Two-point boundary conditions linear in a parameter
Hyman J. Zimmerberg · Pacific Journal of Mathematics · 1962
In considering various classes of two-point boundary problems, whose boundary conditions involve the characteristic parameter linearly, both Bobonis [4] and the author [7] have imposed a condition on the coefficients of the boundary conditions in addition to requiring that the boundary conditions be linearly independent.If the boundary conditions are written in vector form (1) s[y; λj = (M o + \M x yu(a) + (N o + XNJyψ) = 0 , where M o , M u N o , N λ are each n x n constant matrices whose elements may be complex-valued, and y(a), y(b) denote the end-values of the ^-dimensional vector y(x), a <Ξ x fg b, and the n x 2n matrix \\M 0 + XM X N o + \Nx\\ has rank n for every complex value of λ, the imposed assumption is CONDITION (A).There exist n x n constant matrices M 2 , N 2 , P 2 , Q. and n x n matrices P(λ), Q(X) such that for all complex values of X the 2n x 2n matrices