Stable invariant manifold for generalized ODEs with applications to measure differential equations

TL;DR

This paper develops a stable invariant manifold theory for generalized ODEs defined via Kurzweil integrals, extending classical results to a broader class of integral equations on Banach spaces.

Contribution

It introduces a generalized Lyapunov-Perron equation and proves a stable manifold theorem for nonlinear generalized ODEs with exponential dichotomy.

Findings

Established a stable invariant manifold theorem for generalized ODEs

Derived existence results for measure and impulsive differential equations

Extended classical invariant manifold theory to Kurzweil integral-based equations

Abstract

This paper establishes the stable invariant manifold for a new kind of differential equations defined by Kurzweil integral, so-called {\em generalized ODEs} on a Banach space. The nonlinear generalized ODEs are formulated as $$ \frac{dz}{d\tau}=D[\Lambda(t)z+F(z,t)], $$ where $\Lambda(t)$ is a bounded linear operator on a Banach space $\mathscr{Z}$ and $F(z,t)$ is a nonlinear Kurzweil integrable function on $\mathscr{Z}$. The letter $D$ represents that generalized ODEs are defined via its solution, and $\frac{dz}{d\tau}$ only a notation. Hence, generalized ODEs are fundamentally a notational representation of a class of integral equations. Due to the differences between the theory of generalized ODEs and ODEs, it is difficult to extended the stable manifold theorem of ODEs to generalized ODEs. In order to overcome the difficulty, we establish a generalized Lyapunov-Perron equation in the frame of Kurzweil integral theory. Subsequently, we present a stable invariant manifold theorem for nonlinear generalized ODEs when their linear parts exhibit an exponential dichotomy. As effective applications, we finally derive results concerning the existence of stable manifold for measure differential equations and impulsive differential equations.