Semiflows In Banach Spaces Research Articles

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 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.

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.

A discrete-time approach in the qualitative theory of skew-product three-parameter semiflows

We obtain a characterization of the uniform exponential stability for the continuous-time skew-product three-parameter semiflows in Banach spaces, using a discrete-time approach. Our technique is based on the classical test-function method of O. Perron and Ta Li.

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 Exponential Trichotomy of Cocycles over Semiflows

AbstractThe paper considers a concept of exponential trichotomy for cocycles over semiflows in Banach spaces and as a particular case the corresponding dichotomy concept. Our main objective is to give a characterization of exponential trichotomy for a cocycle over a semiflow in terms of exponential dichotomy of two associated cocycles over the same semi ow.

Hopf Bifurcation for Retarded Functional Differential Equations and for Semiflows in Banach Spaces

We indicate how a Hopf bifurcation theorem for retarded functional differential equations can be proved correctly, following an approach that is described, but not carried out properly in the literature. Then we state and prove a Hopf bifurcation theorem for general semiflows in Banach spaces, using different methods. The general statement includes the case of retarded functional differential equations.

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.

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.

On uniform exponential stability of linear skew-product semiflows in Banach spaces

On uniform exponential stability of linear skew-product semiflows in Banach spaces

Inertial manifolds for dissipative semiflows in banach spaces

A generalization to Banach spaces of a method of construction of Inertial Manifolds is provided. The generalized Spectral Gap Conditions are given and a careful atention is taken of the constants and parameters involved.Conditions known for a class of flows are recovered albeit less restrictive in our approach. The nesting of Inertial Manifolds and other related dynamical properties are also discussed

Approximation methods and the generalised Fuller index for semi-flows in Banach spaces

In [3] Fuller introduced an index (now called the Fuller index) in order to study periodic solutions of ordinary differential equations. The objective of this paper is to give a simple generalisation of the Fuller index which can be used to study periodic points of flows in Banach spaces. We do not claim any significant breakthrough but merely suggest that the simplistic approach, presented here, might prove useful for the study of non-linear differential equations. We show our results can be used to study functional differential equations.