Affiliation:
1. Institute of Mathematics , Faculty of Mechanical Engineering , Brno University of Technology , Technická 2, 616 69 Brno , Czech Republic
Abstract
Abstract
We study the existence and multiplicity of positive solutions to the periodic problem
u
′′
=
p
(
t
)
u
-
q
(
t
,
u
)
u
+
f
(
t
)
;
u
(
0
)
=
u
(
ω
)
,
u
′
(
0
)
=
u
′
(
ω
)
,
u^{\prime\prime}=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\quad u^{\prime}(0)=u^{\prime}(\omega),
where
p
,
f
∈
L
(
[
0
,
ω
]
)
p,f\in L([0,\omega])
and
q
:
[
0
,
ω
]
×
R
→
R
q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R}
is a Carathéodory function.
By using the method of lower and upper functions, we show some properties of the solution set of the considered problem and, in particular, the existence of a minimal positive solution.
Reference5 articles.
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Two-Point Boundary Value Problems: Lower and Upper Solutions,
Math. Sci. Eng. 205,
Elsevier, Amsterdam, 2006.
2. A. Fonda,
Playing Around Resonance. An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations,
Birkhäuser Adv. Texts Basler Lehrbücher,
Birkhäuser/Springer, Cham, 2016.
3. A. Lomtatidze,
Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations,
Mem. Differ. Equ. Math. Phys. 67 (2016), 1–129.
4. A. Lomtatidze,
On periodic boundary value problem for second-order ordinary differential equations,
Commun. Contemp. Math. 22 (2020), no. 6, 1950049.
5. J. Šremr,
Positive periodic solutions to the forced non-autonomous Duffing equations,
Georgian Math. J., to appear.
Cited by
1 articles.
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