Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 624631, 30 pages
doi:10.1155/2009/624631
Research Article

Complementary Lidstone Interpolation and Boundary Value Problems

Ravi P. Agarwal,1,2 Sandra Pinelas,3 and Patricia J. Y. Wong4

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
2Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3Department of Mathematics, Azores University, R. Mãe de Deus, 9500-321 Ponta Delgada, Portugal
4School of ELectrical & Electronic Engineering, Nanyang Technological University, 639798, Singapore

Received 21 August 2009; Revised 5 November 2009; Accepted 6 November 2009

Academic Editor: Donal O'Regan

Copyright © 2009 Ravi P. Agarwal et al. 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 shall introduce and construct explicitly the complementary Lidstone interpolating polynomial P2m(t) of degree 2m, which involves interpolating data at the odd-order derivatives. For P2m(t) we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a (2m+1)th order differential equation and the complementary Lidstone boundary conditions.