Theory Research On The Asymptotic Behaviors Of Several Kinds Evolution Type Operators

Firstly, This paper reviews the development history of asymptotic behaviors ofevolution type operators and the experts home and abroad status, and development inthis field of research. The second chapter is based on the concept and properties ofn-times integrated C-semigroups, exponential stability of its relevant abstract Cauchyproblem is discussed in the light of functional analysis methods and operator theoriesin Hilbert space, and the necessary and sufficient conditions concerning theexponential stability are given. The well-known results for exponential stability ofstrongly continuous semigroups are extended to the case of n-times integratedC-semigroups. In chapter3, we introduce the concept of weak exponentialexpansiveness for evolution families which is an extension of classical concept ofexponential expansiveness. Our main objective is to give some characterizations forweak exponential expansiveness properties of evolution families in Banach spaces,and variants for weak exponential expansiveness of some well-known results instability theory are obtained. In chapter4, we introduce the concept of weakexponential expansiveness for skew-evolution semiflows which is an extension ofclassical concept of exponential expansiveness. Our main objective is to give somecharacterizations for weak exponential expansiveness properties of skew-evolutionsemiflows in Banach spaces, and variants for weak exponential expansiveness ofsome well-known results in uniform exponential stability theory and exponentialinstability theory are obtained. As applications we obtain characterizations of theconcepts in terms of Lyapnov functions. In chapter5, we introduce the concept ofnonuniform exponential trichotomy for linear difference equations which is anextension of classical concept of uniform exponential trichotomy. Our main objectiveis to give some characterizations for nonuniform exponential trichotomy properties oflinear difference equations in Banach spaces, and generalizations for nonuniformexponential trichotomy of some results in exponential dichotomy theory are obtained.In chapter6, the conclusions and prospects of this paper are summarized.