Abstract
In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is,
x
̇
(
t
)
∈
F
(
t
,
x
(
t
)
)
a.e. on
I
,
x
(
t
)
∈
S
,
∀
t
∈
I
,
x
(
0
)
=
x
0
∈
S
, (*), where
S
is a closed subset in a Banach space
,
I
=
[
0
,
T
]
,
(
T
>
0
)
,
F
:
I
×
S
→
, is an upper semicontinuous set-valued mapping with convex values satisfying
F
(
t
,
x
)
⊂
c
(
t
)
x
+
x
p
,
∀
(
t
,
x
)
∈
I
×
S
, where
p
∈
ℝ
, with
p
≠
1
, and
c
∈
C
(
[
0
,
T
]
,
ℝ
+
)
. The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.