A Scale Of Almost Periodic Functions Spaces

A scale of almost periodic functions space is proposed by C. Corduneanu in recent years, it is a special class of almost periodic function space, for which we have scattered a number of research articles, this article is to summarize and analyze them, according to own understanding of the space to make a survey.This paper consists of two parts, in the first part, we discuss properties of function in a scale of almost periodic functions spaces. First, we verify whether some classic and important properties in AP_ space, such as Bochner property and Bohr property, are established in AP_r space, Second, we investigate some exclusive properties in AP_r space. Compared with function in AP_ space, the expression of function in AP_r space is more specific and intuitive, so it certainly has some exclusive properties. According to the link between the coefficient of AP_r (1≤r≤2) and l rspace, we can use the conclusions in l rto solve problems in AP_r (1≤r≤2) space. In this article we discuss the coefficient of function in AP_r (1≤r≤2) space, the compactness of sequence, and mapping between them.In the second part, we study the solution of differential equation and its existence and uniqueness of the solutions in AP_r (1≤r≤2) space. First we give definitions of integral and differential and focus on the conditions of existence and uniqueness of solutions of differential equations. For equation =x '(t ) Ax (t ) + f (t ), we change the condition of det ( A ? iωI)≠0 to require that the matrix A has no eigenvalues with real part zero. For the linear system of differential equations, we can make use of the conclusion of the simple one-dimensional equation to get the conclusion of multi-dimensional equations. According to the specific expression of AP_r (1≤r≤2), we have condition that the matrix A has no eigenvalues with real part zero. For nonlinear system, we can use conclusions of linear system, then the conditions are that the matrix A has no eigenvalues with real part zero and operator satisfies Lipschitz condition.