Affiliation:
1. College of Mathematics and Statistics, Northwest Normal University , Lanzhou 730070 , Gansu , China
Abstract
Abstract
In this paper, we use the shooting method to study the solvability of the boundary value problem of differential equations with sign-changing weight function:
u
″
(
t
)
+
(
λ
a
+
(
t
)
−
μ
a
−
(
t
)
)
g
(
u
)
=
0
,
0
<
t
<
T
,
u
′
(
0
)
=
0
,
u
′
(
T
)
=
0
,
\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}^{^{\prime\prime} }\left(t)+\left(\lambda {a}^{+}\left(t)-\mu {a}^{-}\left(t))g\left(u)=0,\hspace{1.0em}0\lt t\lt T,\\ u^{\prime} \left(0)=0,\hspace{1.0em}u^{\prime} \left(T)=0,\end{array}\right.
where
a
∈
L
[
0
,
T
]
a\in L\left[0,T]
is sign-changing and the nonlinearity
g
:
[
0
,
∞
)
→
R
g:{[}0,\infty )\to {\mathbb{R}}
is continuous such that
g
(
0
)
=
g
(
1
)
=
g
(
2
)
=
0
g\left(0)=g\left(1)=g\left(2)=0
,
g
(
s
)
>
0
g\left(s)\gt 0
for
s
∈
(
0
,
1
)
s\in \left(0,1)
,
g
(
s
)
<
0
g\left(s)\lt 0
for
s
∈
(
1
,
2
)
s\in \left(1,2)
.