Abstract
In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions:
{
(
y
′
−
A
y
)
(
t
)
∈
F
(
t
,
y
(
t
)
)
,
a.e.
t
∈
J
∖
{
t
1
,
…
,
t
m
}
,
y
(
t
k
+
)
−
y
(
t
k
−
)
=
I
k
(
y
(
t
k
−
)
)
,
k
=
1
,
…
,
m
,
y
(
0
)
=
y
(
b
)
where
J
=
[
0
,
b
]
and
0
=
t
0
<
t
1
<
⋯
<
t
m
<
t
m
+
1
=
b
(
m
∈
N
∗
)
A
is the infinitesimal generator of a
C
0
-semigroup
T
on a separable Banach space
E
and
F
is a multi-valued map. The functions
I
k
characterize the jump of the solutions at impulse points
t
k
(
k
=
1
,
…
,
m
). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of
T
(
b
)
. Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined.