On nonuniform exponential stability for skew-evolution semiflows on Banach spaces

The paper considers some concepts of nonuniform asymptotic stability for skew-evolution semiflows on Banach spaces. The obtained results clarify differences between the uniform and nonuniform cases. Some examples are included to illustrate the results.Comment: 11 page

Dichotomies for evolution equations in Banach spaces

The aim of this paper is to emphasize various concepts of dichotomies for evolution equations in Banach spaces, due to the important role they play in the approach of stable, instable and central manifolds. The asymptotic properties of the solutions of the evolution equations are studied by means of the asymptotic behaviors for skew-evolution semiflows.Comment: 22 page

THE UNIFORM EXPONENTIAL STABILITY OF LINEAR SKEW-PRODUCT SEMIFLOWS ON REAL HILBERT SPACE

The goal of the paper is to present some characterizations for the uniform exponential stability of linear skew-product semiflows on real Hilbert space

Pointwise Trichotomy for Skew-Evolution Semiflows on Banach Spaces

The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows

Exponential dichotomies of evolution operators in Banach spaces

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid

A General Framework for Splitting Concepts for Cocycles over Generalized Nonautonomous Dynamical Systems

While the classic theory of exponential dichotomy deals with differential and difference equations with uniquely determined forward and backward solutions, nowadays, applications in engineering require having corresponding theory for equations whose backward solutions are not guaranteed to exist or to be unique. To this goal, we will consider general dichotomic behaviors that consist in assuming the existence of splitting into invariant subspaces, where the norms of the evolution trajectories are bounded by some functions that depend on the initial and final times