© Hindawi Publishing Corp. METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR INTEGRODIFFERENTIAL EQUATIONS

We consider a class of quasilinear integrodifferential equations in a reflexive Banach space. We apply the method of semidiscretization in time to establish the existence, uniqueness, and continuous dependence on the initial data of strong solutions. 2000 Mathematics Subject Classification: 34G20, 47D06, 35L40, 35B30, 47H06. 1. Introduction. Let X and Y be two real reflexive Banach spaces such that the em-bedding Y ↩ X is dense and continuous. We consider the following quasilinear evolu-tion equation: u′(t)+A(u(t))u(t) = f (t,u(t),G(u)(t)), 0< t ≤ T