 < (x, h(x, ;t))2 < Cl 1.

Now let us suppose that Ixmlco > 2Ix - x,,Ico (otherwise, lXlc o < 3Ix - x,,,IcO < 3Clo). Then,

�9 (t) = x.(t) + 4 0 - x.(t) _>_ x.(t) - I x - X.,cO

_-> Ix,,lcO #xml x~,(t) - = Ixmlco s i n ( a t + 0) - ~ ,

for every t E I'0, 2zr]. Hence, using y - non-decreasing and the counterpart of (32), we obtain

[sin(,~ + 0) > 2 l

>f -{ ( = 1 y r o sin(ms + 0)-- _~ ds [sin(ms + O) > ~ ]

>=7-('X.lcO"f[ 1 . ( s i n ( m s + O ' - ~ ) d s = ( x / ~ - x / 4 " - ( l X . ' c O ' �9 sin(.~ + 0) > 2 ]

Hence, by (39), we get

~'-(Ix,,}cO ) _-< (.v/2 - at/4)- l(Clx + C,2).

Thus, by (38), we can find a constant C13 > 0 such that

and then, by (35),

lx,,,Ico < C13,

Ixlco < Clo + Cls : = C14,

with C14 > 0 independent of x and 2. Hence (13) follows for any R > C14. This concludes the

proof.

Final Remarks Let us consider the equation

+ g(t , ~ ) = p(t) ,

where an explicit dependence of g on the t-variable is assumed. In this context, we suppose that g :

I-0, 2rt] • R.---, R verifies the following (Carat6odory) conditions: g(t~') is continuous for a.e.

t ~ [0, 2rt], g(., x) is measurable for every x ~ R , and, for each r > 0, there is a function 3, e L2(0,

2r~) such that Ig(t, x)l < cS,(t), for a.e. t~ [0, 2~] and lxf < r. Then replacing g(x) with g(t, x) in condition (4), supposed to hold for a.e. t ~ [0, 2rt], all the conclusions stated in our theorem are still true.

Moreover, from the proof of the theorem (see step (9)), it is clear that condition (4)-0) can be

360 Pierpaolo Cmari & Fabio Zanolin

sharpened by repalacing the assumption

lim

I,d-* +cr with the weaker one

xZa-(x) = + oo (resp. lira x2~t+(x) = + oo)

r

o r

a- (x ) ~ O, for x < O, and lim x2ot-(x) = +

(resp. a+(x) >= O, for x =< O, and lira x2ct+(x) = + oo) X"* § r

(j') *t-(x) > 0, for x >= 0, and lira x2a-(x) = + oo x oo

(resp. ~t+(x)>= O, for x >-O, and lim xZa+(x )= + o r ) x

Conditions (j) and (j') say that At, x ) = m2x, on a x-semiaxis, is allowed.

Such a remark seems of interest, when g explicitly depends on t, in the light of an example

proposed in [121 which shows that, even if p ~ P . , for a nonlinear g, the condition (alone)

m2_-- < g (t, x)/x, for a.e. t~ [0, 27t] and every x # 0,

is not sufficient for the existence of 27t-periodic solutions. Note that g(t, x) = mZx, for x _-< 0, was

already considered in [4], but under further monotonicity conditions on g.

Extension of our theorem can be obtained for the differential-delay equation

+ =p(t),

with r ~ ]0, 2~r[ and x(t - T) defined as usual when dealing with the periodic problem for DDEs.

Such a result follows from some arguments proposed in [19], [20 1 together with a careful reading

of the above proof.

Finally, the existence of classical (i.e. of class C 2) 2~t-periodic solutions is provided by our

theorem, whenever p : R --* R and ~ : R x R ~ R are continuous and 27t-periodic in the t -

variable.

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A note on nonlinear oscillations at resonance

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