﻿ 一类非局部边值问题正解的存在性

# 一类非局部边值问题正解的存在性The Existence of Positive Solutions for a Nonlocal Boundary Value Problem

Abstract: This paper discusses the boundary value problem with nonlocal integral boundary conditions

Abstract:

Abstract: where , ; A and B are functions of bounded variation; , ; the nonlinearity is continuous and is allowed to change sign. According to the fixed point theorem in double cones, we obtain that there exists at least two positive solutions. And according to three-solution theorem, we obtain that there exists at least three positive solutions.

Abstract:

1. 引言

$\left\{\begin{array}{l}{u}^{″}\left(t\right)+q\left(t\right)f\left(t,u\left(t\right)\right)=0,0 (1.1)

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

2000年，Guidotti和Merino在文献 [1] 中研究了

$\left\{\begin{array}{l}-{u}^{″}\left(t\right)=y\left(t\right),t\in \left(0,1\right),\hfill \\ {u}^{\prime }\left(0\right)=0,{u}^{\prime }\left(1\right)+\beta u\left(0\right)=0\left(\beta >0\right).\hfill \end{array}$

2006年，Webb和Infante在文献 [2] 中研究了以下方程正解的存在性问题：

$\left\{\begin{array}{l}-{u}^{″}\left(t\right)=g\left(t\right)f\left(t,u\right),0

2012年，Webb在文献 [3] 中研究了温控器模型正解的存在性：

$\left\{\begin{array}{l}-{u}^{″}\left(t\right)=g\left(t\right)f\left(t,u\right),0

2015年，Infante在文献 [4] 中研究了Neumann型非局部边值问题

$\left\{\begin{array}{l}{u}^{″}\left(t\right)+f\left(t,u\left(t\right)\right)=0,t\in \left[0,1\right],\hfill \\ {u}^{\prime }\left(0\right)=\alpha \left[u\right],\hfill \\ {u}^{\prime }\left(1\right)=\beta \left[u\right].\hfill \end{array}$

$\left(Tu\right)\left(t\right)=\gamma \left(t\right)\alpha \left[u\right]+\delta \left(t\right)\beta \left[u\right]+{\int }_{0}^{1}k\left(t,s\right)g\left(s\right)f\left(s,u\left(s\right)\right)\text{d}s,$

2. 预备知识和引理

(H1) $a>0,b>0$

(H2) $f\in C\left(\left[0,1\right]×\left[0,\infty \right),R\right),f\left(t,0\right)>0,\forall t\in \left[0,1\right]$

(H3) $A\left(t\right),B\left(t\right)$ 是有界变差函数，且 ${\int }_{0}^{1}\text{d}A\left(t\right)\triangleq \alpha \left[\stackrel{^}{1}\right]>0,{\int }_{0}^{1}\text{d}B\left(t\right)\triangleq \beta \left[\stackrel{^}{1}\right]>0$

$\left\{\begin{array}{l}{u}^{″}\left(t\right)=0,t\in \left(0,1\right),\hfill \\ au\left(0\right)-b{u}^{\prime }\left(0\right)=0,\hfill \\ {u}^{\prime }\left(1\right)=0.\hfill \end{array}$

$G\left(t,s\right)=\left\{\begin{array}{l}\frac{1}{a}\left(as+b\right),s\le t,\hfill \\ \frac{1}{a}\left(at+b\right),t\le s.\hfill \end{array}$

${G}_{t}\left(t,s\right)=\left\{\begin{array}{l}0,s\le t,\hfill \\ 1,t\le s,\hfill \end{array}$

$\frac{b}{a+b}G\left(s,s\right)\le G\left(t,s\right)\le G\left(s,s\right),\forall t,s\in \left[0,1\right].$

$X=C\left[0,1\right]=\left\{u:u为\left[0,1\right]上的连续函数\right\}$ 。对 $u\in X$ ，定义

$‖u‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}|u\left(t\right)|,$

$\left(X,‖\cdot ‖\right)$ 为一Banach空间。 $K=\left\{\mathrm{u\in X}:\mathrm{u \left(t \right)}\ge 0\right\}$

${K}^{\prime }=\left\{u\in X:u\left(t\right)\ge \frac{b}{a+b}‖u‖\right\},$

$\theta \left(u\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}|u\left(t\right)|.$

${\left(\cdot \right)}^{+}=\mathrm{max}\left\{\cdot ,0\right\}$ ，我们定义算子 $T,A,{T}^{*}$ 如下：

$T:K\to K,A:K\to X,{T}^{*}:{K}^{\prime }\to {K}^{\prime },$

$\left(Tu\right)\left(t\right)={\left[\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right)f\left(s,u\left(s\right)\right)\text{d}s\right]}^{+},t\in \left[0,1\right],$

$\left({T}^{*}u\right)\left(t\right)=\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s,t\in \left[0,1\right].$

$\forall u\in {K}^{\prime }$ ，有 ${T}^{*}u\in C\left[0,1\right]$ ，且

$\begin{array}{c}\left({T}^{*}u\right)\left(t\right)=\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s\\ \ge \frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+\frac{b}{a+b}{\int }_{0}^{1}G\left(s,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s\\ \ge \frac{b}{a+b}\left(\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+\frac{b}{a+b}{\int }_{0}^{1}G\left(s,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s\right)\\ \ge \frac{b}{a+b}‖{T}^{*}u‖.\end{array}$

$\left(Au\right)\left(t\right)\ge 0,\forall t\in \left[0,1\right].$

${\left(Au\right)}^{\prime \text{​}\prime }\left(t\right)=-q\left(t\right)f\left(t,0\right)<0,t\in \left({t}_{1},{t}_{2}\right).$

1) ${t}_{1}=0$

${t}_{2}<1$ ，则 $\left(Au\right)\left({t}_{2}\right)=0$ ，且 $\left(Au\right)\left(t\right)<0,t\in \left({t}_{1},{t}_{2}\right)$ ，而时， ${\left(Au\right)}^{\prime \text{​}\prime }\left(t\right)<0$ ，即 $Au$$\left[{t}_{1},{t}_{2}\right]$ 上为凹函数，所以 ${\left(Au\right)}^{\prime }\left(t\right)>0$ 。但又

$\alpha \left(Au\right)\left(0\right)-b{\left(Au\right)}^{\prime }\left(0\right)=\alpha \left[u\right],$

$\left(Au\right)\left(0\right)=\frac{b{\left(Au\right)}^{\prime }\left(0\right)+\alpha \left[u\right]}{b}>0.$

${t}_{2}=1$ ，此时 ，且 ，故

$\left(Au\right)\left(t\right)=\frac{1}{a}\alpha \left[0\right]+\left(\frac{b}{a}+t\right)\beta \left[0\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right)f\left(s,0\right)\text{d}s,t\in \left[0,1\right].$

2) ${t}_{1}>0$

${t}_{2}<1$ ，则 $\left(Au\right)\left({t}_{1}\right)=0,\left(Au\right)\left({t}_{2}\right)=0$ ，且 $\left(Au\right)\left(t\right)<0,t\in \left({t}_{1},{t}_{2}\right)$ 于是

${\left(Au\right)}^{\prime \text{​}\prime }\left(t\right)=-q\left(t\right)f\left(t,0\right)<0,t\in \left({t}_{1},{t}_{2}\right)$

$\left(Au\right)\left(t\right)>\left(Au\right)\left({t}_{1}\right)=\left(Au\right)\left({t}_{2}\right)=0.$

${t}_{2}=1$ ，则 $\left(Au\right)\left({t}_{1}\right)=0,\left(Au\right)\left(t\right)<0,t\in \left({t}_{1},{t}_{2}\right)$ ，于是

$u\left(t\right)=0,t\in \left[{t}_{1},{t}_{2}\right],\text{\hspace{0.17em}}{\left(Au\right)}^{\prime \text{​}\prime }\left(t\right)=-q\left(t\right)f\left(t,0\right)<0,t\in \left({t}_{1},{t}_{2}\right).$

(C1) 当 $\left\{u\in P\left(a,b,d\right):\alpha \left(u\right)>b\right\}\ne \varnothing$ ，且 $u\in P\left(a,b,d\right)$ 时，恒有 $\alpha \left(Au\right)>b$

(C2) 当 $u\in {\stackrel{¯}{P}}_{a}$ 时，恒有 $‖Au‖

(C3) 当 $u\in P\left(a,b,c\right)$$‖Au‖>d$ 时，恒有 $\alpha \left(Au\right)>b$

(M1) $‖Tx‖

(M2) $‖{T}^{*}x‖ ，且 $\theta \left({T}^{*}x\right)>b,x\in \partial {K}^{\prime }\left(b\right)$

(M3) $Tx={T}^{*}x,x\in {{K}^{\prime }}_{a}\left(b\right)\cap \left\{x:{T}^{*}x=x\right\}$

$0\le ‖{x}_{1}‖

3. 主要结果

(H4) $f\left(t,u\right)\ge 0,u\in \left[d,R\right]$

(H5) $|f\left(t,u\right)|\le Mr,\left(t,u\right)\in \left[0,1\right]×\left[0,r\right]$

(H6) $|f\left(t,u\right)|\ge mr,\left(t,u\right)\in \left[0,1\right]×\left[\frac{b}{a+b}\mathrm{R.R}\right]$

(H7) $\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]+M{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s<1$

(H8) $\frac{b}{a+b}\left(\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\frac{b}{a}\beta \left[\stackrel{^}{1}\right]+m{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\right)>1$

(H9) $d<\frac{b}{a+b}R$

$0<‖{u}_{1}‖

$‖Tu‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}{\left[\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right)f\left(s,u\left(s\right)\right)\text{d}s\right]}^{+},$

$\begin{array}{c}‖Tu‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}\mathrm{max}\left\{\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s,0\right\}\\ \le \frac{\alpha \left[\stackrel{^}{1}\right]}{a}r+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]r+Mr{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\\ =\left(\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]+M{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\right)r\\

$\begin{array}{c}‖{T}^{*}u‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}\left\{\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s,0\right\}\\ \le \frac{\alpha \left[\stackrel{^}{1}\right]}{a}r+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]r+Mr{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\\

$u\in \partial {K}^{\prime }\left(\frac{b}{a+b}R\right)$ ，即 $u\in {K}^{\prime }$ ，且 $\theta \left(u\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}u\left(t\right)=\frac{b}{a+b}R$ ，我们有

$\frac{b}{a+b}R=\theta \left(u\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}u\left(t\right)\ge \frac{b}{a+b}‖u‖,$

$‖u‖\le R.$

$u\left(t\right)\ge \frac{b}{a+b}R,\forall t\in \left[0,1\right],$

$\frac{b}{a+b}R\le u\left(t\right)\le ‖u‖\le R,\forall t\in \left[0,1\right].$

$\begin{array}{c}\left({T}^{*}u\right)\left(t\right)=\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s\\ \ge \frac{\alpha \left[\stackrel{^}{1}\right]}{a}\frac{b}{a+b}R+\frac{b}{a}\beta \left[\stackrel{^}{1}\right]\frac{b}{a+b}R+mR{\int }_{0}^{1}\frac{b}{a+b}G\left(s,s\right)q\left(s\right)\text{d}s\\ =\frac{b}{a+b}\left(\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]+m{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\right)R\\ >R.\end{array}$

$\theta \left({T}^{*}u\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}\left({T}^{*}u\right)\left(t\right)\ge \frac{b}{a+b}‖{T}^{*}u‖>\frac{b}{a+b}R,$

$\theta \left({T}^{*}u\right)>\frac{b}{a+b}R.$

$u\in \partial {{K}^{\prime }}_{r}\left(\frac{b}{a+b}R\right)\cap \left\{u:{T}^{*}u=u\right\}$ ，即 $u\in {K}^{\prime }$ ，且 $‖u‖>r,\theta \left(u\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}u\left(t\right)<\frac{b}{a+b}R$

$u\left(t\right)\ge \frac{b}{a+b}‖u‖,$

$\frac{b}{a+b}‖u‖\le \underset{t\in \left[0,1\right]}{\mathrm{min}}u\left(t\right)<\frac{b}{a+b}R,$

$‖u‖

$\frac{b}{a+b}r\le \frac{b}{a+b}‖u‖\le u\left(t\right)\le ‖u‖

$d

${f}^{+}\left(t,u\left(t\right)\right)=f\left(t,u\left(t\right)\right).$

$Tu={T}^{*}u.$

$0\le ‖{u}_{1}‖

( ${H}^{\prime }$ ) 存在正数 $k,l$ 满足 $d ，使得

$f\left(t,u\right)\ge mk,\left(t,u\right)\in \left[0,1\right]×\left[k,R\right];$

( ${H}^{\prime }$ ) $|f\left(t,u\right)|\le MR,\left(t,u\right)\in \left[0,1\right]×\left[0,R\right]$

( ${H}^{\prime }$ ) $|f\left(t,u\right)|\le Md,\left(t,u\right)\in \left[0,1\right]×\left[0,d\right]$

( ${H}^{\prime }$ ) $\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\frac{b}{a}\beta \left[\stackrel{^}{1}\right]+m{\int }_{0}^{1}\frac{b}{a+b}G\left(s,s\right)q\left(s\right)\text{d}s>1$

( ${H}^{\prime }$ ) $\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]+M{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s<1$

$\theta \left(u\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}u\left(t\right),u\in K.$

$\begin{array}{c}‖Tu‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}{\left[\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s\right]}^{+}\\ \le \left(\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\left(1+\frac{b}{a}\right)\beta \left[\stackrel{^}{1}\right]+M{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\right)R\\

$T:{\stackrel{¯}{K}}_{R}\to {\stackrel{¯}{K}}_{R}.$

1) 取 ${u}_{1}=1$ , t∈[0.1], ${u}_{1}\in K\left(\theta ,k,l\right)，\theta \left({u}_{1}\right)=l>k.$

$\left\{u\in K\left(\theta ,k,l\right):\theta \left(u\right)>k\right\}\ne \varnothing .$

$k\le u\left(t\right)\le l,t\in \left[0,1\right].$

$\begin{array}{c}\theta \left(Tu\right)=\underset{t\in \left[0,1\right]}{\mathrm{min}}{\left[\frac{1}{a}\alpha \left[u\right]+\left(\frac{b}{a}+t\right)\beta \left[u\right]+{\int }_{0}^{1}G\left(t,s\right)q\left(s\right){f}^{+}\left(s,u\left(s\right)\right)\text{d}s\right]}^{+}\\ \ge \frac{\alpha \left[\stackrel{^}{1}\right]}{a}k+\frac{b}{a}\beta \left[\stackrel{^}{1}\right]k+\frac{b}{a+b}{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\cdot mk\\ =\left(\frac{\alpha \left[\stackrel{^}{1}\right]}{a}+\frac{b}{a}\beta \left[\stackrel{^}{1}\right]+m\frac{b}{a+b}{\int }_{0}^{1}G\left(s,s\right)q\left(s\right)\text{d}s\right)k\\ >k.\end{array}$

2) 对，由假设( ${H}^{\prime }$ )、( ${H}^{\prime }$ )有

3) 取 $u\in K\left(\theta ,k,R\right)$ ，且 $‖Tu‖>l$ ，则

$\theta \left(Tu\right)>k.$

$‖{u}_{1}‖

[1] Guidotti, P. and Merino, S. (2000) Gradual Loss of Positivity and Hidden Invariant Cones in a Scalar Heat Equation. Differential Integral Equations, 13, 1551-1568.

[2] Infante, G. and Webb, J.R.L. (2006) Loss of Positivity in a Nonlinear Scalar Heat Equation. NoDEA Nonlinear Differential Equations Applications, 13, 249-261.
https://doi.org/10.1007/s00030-005-0039-y

[3] Webb, J.R.L. (2012) Existence of Positive Solutions for a Thermostat Model. Nonlinear Analysis: Real World Applications, 13, 923-938.
https://doi.org/10.1016/j.nonrwa.2011.08.027

[4] Infante, G. (2015) Nontrival Solutions of Local and Nonlocal Neumann Boun-dary Value Problems. Classical Analysis and ODEs.

[5] Leggett, R. and Williams, L. (1979) Multiple Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces. Indiana University Mathematics Journal, 28, 673-688.
https://doi.org/10.1512/iumj.1979.28.28046

[6] Ge, W.G. and Ren, J.L. (2006) Fixed Point Theorems in Double Cones and Their Applications to Non-Linear Boundary Value Problems. Chinese Annals of Mathematics, 27, 155-168.

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