Multiple Solutions of Second-Order Damped Impulsive Differential Equations with Mixed Boundary Conditions

Abstract

We use variational methods to investigate the solutions of damped impulsive differential equations with mixed boundary conditions. The conditions for the multiplicity of solutions are established. The main results are also demonstrated with examples.

Periodic solutions to singular damped delay differential equations with impulses

“…For a second-order differential equation u ′′ = f (t, u, u ′ ), one usually considers impulses in the position u and the velocity u ′ . However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that results in jump discontinuities in velocity, but with no change in the position (Carter, 2000;Prado, 2005;Liu and Yan, 2014).…”

“…To study the solvability of impulsive differential equations, various methods have been used: fixed point theorems (Chen et al, 2007;Abdel-Rady et al, 2012), topological degree theory (Qian and Li, 2005;Saker and Alzabut, 2007), lower and upper solution method with monotone iterative technique (Liu, 1997;Chen and Sun, 2006) and variational method (Nieto and O'Regan, 2007;Tian et al, 2009;Chen and Li, 2010;Zhang and Li, 2010). The periodic boundary value problems (PBVP for short) associated to impulsive differential equations, which are involved in various fields of applied mathematics, have become an important field of investigation in recent years (Liu, 2009;Sun and Chen, 2009;Sun et al, 2013), many authors have used a variational method and critical point theory to study the existence and multiplicity of solutions for damped boundary value problems with or without singularity (Wu et al, 2008;Nieto, 2010;Liu and Yan, 2014;Li et al, 2015), for PBVP with delay see (Guo and Yu, 2005;Shu and Xu, 2006). But, this method has rarely been used for damped PBVP associated to delay differential equations.…”