ON ${\bf (a,b)}$ -DICHOTOMY FOR EVOLUTIONARY PROCESSES ON A HALF-LINE

Abstract

Abstract. In this paper we investigate the most general dichotomy concept of evolutionary processes. This dichotomy concept includes many interesting situations, among them we note the nonuniform dichotomy. We characterize the (a, b)-dichotomy in terms of the admissibility of the pair (

Admissibility and roughness of nonuniform exponential dichotomies for difference equations

“…Because of its fundamental importance in determining qualitative properties of systems, indispensable role in discussing invariant manifolds and homoclinic bifurcations, and subtle difficulties in estimating its bound and exponents, over the years, its many properties and corresponding generalizations have been established for ODEs, FDEs, PDEs and evolution equations (see e.g. [7][8][9]11,12,14,15,19,21,23,25,27,28,31,33,[35][36][37][42][43][44][45] and references therein). In order to investigate weaker hyperbolicity, efforts (e.g.…”

“…In order to investigate weaker hyperbolicity, efforts (e.g. [4,5,26,33] and references therein) have been made to extension of dichotomies. In this paper we will consider the following nonuniform exponential dichotomy [4] U (n, m)P (m) ≤ Ke −α(n−m)+δ|m| for n ≥ m, ∈ J, (1.6) U (n, m)Q(m) ≤ Ke −α(m−n)+δ|m| for n ≤ m, ∈ J, (1.7) where α > 0, K > 0 and δ ≥ 0 are constants, called the exponent, the bound and the nonuniformity degree.…”

Robustness of nonuniform mean-square exponential dichotomies

“…In order to investigate more general hyperbolicity, many attempts (see e.g. [37,38,46]) have been made to extend the concept of classical dichotomies. Inspired by the work of Barreira and Pesin on the notion of nonuniformly hyperbolic trajectory [1,2], Barreira and Valls extended the concept of exponential dichotomy 526 H. Zhu to the nonuniform ones and investigated some related problems, see for examples, the works [3][4][5][6][7] and the references therein.…”

Nonuniform mean-square exponential dichotomies and mean-square exponential stability

“…However, dynamical systems exhibit various different kinds of dichotomic behaviors, and the notion of classical exponential dichotomy cannot contain all possible dichotomic behaviors, as Barreira and Valls mentioned in [7], "the notion of exponential dichotomy demands considerably from the dynamics and it is of considerable interest to look for more general types of hyperbolic behavior". In these years, many attempts have been made (see, e.g., [36,37,41]) to extend the concept of the classical dichotomies. For more recent works we mention in particular the papers [4][5][6][7][8][9][10], which, inspired by the fundamental work of nonuniformly hyperbolic trajectory introduced in [2,3], extend the concept of exponential dichotomy to the nonuniform ones and investigate some related problems.…”