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Complementary Lidstone interpolation and boundary value problems

Agarwal, R.P.; Pinelas, Sandra; Wong, P.J.Y.

Journal of Inequalities and Applications, (2009), 624-631

We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial
P2mt of degree 2m, which involves interpolating data at the odd-order derivatives. For P2mt
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 iterativemethods for the complementary Lidstone boundary value
problems which consist of a 2m1th order differential equation and the complementary Lidstone
boundary conditions.