Boundary Value Problems

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

Boundary Value Problems20062006:37524

https://doi.org/10.1155/BVP/2006/37524

©  Denche and Djezzar 2006

Received: 14 October 2004

Accepted: 9 August 2005

Published: 13 February 2006

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.

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Authors’ Affiliations

(1)
Laboratoire Equations Differentielles, Département de Mathématiques, Faculté des Sciences, Université Mentouri Constantine

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Copyright

© Denche and Djezzar 2006

This article is published under license to BioMed Central Ltd. 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.