Remotely Almost Periodic Function In Differential Equations

This paper is concerned with the remotely almost periodic solutions of some differential equations by the fundamental theory and quality of the remotely almost periodic functions and Banach contraction mapping principle.The introduction introduces the development and current situation of the almost periodic theory, and the background, and the newest results.First we introduce the fundamental theory and quality of almost periodic and remotely almost periodic functions. Then we do some work about remotely almost periodic solutions of linear and quasi-linear ordinary differential equations.In Chapter 2 we discuss two existences of the remotely almost periodic solutions. We find the existence and uniqueness in region of the equation(?), then find the forced pendulum equation have a remotelyalmost periodic solution respect in(?)and(?)In Chapter 3 we study the remotely almost periodic problems of the forced pendulum equation. We study the remotely almost periodic solutions of the forced pendulumequation (y|¨) + c(?) + a sin y = p(t) by studying the existence and uniqueness of theremotely almost periodic solutions of Duffing equation(y|¨)+ c(?) -λy = p(t), though the quality of the remotely almost periodic functions andBanach contraction mapping principle, finding the existence and the uniqueness under some condition.