On the Cauchy problem for a class of differential inclusions with applications

ABSTRACT

Our main result is the following: let $F:\mathbf{R}\times {\mathbf{R}}^n\to 2^{{\mathbf{R}}^n}$ be a multifunction, and assume that there exists a neglegible subset $U\subseteq \mathbf{R}\times {\mathbf{R}}^n$, satisfying a certain geometrical condition, such that the restriction of F to $(\mathbf{R}\times {\mathbf{R}}^n)\setminus U$ is bounded, lower semicontinuous with non-empty closed values, and its range belongs to a certain family ${\mathcal{A}}_n$ defined below. Then, there exists a bounded multifunction $G:\mathbf{R}\times {\mathbf{R}}^n\to 2^{{\mathbf{R}}^n}$ such that G is upper semicontinuous with non-empty compact convex values, and every generalized solution of $u^{\mathrm{\prime }}\left(t\right)\in G(t,u(t\left)\right)$ is a solution of $u^{\mathrm{\prime }}\left(t\right)\in F(t,u(t\left)\right)$. Such a result improves a celebrated result by A. Bressan, valid for lower semicontinuous multifunctions. We point out that a multifunction F satisfying our assumptions can fail to be lower semicontinuous even at all points $(t,x)\in \mathbf{R}\times {\mathbf{R}}^n$. We derive some existence and qualitative results for the Cauchy problem associated to such a class of multifunctions. As an application, we prove existence and qualitative results for the implicit Cauchy problem $g\left(u^{\mathrm{\prime }}\right)=f(t,u)$, $u\left(0\right)=\xi$, with f discontinuous in u.

KEYWORDS:

• Differential inclusions
• Cauchy problem
• generalized solutions
• discontinuous selections
• implicit discontinuous differential equations

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 34A60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by the Grant MOST 106-2923-E-039- 001-MY3.