| Since the theory of almost periodic type functions have been proposed, many mathematical workers apply it to the other branches of mathematics, such as differential equations, integral equations, control theory and so on. In this paper, almost periodic type functions are applied to differential equations, the existence of asymptotically almost periodic solutions of first-order and second-order nonlinear differential equation is discussed, the conditions of asymptotically almost periodic for several types of first-order and second-order nonlinear differential equation are given. The main contents are given as follows:First, combining the dissipative type condition, existence of asymptotically almost periodic solutions for a class of nonlinear differential equations is discussed. By using of this kind of functional of the three functional properties specifically, combining with the definition of asymptotically almost periodic function, the existence of asymptotically almost periodic solutions for a class of nonlinear equations is proved.Second, by the use of contraction mapping fixed point theory, the relations of asymptotically almost periodic functions and asymptotically almost periodic sequence and the nature of the asymptotically almost periodic sequence, by constructing the corresponding differential equations asymptotically almost periodic sequence solution, first the existence of asymptotically almost periodic solutions for a class of second order differential equations is discussed, then the sufficient conditions of asymptotically almost periodic solutions for this kind of equation are given. And on this basis for further promotion, asymptotically almost periodic solutions of such equations corresponding to the second-order nonlinear differential equation are discussed, and the sufficient condition of asymptotically almost periodic solutions for this equation is obtained. The equations in this part of the study is more complicated, which contain the integral term and also piecewise constant argument.Some results obtained in this paper some are the result of the promotion, which make relevant conclusions apply more widely, while others are new have great value for solving practical problems. |
