Multipoint boundary value problems for ordinary differential systems (Q1338304)

The authors consider the ordinary differential equation (1) \(du/dx= f(x,u)\) together with a multipoint boundary value condition of the form (2) \(\sum^ k_{j=1} M_ j u(x_ j)= r\). Here \(r\in \mathbb{R}^ n\), \((x,u)\in (a,b)\times \mathbb{R}^ n\); \(\{x_ j\}\in (a,b)\) is a strictly increasing sequence; \(M_ j\) are constant \(n\times n\) matrices. Assuming some uniqueness conditions, the authors prove the differentiability of a solution \(u(x)= u(x,x_ 1,\dots,x_ k,r)\) of (1), (2) with respect to \(x_ j\) and \(r\). The corresponding variational boundary value problem along \(u(x)\) is also derived.