On (h; k)-Dichotomy and (h; k)-Trichotomy of Noninvertible Evolution Operators in Banach Spaces

Abstract

Abstract. The paper considers some concepts of (h, k)-dichotomy and (h, k)-trichotomy for noninvertible evolution operators in Banach spaces. A characterization of the (h, k)-trichotomy of an evolution operator in terms of (h, k)-dichotomy for two associated evolution operators is given. As applications of this result, characterizations for nonuniform exponential trichotomy and nonuniform polynomial trichotomy are obtained.

Characterizations of the (h,k,μ,ν)−trichotomy for linear time‐varying systems

“…Combining Theorems , , and , we have the following: Theorem The LTV system is ( h , k , μ , ν )−trichotomic with projection sequences , i ∈{1,2,3} if and only if if FP –dichotomic with projection sequences , i ∈{1,2}; if FP –dichotomic with projection sequences , i ∈{1,2}. Remark The theorems from this article include and generalize the case considered in . For the continuous case of evolution operators, we can refer to .…”

On uniform logarithmic dichotomy of discrete skew-evolution semiflows

“…One of the most important topic studied for dynamical systems is represented by the property of dichotomy, approached from uniform point of view in: [2], [5], [19], respectively nonuniform in [3], [4], [6], [8], [10], [17], [20], [22], [23].…”

On some dichotomy concepts with different growth rates for evolution operators in Banach spaces

“…In some situations, particularly in the nonautonomous setting, the concept of uniform exponential dichotomy is too restrictive and it is important to consider more general behaviors. A natural generalization of both the uniform and nonuniform (exponential and polynomial) dichotomy is successfully modeled by the concept of (h, k)-dichotomy, where a significant number of papers containing many interesting and recent results were published, from which we mention the papers of A. J. G. Bento and C. Silva [5], M. I. Kovacs, M.-G. Babuţia, M. Megan [7], M. I. Kovacs, M. Megan, C. L. Mihiţ [8], M. Megan [10], R. Naulin and M. Pinto [12].…”