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
Ã 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 Â 2 Œ0; ;
then problem (Q) has a solution.
Proof Assume, for instance, that ˆ.Â/ > 0 for every Â 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 Â 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// :
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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
xÄ
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;
2
1; 1Œ g < 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