Differential inclusions with nonlocal conditions: existence results and topological properties of solution sets

John R. Graef, Johnny Henderson, Abdelghani Ouahab

Abstract


In this paper, we study the topological structure of solution sets
for the first-order differential inclusions with nonlocal
conditions:
$$
\cases
y'(t) \in F(t,y(t)) &\text{a.e } t\in [0,b],\\
y(0)+g(y)=y_0,
\endcases
$$
where $F\colon [0,b]\times \mathbb{R}^n\to{\mathcal P}(\mathbb{R}^n)$ is a multivalued map.
Also, some geometric properties of solution sets, $R_{\delta}$,
$R_\delta$-contractibility and acyclicity, corresponding to
Aronszajn-Browder-Gupta type results, are obtained. Finally, we
present the existence of viable solutions of differential
inclusions with nonlocal conditions
and we investigate the topological properties of the set constituted
by these solutions.

Keywords


Differential inclusions; nonlocal conditions; solution set; compactness; $R_{\delta}$; $R_\delta$-contractibility; acyclicity; proximate retract; tangential conditions; viable solutions

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