On the solvability of the multipoint boundary value problem of calculating the loads in a continuous casting machine

The solvability of a multipoint nonlinear boundary value problem arising in a simulated calculation of the loads in a continuous casting machine is investigated. A solution of the boundary value problem is defined by a cubic spline. To find the solution, we deduce a system of nonlinear algebraic equations and prove that the set of solutions of this system is bounded (by an a priori estimate). We establish the existence of a solution by applying vector field techniques.