Abstract
In this paper, we study the existence of periodic solutions of second order impulsive differential equations at resonance with impulsive effects. We prove the existence of periodic solutions under a generalized Lazer-Leach type condition by using variational method. The impulses can generate a periodic solution.
Highlights
1 Introduction We are concerned with the periodic boundary value problem of second order impulsive differential equations at resonance
We investigate problem ( . ) under a more general Lazer-Leach type condition
We prove that ( . ) has no π -periodic solution by contradiction
Summary
Where m ∈ N, g : R → R is a continuous function, e ∈ L ( , π ), < t < t < · · · < tp < π , and Ij : [ , π] × R → R is continuous for every j. ) becomes the well-known periodic boundary value problem at resonance x (t) + m x(t) + g(x(t)) = e(t), a.e. t ∈ [ , π], ) has at least one π -periodic solution provided that the following condition holds: g(+∞) – g(–∞) = e(t) sin(mt + θ ) dt, ∀θ ∈ R. The periodic problem of the second order differential equation with impulses has been widely studied because of its background in applied sciences (see [ – ] and the references cited therein). (H ) There exist continuous, π -periodic functions K (t), K (t), . Assume that conditions (H ) and (H ) for all θ ∈ R, G(+∞) – G(–∞) = e(t) sin(mt + θ ) dt hold. Since we consider the problem with impulses, Theorem .
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