Uniqueness implies uniqueness and existence for nonlocal boundary value problems for third order ordinary differential equations.

Abstract

For the third order ordinary differential equation, y‴=f(x,y,y′,y″), it is assumed that, for some m≥4, solutions of nonlocal boundary value problems satisfying [y(x_1)=y_1,\ y(x_2)=y_2,] [y(x_m)-\sum_{i=3}^{m-1} y(x_{i})=y_3,] a<x1<x2<⋯<xm<b, and y1,y2,y3∈R, are unique when they exist. It is proved that, for all 3≤k≤m, solutions of nonlocal boundary value problems satisfying [y(x_1)=y_1,\ y(x_2)=y_2,] [y(x_k)-\sum_{i=3}^{k-1} y(x_{i})=y_3,] a<x1<x2<⋯<xk<b, and y1,y2,y3∈R, are unique when they exist. It is then shown that solutions do indeed exist.