Differential Equation over Banach Algebra

TL;DR

This paper explores differential equations over Banach algebras, including quaternion algebra examples, and develops theory for linear systems, eigenvalues, and solutions in non-commutative algebraic contexts.

Contribution

It extends differential equation theory to Banach and quaternion algebras, introducing methods for solving linear systems and analyzing eigenvalues in non-commutative division algebras.

Findings

Solutions for differential equations over quaternion algebra

Development of eigenvalue theory in non-commutative division algebras

Framework for linear systems over Banach D-algebras

Abstract

In the book, I considered differential equations of order $1$ over Banach $D$\Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of differential equations in quaternion algebra. In order to study homogeneous system of linear differential equations, I considered vector space over division $D$-algebra, solving of linear equations over division $D$-algebra and the theory of eigenvalues in non commutative division $D$-algebra.