Pseudo Almost Periodic Mild Solutions And Asymptotically Almost Automorphic Mild Solutions On Two Classes Of Differential Equations

The theories of almost periodic functions are introduced by the Danishmathematician H. Bohr during1925-1926years, after that, almost periodic typefunctions and almost automorphic type functions have been put forward because ofthe need of practical problems. Since almost periodic type functions and almostautomorphic functions are presented, they are widely used in all kinds of equations.The existence and uniqueness of the almost periodic type solutions and almostautomorphic type solutions of differential equations are important research contentsof differential equations.A class of neutral differential equations and a class of semilinear differentialequations are mainly studied in this paper. The existence and uniqueness of the twokinds of differential equations of pseudo almost periodic mild solutions andasymptotically almost automorphic mild solutions are respectively discussed.The contents of this paper are divided into four parts:In the first part, almost periodic type functions theories and almost automorphictype solutions theories’ historical development and their actual situation areintroduced. In addition, the source of the topic is also introduced.In the second part, the concepts and some properties of almost periodicfunctions, pseudo almost periodic functions, almost automorphic functions andasymptotically almost automorphic functions are introduced. In addition, theconcepts and some knowledge of operator semigroup and the evolution family arealso introduced.In the third part, the existence and uniqueness of pseudo almost periodic mildsolutions of a class of neutral differential equations are researched. It is proved thatcomposite functions are pseudo almost periodic functions by use of onlydecomposition properties of pseudo almost periodic functions. Then by using Banachfixed point theorem and some knowledge on operator semigroup, combining with the research of the almost periodic mild solutions of some literature on differentialequations, the existence and uniqueness of pseudo almost periodic mild solutions of aclass of neutral differential equations are discussed.In the fourth part, the existence and uniqueness of almost asymptoticallyautomorphic mild solutions of a class of semilinear differential equations are studiedby using the fixed point theory, the theories of the evolution family and the propertiesof asymptotically almost periodic functions.