| We will be concerned with Lidstone boundary value problems for the 2mth order nonlinear differential equation y2m=f x,y,y',...,y2m-1 which arise in approximation theory and in the modeling of plate and beam deflections. In particular, the differentiability of solutions with respect to linear Lidstone boundary conditions is established. We then generalize these results by showing that such solutions can in fact be differentiated with respect to general nonlinear Lidstone boundary data via function analytic techniques. In each of these cases, the partial derivatives we obtain are shown to be solutions of a related linear boundary value problem. We conclude by providing sufficient conditions for the existence of at least three positive symmetric solutions by implementing a multiple fixed point theorem for operators defined on cones in ordered Banach spaces. |
