Boundary Value Problems
Volume 2006 (2006), Article ID 37524, 8 pages

A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems

M. Denche and S. Djezzar

Laboratoire Equations Differentielles, Département de Mathématiques, Faculté des Sciences, Université Mentouri Constantine, Constantine 25000, Algeria

Received 14 October 2004; Accepted 9 August 2005

Copyright © 2006 M. Denche and S. Djezzar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.