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

Abstract

The n-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 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(n−υ)(t)](υ)=q(t)x−λ, λ>0is included as a special case.

Keywords

• functional differential equations
• discontinuous right hand side
• positive solutions
• asymptotic behavior.