On ( )-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces

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. 1. Introduction In the last few years, the theory of difference equations has witnessed an impressive development. The notion of (uniform) exponential dichotomy introduced by Perron in [1] for differential equations and by Li in [2] for difference equations plays a central role in a substantial part of the theory of differential equations and dynamical systems. The notion of dichotomy for differential equations has gained prominence since the appearance of two fundamental monographs of Massera and Sch？ffer [3] and Daleckii and Krein [4]. These were followed by the important book of Coppel [5] who synthesized and improved the results that existed in the literature up to 1978. Early works in the counterpart results of difference equations appeared in the paper of Coffman and Sch？ffer [6] and later, in 1981, when Henry included discrete dichotomies in his book [7]. This was followed by the classical monographs due to Agarwal [8] where ordinary and exponential dichotomy properties of difference equations are studied and various applications are provided. Significant work was reported by P？tzsche in [9]. One important and useful concept of dichotomy in the study of difference equations is the so-called (uniform) -dichotomy concept introduced by Pinto in [10]. Since then, this concept has been extensively studied and applied; see, for example, Naulin and Pinto [11], Megan [12], Fenner [13], and Lin [14] where several examples are presented. A new notion called nonuniform -dichotomy is proposed by Bento and Silva in [15] for the continuous case and in [16, 17] for discrete time settings, for invertible systems, with growth rates given by increasing functions (sequences) which go to infinity. In the last years, this subject (resp., nonuniform exponential dichotomy and nonuniform polynomial dichotomy) became one of the subjects of large interest, significant results being obtained (see, e.g., [18–24]). On the other hand, we can consider the case of noninvertible