A compactness condition for solutions of ordinary differential equations

Jackson, L. K.

doi:10.1090/S0002-9939-1976-0404743-1

A compactness condition for solutions of ordinary differential equations
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Proc. Amer. Math. Soc. 57 (1976), 89-92 Request permission

Abstract:

It is proven that a sequence $\{ {y_k}(x)\}$ of solutions of ${y^{(n)}} = f(x,y,y’, \ldots ,{y^{(n - 1)}})$ with $\{ {y_k}(x)\}$ uniformly bounded on a compact interval $[c,d]$ has a bounded total variation sequence $\{ V_c^d({y_k})\}$ provided solutions of the differential equation extend and $n$-point boundary value problems have at most one solution.

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Continuous dependence for $n$-point boundary value problems

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