We study the periodic solutions of Duffing equations with singularitiesx′′+g(x)=p(t). By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided thatgsatisfies the singular condition and the time map related to autonomous systemx′′+g(x)=0tends to zero.
Abstract
We study the existence and multiplicity of positive solutions to the periodic problem for a forced non-autonomous Duffing equation
u
′′
=
p
(
t
)
u
-
h
(
t
)
|
u
|
λ
sgn
u
+
f
(
t
)
;
u
(
0
)
=
u
(
ω
)
,
u
′
(
0
)
=
u
′
(
ω
)
,
u^{\prime\prime}=p(t)u-h(t)\lvert u\rvert^{\lambda}\operatorname{sgn}u+f(t);\quad u(0)=u(\omega),\ u^{\prime}(0)=u^{\prime}(\omega),
where
p
,
h
,
f
∈
L
(
[
0
,
ω
]
)
p,h,f\in L([0,\omega])
and
λ
>
1
\lambda>1
.
The obtained results are compared with the results known for the equations with constant coefficients.
Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map
