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6 Lazer–Leach Conditions for the Asymmetric Oscillator

6 Lazer–Leach Conditions for the Asymmetric Oscillator

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150



6 Playing Around Resonance



extended by -periodicity to the whole R. Let us define the -periodic continuous

function

à Z T

Â

h.C1/ h. 1/

ˆ.Â/ D 2N

e.t/ .t C Â/ dt :

(6.3)

0



The Lazer–Leach condition is generalized in the following corollary.

Corollary 6.6.1 (Dancer, 1976) If

./ Ô 0 ;



for every  2 Œ0;  ;



then problem (Q) has a solution.

Proof Assume, for instance, that ˆ.Â/ > 0 for every  2 Œ0; . Writing v.t/ D

.t C Â/, it can be seen that

Z

fv>0g



vD



2N



Z

;



fv<0g



. v/ D



2N



:



(6.4)



Hence,

Z

lim sup .h.x/



e.t//v.t/ dt D



fv<0g x! 1



D



2N



Z

h. 1/

fv<0g



e.t/ .t C Â/ dt ;



and

Z

lim inf .h.x/



e.t//v.t/ dt D



fv>0g x!C1



D



2N



Z

h.C1/

fv>0g



whence, setting h.t; x/ D h.x/



e.t/ .t C Â/ dt ;



e.t/,

Z



Z

lim sup h.t; x/v.t/ dt C



fv<0g x! 1



Â

D 2N



h.C1/



lim inf h.t; x/v.t/ dt D



fv>0g x!C1



h. 1/



Ã



Z



T

0



e.t/ .t C Â/ dt > 0 ;



so that Theorem 6.5.1 applies.

In the case when ˆ.Â/ < 0 for every  2 Œ0; , one sees analogously that

Theorem 6.5.2 can be applied.



6.7 More Subtle Nonresonance Conditions



151



6.7 More Subtle Nonresonance Conditions

We now assume that h is locally Lipschitz continuous and has finite limits

h. 1/ D lim h.x/ ;

x! 1



h.C1/ D lim h.x/ :

x!C1



Writing the equivalent system

x0 D y ;

xC C x

y0 D



h.x/ C e.t/ ;



we denote by .x.tI x0 ; y0 /; y.tI x0 ; y0 // the solution with initial values

x.0I x0 ; y0 / D x0 ;



y.0I x0 ; y0 / D y0 :



The Poincaré map P W R2 ! R2 is well-defined as

P.x0 ; y0 / D .x.TI x0 ; y0 /; y.TI x0 ; y0 // :

In order to detect a periodic solution of problem (Q), we will look for a fixed point

of P. To this aim, we will compute the Brouwer degree of P I on the set

"



D



r .s/; r 0 .s/ W 0 Ä r <



1

; s 2 Œ0;  ;

"



with " > 0 sufficiently small, where is that function defined in Sects. 5.4 and 6.6.

Passing to the generalized polar coordinates

x.t/ D



.t/

.t C Â.t// ;

"



we see that

8 0

.t C Â/ C 0 .t C Â/.1 C Â 0 / D

ˆ

<

0 0

.t C Â/ C 00 .t C Â/.1 C Â 0 / D

ˆ

C

:

D

.t C Â/ C



.t/

"



y.t/ D



0



0



.t C Â.t// ;



.t C Â/ ;

.t C Â/



"h



"



Á

.t C Â/ C "e.t/ ;



i.e.,

(



0



.t C Â/ C



0 0



.t C Â/ C



0



.t C Â/Â 0 D 0 ;

00



.t C Â/Â 0 D "h



"



Á

.t C Â/ C "e.t/ :



152



6 Playing Around Resonance



Multiplying the first equation by

subtracting, we get

Œ 0 .t C Â/2



.t C Â/



00



0



.t C Â/, the second one by



.t C Â/Â 0 D "h



"



.t C Â/



Á



.t C Â/



Going back to the system, multiplying the first equation by

one by 0 .t C Â/ and subtracting again, we have

0



Œ .t C Â/



00



.t C Â/



0



.t C Â/2  D "h



.t C Â/



"



Á



0



00



.t C Â/, and



"e.t/ .t C Â/ :



.t C Â/, the second



.t C Â/



"e.t/ 0 .t C Â/ :



Since, for every s 2 R,

0



.s/2



.s/



00



.s/ D 1 ;



we have

Á

i

8

"h

< Â0 D

h

.t C Â/ .t C Â/ e.t/ .t C Â/ ;

Á

h "

i

: 0D " h

.t C Â/ 0 .t C Â/ e.t/ 0 .t C Â/ :

"

Let .Â.tI Â0 /; .tI Â0 // be the solution with initial values

Â.0I Â0 / D Â0 2 Œ0;  ;



.0I Â0 / D 1 :



We see that

lim Â.tI Â0 / D Â0 ;



"!0C



lim



"!0C



.tI Â0 / D 1 ;



(6.5)



uniformly in t 2 Œ0; T. Hence, if " > 0 is small, then .tI Â0 / > 0, for every

t 2 Œ0; T.

Lemma 6.7.1 We have the following equalities1 :

Â.TI Â0 / D Â0 C "ˆ.Â0 / C o."/ ;

.TI Â0 / D 1 "ˆ0 .Â0 / C o."/ ;

where ˆ is the function defined in (6.3).

1



The notation o."/ used here has the following meaning: for some function R."I Â0 /,

R."I Â0 / D o."/







lim



"!0C



1

R."I Â0 / D 0 ;

"



uniformly in Â0 2 Œ0; :



6.7 More Subtle Nonresonance Conditions



153



Proof We have to prove that

Z



1 h

h

.t/



T



lim



"!0C



0



Á

.t/

.t C Â.t// .t C Â.t//

"



i

e.t/ .t C Â.t// dt D ˆ.Â0 / ;



and

Z



T



h



Á

.t/

.t C Â.t// 0 .t C Â.t//

"



h



lim



"!0C



0



i

e.t/ 0 .t C Â.t// dt D ˆ0 .Â0 / ;



uniformly with respect to Â0 2 Œ0; . Using (6.5), we have

Z



T



lim



"!0C



Z



0

T



lim



"!0C



0



Z

e.t/ .t C Â.t// dt D



T



e.t/ .t C Â0 / dt ;



0



Z



0



e.t/ .t C Â.t// dt D



T

0



e.t/ 0 .t C Â0 / dt :



On the other hand, using (6.4), we get

Z



T



lim



"!0C



0



1

h

.t/

Z

D



Á

.t/

.t C Â.t// .t C Â.t// dt D

"

Z

h. 1/ .t C Â0 / dt C



f . CÂ0 /<0g



Z

"!0C



2N



C h.C1/



2N



h. 1/



h



Á

.t/

.t C Â.t// 0 .t C Â.t// dt D

"



0



Z

D



f . CÂ0 /<0g



h.C1/ .t C Â0 / dt



;



D

T



lim



f . CÂ0 />0g



h. 1/ 0 .t C Â0 / dt C



Z

f . CÂ0 />0g



h.C1/ 0 .t C Â0 / dt



D 0:

All the above limits are uniform in Â0 2 Œ0; , whence the conclusion of the proof

of the lemma.

Let us introduce the vector

'.t/ D . .t/;



0



.t// :



154



6 Playing Around Resonance



We deduce from Lemma 6.7.1 that, if .x0 ; y0 / D 1" '.Â0 /, then

P



Á

.TI Â0 /

1

'.Â0 / D

'.Â.TI Â0 //

"

"

i

1h

D .1 "ˆ0 .Â0 //'.Â0 C "ˆ.Â0 // C o."/

"

i

1h

D .1 "ˆ0 .Â0 //.'.Â0 / C "ˆ.Â0 /' 0 .Â0 // C o."/

"

h

i 1

1

D '.Â0 / C ˆ.Â0 /' 0 .Â0 / ˆ0 .Â0 /'.Â0 / C o."/ ;

"

"



so that

.P



I/



Á

1

'.Â0 / D ˆ.Â0 /' 0 .Â0 /

"



1

ˆ0 .Â0 /'.Â0 / C o."/ :

"



Notice that the two vectors ' 0 .Â0 /; '.Â0 /, being linearly independent, make up a

basis of R2 , which rotates, performing a complete clockwise rotation in the time .

With respect to this basis, the coordinates of .P I/. 1" '.Â0 // are

Á

1

1

ˆ.Â0 / C o."/; ˆ0 .Â0 / C o."/ :

"

"

We now make the following assumption:

.0 /2 C 0 .0 /2 Ô 0 ;



for every Â0 2 Œ0;  :



We are thus assuming that the function ˆ only has simple zeros. Then, in the time

, the curve

Â0 7! .ˆ.Â0 /; ˆ0 .Â0 //

makes a certain integer number of counter-clockwise rotations around the origin.

If " is small, the same will be true for the curve

Á

1

1

Â0 7! ˆ.Â0 / C o."/; ˆ0 .Â0 / C o."/ ;

"

"

according to Rouché’s property. Recalling that, when Â0 varies from 0 to , the

vectors ' 0 .Â0 /; ' 0 .Â0 / make a complete clockwise rotation, we conclude that the

curve

Â0 7! .P



I/



Á

1

'.Â0 /

"



6.8 Concluding Remarks



155



makes exactly 1

clockwise rotations around the origin, for " sufficiently small.

We have thus computed the degree:

d.P



I;



"/



D1



;



and we can therefore conclude with the following.

Theorem 6.7.2 (Fabry–Fonda, 1998) If the function ˆ has only simple zeros, and

their number in Œ0; Œ is not exactly 2, then problem (Q) has a solution.

Proof From the above, the function ˆ vanishes exactly 2 times in the interval

0; . If Ô 1, the Brouwer degree of P I with respect to the set " is different

from zero, provided that " > 0 is sufficiently small. Therefore, there is an x 2 "

such that .P I/.x/ D 0, so that x is a fixed point of P.

If D 0, the function ˆ has constant sign, and we find again the result of

Corollary 6.6.1: in this case, the degree is equal to 1.



6.8 Concluding Remarks

The Lazer–Leach condition was introduced in [151], and then adapted to a Dirichlet

problem for an elliptic equation by Landesman and Lazer in [148]. The first

proofs made use of the Schauder Theorem. Since then, many generalizations

have been proposed, see e.g. [31, 63, 64, 137, 139, 167, 180, 213, 214, 221].

In [69, 70], a “double resonance” situation was considered, leading to the following

generalization of Theorem 6.5.1.

Theorem 6.8.1 (Fabry, 1995) Assume that there are two constants C > 0 and

d > 0 such that

x



H)



d



1x



C Ä g.t; x/ Ä



2x



CC;



1x



CC;



and



where



1,



2,



1,



p



2



d



H)



2x



C Ä g.t; x/ Ä



are positive constants such that



1



Cp



1



D



T

;

N



p



2



Cp



2



D



T

:

N C1



156



6 Playing Around Resonance



If, moreover, for any non-zero v satisfying

v 00 C 1 v C

1v D 0 ;

v.0/ D v.T/ ; v 0 .0/ D v 0 .T/

one has

Z



Z

lim sup.g.t; x/



fv<0g x! 1



1 x/v.t/ dt



C



lim inf.g.t; x/



fv>0g x!C1



1 x/v.t/ dt



> 0;



2 x/w.t/ dt



< 0;



and for any non-zero w satisfying

w00 C 2 wC

2w D 0 ;

w.0/ D w.T/ ; w0 .0/ D w0 .T/

one has

Z



Z

lim inf.g.t; x/



fw<0g x! 1



2 x/w.t/ dt



C



lim sup.g.t; x/



fw>0g x!C1



then problem (P) has a solution.

The above theorem has been further generalized in [83], in the case of Hamiltonian-like systems in the plane. See also [121].

The result in Theorem 6.7.2, first proved in [71], can be stated without assuming

the nonlinearity to be locally Lipschitz continuous. Indeed, one can approximate the

continuous function h.x/ by a sequence of locally Lipschitz continuous functions

and then pass to the limit (see [71]). It can be seen that, in the situation of

Remark 5.5.5, where no T-periodic solutions exist, if

Ô , the function

ˆ.Â/ has only simple zeros, and their number in Œ0; Œ is exactly 2 (cf. [80]).

Theorem 6.7.2 has been extended in [38, 39] for equations of Liénard type, and

in [72, 73, 75, 79, 80, 97] for planar systems. See also [186], where periodic

perturbations of an isochronous center are considered.

It is not at all clear whether Theorem 6.7.2 could be extended to deal with a

“double resonance” situation.



Chapter 7



The Variational Method



In this chapter we introduce some variational techniques, with the aim of obtaining

further existence results for the periodic problem (P). We use some known results

of differential calculus in normed vector spaces, which are collected in Appendix B.



7.1 Definition of the Functional

We consider the Hilbert space

HT1 D fx 2 W 1;2 .0; T/ W x.0/ D x.T/g ;

with the scalar product

Z



T



hx; yi D



Z



T



x.t/y.t/ dt C



0



0



x0 .t/y0 .t/ dt ;



and associated norm

kxk D hx; xi1=2 D .kxk22 C kx0 k22 /1=2 :

We recall the periodic problem

.P/



x00 C g.t; x/ D 0 ;

x.0/ D x.T/ ; x0 .0/ D x0 .T/ ;



© Springer International Publishing AG 2016

A. Fonda, Playing Around Resonance, BirkhRauser Advanced Texts Basler

LehrbRucher, DOI 10.1007/978-3-319-47090-0_7



157



158



7 The Variational Method



where g W Œ0; T



R ! R is a continuous function. Setting

Z

G.t; x/ D



x



0



g.t; u/ du ;



let F W HT1 ! R be the function defined as

Z



T



F.x/ D

0



1

2



x0 .t/2



G.t; x.t// dt :



Since its values are in R, we say that F is a functional on H. In particular, this is the

so called action functional.

Lemma 7.1.1 The functional F is continuously differentiable and, for every h 2

HT1 , one has

F 0 .x/h D



Z



T

0



Œx0 .t/h0 .t/



g.t; x.t// h.t/ dt :



Moreover, if F 0 .x/ D 0, then x is a solution of problem (P).

Proof For any h 2 HT1 with khk D 1, we have that

F.x C h/



F.x/



1 1

2



D



Z

0



Z

Z

0



T



Œ.x0 .t/ C h0 .t//2



x0 .t/2  dt

!



T



ŒG.t; x.t/ C h.t//



0



D



T



x0 .t/h0 .t/ dt C

Z



T



Z



T

0



G.t; x.t// dt



h0 .t/2 dt



G.t; x.t/ C h.t//



G.t; x.t//



0



dt ;



for every 2 Œ 1; 1 n f0g. By the Lagrange Mean Value Theorem, there is a

2 0; Œ such that

G.t; x.t/ C h.t//



G.t; x.t//



D g.t; x.t/ C h.t// h.t/ ;



and since all the functions involved here are continuous, it is

supfjg.t; x.t/ C h.t// h.t/j W t 2 Œ0; T;







1; 1Πg < C1 :



7.1 Definition of the Functional



159



Hence, by the Dominated Convergence Theorem,

lim



F.x C h/



Z



F.x/



T



D



!0



Z



x0 .t/h0 .t/ dt



0



T

0



g.t; x.t// h.t/ dt :



We thus proved that F is G-differentiable. Since the function dG F W HT1 ! L.HT1 ; R/,

defined by

Z

dG F.x/.h/ D



T

0



0



Z



0



x .t/h .t/ dt



T

0



g.t; x.t// h.t/ dt ;



is continuous, we really have that F is continuously differentiable.

Assume now that F 0 .x/ D 0, for some x 2 HT1 . Then,

Z



T

0



0



Z



0



x .t/h .t/ dt D



for every h 2 HT1 . Setting v.t/ D

derivative v 2 C.Œ0; T/, hence

0



0



Z



x .t/ D x .0/ C



0



g.t; x.t// h.t/ dt ;



(7.1)



g.t; x.t//, we thus have that x0 has a weak



t

0



T



0



v.s/ ds D x .0/



Z

0



t



g.s; x.s// ds ;



for every t 2 Œ0; T. As a consequence, x0 is continuously differentiable, and

x00 .t/ D



g.t; x.t// ;



for every t 2 Œ0; T. Moreover, taking the constant function h.t/ D 1 in (7.1), we see

RT

that 0 g.t; x.t// dt D 0, whence x0 .0/ D x0 .T/.

We recall that, for any x 2 HT1 , the gradient of F at x is the vector rF.x/ 2 HT1

such that

F 0 .x/.h/ D hrF.x/; hi ;

for every h 2 HT1 . We say that x is a critical point of F if F 0 .x/ D 0 (or, equivalently,

rF.x/ D 0); in that case, F.x/ is said to be a critical value of F.

By Lemma 7.1.1 above, the search of solutions of problem .P/ is reduced to the

search of critical points of the functional F. We will see next that the easiest critical

points to look at are the minimum points. In order to find them, we need the fact that

the action functional F is weakly lower semicontinuous, as stated below.



160



7 The Variational Method



Lemma 7.1.2 If .xn /n is a sequence in HT1 which weakly converges to some function

x 2 HT1 , then

F.x/ Ä lim inf F.xn / :

n



Proof By definition, F.x/ D F1 .x/ C F2 .x/, with

1

F1 .x/ D

2



Z



T

0



0



Z



2



x .t/ ; dt ;



F2 .x/ D



T



G.t; x.t// dt :



0



We can easily see that F1 is a convex functional, i.e., if x; y 2 H and

F1 . x C .1



/y/ Ä F1 .x/ C .1



2 Œ0; 1, then



/F1 .y/ :



In this case, rF1 W H ! H has to be monotone. Indeed, writing (for

above inequality as

F1 .y C .x

passing to the limit as



F1 .y/



y//



Ä F1 .x/



Ô 0) the



F1 .y/ ;



! 0 we have that

hrF1 .y/; x



yi Ä F1 .x/



F1 .y/ :



xi Ä F1 .y/



F1 .x/ ;



Symmetrically, we can also prove that

hrF1 .x/; y

so that, summing up, we get

hrF1 .y/



rF1 .x/; y



xi



0:



Let .xn /n be such that xn * x, weakly in HT1 . Using the monotonicity of rF1 , we

can write

Z

F1 .xn / F1 .x/ D

Z



1



D

Z



0

1



D

Z



0

1

0



0



1



d

F1 . xn C .1

d



hrF1 . xn C .1



/x/ d



/x/; xn



xi d

Z



hrF1 . xn C .1



hrF1 .x/; xn



/x/



xi d ;



rF1 .x/; xn



xi d C

0



1



hrF1 .x/; xn



xi d



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