Affiliation:
1. Department of Mathematics, Northwest Normal University , Lanzhou 730070 , People’s Republic of China
Abstract
Abstract
This article discusses the existence of positive solutions for the system of second-order ordinary differential equation boundary value problems
−
u
″
(
t
)
=
f
(
t
,
u
(
t
)
,
v
(
t
)
,
u
′
(
t
)
)
,
t
∈
[
0
,
1
]
,
−
v
″
(
t
)
=
g
(
t
,
u
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
,
t
∈
[
0
,
1
]
,
u
(
0
)
=
u
(
1
)
=
0
,
v
(
0
)
=
v
(
1
)
=
0
,
\left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(t)=f\left(t,u\left(t),v\left(t),u^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ -{v}^{^{\prime\prime} }\left(t)=g\left(t,u\left(t),v\left(t),v^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ u\left(0)=u\left(1)=0,\hspace{1em}v\left(0)=v\left(1)=0,\end{array}\right.
where
f
,
g
:
[
0
,
1
]
×
R
+
×
R
+
×
R
→
R
+
f,g:\left[0,1]\times {{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{+}\times {\mathbb{R}}\to {{\mathbb{R}}}^{+}
are continuous. Under the related conditions that the nonlinear terms
f
(
t
,
x
,
y
,
p
)
f\left(t,x,y,p)
and
g
(
t
,
x
,
y
,
q
)
g\left(t,x,y,q)
may be super-linear growth or sub-linear growth on
x
,
y
,
p
x,y,p
, and
q
q
, we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition
(
F
3
)
\left(F3)
is presented to restrict the growth of
f
(
t
,
x
,
y
,
p
)
f\left(t,x,y,p)
and
g
(
t
,
x
,
y
,
q
)
g\left(t,x,y,q)
on
p
p
and
q
q
. The super-linear growth or sub-linear growth of the nonlinear terms
f
f
and
g
g
is described by related inequality conditions instead of the usual independent inequality conditions about
f
f
and
g
g
. The discussion is based on the fixed point index theory in cones.