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: $$ \begin{cases} y'(t) \in F(t,y(t)) &\text{a.e. } t\in [0,b],\\ y(0)+g(y)=y_0, \end{cases} $$ 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.