Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair As a consequence, we prove that a linear skew-product flow on is uniformly exponentially dichotomic if and only if the pair is uniformly admissible for .     

Exponential Expansiveness and Complete Admissibility For Evolution Families

Connections between uniform exponential expansiveness and complete admissibility of the pair are studied. A discrete version for a theorem due to Van Minh, Räbiger and Schnaubelt is presented. Equivalent characterizations of Perron type for uniform exponential expansiveness of evolution families in terms of complete admissibility are given.     

Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications

The purpose of this paper is to provide a new, unified and complete study for uniform dichotomy and exponential dichotomy on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over _+{\mathbb {R}_+} which are invariant under translations. After that, we obtain a classification of the main classes of function spaces over \mathbb R₊{\mathbb {R}_+}, in order to deduce necessary and sufficient conditions for the existence of exponential dichotomy, emphasizing on the main technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years.     

Exponential dichotomy and dichotomy radius for difference equations

The aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces. We classify the classes of input and output spaces, respectively, and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems. By an example we show that the obtained results are the most general in this topic. As an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input-output operators acting on sequence spaces which are invariant under translations.     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows. Received August 10, 2001; in revised form June 25, 2002     

The Anti-Leukemic Activity of Natural Compounds

The use of biologically active compounds has become a realistic option for the treatment of malignant tumors due to their cost-effectiveness and safety. In this review, we aimed to highlight the main natural biocompounds that target leukemic cells, assessed by in vitro and in vivo experiments or clinical studies, in order to explore their therapeutic potential in the treatment of leukemia: acute myeloid leukemia (AML), chronic myeloid leukemia (CML), acute lymphocytic leukemia (ALL), and chronic lymphocytic leukemia (CLL). It provides a basis for researchers and hematologists in improving basic and clinical research on the development of new alternative therapies in the fight against leukemia, a harmful hematological cancer and the leading cause of death among patients.     

A lower bound for the stability radius of time-varying systems

We introduce and characterize the stability radius of systems whose state evolution is described by linear skew-product semiflows. We obtain a lower bound for the stability radius in terms of the Perron operators associated to the linear skew-product semiflow. We generalize a result due to Hinrichsen and Pritchard.

     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows.     

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ={Φ(m,n)}_(m,n)∈Δ the subspace . Supposing that X₁ is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on X₁, we obtain that the admissibility of the pair with p∈(1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.     

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with