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International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 263-273
http://dx.doi.org/10.1155/S0161171282000234

On the positive solutions of a higher order functional differential equation with a discontinuity

Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762, USA

Received 7 August 1981

Copyright © 1982 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

The n-th order nonlinear functional differential equation [r(t)x(nυ)(t)](υ)=f(t,x(g(t)))is considered; necessary and sufficient conditions are given for this equation to have: (i) a positive bounded solution x(t)B>0 as t; and (ii) all positive bounded solutions converging to 0 as t. Other results on the asymptotic behavior of solutions are also given. The conditions imposed are such that the equation with a discontinuity [r(t)x(nυ)(t)](υ)=q(t)xλ,λ>0is included as a special case.