Existence and asymptotic behavior of solutions of nonlinear neutral differential equations

The nonlinear neutral differential equation dⁿ/dt ⁿ[x(t)+h(t)x(τ(t))]+f(t,x(g(t)))=q(t), is considered under the following conditions: n∈N; h∈C[t0,∞); τ∈C[t0,∞) is strictly increasing, limt→∞τ(t)=∞ and τ(t)<t for t>t₀; g∈C[t₀,∞) and lim _t→∞g(t)=∞; f∈C([t₀,∞)×R) ; q∈C[t₀,∞). It is shown that if f is small enough in some sense, Eq. (1) has a solution x(t) which behaves like the solution of the unperturbed equation dⁿ/dtⁿ[ω(t)+h(t) ω(τ(t))]=q(t). Several known results in the literature can be obtained from our results.