A New Approach on Datko–Zabczyk Method for Nonuniform Exponential Stability

We provide a sequence of projections on the linear space of all sequences and connect the existence of nonuniform exponential stability to the restrictions of these projections on a class of Banach sequence spaces defined by a discrete dynamics. As a consequence, we obtain a Datko–Zabczyk type characterization of nonuniform exponential stability. We develop our analysis without any assumption on the invertibility of the dynamics, thus our results are applicable to a large class of difference equations.

A NOTE ON SOME THEOREMS OF R. DATKO

The asymptotic behavior of the evolution families is a widely interesting topic in mathematics over time. In 1930, O. Perron was the first one who established the connection between the asymptotic behavior of the solution of the homogenous differential equation and the associated non-homogeneous equation, in finite dimensional spaces. Further, the result was extended for infinite dimensional spaces. The case of dynamical systems described by evolution processes was studied by C. Chicone and Y. Latushkin. One of the most remarkable results in the theory of stability of dynamical systems has been obtained by R. Datko in 1970 for the particular case of C0-semigroups. Practically, R. Datko defines a characterization for uniform exponential stability of the C0-semigroups. Later, it was proved that a similar characterization is also valid for two-parameter evolution families.In this paper we obtain different versions of a well-known theorem of R. Datko for uniform and nonuniform exponential bounded evolution families. More precisely, we obtain theorems that characterize the nonuniform and uniform exponential stability of evolution families with uniform and nonuniform exponential growth. We show that, if we choose K dependent of t0 in the form of Datko's theorem used by C. Stoica and M. Megan, we obtain a result of nonuniform exponential stability, which is no longer possible in the original form of Datko's theorem.In conclusion, we generalize the results initially obtained by Datko (1972) and Preda and Megan (1985), by presenting some sufficient conditions for the nonuniform exponential stability of evolution families with nonuniform exponential growth.

Nonuniform exponential stability for evolution families on the real line

The purpose of the present paper is to investigate the problem of nonuniform exponential stability of evolution families on the real line using the input-output technique known in the literature as the Perron method for the study of exponential stability. In this manuscript we describe an evolution family on the real line and we present sufficient conditions for the nonuniform exponential stability of an evolution family on the real line that does not have exponential growth.

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.