The existence of almost periodic solution: via coincidence degree theory

In the present paper, a new method is developed to study the existence of an almost periodic solution for the ordinary or functional differential equations. The approaches are based on topological degree and novel estimation techniques for the a priori bounds of unknown solutions for Lx = λNx. To investigate the existence of an almost periodic solution, a few good methods have been presented in the previous literature (such as using the Lyapunov function, averaging, exponential dichotomy, stability, separate conditions, and so on). But topological degree theory was never employed to study the almost periodic differential equations. Though Mawhin’s coincidence degree is employed to study the existence of periodic differential equations extensively, it cannot be applied to study the almost periodic systems immediately. Some essentially new and interesting lemmas should be proved before applying topological degree theory to almost periodic systems. To the best knowledge of the authors’, it is the first time that topological degree theory is employed to study the existence of almost periodic solution and this method can be seen as a good supplement to the known methods. Therefore, it will be of great significance to study the almost periodic systems by using this method. The approach followed in the paper could be further generalized to investigate the existence of almost periodic oscillatory in some other nonlinear dynamical systems. It is believed that it can be applied to image patterns, digital image processing, data processing, signal sparse decomposition and information technology, etc.