Abstract
We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows:
D
c
α
x
t
-
f
t
,
x
t
=
g
t
,
y
t
,
I
α
y
t
,
t
∈
J
=
[
0,1
]
,
D
c
α
y
t
-
f
t
,
y
t
=
g
t
,
x
t
,
I
α
x
t
,
t
∈
J
=
0,1
,
D
c
p
x
0
=
ψ
x
η
1
,
x
′
0
=
0
,
…
,
x
n
-
2
0
=
0
,
D
c
p
x
1
=
ψ
x
η
2
,
D
c
p
y
0
=
ψ
y
η
1
,
y
′
0
=
0
,
…
,
y
n
-
2
0
=
0
,
D
c
p
y
1
=
ψ
y
η
2
, where
p
,
η
1
,
η
2
∈
0,1
,
ψ
is linear,
D
c
α
is Caputo fractional derivative of order
α
, with
n
-
1
<
α
≤
n
,
n
∈
N
, and
I
α
is fractional integral of order
α
. The nonlinear functions
f
,
g
are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.