Abstract
AbstractIn this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type
$$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$
(
x
(
t
)
−
c
x
(
t
−
σ
)
)
″
+
f
(
x
(
t
)
)
x
′
(
t
)
−
φ
(
t
)
x
μ
(
t
)
+
α
(
t
)
x
γ
(
t
)
=
e
(
t
)
,
where $f:(0,+\infty)\rightarrow R$
f
:
(
0
,
+
∞
)
→
R
, $\varphi(t)>0$
φ
(
t
)
>
0
and $\alpha(t)>0$
α
(
t
)
>
0
are continuous functions with T-periodicity in the t variable, c, γ are constants with $|c|<1$
|
c
|
<
1
, $\gamma\geq1$
γ
≥
1
. Many authors obtained the existence of periodic solutions under the condition $0<\mu\leq1$
0
<
μ
≤
1
, and we extend the result to $\mu>1$
μ
>
1
by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
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