Almost Periodic Type Solutions Of Differential Equations With Piecewise Constant Argument

In this paper, almost periodic type functions are applied to several types of differential equations with piecewise constant argument. The meaning of research lies in: the periodic function can not constitute Banach space on R , however, with the supermum norm almost periodicity type functions constitute Banach space on R , so that the almost periodic is the expansion of the periodici- ty. The periodic function is a kind of theoretically ideal appearance, but the data may have error in the actual problem, the almost period phenomenon is a more universal phenomenon. Almost periodic type functions have been one of the most popular topics in studying of many qualitative theories such as ordinary differen- tial equations.This article focuses on the existence and uniqueness of the almost periodic type solutions for several classes of a first-order linear differential equations with piecewise constant argument, and the main contents are as follows:first, by using of exponential dichotomy of differential equations, and combining the relations between the asympototically almost periodic functions and asympototically almost periodic sequence, and when the coefficients and the non-homogeneous term satisfy some different conditions, the existence and uniqueness of the asympototically almost periodic solutions of a class of variable coefficient differential equation with piecewise constant argument are discussed.Second, the form of the piecewise constant variable of the first part is changed and the asympototically almost periodic solutions of another class of differential equations with piecewise constant argument are discussed.Finally, by using of the asympototically theory of almost periodic sequences and the fixed point theory, the existence of asympototically almost periodic solutions of a class of differential equations with piecewise constant argument and the initial conditions is discussed.The result in this paper is the promotion of the existing results, which make the relevant conclusions be applied more widely or give another proof method for the existing questions. It can also provide some methods and references for the research of other issues.