The Study On The Theory Of Almost Periodic Time Scales And Some Of Its Applications

In this thesis, we propose some new concepts of almost periodic time scales, almost periodic functions on time scales, almost automorphic functions on time scales, and give some basic properties of these new types of almost periodic time scales and almost periodic functions on time scales as well as almost automorphic functions on time scales. We prove a result ensuring the existence of almost automorphic solutions or almost periodic solutions for almost periodic or almost automorphic linear nonhomogencous dynamic equations on time scales, assuming that the associated homogeneous equations admit an exponential dichotomy. As applications of our obtained results, (ⅰ) we establish the existence and global exponential stability of almost automorphic solutions to a class of shunting inhibitory cellular neural networks with time-vary ing delays on time scales. Our results about the shunting inhibitory cellular neural network with time-varying delays on time scales are new even for both cases of differential equations (the time scale T=R) and difference equations (the time scale T=Z). (ⅱ) we obtain some sufficient conditions for the existence and exponential stability of positive almost periodic solutions for a class of Nicholson’s blowflies models on time scales.Our results are new even for the cases of T=R and T=Z and improve some known results. Our results show that under a simple condition the continuous-time Nicholson’s blowflies models and their discrete-time analogue have the same dynamical behaviors. We also present an illustrative example to show the effectiveness of obtained results. (ⅲ) we establish the existence and global exponential stability of almost periodic solutions for this class of neutral type competitive neural networks with mixed time-varying delays and leakage delays on time scales. The obtained results about this class of neutral type competitive neural networks are completely new. We also give an example to show the effectiveness of the obtained results.