具P-Laplace算子Sturm-Liouville型边值问题多个正解的存在性

Multiple Positive Solutions of Discrete Sturm-Liouville-Like P-Laplacian Boundary Value Problems

摘要: 利用Leggett-Williams不动点定理研究一类离散具p-Laplace算子Sturm-Liouville型边值问题，给出问题至少存在三个正解的几个充分条件。

Abstract: By using the Leggett-Williams fixed point theorem, a class of discrete Sturm-Liouville-like p-Laplacian boundary value problem is studied and some sufficient conditions for the existence of at least three positive solutions for the boundary value problem are obtained.

1. 引言

离散型边值问题正解的存在性因其具有实际应用背景而受到学者广泛关注和研究 [1] [2] [3] [4] [5] 。Zhang Meng，Sun Shurong [4] 等人研究一类具p-Laplace算子离散Sturm-Liouville型边值问题

$\Delta \left({\phi }_p\left(\Delta u\left(n\right)\right)\right)+h\left(n\right)f\left(u\left(n\right)\right)=0$， $n\in \mathbb{N}\left[0,N\right]$， (1)

$\alpha u\left(0\right)-\beta \Delta u\left(\xi \right)=0$， $\gamma u\left(N+2\right)+\sigma \Delta u\left(\eta \right)=0$。 (2)

其中， $\Delta$ 表示步差为1的向前差分算子， ${\phi }_p\left(u\right)={\left|u\right|}^{p-2}u$， $p>1$， $\mathbb{N}\left[0,N\right]=\left\{0,1,\cdots ,N\right\}$， $\alpha >0$， $\beta >0$， $\gamma >0$， $\sigma >0$， $\xi$， $\eta$，N为正整数，满足

$0<\xi \le \frac{\delta }{\gamma }<\max \left\{\frac{2\delta }{\gamma },N+2-\frac{\beta }{\alpha },\frac{N+2}{2}+\frac{\delta }{2\gamma }\right\}\le \eta <N$。

记 ${\left({\phi }_p\right)}^{-1}={\phi }_q$， $\frac{1}{p}+\frac{1}{q}=1$。函数h，f满足下列条件：

(C₁)： $f:\left[0,\infty \right)\to \left(0,\infty \right)$ 连续；

(C₂)：h为定义在 $N\left[0,N+2\right]$ 上的非负函数。

他们借助Guo-krasnosel’ skii’s不动点定理，给出问题至少一个正解存在的若干充分条件。2012年针对该问题，张萌 [5] 等人利用Avery-Henderson不动点定理给出至少存在两个正解的充分条件。

受到前面研究启发，我们将利用Leggett-Williams不动点定理继续研究问题(1)和(2)，给出至少三个正解的几个充分条件。

2. 引理

引理1：若 $u\left(n\right)$ 是边值问题(1)和(2)的解，则存在唯一的 $n_0$ 使得 $\Delta u\left(n_0\right)\ge 0$， $\Delta u\left(n_0+1\right)\le 0$。

证明：因为 $u\left(n\right)$ 边值问题(1)和(2)的解，所以 $\Delta u\left(n\right)$ 在 $N\left[0,N+1\right]$ 上递减。若对所有的 $n\in N\left[0,N+1\right]$ 都有 $\Delta u\left(n\right)>0$，则由边值条件(2)可知 $u\left(0\right)>0$， $u\left(N+2\right)<0$，矛盾。

类似可证对 $n\in N\left[0,N+1\right]$， $\Delta u\left(n\right)<0$ 也不成立。因此，存在 $n_0\in N\left[0,N\right]$ 使得 $\Delta u\left(n_0\right)\ge 0$， $\Delta u\left(n_0+1\right)\le 0$。

唯一性若还存在 $n_1\in N\left[0,N\right]$ (不失一般性，不妨 $n_1+1<n_0$ )满足 $\Delta u\left(n_1\right)\ge 0$， $\Delta u\left(n_1+1\right)\le 0$，则对 $n\in N\left[n_1+1,n_0\right]$ 有 $\Delta u\left(n\right)\equiv 0$，从而 $\Delta \left({\phi }_p\left(\Delta u\left(n\right)\right)\right)=-h\left(n\right)f\left(u\left(n\right)\right)\equiv 0$， $n\in N\left[n_1+1,n_0\right]$，与条件(C₂)矛盾。

为研究方便，令

$M=\max \left\{{\displaystyle \underset{i=0}{\overset{\xi }{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{\xi }{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right),{\displaystyle \underset{i=\eta }{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}}\right\}$ ；

$m=\min \left\{{\displaystyle \underset{i=\xi +1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)},{\displaystyle \underset{i=0}{\overset{\eta -1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\right\}$

$L_1=\frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{n_0}{\sum }}h\left(j\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q({\displaystyle \underset{j=i}{\overset{n_0}{\sum }}h(j)})}$

$L_2=\frac{\sigma }{\gamma }{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{\eta -1}{\sum }}h\left(j\right)}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q({\displaystyle \underset{j=n_0}{\overset{i-1}{\sum }}h(j)})}$

引理2：若条件(C₁)~(C₂)成立，对任意的 $u\in \left[0,+\infty \right)$， $M\le m$ 则

$u\left(n\right)=\{\begin{array}{l}\frac{\beta }{\alpha }{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n-1}{\sum }}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)},n\in N\left[0,n_0+1\right]\\ \frac{\delta }{\gamma }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-A_u\right)+{\displaystyle \underset{i=0}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-A_u\right)},n\in N\left[n_0+1,N+2\right]\end{array}$

为边值问题(1)和(2)的解，这里的 $A_u$ 是下列方程的唯一解

$\frac{\beta }{\alpha }{\phi }_q\left(y-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)=\frac{\sigma }{\gamma }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-y\right)+{\displaystyle \underset{i=0}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-y\right)}$

并且 $u\left(n\right)$ 在 $N\left[0,\xi +1\right]$ 递增， $N\left[\eta ,N+2\right]$ 递减。

证明：我们仅证明 $u\left(n\right)$ 在 $N\left[0,\xi +1\right]$ 递增， $N\left[\eta ,N+2\right]$ 递减，其他可见参考文献 [4] [5] 。

若 $u\left(n\right)$ 在 $N\left[0,\xi +1\right]$ 非增，则存在 $n_0\in N\left[0,\xi -1\right]$ 使得 $\Delta u\left(n_0\right)\ge 0$， $\Delta u\left(n_0+1\right)\le 0$。从而 $u\left(0\right)=\frac{\beta }{\alpha }\Delta u\left(\xi \right)\le 0$， $u\left(N+2\right)=\frac{\sigma }{\gamma }\Delta u\left(\eta \right)\ge 0$。由u的表达式，知

$\begin{array}{c}u\left(n_0+1\right)=\frac{\beta }{\alpha }{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ <{\displaystyle \underset{i=0}{\overset{\xi }{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\xi }{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ ={\displaystyle \underset{i=0}{\overset{\xi }{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{\xi }{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\le M\end{array}$，

另一方面

$\begin{array}{c}u\left(n_0+1\right)=\frac{\sigma }{\gamma }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-A_u\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-A_u\right)}\\ \ge {\displaystyle \underset{i=\xi }{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ ={\displaystyle \underset{i=\xi +1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\ge m\end{array}$。

因此， $M>m$，矛盾，故 $u\left(n\right)$ 在 $N\left[0,\xi +1\right]$ 递增。

类似可证 $u\left(n\right)$ 在 $N\left[\eta ,N+2\right]$ 递减。

设空间 $E=\left\{u:\mathbb{N}\left[0,N+2\right]\to R^{+}\right\}$，如果对所有的 $$，都成立 $u_1\left(n\right)\le u_2\left(n\right)$，则 $u_1\le u_2$，定义范数 $\Vert u\Vert =\underset{n\in \mathbb{N}\left[0,N+2\right]}{\max }\left|u\left(n\right)\right|$，则E为Banach空间。

在空间E上定义锥 $P:\{u:u\in E,u\left(n\right)\ge 0,n\in \mathbb{N}\left[0,N+2\right],{\Delta }^{2}u\left(n\right)\le 0,n\in \mathbb{N}\left[0,N\right],\exists n_0使得$

$\Delta u\left(n\right)\ge 0,n\in \mathbb{N}\left[0,n_0\right],\Delta u\left(n\right)\le 0,n\in \mathbb{N}\left[n_0+1,N+1\right]\}.$

对 $u\in P$，易知 $\Vert u\Vert =u\left(n_0+1\right)$。

定义算子 $A:P\to E$

$Au\left(n\right)=\{\begin{array}{l}\frac{\beta }{\alpha }{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n-1}{\sum }}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)},n\in N\left[0,n_0+1\right]\\ \frac{\delta }{\gamma }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-A_u\right)+{\displaystyle \underset{i=0}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-A_u\right)},n\in N\left[n_0+1,N+2\right]\end{array}$

对 $u\in P$，下面说明 $Au\in P$ 且 $\Vert Au\Vert =Au\left(n_0+1\right)$。

事实上

$\Delta Au\left(n\right)=\{\begin{array}{l}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{n-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)\ge 0,n\in N\left[0,n_0\right]\\ {\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{n-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)\le 0,n\in N\left[n_0,N+1\right]\end{array}$，

${\Delta }^{2}Au\left(n\right)=-h\left(n\right)f\left(u\left(n\right)\right)\le 0$，故易知 $A:P\to P$ 且全连续。

引理3： [5] 若非负函数 $h\left(i\right)$ 满足 ${\displaystyle \underset{i=0}{\overset{N+2}{\sum }}h\left(i\right)}<\infty$，则存在整数 $w\in \left(0,\frac{N+2}{2}\right)$，使得 $\left(w,N+2-w\right)\ne \varnothing$，且 ${\displaystyle \underset{i=w}{\overset{N+2-w}{\sum }}h\left(i\right)}<\infty$，更进一步 $B\left(n\right)={\phi }_q\left({\displaystyle \underset{i=w-1}{\overset{t-2}{\sum }}h\left(i\right)}\right)+{\phi }_q\left({\displaystyle \underset{i=n}{\overset{N+1-w}{\sum }}h\left(i\right)}\right)$， $n\in \left[w,N+2-w\right]$ 有最小值l。

引理4： [5] 若 $u\in p$，w为引理3中给出的参数，则对所有 $t\in \left[w,N+2-w\right]$，有 $u\left(n\right)\ge \frac{w}{N+2}\Vert u\Vert$。

引理5： [6] 对于 $0<a<b$ 及锥P上的非负连续凹泛函 $\psi$，定义凸子集 $P_r$ 和 $P\left(\psi ,a,b\right)$ 如下： $P_r=\left\{x\in P:\Vert x\Vert <r\right\}$， $P\left(\psi ,a,b\right)=\left\{x\in P:a\le \psi \left(x\right),\Vert x\Vert \le b\right\}$。设 $A:{\bar{P}}_c\to {\bar{P}}_c$ 是全连续的， $\psi$ 是锥P上的非负连续凹泛函，且 $\psi \left(x\right)\le \Vert x\Vert$， $x\in {\bar{P}}_c$。如果存在常数满足 $0<a<b<d\le c$ 使得

(i) $\left\{x\in P\left(\psi ,a,b\right):\psi \left(x\right)>a\right\}\ne \varnothing$ 且 $\psi \left(Ax\right)>a$， $x\in P\left(\psi ,a,b\right)$ ；

(ii) $\Vert Ax\Vert <d$， $\Vert x\Vert \le d$ ；

(iii) 对于 $x\in P\left(\psi ,a,c\right)$ 且 $\Vert Ax\Vert >b$，有 $\psi \left(Ax\right)>a$。

则A至少有三个不动点 $x_1$， $x_2$ 与 $x_3$，且满足

$\Vert x_1\Vert <d$， $a<\psi \left(x_2\right)$， $\Vert x_3\Vert >d$ 和 $\psi \left(x_3\right)<a$。

3. 主要结果

定义非负连续凹泛函 $\varphi :P\to \left[0,\infty \right)$

$\varphi \left(u\right)=\frac{1}{2}\left(u\left(w\right)+u\left(N+2-w\right)\right)$， $u\in P$，

则对 $u\in P$，有 $\varphi \left(u\right)\le \Vert u\Vert$。

定理1：假设存在常数 $0<a^{\prime }<\frac{w}{N+2}{b}^{\prime }<b^{\prime }<\frac{N+2}{w}{b}^{\prime }\le d^{\prime }$，使得下列条件成立

(H₁)： $f\left(u\right)<{\phi }_p\left(\frac{a^{\prime }}{L_i}\right)$， $i=1,2$， $0\le u\le a^{\prime }$ ；

(H₂)：存在常数 $e^{\prime }>d^{\prime }$，使得 $f\left(u\right)<{\phi }_p\left(\frac{e^{\prime }}{L_i}\right)$， $i=1,2$， $0\le u\le e^{\prime }$ ；

(H₃)： $f\left(u\right)>{\phi }_p\left(\frac{2b^{\prime }}{wl}\right)$， $\frac{w}{N+2}{b}^{\prime }\le u\le d^{\prime }$ ；

则边值问题(1)和(2)至少存在三个正解 $u_1$， $u_2$， $u_3$ 满足 $0\le \Vert u_1\Vert <a^{\prime }$， $\varphi \left(u_2\right)>b^{\prime }$， $\Vert u_3\Vert >a^{\prime }$， $\varphi \left(u_3\right)<b^{\prime }$。

证明：由算子A的定义及性质，只需证明算子A满足引理 [6] 条件。

首先，若 $\Vert u\Vert <a^{\prime }$，则 $\Vert Au\Vert <a^{\prime }$。事实上，若 $u\in \overline{P_{a^{\prime }}}$，则由(H₁)知，

$\begin{array}{c}\Delta Au\left(n\right)=\{\begin{array}{l}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{n-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)\ge 0,n\in N\left[0,n_0\right]\\ {\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{n-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)\le 0,n\in N\left[n_0,N+1\right]\end{array}\Vert Au\Vert =Au\left(n_0+1\right)\\ =\frac{\beta }{\alpha }{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \le \frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{n_0}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{n_0}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \le \frac{a^{\prime }}{L_1}\left[\frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{n_0}{\sum }}h\left(j\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{n_0}{\sum }}h\left(j\right)}\right)}\right]=a^{\prime }\end{array}$，

和

$\begin{array}{c}\Vert Au\Vert =Au\left(n_0+1\right)\\ =\frac{\delta }{\gamma }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)-A_u}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)-A_u}\right)}\\ \le \frac{\delta }{\gamma }{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \le \frac{a^{\prime }}{L_2}\left[\frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{\eta -1}{\sum }}h\left(j\right)}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{i-1}{\sum }}h\left(j\right)}\right)}\right]<a^{\prime }\end{array}$。

同理，条件(H₂)保证，存在常数 $c^{\prime }>d^{\prime }$，使得

$A:\overline{P_{c^{\prime }}}\to \overline{P_{c^{\prime }}}$。 (3)

由条件(H₂)和(3)式易知，取 $e^{\prime }=c^{\prime }$ 即可。

其次证明引理 [6] 中的条件(i)满足。为了验证条件成立，假设 $u\left(t\right)\equiv \frac{b^{\prime }+d^{\prime }}{2}$。

注意到 $\varphi \left(u\left(t\right)\right)=\varphi \left(\frac{b^{\prime }+d^{\prime }}{2}\right)=\frac{b^{\prime }+d^{\prime }}{2}>b^{\prime }$，因此 $\left\{u\in P\left(\varphi ,b^{\prime },d^{\prime }\right):\varphi \left(u\right)>b^{\prime }\right\}\ne \varnothing$。

令 $u\in P\left(\varphi ,b^{\prime },d^{\prime }\right)$，则 $\varphi \left(u\right)>b^{\prime }$，因此有 $b^{\prime }\le \Vert u\Vert \le d^{\prime }$，由引理 [4] 知， $\frac{w}{N+2}{b}^{\prime }\le \frac{w}{N+2}\Vert u\Vert \le u\left(n\right)\le d^{\prime }$， $n\in \left[w,N+2-w\right]$，从而 $f\left(u\right)>{\phi }_p\left(\frac{2b^{\prime }}{wl}\right)$。

若 $n_0+1\in \left(0,w\right]$，则对任意的 $u\in P\left(\varphi ,b^{\prime },d^{\prime }\right)$，有

$\begin{array}{c}\varphi \left(Au\right)=\frac{1}{2}\left(Au\left(w\right)+Au\left(N+2-w\right)\right)\ge Au\left(N+2-w\right)\\ \ge \frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=n_0+1}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=n_0+1}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \ge {\displaystyle \underset{N+2-w}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=n_0+1}{\overset{N+1-w}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}>\frac{2b^{\prime }}{wl}w{\phi }_q\left({\displaystyle \underset{j=n_0+1}{\overset{N+1-w}{\sum }}h\left(j\right)}\right)>b^{\prime }\end{array}$，

若 $n_0+1\in \left(w,N+2-w\right)$，则对 $u\in P\left(\varphi ,b^{\prime },d^{\prime }\right)$，有

$\begin{array}{c}2\varphi \left(Au\right)=Au\left(w\right)+Au\left(N+2-w\right)\\ \ge {\displaystyle \underset{i=0}{\overset{w-1}{\sum }}{\phi }_q}\left(A_u-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=N+2-w}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)-A_u}\right)}\\ \ge {\displaystyle \underset{i=0}{\overset{w-1}{\sum }}{\phi }_q}\left({\displaystyle \underset{j=i}{\overset{n_0-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=N+2-w}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=n_0+1}{\overset{N+1-w}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ >\frac{2b^{\prime }}{wl}w\left[{\phi }_q\left({\displaystyle \underset{j=w-1}{\overset{n_0-1}{\sum }}h\left(j\right)}\right)+{\phi }_q\left({\displaystyle \underset{j=n_0+1}{\overset{N+1-w}{\sum }}h\left(j\right)}\right)\right]>2b^{\prime }\end{array}$，

从而 $\varphi \left(Au\right)>b^{\prime }$。

若 $n_0+1\in \left[N+2-w,N+2\right]$，则对任意的 $u\in P\left(\varphi ,b^{\prime },d^{\prime }\right)$，有

$\begin{array}{c}\varphi \left(Au\right)=\frac{1}{2}\left(Au\left(w\right)+Au\left(N+2-w\right)\right)\ge Au\left(w\right)\\ \ge {\displaystyle \underset{i=0}{\overset{w-1}{\sum }}{\phi }_q}\left({\displaystyle \underset{j=0}{\overset{n_0-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)\\ ={\displaystyle \underset{i=0}{\overset{w-1}{\sum }}{\phi }_q}\left({\displaystyle \underset{j=i}{\overset{n_0-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)\\ >\frac{2b^{\prime }}{wl}w{\phi }_q\left({\displaystyle \underset{j=w-1}{\overset{n_0-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)>b^{\prime }\end{array}$，

因此，对任意的 $u\in P\left(\varphi ,b^{\prime },d^{\prime }\right)$，有 $\varphi \left(Au\right)>b^{\prime }$。

最后验证引理 [6] 中的条件(iii)成立。设 $u\in P\left(\varphi ,b^{\prime },c^{\prime }\right)$，且 $\Vert Au\Vert >d^{\prime }$，由条件 $d^{\prime }\ge \frac{N+2}{w}{b}^{\prime }$，知

$\varphi \left(Au\right)=Au\left(w\right)+Au\left(N+2-w\right)\ge \frac{w}{N+2}\Vert Au\Vert >\frac{w}{N+2}{d}^{\prime }\ge b^{\prime }$。

综上所述边值问题(1)和(2)至少存在三个正解 $u_1$， $u_2$， $u_3$ 满足 $0\le \Vert u_1\Vert <a^{\prime }$， $\varphi \left(u_2\right)>b^{\prime }$， $\Vert u_3\Vert >a^{\prime }$， $\varphi \left(u_3\right)<b^{\prime }$。

定理2：如果把定理1的条件(H₂)替换成

( ${H^{\prime }}_{\text{2}}$ )： $\underset{u\to \infty }{\lim \sup }<{\phi }_p\left(\frac{1}{L_i}\right)$， $i=1,2$。

则定理1的结论依然成立。

证明：只需证明存在常数 $c^{\prime }$ 使得 $c^{\prime }>d^{\prime }$， $A:\overline{P_{c^{\prime }}}\to \overline{P_{c^{\prime }}}$。

由条件( ${H^{\prime }}_{\text{2}}$ )知存在 $M^{\prime }$， $\epsilon <{\phi }_p\left(\frac{1}{L_i}\right)$ 使得

$\frac{f\left(u\right)}{{\phi }_p\left(u\right)}\le \epsilon$， $u\ge M^{\prime }$。 (4)

令 $G=\underset{u\in \left[0,M^{\prime }\right]}{\max }f\left(u\right)$，由(4)式易得

$f\left(u\right)\le G+\epsilon {\phi }_p\left(u\right)$， $u\ge 0$。 (5)

取 $c^{\prime }$ 满足

${\phi }_p\left(c^{\prime }\right)>\max \left\{{\phi }_p\left(d^{\prime }\right),G{\left({\phi }_p\left(\frac{1}{L_1}-\epsilon \right)\right)}^{-1},G{\left({\phi }_p\left(\frac{1}{L_2}-\epsilon \right)\right)}^{-1}\right\}$。 (6)

则对任意的 $u\in \overline{P_{c^{\prime }}}$，由(5)式和(6)式可得

$\begin{array}{c}\Vert Au\Vert =Au\left(n_0+1\right)=\frac{\beta }{\alpha }{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left(A_u-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \le \frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{n_0}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}-{\displaystyle \underset{j=0}{\overset{\xi -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{n_0}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)-{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}}\right)}\\ \le \frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{n_0}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{n_0}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \le \frac{\beta }{\alpha }{\phi }_q\left({\displaystyle \underset{j=\xi }{\overset{n_0}{\sum }}h\left(j\right)\left(G+\epsilon {\varphi }_p\left(u\right)\right)}\right)+{\displaystyle \underset{i=0}{\overset{n_0}{\sum }}{\phi }_q\left({\displaystyle \underset{j=i}{\overset{n_0}{\sum }}h\left(j\right)\left(G+\epsilon {\varphi }_p\left(u\right)\right)}\right)}\\ \le {\phi }_q\left(G+\epsilon {\phi }_p\left(c^{\prime }\right)\right)L_1<L_1{\phi }_q\left({\phi }_p\left(c^{\prime }\right)\left({\varphi }_p\left(\frac{1}{L_1}\right)-\epsilon \right)+\epsilon {\phi }_p\left(c^{\prime }\right)\right)\le c^{\prime }\end{array}$，

及

$\begin{array}{c}\Vert Au\Vert =Au\left(n_0+1\right)\\ =\frac{\delta }{\gamma }{\phi }_q\left({\displaystyle \underset{j=0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)-A_u}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)-A_u}\right)}\\ \le \frac{\delta }{\gamma }{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{\eta -1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)+{\displaystyle \underset{i=n_0+1}{\overset{N+1}{\sum }}{\phi }_q\left({\displaystyle \underset{j=n_0}{\overset{i-1}{\sum }}h\left(j\right)f\left(u\left(j\right)\right)}\right)}\\ \le L_2{\phi }_q\left({\phi }_p\left(c^{\prime }\right)\left({\phi }_p\left(\frac{1}{L_2}\right)-\epsilon \right)+\epsilon {\phi }_p\left(c^{\prime }\right)\right)\le c^{\prime }\end{array}$。

由上面的证明可知 $A:\overline{P_{c^{\prime }}}\to \overline{P_{c^{\prime }}}$。定理证毕。

定理3：假设存在常数满足 $0<{a^{\prime }}_1<\frac{w}{N+2}{b^{\prime }}_1<{b^{\prime }}_1<\frac{N+2}{w}{b^{\prime }}_1\le {d^{\prime }}_1<{a^{\prime }}_2<\frac{w}{N+2}{b^{\prime }}_2<{b^{\prime }}_2<\frac{N+2}{w}{b^{\prime }}_2\le {d^{\prime }}_2<\cdots <{a^{\prime }}_n$， $n\in N$ 使得

(H₄)： $f\left(u\right)<{\phi }_p\left(\frac{{a^{\prime }}_i}{L_j}\right)$， $j=1,2$， $0\le u\le {a^{\prime }}_i$ ；

(H₅)： $f\left(u\right)>{\phi }_p\left(\frac{2{b^{\prime }}_i}{wl}\right)$， $\frac{w}{N+2}{b^{\prime }}_i\le u\le {d^{\prime }}_i$。

则边值问题(1)和(2)至少存在 $2n-1$ 个正解。

证明：当n = 1时，由条件(H₄)可知A： $\stackrel{¯}{P_{{a^{\prime }}_1}}\to \stackrel{¯}{P_{{a^{\prime }}_1}}\subset \stackrel{¯}{P_{{a^{\prime }}_1}}$，根据Schauder不动点定理可知A至少有一个不动点，即问题至少存在一个正解。

当n = 2时，取 $c^{\prime }={a^{\prime }}_2$，则定理1条件满足，从而至少存在三个不同正解。

以此类推，用归纳法可知问题至少存在 $2n-1$ 个正解。

致谢

作者感谢编辑和审稿人给予的指导和帮助。

基金项目

济南大学泉城学院科研项目(18JDQYKY17)。