Study On The Polynomial Asymptotic Behaviors Of Several Kinds Evolution Type Operators

Firstly, This paper reviews the development of asymptotic behaviors of evolution type operators and development in this field of research. In the second chapter, we study four concepts of polynomial stability for difference equations in Banach space. Characterizations of these concepts are given and the illustrative examples clarifies the relations between these concepts. Based on the extension of techniques for exponential stability to the case of polynomial stability ,we discuss discrete characterizations of polynomial stability. The obtained results are generalizations of well-known theorems about the exponential stability. The aim of the 3 chapter is to give the concepts of weak polynomial expensiveness for skew-evolution semiflows in Banach spaces ,we discussed the relations of them and the integrability conditions and get some necessary and sufficient conditions for the weak polynomial expensiveness. As applications we obtain characterizations of the concepts in terms of Lyapunov functions. In chapter 4,We give necessary and sufficient conditions for uniform polynomial dichotomy of evolution families in terms of the admissibility of the pair (Lp(R,X),Lq(R,X)) .we discussed the relations of uniform polynomial dichotomy and the admissibility .At the last chapter ,the conclusions and prospects of this paper are summarized.