Module-valued ordinary differential equations and structure of solution spaces

TL;DR

This paper introduces a framework for ODEs with solutions in Banach modules over finite-dimensional algebras, revealing that homogeneous linear solutions form finitely generated modules.

Contribution

It defines module-valued ODEs using tensor products and proves the solution space is finitely generated over the algebra, extending classical theory.

Findings

Solution spaces are finitely generated modules over the algebra.

The paper establishes a new structure theory for module-valued ODEs.

Abstract

We define and study ordinary differential equations (ODEs) for functions valued in a Banach module $V$ over a finite-dimensional $\Bbbk$-algebra $\mathit{\Lambda}$ by using the tensor of Banach modules. Furthermore, we show that the solution space of a homogeneous linear ODE as above is shown to be a finitely generated $\mathit{\Lambda}$-submodule.