Integral conditions for nonuniform $\mu$-dichotomy on the half-line

TL;DR

This paper establishes integral conditions for the existence of a generalized nonuniform ichotomy for evolution operators on the half-line, unifying and extending known concepts like exponential and polynomial dichotomies.

Contribution

It introduces necessary and sufficient integral conditions for ichotomies, including new cases, and develops an adapted Lyapunov function approach for these conditions.

Findings

Derived integral conditions for ichotomy existence.

Unified framework encompassing exponential, polynomial, and new ichotomy cases.

Provided Lyapunov function criteria for nonuniform ichotomies.

Abstract

We give necessary integral conditions and sufficient ones for the existence of a general concept of $\mu$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $\mu$-dichotomies using these Lyapunov functions.