Monotone ODEs with Discontinuous Vector Fields in Sequence Spaces

TL;DR

This paper proves the existence of solutions for monotone ordinary differential equations with discontinuous vector fields in sequence spaces, introducing a novel partial order approach in the context of Fréchet spaces.

Contribution

It establishes an existence theorem for such ODEs using a new partial order method, extending the theory to discontinuous vector fields in infinite-dimensional spaces.

Findings

Existence of solutions under monotonicity conditions

Introduction of a new partial order technique

Application to ODEs in Fréchet spaces

Abstract

We consider a system of ODE in a Fr\'echet space with unconditional Schauder basis. The right side of the ODE is a discontinuous function. Under certain monotonicity conditions we prove an existence theorem for the corresponding initial value problem. We employ an idea of the partial order which seems to be new in this field.