王 静



时间测度链上三点边值问题对称正解的存在性

王 静*

(兰州文理学院 数学系, 甘肃 兰州, 730000)

利用锥上的压缩与拉伸不动点定理研究了测度链上一类二阶动力方程三点边值问题至少一个对称正解的存在性. 并且给出了与之相关联的线性动力方程三点边值问题的格林函数及格林函数的一些性质.

时间测度链; 边值问题; 不动点定理; 锥; 对称正解

1990年, 德国数学家Hilger发表了时间测度链分析理论[1], 为微分、差分方程的研究提供了强有力的工具. 此后, 时间测度链上动力方程不仅在理论研究中占据非常重要的地位, 而且在应用数学、物理领域及其它边缘学科中亦有着极为广泛的应用背景, 特别是在种群动力学、神经网络系统等学科领域中应用更为普遍[2]. 近年来, 文献[3—7]得到了一些关于时间测度链上二阶动力方程的很好结果, 引起了很多学者的高度关注.

文献[7]运用锥上的压缩与拉伸不动点定理研究了三点边值问题对称正解的存在性, 受其启发, 本文考虑时间测度链上二阶动力方程三点边值问题

为后面推理的需要, 做如下记号:

本文所用的工具为如下的压缩与拉伸不动点定理.

1 预备引理

存在唯一解

从而可得:

再次积分, 得:

因此, 边值问题(2)有唯一对称解.

即有:

2 定理及其证明

[1] Hilger S. Analysis on measure chains: A unified approach to continuous and discrete calculus[J]. Results Math, 1990, 18: 18—56.

[2] Agarwal R P, Bohner M, Li W T. Nonoscillation and oscillation theory for functional differential equations, pure and applied mathematics series[M]. New York: Marcel Dekker, 2004.

[3] Bai D L, Feng H Y. Eigenvalue for a singular second order three-point boundary value problem[J]. J Appl Math Comp, 2012, 38: 443—452.

[4] Kosmatov N. Symmetric solutions of a multi-point boundary value problem[J]. J Math Anal Appl, 2005, 309: 25—36.

[5] Zhao J F, Lian H R, Ge W G. Existence of positive solutions for nonlinear m-point boundary problems on time scales[J]. Boundary Value Problems, 2012(1): 1—15.

[6] 邹序焱, 惠远先. 三阶二点边值问题三个正解的存在性[J]. 湖南文理学院学报: 自然科学版, 2009, 21(2): 4—6.

[7] Sun Y. Existence and multiplicity of symmetric positive solutions for three-point boundary value problem[J]. J Math Anal Appl, 2007, 329: 998—1009.

[8] Guo D, Lakshmikantham V. Nonlinear Problems in Abstract Cones[M]. New York: Academic Press, 1988.

The existence of symmetric positive solution for triple-point boundary value problem on time scales

WANG Jing

(Department of Mathematics, Lanzhou University of Arts and Science, Lanzhou 730000, China)

By the fixed point theorems of cone expansion and compression,the existence of at least one positive solution of triple-point boundary value problem for second order dynamic equation on time scales was investigated. The Green’s function and some of its properties of the linear three-point dynamic equation boundary value problem related to the nonlinear boundary value problem were put foeward.

time scales; boundary value problem; fixed point theorem; cone; symmetric positive solutions

10.3969/j.issn.1672-6146.2013.03.002

O 175

1672-6146(2013)03-0006-04

email: wangjing7723@163.com.

2013-08-06

甘肃省自然科学基金项目(3ZS042-B26-021); 甘肃省教育厅科研项目(1013B-03).

(责任编校:刘晓霞)