Eventually positive and bounded solutions of even-order nonlinear neutral differential equations

Consider the even-order nonlinear neutral differential equation [a(t)x(t)−∑i=1mbi(t)x(t−τi)](n)−∑j=1lpj(t)fj(t,x(t−σj))=0,t≥0, and the associated differential inequality [a(t)x(t)−∑i=1mbi(t)x(t−τi)](n)−∑j=1lpj(t)fj(t,x(t−σj))≥0,t≥0. Using Lebesgue’s dominated convergence theorem, a necessary and sufficient condition for the existence of eventually positive and bounded solutions is obtained.