Differentiable mappings on products with different degrees of differentiability in the two factors

TL;DR

This paper develops a differential calculus framework for $C^{r,s}$-mappings on product spaces, establishing exponential laws and applying these to parameter-dependent differential equations in Banach spaces.

Contribution

It introduces a new calculus for $C^{r,s}$-mappings on product spaces and proves exponential laws, extending differential calculus to more general settings.

Findings

Established exponential laws for $C^{r,s}$-mappings.

Applied the calculus to parameter-dependent differential equations.

Proved regularity properties of flows in Banach spaces.

Abstract

We develop differential calculus of $C^{r,s}$-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class $C^{r,s}$.