Ciprian Preda

Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

We analyze the existence of (no past) exponential dichotomies for a well-posed autonomous differential equation (that generates a C0-semigroup {𝑇(𝑡)}𝑡≥0). The novelty of our approach consists in the fact that we do not assume the T(t)-invariance of the unstable manifolds. Roughly speaking, we prove that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the autonomous homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the aforementioned condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical l𝑝 spaces, sequence Orlicz spaces, etc.) and that from discrete-time conditions we get information about the continuous-time behavior of the solutions.

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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

{\bf (a,b)}

(a,b)

-DICHOTOMY FOR EVOLUTIONARY PROCESSES ON A HALF-LINE

-dichotomy in terms of the admissibility of the pair

Lyapunov operator inequalities for exponential stability of Banach space semigroups of operators

(L^1_a, L^{\infty}_b)

A new version of a theorem of Minh–Räbiger–Schnaubelt regarding nonautonomous evolution equations

. Also, generalizations of the results of [ 20 ], [ 23 ] are obtained.

The Lyapunov operator equation for the exponential dichotomy of one-parameter semigroups

Abstract We obtain continuous-time and discrete-time Lyapunov operator inequalities for the exponential stability of strongly continuous, one-parameter semigroups acting on Banach spaces. Thus we extend the classic result of Datko (1970) [2] from Hilbert spaces to Banach spaces.

On the connection between the exponential stability of C0-semigroups and the admissibility of a certain Sobolev space

We develop a new version of a known theorem obtained by Van Minh, Räbiger, Schnaubelt in [N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equ. Oper. Theory 32 (1998) pp. 332–353]. We rely completely on the classical ‘test functions’ method designed by Perron in 1930. The advantage of such a version is that is more readable since the classical method of Perron have been known for decades and that we do not involve a sophisticated mathematical machinery. Our approach is in contrast with the general philosophy of ‘autonomization’ the nonautonomous system, since we do not require to attach the evolution semigroup. Also we point out a discrete-time version of our approach extending some known results given by Li and Henry.

Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line

Abstract We point out a Lyapunov-type operator equation that shows the existence of an exponential dichotomy for an arbitrary C 0 -semigroup T = { T ( t ) } t ≥ 0 generated by a (possible) unbounded A . Thus, we extend some known results given by Chicone and Latushkin, Daletskii and Krein, and Datko.

On the uniform exponential stability of linear skew-product semiflows ∗

Abstract Let T be a strongly continuous semigroup on a Banach space X and A its infinitesimal generator. We will prove that T is exponentially stable, if and only if, there exist p∈[1,∞) such that the space W p,1 0 ( R + ,X) is admissible to the system Σ(A,I,I), defined below (i.e for all f belonging to the Sobolev space W p,1 0 ( R + ,X), the convolution T ∗f lies in W p,1 0 ( R + ,X) .