arXiv subjects
Publications and source records attributed to Codruta Stoica.
The aim of this paper is to emphasize various concepts of dichotomies for evolution equations in Banach spaces, due to the important role they play in the approach of stable, instable and central manifolds. The asymptotic properties of the solutions of the evolution equations are studied by means of the asymptotic behaviors for skew-evolution semiflows.
The paper considers some concepts of nonuniform asymptotic stability for skew-evolution semiflows on Banach spaces. The obtained results clarify differences between the uniform and nonuniform cases. Some examples are included to illustrate the results.
The paper emphasizes asymptotic behaviors, as stability, instability, dichotomy and trichotomy for skew-evolution semiflows, defined by means of evolution semiflows and evolution cocycles and which can be considered generalizations for evolution operators and skew-product semiflows. The definition are given in continuous time, but the unified treatment for the characterization of the studied properties in the nonuniform case is given in discrete time. The property of trichotomy, introduced in finite dimension by S. Elaydi and O. Hajek in 1988 as a natural generalization for the dichotomy of linear time-varying differential systems, was studied by us in continuous time and from uniform point of view and in discrete time and from nonuniform point of view but for a particular case of one-parameter semiflows.
The aim of this paper is to give necessary and sufficient conditions for the uniform exponential trichotomy property of nonlinear evolution operators in Banach spaces. The obtained results are generalizations for infinite-dimensional case of some well-known results of Elaydi and Hajek on exponential trichotomy of differential systems.
The paper emphasizes some asymptotic behaviors for skew-evolution semiflows in Banach spaces. These are defined by means of evolution semiflows and evolution cocycles. Some characterizations which generalize classical results are also provided. The approach is from nonuniform point of view.
The paper emphasizes the properties of exponential dichotomy and exponential trichotomy for skew-evolution semiflows in Banach spaces, by means of evolution semiflows and evolution cocycles. The approach is from uniform point of view. Some characterizations which generalize classic results are also provided.
The paper emphasizes the property of stability for skew-evolution semiflows on Banach spaces, defined by means of evolution semiflows and evolution cocycles and which generalize the concept introduced by us in a previous paper. There are presented several general characterizations of this asymptotic property out of which can be deduced well known results of the stability theory. A unified treatment in the uniform and in the nonuniform setting is given. The main results are also formulated in discrete time.
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 classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows.