﻿ 离散半正边值问题正解的存在性及多解性 Existence and Multiplicity of Semipositone Discrete Boundary Value Problems

Advances in Applied Mathematics
Vol.05 No.02(2016), Article ID:17554,10 pages
10.12677/AAM.2016.52030

Existence and Multiplicity of Semipositone Discrete Boundary Value Problems

Yunxia Zeng

School of Mathematics and Information, Guangzhou University, Guangzhou Guangdong

Received: Apr. 25th, 2016; accepted: May 10th, 2016; published: May 13th, 2016

Copyright © 2016 by author and Hans Publishers Inc.

ABSTRACT

By using the Guo-Krasnosel’skii fixed point theorem, a Dirichlet boundary value problem with sign-changing nonlinearity is discussed and some results of existence and multiplicity of positive solutions are established.

Keywords:Positive Solution, Green Function, Fixed Point Theorem, Semipositone Problem

1. 引言

2011年，文 [12] 考虑了如下半正二阶多点边值问题正解的存在性

(1.1)

(C1)：连续；

(C2)：存在，使得

(C3)：存在函数，使得

2. 准备工作

。设

，则由边界条件知，，这与为正解矛盾。

，则

(2.1)

(2.2)

，且

(2.3)

(i)

(ii)

3. 主要结果

(C4)

(C5)最终非正，即，当时，对所有

(C6)

(3.1)

(3.2)

(3.3)

，我们有。则

。又。因此由(C4)得到

(3.4)

。对任意的，类似于(3.4)，我们有

(3.5)

。因此，由引理2.6知，算子有一个不动点，且满足。所以是满足边值问题(1.1)的另外一个正解。

(C4)*：

(C5)*：最终非负，即，当时，对所有

(C6)*：

Existence and Multiplicity of Semipositone Discrete Boundary Value Problems[J]. 应用数学进展, 2016, 05(02): 232-241. http://dx.doi.org/10.12677/AAM.2016.52030

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