Skew-evolution Semiflows Research Articles

On uniform instability in mean of stochastic skew-evolution semiflows

In this paper we study three concepts of uniform instability in mean for stochastic skew-evolution semiflows: uniform exponential instability in mean, uniform polynomial instability in mean and uniform \(h\)-instability in mean. These concepts are natural generalizations from the deterministic case. Connections between these concepts are presented. Additionally, some expansion properties, logarithmic criteria and majorization criteria of these concepts are given, respectively.

On uniform dichotomy in mean of stochastic skew-evolution semiflows in Banach spaces

Abstract In this paper we consider three concepts of uniform dichotomy in mean for stochastic skew-evolution semiflows: uniform exponential dichotomy in mean, uniform polynomial dichotomy in mean and uniform h- dichotomy in mean. Some characterizations of these notions and connections between these concepts are given. The obtained results can be considered generalizations to the dichotomy case of the results from [29].

On uniform logarithmic dichotomy of discrete skew-evolution semiflows

Abstract The paper considers two notions of logarithmic dichotomy for discrete skew-evolution semiflows in Banach spaces. We establish the relation between them, we give a characterization for the uniform logarithmic dichotomy of Zabczyk type and a sufficient criteria for the uniform logarithmic dichotomy.

On uniform polynomial trichotomy of skew-evolution semiflows

"The paper treats two concepts of uniform polynomial trichotomy for the skew-evolution semi- flows in Banach spaces. We obtain the connection between them, a characterization for a property of uniform polynomial growth and a sufficient criteria for the uniform polynomial trichotomy."

On Uniform Stability with Growth Rates of Stochastic Skew-Evolution Semiflows in Banach Spaces

The main purpose of this paper is to study a more general concept of uniform stability in mean in which the uniform behavior in the classical sense is replaced by a weaker requirement with respect to some probability measure. This concept includes, as particular cases, the concepts of uniform exponential stability in mean and uniform polynomial stability in mean. Extending techniques employed in the deterministic case, we obtain variants of some results for the general cases of uniform stability in mean for stochastic skew-evolution semiflows in Banach spaces.

Polynomial Stability and Polynomial Instability for Skew-Evolution Semiflows

In this paper, we give characterizations for a polynomial stability of skew-evolution semiflows, using Banach spaces of sequences or functions. We obtain variants of some theorems due to Datko, for the case of a polynomial stability. In parallel, similar characterizations for a polynomial instability are also carried out.

Splitting with Different Growth Rates for Linear Discrete-time Skew-evolution Semiflows in Banach Spaces

Abstract In this paper we intend to study three concepts of (h, k)-splitting for skew-evolution semiflows, which model discrete-time variational systems in Banach spaces. We also aim to give connections between them, emphasized by counterexamples and we propose an open problem.

Integral characterizations for the (h,k)-splitting of skew-evolution semiflows

The main aim of this paper is to give integral characterizations for a general concept of (h,k)-splitting for skew-evolution semiflows in Banach spaces. As consequences, criteria for the properties of (h,k)-dichotomy, nonuniform exponential splitting and exponential splitting are obtained.

Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.

On weak exponential expansiveness of skew-evolution semiflows in Banach spaces

Abstract The aim of this paper is to give several characterizations for weak exponential expansiveness properties of skew-evolution semiflows in Banach spaces. Variants for weak exponential expansiveness of some well-known results in uniform exponential stability theory (Datko (1973)) and exponential instability theory (Lupa (2010), Megan et al. (2008)) are obtained. MSC:93D20, 34D20.

On nonuniform exponential stability for skew-evolution semiflows in Banach spaces

The paper considers some concepts of nonuniform asymptotic stability for skew-evolution semiflows in Banach spaces, which we have introduced in [Megan, M. and Stoica, C., Exponential instability of skew-evolution semiflows in Banach spaces, Stud. Univ. Babes-Bolyai Math., LIII (2008), No. 1, 17–24] and for which we present equivalent definitions, as well as integral characterizations in a nonuniform setting. Some examples are included to illustrate the results and to clarify the differences between the uniform and nonuniform cases.

Trichotomy for Dynamical Systems in Banach Spaces

We construct a framework for the study of dynamical systems that describe phenomena from physics and engineering in infinite dimensions and whose state evolution is set out by skew-evolution semiflows. Therefore, we introduce the concept of ω-trichotomy. Characterizations in a uniform setting are proved, using techniques from the domain of nonautonomous evolution equations with unbounded coefficients, and connections with the classic notion of trichotomy are given. The statements are sustained by several examples.

On nonuniform dichotomy for stochastic skew-evolution semiflows in Hilbert spaces

In this paper we study a general concept of nonuniform exponential dichotomy in mean square for stochastic skew-evolution semiflows in Hilbert spaces. We obtain a variant for the stochastic case of some well-known results, of the deterministic case, due to R. Datko: Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3(1972), 428–445. Our approach is based on the extension of some techniques used in the deterministic case for the study of asymptotic behavior of skew-evolution semiflows in Banach spaces.

Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows

This article is concerned with a well-known theorem of Barbashin which states that an evolution family {U(t, s)} t≥s≥0, or simple 𝒰, is uniformly exponentially stable if and only if 𝒰 satisfies the integral condition . In fact, the author formulated the above result for non-autonomous differential equations in the frame work of finite-dimensional spaces. The aim of this article is to give discrete and continuous versions of Barbashin-type theorem for the case linear skew-evolution semiflows. Giving up disadvantages in Barbashin's proof, we shall extend this problem, based on the recent methods. Thus we obtain necessary and sufficient conditions for uniform exponential stability, generalizing a classical stability theorem due to Barbashin.

Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows

In this paper, we will consider the concept “linear skew-evolution semiflows” and extend theorems of R. Datko, S. Rolewicz, Zabczyk and J.M.A.M van Neerven for this case [15].

On uniform exponential stability for skew-evolution semiflows on Banach spaces

The aim of this paper is to give several characterizations for the property of stability for skew-evolution semiflows on Banach spaces. Thus, we obtain generalizations of some well known results due to Barbashin, Datko and Rolewicz in the case of evolution equations in Banach spaces. A unified treatment in the uniform setting is provided.

Pointwise trichotomy for skew-evolution semiflows on Banach spaces

The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classical notion of trichotomy presented in [8] for evolution operators is also emphasized, and some characterizations are given. The approach of the theory is from a uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows.