Nonlinear singular multipoint boundary value problems

Let Λ be a linear differential operator on (a, b) of order 2m with leading coefficient possessing an integrable reciprocal. Global C^m‐1(a, b) solutions are shown to exist for certain nonlinear multipoint boundary value problems of the form A u = F(u, …, u^m‐1, where linear combinations of derivatives through order m – 1 are specified at points interior to (a, b). The associated linear problem is completely solved through the construction of an appropriate Green's function which determines a compact operator G = A⁻¹. The linear eigenvalue problem is examined and a complete orthonormal system exhibited in L²(a, b). Solutions of the nonlinear problem are obtained by exhibiting fixed points for the operator GF.