S-asymptotically ω-periodic Solutions Of Two Classes Of Fractional Differential Equations

Fractional calculus is the promotion of integral calculus, and it has stronger ob-jects Richard background than integral. Because of fractional order derivative memoryproperties, we can apply it more e?ciently in physical memory and genetic properties,and it can better simulate natural physical process and power system process. Thusfractional calculus is used wider and wider in control theory, biological engineering,electrochemical processes, semiconductor physics, mechanical engineering, condensedmatter physics.In this paper, we mainly study two types of problems involving in fractional orderdi?erential equations of S-asymptoticallyω-periodic solutions. The first kind is semi-linear the fractional di?erential equation, this paper mainly uses Laplace transform andsemigroup operator principle to obtain reasonable mild solutions of given equations,and then, this paper applies the contraction mapping principle to verify the system hasthe only period solution to S-asymptoticallyω-periodic. Under su?cient conditions, ithas di?erent points comparing with the first kind of reasoning proof. Specific points isdiscussed in the chapter 4.This paper also introduces the developmental background of fractional calculusand some basic definition and theorem about fractional calculus at start. In chaptertwo, special kinds fractional order di?erential equations of S-asymptoticallyω-periodicsolutions are discussed, Mittag-Le?er function is applied to express mild solution tothe system, and this solution is the one to S-asymptoticallyω-periodic is proved.