Note on the existence theory for non-induced evolution equations

TL;DR

This paper develops a theoretical framework for proving existence of solutions to non-linear evolution equations involving non-induced operators, introducing new concepts like $C^0$-Bochner pseudo-monotonicity and coercivity.

Contribution

It introduces a novel abstract existence framework for non-induced evolution equations and adapts key monotonicity and coercivity notions for these problems.

Findings

Established an abstract existence theorem for non-induced evolution equations.

Introduced $C^0$-Bochner pseudo-monotonicity and coercivity concepts.

Provided a foundation for analyzing non-linear evolutionary problems with non-standard operators.

Abstract

In this note we develop a framework which allows to prove an abstract existence result for non-linear evolution equations involving so-called non-induced operators, i.e., operators which are not prescribed by a time-dependent family of operators. Apart from this, we introduce the notion of $C^0$-Bochner pseudo-monotonicity, and $C^0$-Bochner coercivity, which are appropriate adaptions of the standard notion to the case of evolutionary problems.