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

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

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.