Mihail Megan

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C0,C0) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.

Nonuniform dichotomy of evolutionary processes in Banach spaces

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and ( B, D ) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C 0 and ( B, D ) = [( L p , L q ).

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.

Generalized Exponential Trichotomies for Abstract EvolutionOperators on the Real Line

This paper considers two trichotomy concepts in the context of abstract evolution operators. The first one extends the notion of exponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators, and it is a natural extension of the generalized exponential dichotomy considered in the paper by Jiang (2006). The second concept is dual in a certain sense to the first one. We prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomy on both half-lines. We emphasize that we do not assume the invertibility of the evolution operators on the whole space X (unlike the case of evolution operators generated by differential equations).

On (h, k)-stability of evolution operators in Banach spaces

Abstract The main aim of this work is to define and exemplify various stability concepts and to emphasize connections between them. These stability concepts are included in a general concept, the so-called ( h , k ) -stability. We motivate our approach with illustrative examples.

On (

This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant dichotomy projections, we present characterizations of these sequences and connections with other dichotomy concepts existent in the literature. Some illustrative examples clarify the implications between these concepts.

h,k

In this paper we investigate four concepts of exponential stability for difference equations in Banach spaces. Characterizations of these concepts are given. They can be considered as variants for the discrete-time case of the classical results due to Barbashin [6] and Datko [5]. An illustrative example clarifies the relations between these concepts.

)-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces

This paper introduces the concept of exponential h-expansiveness for semigroups of nonlinear operators, which is an extension of classical concept of exponential expansiveness. Following the idea of obtaining an unitary treatment for stability and expansiveness, necessary and sufficient conditions for exponential h-expansiveness are given. As particular cases, the variants for exponential expansiveness of some well-known stability results due to Datko, Pazy, Ichikawa, Rolewicz and Neerven are obtained.

On exponential stability for linear discrete-time systems in Banach spaces

This paper introduces a concept of exponential dichotomy for a general class of nonlinear evolution operators. Necessary and sufficient conditions for exponential dichotomy are given. Obtained results are generalizations of well-known results of R. Datko (1973) and A. Ichikawa (1984) about exponential stability.

On exponential h-expansiveness of semigroups of operators in Banach spaces

The aim of this paper is to give characterizations in terms of Lyapunov functions for nonuniform exponential dichotomies of nonautonomous and noninvertible discrete-time systems.