Search

The paper considers the general concept of uniform exponential splitting as a generalization of uniform exponential dichotomy property for evolution operators in Banach spaces.

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples. [ABSTRACT FROM AUTHOR]

In this paper we define the notion of (h, k)-trichotomy for skew-evolution semiflows and we emphasize connections between various other concepts of trichotomy on infinite dimensional spaces, as uniform exponential trichotomy, exponential trichotomy and Barreira-Valls exponential trichotomy. The approach is motivated by various examples. Some characterizations for the newly introduced concept are also provided. [ABSTRACT FROM AUTHOR]