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

Eloe, Paul; Henderson, Johnny

doi:10.1090/proc/15115

Two-point boundary value problems for ordinary differential equations, uniqueness implies existence
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by Paul W. Eloe and Johnny Henderson PDF

Proc. Amer. Math. Soc. 148 (2020), 4377-4387 Request permission

Corrigendum: Proc. Amer. Math. Soc. 150 (2022), 3649-3654.

Abstract:

We consider a family of two-point $n-1 ,1$ boundary value problems for $n$th order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions that 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 shall replace that standard hypothesis with one in which we assume uniqueness of solutions of a large family of two-point boundary value problems. We then obtain readily verifiable conditions on the nonlinear term that in fact imply the uniqueness of solutions of the large family of two-point boundary value problems.

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