%0 Journal Article
%T Discrete -Dichotomy and Remarks on the Boundedness of the Projections
%A Mihai-Gabriel Babu£¿ia
%A Monteola Ilona Kov¨¢cs
%A M£¿rioara L£¿p£¿dat
%A Mihail Megan
%J Journal of Operators
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/196345
%X The present paper treats a concept of -dichotomy for linear discrete systems. Sufficient conditions for the -boundedness of the projection sequences that give the dichotomy are presented and an illustrative example shows the connection between the growth of the system and the bound of the sequence of projections. Thus the growth of the system that is assumed in the theorems is essential. 1. Introduction Among the asymptotic behaviors of discrete linear systems, an important role is played by the dichotomy property (see [1¨C12] and the references therein). A natural generalization of both the uniform and nonuniform dichotomy is successfully modeled by the concept of -dichotomy, where a significant number of papers containing many interesting results were published, from which we mention [5, 13¨C15]. In recent years, several papers, also dedicated to asymptotic behavior of discrete dynamical systems, appeared and hence the generalization to -behavior is present even for the trichotomy property (see [3, 8, 9, 15¨C17]). Among the results that were obtained in this field, we concentrate our attention on the property of boundedness of the sequence of the projections that give the dichotomic behavior. For example, in [10], in the case of discrete variational systems, the uniform exponential dichotomy property of a discrete cocycle which has a uniform exponential growth implies the boundedness of the dichotomic sequence of projections (here, the proof being more direct, compared with those found in the literature, without using the angular distance between the dichotomy subspaces). Further results concerning the boundedness of the dichotomy sequence of the projections are also presented in [4]. In [12], such results are proved both in the case of uniformly bounded coefficients of the discrete linear system (see Remark 3.3) and through techniques of admissibility of pairs of function spaces to the associated control system of the discrete linear system (Theorem 3.3). This is the direction in which the present paper intends to state the results, by defining the concept of -dichotomy for linear discrete systems and showing that, under similar hypotheses as stated in [10], we obtain the -boundedness of the sequence of projections that give the dichotomic behavior. Bounded and exponentially bounded sequences of projections, although not explicitly stated, are widely used in the study of the exponential dichotomy, and this intrinsic property is assumed from the beginning, in the definition of the asymptotic behavior (see, e.g., [2]). Having in mind the above reasons, we
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