Bounded solutions of delay nonlinear evolutionary equations

We consider the initial value problem {dudt+Au(t)=f(u(t),u(t−w)),t≥0,u(t)=φ(t),−w≤t≤0 in a Banach space E with the positive operator A. Theorem on the existence and uniqueness of a bounded solution of this problem is established for a nonlinear evolutionary equation with time delay. The application of the main theorem for four different nonlinear partial differential equations with time delay is shown. The first and second order of accuracy difference schemes for the solution of one dimensional nonlinear parabolic equation with time delay are presented. Numerical results are provided.