Eigenvalue problems of second order impulsive differential equations

Abstract

a b s t r a c tThis paper is concerned with an eigenvalue problem for second order differential equations with impulse. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved.

Eigenvalue problem of an impulsive differential equation governed by the one-dimensional p-Laplacian operator

“…On the other hand, many researchers have used a variational method to the existence and multiplicity of solutions with impulsive effect (see [12], [13], [14], [15], [16] and [17]). The spectrum of the equations that involve the one-dimensional p-Laplacian operator with the different boundary condition, has been studied by several authors, for literature we quote here some works [18], [19], [20] and [21].…”

“…< t i < t i+1 = 1, u 0 (t + k ) and u 0 (t − k ) represents the right limit also the left one of u 0 (t) at t k respectively. In [21], the authors characterized the first eigenvalue λ 1 related to the problem (1.1) − (1.3), by…”

“…Motivated by [21], we will establish some new properties for the eigenvalue problem (1.10)−(1.12). This paper is organized as follows.…”

Variational analysis to fourth-order impulsive differential equations

“…Recently, the existence and multiplicity of solutions for impulsive boundary value problems by using variational methods and critical point theory has been considered and here we cite the papers [1,2,10,14,15,16,17,18,19]. In this paper, motivated by the above facts and the recent paper [4], we consider the fourth-order boundary value problem with impulsive effects…”

Positive periodic solutions to impulsive delay differentialequations

“…Recently, variational methods and critical point theory have been successfully employed to investigate impulsive differential equations when the nonlinearity is regular; the existence and multiplicity of solutions for impulsive boundary value problems have been considered in [9,16,[19][20][21][22][23][24][25][26][27][28].…”