AbstractWe show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$
t
0
≥
0
and $M>0$
M
>
0
such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$
{
(
i
m
)
α
}
|
m
|
>
t
0
⊂
ρ
(
A
)
, the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$
∥
(
i
m
)
α
(
A
+
(
i
m
)
α
I
)
−
1
∥
≤
M
for all
|
m
|
>
t
0
,
m
∈
Z
,
then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$
1
p
<
α
≤
2
p
and $1< p < 2$
1
<
p
<
2
, the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$
{
D
t
α
G
L
u
(
t
)
+
A
u
(
t
)
=
f
(
t
)
,
t
∈
(
0
,
2
π
)
;
u
(
0
)
=
u
(
2
π
)
,
where $_{GL}D^{\alpha }$
D
α
G
L
denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$
f
∈
L
2
π
p
(
R
,
X
)
that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$
ϕ
A
∈
(
0
,
α
π
/
2
)
and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$
∫
0
2
π
f
(
t
)
d
t
∈
Ran
(
A
)
.
Analyticity of the Resolvent Function