Some Results Of Almost Periodic Type Sequences

Almost periodic type sequences are closely connected with almost periodic type functions and have important applications in discussing the solutions of differential and difference equations. Difference equations arise as variational equations along the orbits of discrete dynamical systems and play a key role in various fields, such as numerical methods of differential equations, finite elements techniques, control theory, and computer science. The existing of some kinds of solutions to these equations and their applications to discrete dynamical systems and differential equations with piecewise constant arguments have been recently studied by several mathematical authors.In this thesis, some results concerning almost periodic type sequences are obtained, and the existence of asymptotically almost periodic sequence solutions to a class of difference equations is discussed. The present paper is organized as follows:Firstly, we investigate some properties of almost periodic type sequence, and show that the space of almost periodic sequence, asymptotically almost periodic sequence and pseudo almost periodic sequence are Banach spaces. We generalize the composition results of almost periodic and give the composition results of asymptotically almost periodic sequence. Then, considering ergodic zero set, we give the definition of ergodic zero integer set, and investigate the composition of pseudo almost periodic sequence using the definition.Secondly, under the assumption of exponential trichotomy, we discuss the existence of asymptotically almost periodic sequence solutions of some difference equations. Using contraction mapping principle, we discuss the existence of asymptotically almost periodic sequence solutions for the corresponding nonlinear difference equations.In conclusion, some of the results in this paper improve and generalize the well-known results, others are totally new.