Complementary Lidstone Interpolation and Boundary Value Problems
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Research Article Complementary Lidstone Interpolation and Boundary Value Problems Ravi P. Agarwal,1, 2 Sandra Pinelas,3 and Patricia J. Y. Wong4 1
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 3 Department of Mathematics, Azores University, R. M˜ae de Deus, 9500-321 Ponta Delgada, Portugal 4 School of ELectrical & Electronic Engineering, Nanyang Technological University, Singapore 639798 2
Correspondence should be addressed to Ravi P. Agarwal, [email protected] Received 21 August 2009; Revised 5 November 2009; Accepted 6 November 2009 Recommended by Donal O’Regan 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 2m1th order differential equation and the complementary Lidstone boundary conditions. Copyright q 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.
1. Introduction In our earlier work 1, 2 we have shown that the interpolating polynomial theory and the qualitative as well as quantitative study of boundary value problems such as existence and uniqueness of solutions, and convergence of various iterative methods are directly connected. In this paper we will extend this technique to the following complementary Lidstone boundary value problem involving an odd order differential equation −1m x2m1 t ft, xt,
t ∈ 0, 1, m ≥ 1,
1.1
and the boundary data at the odd order derivatives x0 α0 ,
x2i−1 0 αi ,
x2i−1 1 βi ,
1 ≤ i ≤ m.
1.2
2
Journal of Inequalities and Applications
Here x x, x , . . . , xq , 0 ≤ q ≤ 2m but fixed, and f : 0, 1 × Rq1 → R is continuous at least in the interior of the domain of interest. Problem 1.1, 1.2 complements Lidstone boundary value problem nomenclature comes from the expansion introduced by Lidstone 3 in 1929, and thoroughly characterized in terms of completely continuous functions in the works of Boas 4, Poritsky 5, Schoenberg 6–8, Whittaker 9, 10, Widder 11, 12, and others which consists of an even-order differential equation and the boundary data at the even-order derivatives −1m x2m t ft, xt, x2i 0
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