Almost Periodic Type Functions And Solutions Of Almost Periodic Type Differential Equations

The paper consists of two parts. One part concerns the properties of almostperiodic type functions, the other part concerns the solutions of almost periodic typedifferential equations.Differential equations with piecewise constant argument have the structure ofcontinuous dynamical systems within intervals of unit length. Continuity of a solu-tion at a point joining any two consecutive intervals implies recursion relations forthe values of the solution at such points. Therefore, they combine the properties ofdifferential equations and difference equations. Some people studied the existenceof almost periodic solutions and pseudo almost periodic solution of some differentialequations with piecewise constant argument.Sarason defined remotely almost periodic functions. To our knowledge, nobodyhas applied such functions to the theory of differential equations. In this paper, westudy the properties of remotely almost periodic functions, give the relation betweenremotely almost periodic functions and remotely almost periodic sequences. Using theremotely almost periodic sequence solutions of difference equations, we investigatethe existence and uniqueness of remotely almost periodic solutions to systems of linearand nonlinear differential equations with piecewise constant argument.Some people discussed the almost periodic type solutions of abstract differentialequations. In this paper, based on the method of studying almost periodic solutionsof ordinary differential equations, we study the existence of remotely almost periodicsolutions of abstract differential equations by using the form of solution of abstractdifferential equations and the structure of remotely almost periodic functions.Stepanov almost periodic functions and equi-Weyl almost periodic functionswere studied and some result was obtained, but Weyl almost periodic functions haveseldom been discussion. In this paper, similar as the definition of almost periodic func-tions, we define Weyl almost periodic functions by trigonometric polynomial, studysome properties of Weyl almost periodic functions and Fourier analysis.The theory of mean-square has applications in random process and stochasticdifferential equations. Some people study the almost periodicity in probability space. We consider the almost periodicity in theory of mean-square, propose the notion ofmean-square almost periodic random process, study some properties of mean-squarealmost periodic random process, and give the existence of mean-square almost peri-odic solutions of stochastic differential equations and Ito? stochastic differential equa-tions.