﻿ 非线性项依赖于导数的奇异半正非局部边值问题正解的存在性 The Existence of Positive Solutions for Singular Semi-Positive Non-Local Boundary Value Problems with Nonlinear Term Depending on Derivatives

Advances in Applied Mathematics
Vol.07 No.08(2018), Article ID:26600,12 pages
10.12677/AAM.2018.78120

The Existence of Positive Solutions for Singular Semi-Positive Non-Local Boundary Value Problems with Nonlinear Term Depending on Derivatives

Yu Zhao, Xiujie Yu

School of Mathematics and Statistics, Shandong Normal University, Jinan Shandong

Received: Aug. 2nd, 2018; accepted: Aug. 20th, 2018; published: Aug. 27th, 2018

ABSTRACT

In this paper, we consider the singular semi-positive non-local boundary value problem

$\left\{\begin{array}{l}{u}^{″}\left(t\right)+q\left(t\right)f\left(t,u\left(t\right),{u}^{\prime }\left(t\right)\right)=0,0

where $\alpha \left[u\right]={\int }_{0}^{1}u\left(t\right)dA\left(t\right)$ , $\beta \left[u\right]={\int }_{0}^{1}u\left(t\right)dB\left(t\right)$ ; A and B are bounded variation functions; $a>0$ , $b>0$ ; the nonlinear item $f:\left[0,1\right]×\left(0,+\infty \right)×\left(-\infty ,+\infty \right)\to R$ is continuous and it is allowed to change the sign. Based on the fixed point index theory, this paper studies the existence of multiple positive solutions to the above problem.

Keywords:Singular Semi-Positive Non-Local Boundary Value Problem, Existence, Fixed Point Theory, Positive Solutions

$\left\{\begin{array}{l}{u}^{″}\left(t\right)+q\left(t\right)f\left(t,u\left(t\right),{u}^{\prime }\left(t\right)\right)=0,0

Copyright © 2018 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

$\left\{\begin{array}{l}{u}^{″}\left(t\right)+q\left(t\right)f\left(t,u\left(t\right),{u}^{\prime }\left(t\right)\right)=0,0 (1)

$\alpha \left[u\right]={\int }_{0}^{1}u\left(t\right)\text{d}A\left(t\right),\beta \left[u\right]={\int }_{0}^{1}u\left(t\right)\text{d}B\left(t\right),$

2006年，G.C.Yang在文献 [1] 中讨论了奇异的Dirichlet边值问题

$\left\{\begin{array}{l}{x}^{″}\left(t\right)+f\left(t,x\left(t\right)\right)=0,0

(H1) $f\left(t,x\right)\in C\left(\left(0,1\right)×\left(0,+\infty \right),\left(-\infty ,+\infty \right)\right)$

(H2) $k\left(t\right),\alpha \left(t\right),b\left(t\right)\in C\left(\left(0,1\right),\left(0,+\infty \right)\right),t\left(1-t\right)k\left(t\right)\in {L}^{1}\left[0,1\right]$

(H3) $F\left(x\right)\in C\left(\left(0,+\infty \right),\left(0,+\infty \right)\right),f\left(t,x\right)\le k\left(t\right)F\left(x\right)$

(H4)对任意 $x\left(t\right)\in C\left[0,1\right]$ ，且 $0 ，有 $f\left(t,x\left(t\right)\right)\ge a\left(t\right)$

(H5)上单调递减下，文章运用了Schauder不动点定理得到上述问题的正解的存在性。

2009年，Gennaro Infante在文献 [2] 中考虑了奇异非局部边值问题

(i) g为非负的函数，

(ii) f是非负的函数，且在第二个变量处是有奇异性的，

(iii)上的有界线性泛函，并且

2011年，C. Ji，D. O’Regan，B.Yan等人在文献 [3] 中考虑了二阶奇异三点边值问题

2014年，B.Yan，O'Regan等人在文献 [4] 中讨论了非线性非局部边值问题

2014年，Zima在文献 [5] 中考虑了

2. 预备知识及引理

，对于，我们定义，其中，则是一个Banach空间。

(C1)

(C2)是连续的函数，

(C3)是有界的变差函数，并且

(C4)

(C5)

(C6)存在上的连续函数，并且在，使得当

，并且

(i)

(ii)

,则对,

(2)

(3)

(4)

.

(5)

3. 主要结果

(A1)，有

(6)

(A2) (7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

The Existence of Positive Solutions for Singular Semi-Positive Non-Local Boundary Value Problems with Nonlinear Term Depending on Derivatives[J]. 应用数学进展, 2018, 07(08): 1028-1039. https://doi.org/10.12677/AAM.2018.78120

1. 1. Yang, G.C. (2006) Positive Solutions of Singular Dirichlet Boundary Value Problems with Sign-Changing Nonlinearities. Computers & Mathematics with Applications, 51, 1463-1470. https://doi.org/10.1016/j.camwa.2006.01.006

2. 2. Infante, G. (2009) Positive Solutions of Nonlocal Boundary Value Problems with Singularities. Discrete & Continuous Dynamical Systems, 377-384.

3. 3. Ji, C., O’Regan, D., Yan, B. and Agarwal, R.P. (2011) Nonexistence and Existence of Positive Solutions for Second Order Singular Three-Point Boundary Value Problems with Derivative Dependent and Sign-Changing Nonlinearities. Journal of Applied Mathematics and Computing, 36, 61-87. https://doi.org/10.1007/s12190-010-0388-5

4. 4. Yan, B.Q., O’Regan, D. and Agarwal, R.P. (2014) Multiplicity and Uniqueness Results for the Singular Nonlocal Boundary Value Problem Involving Nonlinear Integral Conditions. Boundary Value Problems, 148. https://doi.org/10.1186/s13661-014-0148-9

5. 5. Zima, M. (2014) Positive Solutions of Second-Order Non-Local Boundary Value Problem with Singularities in Space Variables. Boundary Value Problems, 200. https://doi.org/10.1186/s13661-014-0200-9

6. 6. Guo, D. and Lakshmikantham, V. (1988) Nonlinear Problems in Abstract Cones. Academic Press.