arXiv subjects
Publications and source records attributed to Liviu Horia Popescu.
We introduce a special class of real semiflows, which is used to define a general type of evolution semigroups, associated to not necessarily exponentially bounded evolution families. Giving spectral characterizations of the corresponding generators, our results directly apply to a wide class of dichotomies, such as those with time-varying rate of change.
For nonuniform exponentially bounded evolution families defined on Banach spaces, we introduce a class of Banach function spaces, whose norms are completely determined by the nonuniform behaviour of the corresponding evolution family. We generalize a classical theorem of Datko on these spaces. In addition, we obtain new criteria for the existence of nonuniform stability.
For nonuniform exponentially bounded evolution families on the half-line we introduce a class of Banach function spaces on which we define nonuniform evolution semigroups. We completely characterize nonuniform exponential stability in terms of invertibility of the corresponding generators. We emphasize that in particular our results apply to all linear differential equations with bounded operator and finite Lyapunov exponent.
For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach sequence spaces. We connect the invertibility of this operator to the existence of a particular type of admissible exponents. For the bounded orbits, exponential stability results from a spectral property. Some adequate examples are presented to emphasize some significant qualitative differences between uniform and nonuniform behavior.