Asymptotic behaviors of nonlinear neutral impulsive delay differential equations with forced term

Abstract

In this paper, we study the asymptotic behavior of solutions of a class of nonlinear neutral impulsive delay differential equations with forced term of the form

$\left\{$ [x(t) + c(t)x(t − τ)]′ + p(t)f(x(t − δ)) = q(t),  t ≥ t₀, t ≠ t_k,
x(t_k) = b_kx(t_k⁻) + (1 − b_k) ∫^tk_tk−δ p(s + δ)f(x(s)) ds + (b_k − 1) ∫^∞_tk q(s) ds,  k ∈ Z₊.

Sufficient conditions are obtained for every solution of the equations that tends to a constant as t → ∞.