Applications of Variational Methods to Boundary-Value Problem for Impulsive Differential Equations

Abstract

In this paper, we investigate the existence of positive solutions to a second-order Sturm–Liouville boundary-value problem with impulsive effects. The ideas involve differential inequalities and variational methods.

On variational impulsive boundary value problems

“…Assume that ∈ H 1 0 (0 π) satisfies (13). Let us take any interval ( +1 ) ⊂ (0 π) and a test function ∈ H 1 0 ( +1 ).…”

“…On the other hand, in [13] the variational framework for the Sturm-Liouville boundary value problem is developed in the case of the second order impulsive ordinary differential equation of a -Laplacian type independently from [10]. The framework from [13], in which the impulsive variational problems can be considered, is different from the approach of [10], especially in the definition of the type of classical solution.…”

“…The framework from [13], in which the impulsive variational problems can be considered, is different from the approach of [10], especially in the definition of the type of classical solution. We shall follow some ideas from [10].…”

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…”

Existence of three solutions for impulsive multi-point boundary value problems

“…There are some common techniques to approach these problems: Fixed point theorems [8,9,31], the method of upper and lower solutions [7], and topological degree theory [37]. In the last few years, variational methods and critical point theory have been used to determine the existence of solutions for impulsive differential equations under certain boundary conditions, see [1,21,43,44,46,48] and the references therein. We note that the difficulties dealing with such problems are that their states are discontinuous.…”