Gorodnii M. The properties of solutions for difference and differential equations and its stochastic analogue in Banach space

We investigate the problem of the existence and uniqueness of bounded, periodic or lp- solutions for some classes of difference equations and the problem of the existence and uniqueness of bounded in the mean of order p or stationary solutions for its stochastic analogue. We prove theorems on the approximation of bounded solutions for such equations by solutions of corresponding boundary problems and a theorem on the approximation of bounded solutions of a linear differential equation with the sectorial operator coefficient by bounded solutions of corresponding difference equations. We give sufficient conditions for the existence of unique bounded solution for the differential equation of second order with the small positive parameter є at the second derivative and the sectorial operator coefficient. We prove uniform converge in R of these solutions to the unique bounded solution of the corresponding differential equation of the first order as є->0 . We also give sufficient conditions for continuous differentiability of almost all trajectories of bounded in the mean of order p solution for the differential-operator equations, wich are perturbed by the stochastic processes.