DOI: 10.1515/gmj-2026-3018 ISSN: 1072-947X
On a periodic problem for second-order Duffing-type equations
Alexander Lomtatidze, Jiří Šremr Abstract
The paper addresses the periodic problem
u
′′
=
p
(
t
)
u
-
q
(
t
,
u
)
u
,
u
(
0
)
=
u
(
ω
)
,
u
′
(
0
)
=
u
′
(
ω
)
,
see text
u^{\prime\prime}=p(t)u-q(t,u)u,\qquad u(0)=u(\omega),\quad u^{\prime}(0)=u^{%
\prime}(\omega),
where
p
:
[
0
,
ω
]
→
ℝ
p:\kern-0.569055pt[0,\omega]\to\mathbb{R}
is a Lebesgue integrable function and
q
:
[
0
,
ω
]
×
ℝ
→
ℝ
q:\kern-0.569055pt[0,\omega]\kern-1.0pt\times\kern-1.0pt\mathbb{R}\to\mathbb{R}
is a function satisfying
Carathéodory conditions. By combining the maximum and antimaximum
principles with the method of non-ordered lower and
upper functions, sufficient and necessary conditions are found for
the existence of a positive solution to the given
problem. The main results are applied and further refined for
the equation
u
′′
=
p
(
t
)
u
-
h
(
t
)
φ
(
u
)
u
,
see text
u^{\prime\prime}=p(t)u-h(t)\varphi(u)u,
whose non-linearity has separated arguments.