| Contents |
| We study fractional integro-differential inclusions of the form
$$
D_c^{\alpha }x(t)\in F(t,x(t),V(x)(t))\quad a.e.\; ([0,T]),\quad
x(0)=x_0,\quad x'(0)=x_1,
$$
where $\alpha \in (1,2]$, $D_c^{\alpha }$ is the Caputo fractional
derivative, $F:[0,T]\times \mathbf{R}\times \mathbf{R}\to
\mathcal{P}(\mathbf{R})$ is a set-valued map and $x_0,x_1\in
\mathbf{R}$, $x_0,x_1\neq 0$. $V:C([0,T],\mathbf{R})\to
C([0,T],\mathbf{R})$ is a nonlinear Volterra integral operator
defined by $V(x)(t)=\int_0^tk(t,s,x(s))ds$ with
$k(.,.,.):[0,T]\times \mathbf{R}\times \mathbf{R}\to \mathbf{R}$ a
given function.
We prove the arcwise connectedness of the solution set this problem
when the set-valued map is Lipschitz in the second and third
variable. Moreover, under such type of hypotheses on the
set-valued map, we establish a more general topological property
of the solution set of our problem. Namely, we prove that the set
of selections of the set-valued map $F$ that correspond to the
solutions of the problem is a retract of $L^1([0,T],\mathbf{R})$.
Both results are essentially based on Fryszkowski, Bressan and
Colombo results concerning the existence of continuous selections
of lower semicontinuous multifunctions with decomposable values. |
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