Asymptotic integration of second-order nonlinear delay differential equations

Abstract

Abstract We study the asymptotic integration problem for second-order nonlinear delay differential equations of the form ( p ( t ) x ′ ( t ) ) ′ + q ( t ) x ( t ) = f ( t , x ( g ( t ) ) ) . It is shown that if u and v are principal and nonprincipal solutions of equation ( p ( t ) x ′ ) ′ + q ( t ) x = 0 , then there are solutions x 1 ( t ) and x 2 ( t ) of the above nonlinear equation such that x 1 ( t ) = a u ( t ) + o ( u ( t ) ) , t → ∞ and x 2 ( t ) = b v ( t ) + o ( v ( t ) ) , t → ∞ .