Two-point boundary value problems for ordinary differential equations, uniqueness implies existence

Abstract

We consider a family of three point n − 2, 1, 1 conjugate boundary value problems for nth order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions imply existence of solutions. A standard hypothesis that has proved effective in uniqueness implies existence type results is to assume uniqueness of solutions of a large family of n−point boundary value problems. Here, we replace that standard hypothesis with one in which we assume uniqueness of solutions of large families of two and three point boundary value problems. We then close the paper with verifiable conditions on the nonlinear term that in fact imply global uniqueness of solutions of the large family of three point boundary value problems.