Abstract
This paper deals with the following boundary value problem
D
α
u
t
=
f
t
,
u
t
,
t
∈
0
,
1
,
u
0
=
u
1
=
D
α
−
3
u
0
=
u
′
1
=
0
,
where
3
<
α
≤
4
,
D
α
is the Riemann-Liouville fractional derivative, and the nonlinearity
f
,
which could be singular at both
t
=
0
and
t
=
1
,
is required to be continuous on
0
,
1
×
ℝ
satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution
u
satisfying
u
t
≤
c
ω
t
,
for
t
∈
0
,
1
and
c
>
0
,
where
ω
t
≔
t
α
−
2
1
−
t
2
.