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International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 267-274
http://dx.doi.org/10.1155/S0161171288000328

Optimality and existence for Lipschitz equations

Department of Algebra, Combinatorics & Analysis, Auburn University, Auburn, Alabama 36849, USA

Received 5 January 1987; Revised 28 July 1987

Copyright © 1988 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Solutions of certain boundary value problems are shown to exist for the nth order differential equation y(n)=f(t,y,y,,y(n1)), where f is continuous on a slab (a,b)×Rn and f satisfies a Lipschitz condition on the slab. Optimal length subintervals of (a,b) are determined, in terms of the Lipschitz coefficients, on which there exist unique solutions.