Integral conditions for nonuniform $μ$-dichotomy on the half-line

Abstract

We give necessary integral conditions and sufficient ones for the existence of a general concept of µ-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 µ-dichotomies using these Lyapunov functions.

Datko-Pazy conditions for nonuniform exponential stability

“…The more recent contributions in this line of research deal with certain stochastic extensions (see [14]), weaker forms of (exponential stability) (see [5,10,11,15,24]) and with various relaxations of Datko's condition that still imply the existence of exponential stability (see [22,26]). …”

On Dichotomy With Differentiable Growth Rates for Skew-Evolution Cocycles in Banach Spaces

“…Throughout the years an important extension of exponential dichotomy and polynomial dichotomy is introduced by Pinto [16] and it is called dichotomy with growth rates or h-dichotomy, where the growth rate is a nondecreasing and bijective function h : R + → [1, ∞). For recent contributions we refer to the works of Bento, Lupa, Megan and Silva [2], Mihit ¸, Borlea and Megan [14], Gȃinȃ [9], Megan and Gȃinȃ [13].…”