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

#### 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.

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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