﻿ 具有超前和滞后的二阶差分方程的边值问题 Boundary Value Problems for Second Order Difference Equations Containing Both Advance and Retardation

Vol.05 No.04(2016), Article ID:19068,10 pages
10.12677/AAM.2016.54081

Boundary Value Problems for Second Order Difference Equations Containing Both Advance and Retardation

Jialin Xu, Zhan Zhou

School of Mathematics and Information Science, Guangzhou University, Guangzhou Guangdong

Received: Nov. 4th, 2016; accepted: Nov. 20th, 2016; published: Nov. 28th, 2016

ABSTRACT

In this paper, the boundary value problems for a class of second order nonlinear difference equations containing both advance and retardation are studied. First, a variational functional corresponding to the boundary value problems as aforementioned is established. Next, the existence of solutions of the boundary value problems is transformed into the existence of critical points for the corresponding functional. Then, by using Mountain Pass Lemma, the existence of critical points of the functional is obtained, and thus the existence of solutions for the initial boundary value problems is also obtained.

Keywords:Second Order Difference Equations, Boundary Value Problems, Mountain Pass Lemma

1. 引言

(1.1)

(1.2)

(1.3)

(1.4)

2003年开始，临界点理论被用来研究了二阶超线性差分方程的周期解和次调和解的存在性，后来临界点理论也被应用于研究差分方程的边值问题，许多学者对方程(1.1)的一些特殊情形进行了深刻的讨论，得出了一系列有意义的结果，参见 [3] - [8] 。然而据我们所知，到目前为止，用临界点方法讨论方程(1.1)边值问题的文献很少(见 [9] [10] [11] [12] [13] )，因为方程(1.1)中依赖于，而在文 [3] - [8] 中建立泛函的方法面对我们的情况则无能为力。

2009年开始，石海平利用临界点理论给出了下列二阶非线性泛函差分方程

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

(1.10)

2. 变分框架及基本引理

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

，其中

(2.6)

(2.7)

(2.8)

3. 主要结论及其证明

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

，边值问题(1.1)~(1.2)有下列形式

4. 例题

(4.1)

(4.2)

Boundary Value Problems for Second Order Difference Equations Containing Both Advance and Retardation[J]. 应用数学进展, 2016, 05(04): 695-704. http://dx.doi.org/10.12677/AAM.2016.54081

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