| Almost automorphic function is an important generalization of almost periodic function,which was proposed by S.Bochner in the study of differential geometry.Almost automorphic function is a wider class of functions than almost periodic function,it has many properties similar to periodic functions and almost periodic functions,and has many important applications in the fields of physics,biology,statistics,etc.Therefore,the research on the property and application of almost automorphic function is of great significance.Also,impulsive effects and stochastic effects often appear in the real world,the research on the solutions of impulsive evolution equations and impulsive stochastic evolution equations has a wider meaning.Therefore,this paper mainly constructs several types of piecewise almost automorphic function spaces,gives their definitions and related properties,and applies them to impulsive evolution equations,and discusses the existence and stability of piecewise almost automorphic type solutions for impulsive evolution equations and impulsive stochastic evolution equations.Firstly,the almost automorphic function is introduced from the continuous function space to the piecewise continuous function space,the concept and properties of the piecewise almost automorphic function are introduced,and applied to the impulsive evolution equation.Using the theory of semigroup of operators and the contraction mapping theorem,the existence and uniqueness of the piecewise almost automorphic mild solution for a class of impulsive evolution equations is discussed,based on this analysis the stability of such solutions.Secondly,by adding the traversal perturbation on the basis of the piecewise almost automorphic function,the piecewise pseudo almost automorphic function is defined,and its compound properties and unique decomposition properties are given.Then,using the theory of semigroup of operators and the contraction mapping theorem,the existence of piecewise pseudo almost automorphic mild solutions for a class of abstract impulsive evolution equations is studied.In addition,by the generalized Gronwall-Bellman inequality,sufficient conditions for their exponential stability are obtained.Finally,the piecewise almost automorphic function is introduced into a stochastic process,the concept and properties of the square-mean piecewise almost auromorphic function are introduced,and applied to impulsive stochastic evolution equations.Using the theory of semigroup of operators and the contraction mapping theorem,the existence of the squaremean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is studied.In addition,by the generalized Gronwall-Bellman inequality,the exponential stability of the square-mean piecewise almost automorphic mild solution for nonlinear impulsive stochastic evolution equation is obtained. |
