Almost Periodic Type Mild Solutions Of Some Evolution Equations

It’s well known that evolution equations have many important applications in biolo-gy, chemistry, physics, engineering, etc. Impulsive evolution equations describe process-es which are subjected to abrupt changes. These phenomena take place instantaneously,whose duration is negligible in comparison with the duration of an entire evolution. Im-pulsive stochastic evolution equations arise from impulsive processes which experiencethe stochastic efects. Impulsive evolution equations combine the properties of diferen-tial equations and diference equations, which more accurately and further reflect the lawof development of things. Therefore, evolution equations have attracted much attentionand considerable interest in the existence and stability of mild solutions for evolutionequations.The notion of almost periodic functions was introduced in1920s by the Danish math-ematician H. Bohr. Because almost periodic functions not only have many properties asperiodic functions but also have some diferent characteristics, which describe the realphenomena more appropriately in a broader sense. So, almost periodic functions haveelicited a great deal of attention from many researchers as soon as they were introduced.The same thing happens to the study of the generalization of almost periodic functionsand their applications in equations, while the latter embodies the study of the existence,uniqueness and stability of almost periodic type mild solutions of equations.For these reasons, more and more people are concerned about the almost periodictype mild solutions for evolution equations, especially for impulsive evolution equationsand impulsive stochastic evolution equations. The main work in this thesis is as follows:Firstly, composition theorems of piecewise almost periodic functions are providedfor applications in impulsive evolution equations, the existence and stability of piecewisealmost periodic mild solutions to impulsive evolution equations is investigated. To ourknowledge, the existence of piecewise almost periodic mild solutions to impulsive evo-lution equations is mostly discussed by virtue of Contraction theorem, while a Lipschitzcontinuity condition of the perturbation function is necessary. In this work, we intro-duce an equivalent definition of a piecewise continuous function set and investigate theexistence of piecewise almost periodic mild solutions by Schauder’s fixed point theorem,which overcomes the limitation of the Lipschitz continuity condition of the perturbation function. In addition, the stability of piecewise almost periodic mild solutions is discussedby the generalized Gronwall-Bellman lemma.Then, the notion of pseudo almost periodic functions is introduced in the piecewisecontinuous function space by defining piecewise pseudo almost periodic functions. Thenthe equivalent definition and properties such as, translation invariance, unique decompo-sition of piecewise pseudo almost periodic functions are studied in detail. Besides, theexistence and stability of piecewise pseudo almost periodic mild solutions to impulsiveevolution equations is also investigated due to their importance in application. In a word,piecewise pseudo almost periodic functions are generalizations of piecewise almost peri-odic functions and pseudo almost periodic functions, which describe the real phenomenamore appropriately in a broader sense.Finally, the composition theorems of piecewise square-mean almost periodic func-tions are presented and the existence and stability of piecewise square-mean almost peri-odic mild solutions to impulsive stochastic evolution equations is investigated by meansof Schauder fixed point theorem and Contraction mapping theorem.In this work, the properties of almost periodic type functions are investigated andapplied to impulsive evolution equations and impulsive stochastic evolution equations,the existence and stability of almost periodic type mild solutions to impulsive evolutionequations and impulsive stochastic evolutions are studied. Therefore, the work enrichesalmost periodic functions theory and extends the applications of almost periodic functionstheory.