Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1⩽k⩽n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1⩽k⩽n−2. These relationships are investigated as well.