Existence and asymptotic behavior of solutions of nonlinear neutral differential equations

The nonlinear neutral differential equation (1) d n d t 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 [ t 0 , ∞ ) ; τ ∈ C [ t 0 , ∞ ) is strictly increasing, lim t → ∞ τ ( t ) = ∞ and τ ( t ) < t for t ≥ t 0 ; g ∈ C [ t 0 , ∞ ) and lim t → ∞ g ( t ) = ∞ ; f ∈ C ( [ t 0 , ∞ ) × R ) ; q ∈ C [ t 0 , ∞ ) . 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 d n d t n [ ω ( t ) + h ( t ) ω ( τ ( t ) ) ] = q ( t ) . Several known results in the literature can be obtained from our results.