On a class of second‐order impulsive boundary valueproblem at resonance

Guolan Cai,Weigao Ge,Zengji Du

doi:10.1155/ijmms/2006/62512

Abstract

We consider the following impulsive boundary value problem, x″(t) = f(t, x, x′), t ∈ J\{t1, t2, …, tk}, Δx(ti) = Ii(x(ti), x′(ti)), Δx′(ti) = Ji(x(ti), x′(ti)), i = 1, 2, …, k, x(0) = (0), . By using the coincidence degree theory, a general theorem concerning the problem is given. Moreover, we get a concrete existence result which can be applied more conveniently than recent results. Our results extend some work concerning the usual m‐point boundary value problem at resonance without impulses.

Highlights

In the few past years, boundary value problems for impulsive differential equation have been studied
Our results extend some work concerning the usual m-point boundary value problem at resonance without impulses
They discussed the existence of solutions for first-order impulsive equations by the use of upper and lower solution methods

Summary

Introduction

In the few past years, boundary value problems for impulsive differential equation have been studied (see [1, 5, 7]). They discussed the existence of solutions for first-order impulsive equations by the use of upper and lower solution methods. 2. A map x : J → R is said to be solution of (1.1)–(1.2), if it satisfies (1) x(t) is twice continuously differentiable for t ∈ J , both x(t + 0) and x(t − 0) exist at t = ti, and x(ti) = x(ti − 0), i = 1, 2, . This paper is motivated by [2, 3, 6, 8, 9]

Preliminary lemmas

Main results