Existence of solutions for second-order differential inclusions involving proximal normal cones

TL;DR

This paper proves the global existence of solutions for second-order differential inclusions involving proximal normal cones, accommodating non-smooth, non-convex, and impact conditions in a broad mathematical framework.

Contribution

It establishes the existence of solutions for complex second-order differential inclusions with impact laws, extending previous results to more general, non-smooth, and non-convex settings.

Findings

Proved existence of solutions under broad conditions.

Handled non-smooth, non-convex set-valued maps.

Included impact laws in the analysis.

Abstract

In this work, we prove global existence of solutions for second order differential problems in a general framework. More precisely, we consider second order differential inclusions involving proximal normal cone to a set-valued map. This set-valued map is supposed to take admissible values (so in particular uniformly prox-regular values, which may be non-smooth and non-convex). Moreover we require the solution to satisfy an impact law, appearing in the description of mechanical systems with inelastic shocks.