Résumé:
In this paper, we prove the existence and uniqueness of solutions for a system for fractional differential
equations with four point boundary conditions. The results are obtained using Banach contraction principle
and Krasnoselkii’s fixed point theorem
D
α
x
(
t
)+
f
(
t
,
y
(
t
)
,
D
δ
y
(
t
)
)
=
0,
t
∈
J
,
D
β
y
(
t
)+
g
(
t
,
x
(
t
)
,
D
σ
x
(
t
))=
0,
t
∈
J
,
x
(
0
)=
y
(
0
)=
0,
x
(
1
)
−
λ
1
x
(
η
)=
0,
y
(
1
)
−
λ
1
y
(
η
)=
0,
x
′′
(
0
)=
y
′′
(
0
)=
0,
x
′′
(
1
)
−
λ
2
x
′′
(
ξ
)=
0,
y
′′
(
1
)
−
λ
2
y
′′
(
ξ
)=
0,
where 3
<
α
,
β
≤
4,
α
−
2
<
σ
≤
α
−
1,
β
−
2
<
δ
≤
β
−
1, 0
<
ξ
,
η
<
1, and
D
α
,
D
β
,
D
δ
and
D
σ
, are the
Caputo fractional derivatives,
J
=[
0, 1
]
,
λ
1
,
λ
2
are real constants with
λ
1
η
6
=
1,
λ
2
ξ
6
=
1 and
f
,
g
continuous
functions on
[
0, 1
]
×
R
2
.
