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

Abstract

Then-th order nonlinear functional differential equation[r(t)x(n−υ)(t)](υ)=f(t,x(g(t)))is considered; necessary and sufficient conditions are given for this equation to have: (i) a positive bounded solutionx(t)→B>0ast→∞; and (ii) all positive bounded solutions converging to0ast→∞. 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(n−υ)(t)](υ)=q(t)x−λ,   λ>0is included as a special case.