A Note on the Paper “Multipoint BVPs for Second-Order Differential Equations with Impulses” by Xuxin Yang, Zhimin He, and Jianhua Shen

Jitai Liang,Yiliang Liu

doi:10.1155/2010/497029

Abstract

We give a counter example to the comparison principle for the multipoint BVPs (by Xuxin Yang, Zhimin He, and Jianhua Shen, in Mathematical Problems in Engineering, Volume 2009, Article ID 258090, doi:10.1155/2009/258090). Then we suggest and prove a corrected version of the comparison principle.

Highlights

Let P C J {x : J → R; x t be continuous everywhere expect for some tk at which x tk and x t−k exist and x tk x t−k, k 1, 2, . . . , m}; P C1 J
{x ∈ P C J : x t is continuous everywhere expect for some tk at which x tk and x t−k exist and x tk Mathematical Problems in Engineering x t−k, k 1, 2, . . . , m}
Let J− J \ {tk, k 1, 2, . . . , m}, and E P C1 J, R ∩ C2 J−, R . a function x ∈ E is called a solution of BVPS 1.1 if it satisfies 1.1

Summary

Introduction and Preliminaries

Consider the following multipoint BVPs 1 :. Let P C J {x : J → R; x t be continuous everywhere expect for some tk at which x tk and x t−k exist and x tk x t−k , k 1, 2, . {x ∈ P C J : x t is continuous everywhere expect for some tk at which x tk and x t−k exist and x tk Mathematical Problems in Engineering x t−k , k 1, 2, . A function x ∈ E is called a solution of BVPS 1.1 if it satisfies 1.1. The purpose of this note is to point out that the results basing on the comparison principle 1, Theorem 2.1 are not true.

Problem and Statement

A Counter Example Let