時(shí)間測(cè)度鏈上三點(diǎn)邊值問題對(duì)稱正解的存在性

王 靜

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時(shí)間測(cè)度鏈上三點(diǎn)邊值問題對(duì)稱正解的存在性

王 靜*

(蘭州文理學(xué)院 數(shù)學(xué)系, 甘肅 蘭州, 730000)

利用錐上的壓縮與拉伸不動(dòng)點(diǎn)定理研究了測(cè)度鏈上一類二階動(dòng)力方程三點(diǎn)邊值問題至少一個(gè)對(duì)稱正解的存在性. 并且給出了與之相關(guān)聯(lián)的線性動(dòng)力方程三點(diǎn)邊值問題的格林函數(shù)及格林函數(shù)的一些性質(zhì).

時(shí)間測(cè)度鏈; 邊值問題; 不動(dòng)點(diǎn)定理; 錐; 對(duì)稱正解

1990年, 德國數(shù)學(xué)家Hilger發(fā)表了時(shí)間測(cè)度鏈分析理論[1], 為微分、差分方程的研究提供了強(qiáng)有力的工具. 此后, 時(shí)間測(cè)度鏈上動(dòng)力方程不僅在理論研究中占據(jù)非常重要的地位, 而且在應(yīng)用數(shù)學(xué)、物理領(lǐng)域及其它邊緣學(xué)科中亦有著極為廣泛的應(yīng)用背景, 特別是在種群動(dòng)力學(xué)、神經(jīng)網(wǎng)絡(luò)系統(tǒng)等學(xué)科領(lǐng)域中應(yīng)用更為普遍[2]. 近年來, 文獻(xiàn)[3—7]得到了一些關(guān)于時(shí)間測(cè)度鏈上二階動(dòng)力方程的很好結(jié)果, 引起了很多學(xué)者的高度關(guān)注.

文獻(xiàn)[7]運(yùn)用錐上的壓縮與拉伸不動(dòng)點(diǎn)定理研究了三點(diǎn)邊值問題對(duì)稱正解的存在性, 受其啟發(fā), 本文考慮時(shí)間測(cè)度鏈上二階動(dòng)力方程三點(diǎn)邊值問題

為后面推理的需要, 做如下記號(hào):

本文所用的工具為如下的壓縮與拉伸不動(dòng)點(diǎn)定理.

1 預(yù)備引理

存在唯一解

從而可得:

再次積分, 得:

因此, 邊值問題(2)有唯一對(duì)稱解.

即有:

2 定理及其證明

[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] 鄒序焱, 惠遠(yuǎn)先. 三階二點(diǎn)邊值問題三個(gè)正解的存在性[J]. 湖南文理學(xué)院學(xué)報(bào): 自然科學(xué)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

甘肅省自然科學(xué)基金項(xiàng)目(3ZS042-B26-021); 甘肅省教育廳科研項(xiàng)目(1013B-03).

(責(zé)任編校:劉曉霞)