Abstract
We study the existence of distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory.
Highlights
Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time
In the last few years, some researchers have used variational methods to study the existence of solutions for boundary value problems 10–16, especially, in 14–16, the authors have studied the existence of infinitely many solutions by using variational methods
As far as we know, few researchers have studied the existence of n distinct pairs of nontrivial solutions for impulsive boundary value problems by using variational methods
Summary
Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Such processes are naturally seen in control theory 1, 2 , population dynamics 3 , and medicine 4, 5. As far as we know, few researchers have studied the existence of n distinct pairs of nontrivial solutions for impulsive boundary value problems by using variational methods. Motivated by the above facts, in this paper, our aim is to study the existence of n distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order impulsive differential equations u t λh t, u t 0, t / tj , a.e. t ∈ 0, T ,. −Δu tj Ij u tj , j 1, 2, . . . , p, 1.1 u 0 u T 0, where 0 t0 < t1 < · · · < tp < tp 1 T , λ > 0, h ∈ C 0, T × R, R , Ij ∈ C R, R , j 1, 2, . . . , p, Δu tj u tj − u t−j , u tj and u t−j denote the right and the left limits, respectively, of u tj at t tj , j 1, 2, . . . , p
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