Some Generalization Of Almost Periodic Functions And The Solution Of Nonlinear Parabolic Equation

Since the Danish mathematician H.Bohr developed the theory of almost periodic functions during 1925-1926, the theory has been improved greatly by the hardworking of many mathematicians for decades. However there are still many problems unsolved.The theory of almost periodic functions was extended to abstract spaces. Pseudo almost periodic functions, as a generalization of almost periodic functions, has important applications in differential equations. However the theory of pseudo almost periodic functions with values in a metric space was not yet developed. In this paper, we first define the pseudo almost periodic functions in metric spaces, and find some properties. We also prove the unique decomposition theorem: the distance between vector-valued pseudo almost periodic functions in metric spaces and almost periodic functions is an ergodic perturbation.It is interesting to find the almost periodic solutions to differential equations, such as ordinary, partial as well as abstract differential equations and smooth dynamical systems. In this paper, we consider the existence and uniqueness of solutions to the nonlinear Cauchy problem. We prove the nonhomogeneous Cauchy problem has a unique bounded solution, then applying contraction fixed point theorem to prove the existence and uniqueness of the nonlinear Cauchy problem, and give the condition when the solution exists.We deal with asymptotically almost periodic functions and semigroups of operators on Fréchet spaces. We apply the obtained results to the abstract Cauchy problem. We prove the existence and uniqueness of asymptotically almost periodic solutions to the abstract Cauchy problem.