Remotely Almost Periodic Functions And Solutions Of Differential Equations

In developing the theory of the almost periodic type functions, a main feature is that the scope is expanding. From Almost periodic function, Uniform almost periodic function, Asymptotically almost periodic function, Weakly almost periodic function to Pseudo almost periodic function proposed by Professor Zhang in 1992. Every extension consumedly enlarges the application of the almost periodic functional theory and solving to the solutions of differential equations. The existence and the uniqueness of almost periodic type solutions of integral equations and differential equations are interesting in the theory of mathematics and have broad prospects in practical applications. We have solved the existence and the uniqueness of almost periodic type solutions of ordinary equations. In this paper, in allusion to above-motioned situations, two aspects of concrete work are done to Vector-valued remotely almost periodic function and oscillate slowly function:1. The properties of Vector-valued remotely almost periodic function and oscillate slowly function are discussed.This paper is mainly based on forefathers investigation on the properties of almost periodic type functions and mainly makes use of the definitions of vector-valued remotely almost periodic function and oscillate slowly function. The natures of vector-valued remotely almost periodic function and oscillate slowly function is investigated, which will propel the development of the almost periodic type functions further.2. The vector-valued remotely almost periodic solutions of some differential equations and integral equations are investigated.I mainly have aid of the study of the almost periodic type solutions of differential equations and mainly make use of the definitions of vector-valued remotely almost periodic function. The vector-valued remotely almost periodic solutions of abstract evolution equations and integral equations are firstly studied, thus the rule of the type of almost periodic function is consummated further.