An existence theorem for boundary value problems of nonlinear ordinary differential equations

Klaasen, Gene A.

doi:10.1090/S0002-9939-1976-0466711-3

An existence theorem for boundary value problems of nonlinear ordinary differential equations
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Proc. Amer. Math. Soc. 61 (1976), 81-84 Request permission

Abstract:

Let $f$ be continuous on $(a,b) \times {R^n}$ and suppose solutions of initial value problems for ${y^{(n)}} = f(t,y, \ldots ,{y^{(n - 1)}})$ exist on $(a,b)$. Relaxing the assumption that solutions of initial value problems are unique, global existence of solutions of the boundary value problem \[ {y^{(n)}} = f(t,y, \ldots ,{y^{(n - 1)}}),y({t_i}) = {\alpha _i}\quad {\text {for }}1 \leqslant i \leqslant n,\] is established assuming uniqueness of solutions of these problems and a compactness property of solutions of the differential equation.

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