Prescribed asymptotic behavior of second-order impulsive differential equations via principal and nonprincipal solutions

Abstract

Finding solutions with prescribed asymptotic behavior is a classical problem for differential equations, which is also known as the asymptotic integration problem for differential equations. Very recent results have revealed that the problem is closely related to principal and nonprincipal solutions of a related homogeneous linear differential equation. Such solutions for second-order linear differential equations without impulse effects, first appeared in [W. Leighton, M. Morse, Trans. Amer. Math. Soc. 40 (1936), 252–286]. In the present work we first establish the concept of principal and nonprincipal solutions for second-order linear impulsive differential equations, and then use them to prove the existence of solutions for a class of second-order nonlinear impulsive differential equations, with prescribed asymptotic behavior at infinity in terms of a linear combination of these principal and nonprincipal solutions. Examples and numerical simulations are provided to illustrate the obtained results.