Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 14, pp. 1-9.

Title: Boundary-value problems for ordinary differential equations with
       matrix coefficients containing a spectral parameter

Authors: Mohamed Denche (Univ. Mentouri, Constantine, Algeria)
         Amara Guerfi   (Univ. of Ouargla, Algeria)

Abstract:
 In the present work, we study a multi-point boundary-value problem
 for an ordinary differential equation with matrix coefficients
 containing a spectral parameter in the boundary conditions.
 Assuming some regularity conditions, we show that the characteristic
 determinant has an infinite number of zeros, and specify their
 asymptotic behavior. Using the asymptotic behavior of Green matrix
 on contours expending at infinity, we establish the series expansion
 formula of sufficiently smooth functions in terms of residuals
 solutions to the given problem. This formula actually gives the
 completeness of root functions as well as the possibility of
 calculating the coefficients of the series.
 
Submitted August 15, 2006. Published January 8, 2007.
Math Subject Classifications: 34L10, 34E05, 47E05.
Key Words: Characteristic determinant; expansion formula;
           Green matrix; regularity conditions.