Some Results for Impulsive Problems via Morse Theory

TL;DR

This paper employs Morse theory to analyze impulsive differential problems, establishing existence results, computing critical groups, and identifying resonance sets, leading to new solutions for piecewise linear impulsive problems.

Contribution

It introduces a novel approach using Morse theory to study impulsive problems, including existence proofs and resonance analysis, with applications to piecewise linear cases.

Findings

Proved existence of solutions for superlinear impulsive problems.

Computed critical groups at zero for asymptotically linear impulses.

Identified a resonance set crucial for solution existence.

Abstract

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.