Abstract
In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation
D
0
+
α
u
(
t
)
+
λ
f
(
t
,
u
(
t
)
,
v
(
t
)
)
=
0
,
0
<
t
<
1
,
D
0
+
α
v
(
t
)
+
μ
g
(
t
,
u
(
t
)
,
v
(
t
)
)
=
0
,
0
<
t
<
1
,
u
(
0
)
=
v
(
0
)
=
0
,
a
1
D
0
+
β
u
(
1
)
=
b
1
D
0
+
β
v
(
ξ
)
,
a
2
D
0
+
β
v
(
1
)
=
b
2
D
0
+
β
u
(
η
)
,
η
,
ξ
∈
(
0,1
)
,
where the coefficients
a
i
,
b
i
,
i
=
1,2
are real positive constants,
α
∈
(
1,2
]
,
β
∈
(
0,1
]
,
D
0
+
α
,
D
0
+
β
are the standard Riemann-Liouville derivatives. Values of the parameters
λ
and
μ
are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.