Asymptotically Almost Periodic Solutions For Two Classes Of Perturbed Systems

The existence problems on all kinds of solutions of differential equations areconcerned by the majority of mathematical works, the problem of the existence ofalmost periodic solution for differential equations was the most studied undoubtedly.In order to solve practical problems, only study of almost periodic solutions ofdifferential equations is not enough, then there are so many mathematical workershave studied asymptotically almost periodic solutions, pseudo almost periodicsolutions, almost automorphic solutions, asymptotically almost automorphicsolutions and pseudo almost automorphic solution of differential equations.By reading some of literatures, and basing on the study of almost periodicsolutions for two classes of perturbed systems of existing literatures, the existenceand uniqueness of asymptotically almost periodic solutions for above two perturbedsystems are discussed. The text is divided into four parts:The purpose and significance of the research is introduced in the first part,meanwhile, the origin of the subject and the research status and development trendin this direction in abroad are also introduced, the main content of this paper is givenat the end of this part.Some related theoretical knowledge of almost periodic type functions andalmost periodic type sequences is introduced in the second part, it can providetheoretical basis for the proof of the main conclusion.The third part is one of the main research work of this paper, the existence ofasymptotically almost periodic solutions for a class of perturbed systems withpiecewise constant argument is investigated. Asymptotically almost periodicsequence solutions of related difference equations are discussed using uniquedecomposition theorem of asymptotically almost periodic functions, the conditionswhich make related equations exist only bounded asymptotically almost periodicsolution are obtained. Then the existence and uniqueness of asymptotically almostperiodic solutions of the system are discussed by using the exponential dichotomyand contraction mapping principle.. The fourth part is another major research work of this paper, the existence anduniqueness of asymptotically almost periodic solutions for a kind of perturbedsystems with two parameters are studied, using contraction mapping principle andexponential dichotomy and roughness theory.