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

摘要

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 yx1=y1 ,yx2=y 2, yxm- i=3m-1yxi =y3, 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 yx1=y1 ,yx2=y 2, yxk- i=3k-1yxi =y3, a < x1 < x2 < ··· < xk < b, and y1, y2, y3 ∈ R , are unique when they exist. It is then shown that solutions do indeed exist.