International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 263-273

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

John R. Graef, Paul W. Spikes, and Myron K. Grammatikopoulos

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

Received 7 August 1981

Copyright © 1982 John R. Graef et al. 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.


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.