# Acyclicity of the solution set of two-point boundary value problems for second order multivalued differential equations

## TL;DR

This paper investigates the topological structure of solution sets for second order differential inclusions with two-point boundary conditions, showing they are nonempty, compact, and acyclic under certain assumptions, with special cases for Lipschitz conditions.

## Contribution

It establishes the acyclicity and topological properties of solution sets for second order differential inclusions, including the Lipschitz case, in Banach spaces.

## Key findings

- Solution set is nonempty and compact.
- Solution set is acyclic in relevant function spaces.
- Lipschitz case yields an absolute retract.

## Abstract

The topological and geometrical structure of the set of solutions of two-point boundary value problems for second order differential inclusions in Banach spaces is investigated. It is shown that under the Carath\'eodory-type assumptions the solution set of the periodic boundary value problem is nonempty compact acyclic in the space of continuously differentiable functions as well as in the Bochner-Sobolev space $\mathbb{H}^2$ endowed with the weak topology. The proof relies heavily on the accretivity of the right-hand side of differential inclusion. The Lipschitz case is treated separately. As one might expect the solution set is, in this case, an absolute retract.