Approximation of solutions to history-valued neutral functional differential equations

K e y w o r d s N e u t r a l functional differential equation, Analytic semigroup, Mild solution, FaedoGalerkin approximation. 1. I N T R O D U C T I O N Let H be a separable Hilbert space and 0 < T, T < ec. For 0 < t < T, Ct := C([-~-, t]; H) be the Banach space of all continuous functions from l--T, t] into H endowed with the supremum norm, Ilullt := sup I lu(s) l l , u • Ct, (1.1) -r<_s<t where I].H is the norm in H. For u • CT and 0 < t < T, us • Co be the function defined by u,(e) = u(t + e), for e • [ -~,0] . We are interested in the following neutral functional differential equation in H, d d--t (u (t) + g (t, u ( t ) , ut)) + A u (t) = f (t, u ( t ) , u s ) , 0 < t < T < co, (1.2) (t) = x (t), t ~ [ -~ , 0], where A : D ( A ) C H --~ H is a closed, densely defined positive definite, self-adjoint linear operator and the nonlinear functions f and g are defined from [0, T] x H x Co into H and X E Co. For the earlier work on existence, uniqueness, regularity, and stability of various types of solutions of differential equations, functional differential equations, and neutral functional differential equations under different conditions, we refer to [1-6] and reference cited in these papers. The authors would like to thank National Board for Higher Mathematics for providing the financial support to carry out this work under its research project No. NBHM/2001/P~D-I]. The authors would like to thank the Professor D. Bahuguna and Professor R.P. Agarwal for his valuable suggestion. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by .AA/cS-'I~X doi: 10.1016/j.camwa.2005.07.013