Boundary Value Problems
Volume 2006 (2006), Article ID 37524, 8 pages
doi:10.1155/BVP/2006/37524
A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems
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.
Abstract
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.