一类分数阶脉冲微分方程边值问题解的存在性

Existence of Solutions to Boundary Value Problems of a Class of Fractional Order Impulsive Differential Equations

摘要

本文基于对一类带p-Laplacian算子的分数阶微分方程的研究，讨论了同时具有瞬时脉冲和非瞬时脉冲时该类分数阶微分方程在Dirichlet边值条件下解的存在性问题。利用Ricceri提出的临界点理论，证明了该类边值问题至少存在三个解，且得到了该类边值问题解的多重性依赖于两个参数的结论。最后举例说明了结果的适用性。

Abstract

In this paper, based on the study of a class of fractional-order differential equations with p-Laplacian operator, we discuss the existence of solutions for the class of fractional-order differential equations with both instantaneous and non-instantaneous pulses under Dirichlet boundary value conditions. By using the critical point theory proposed by Ricceri, it is proved that there are at least three solutions of this kind of boundary value problem, and the multiplicity of solutions of this kind of boundary value problem depends on two parameters. Finally, an example is given to illustrate the applicability of the results.

关键词

变分法 / 瞬时和非瞬时脉冲 / p-Laplacian算子 / 临界点理论

Key words

variational method / instantaneous and non-instantaneous pulses / p-Laplacian operator / critical point theory

0 引言

分数阶微分方程是近年来备受关注的一个领域。这既是由于分数阶微积分理论本身的深入发展，也是由于分数阶微积分在物理、力学、化学、工程等各个科学领域的应用^［1-4］。

脉冲微分方程是一类用来描述具有跳跃点（脉冲）的不连续发展过程的微分方程，它有着十分广泛的应用背景，如控制论、医学和生物学、火箭与宇宙飞船运动以及力学等领域中都会涉及此类方程^［5-8］。关于含有瞬时脉冲的分数阶微分方程解的存在性问题的研究已经取得了很大的进展。然而，许多现象不能用分数阶瞬时脉冲微分方程解的存在性问题来模拟，例如在某些演化过程的动力学中。2013年，Hern􀅡ndez和O’Regan首次提出了非瞬时脉冲的概念^［9］，它可以刻画在任意固定点处突然开始并在有限时间区间内持续的脉冲跳跃。这种类型的脉冲更多是在模拟演化过程的动力学方面具有优势，因此，涉及这种脉冲的微分方程受到了广泛的关注^［10-11］。Khaliq和Rehman在文献［12］中利用Lax-Milgram定理研究了下面一类分数阶非瞬时脉冲微分方程边值问题的弱解存在性，

\left\{\begin{array}{ll}{}_t{}^{}{D}_T^{\alpha }\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)=f_i\left(t\right),& t\in \left.\left(s_i,t_{i+\mathrm{1}}\right.\right],i=\mathrm{0,1},\cdots ,n,\\ {}_t{}^{}{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)=c_i,& t\in \left.\left(t_i,s_i\right.\right],i=\mathrm{1,2},\cdots ,n,\\ {}_t{}^{}{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_i^{-}\right)\right)={}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_i^{+}\right)\right),& i=\mathrm{1,2},\cdots ,n,\\ {}_t{}^{}{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_i^{-}\right)\right)=c_{\mathrm{0}},& u\left(\mathrm{0}\right)=u\left(T\right)\mathrm{。}\end{array}\right.（1）

文献［13］中利用变分法得到了同时具有瞬时脉冲和非瞬时脉冲的分数阶脉冲微分方程边值问题的一个经典解，

\left\{\begin{array}{ll}{}_t{}^{}{D}_T^{\alpha }\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)=f_i\left(t,u\left(t\right)\right),& t\in \left.\left(s_i,t_{i+\mathrm{1}}\right.\right],i=\mathrm{0,1},\cdots ,n,\\ \mathrm{\Delta }\left({}_t{}^{}{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\right)\right)\left(t_i\right)=I_i\left(u\left(t_i\right)\right),& i=\mathrm{1,2},\cdots ,n,\\ {}_t{}^{}{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)={}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t_i^{+}\right)\right),& t\in \left.\left(t_i,s_i\right.\right],i=\mathrm{1,2},\cdots ,n,\\ {}_t{}^{}{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_i^{-}\right)\right)={}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_i^{+}\right)\right),& i=\mathrm{1,2},\cdots ,n;u\left(\mathrm{0}\right)=u\left(T\right)\mathrm{。}\end{array}\right.（2）

受上述工作的启发，本文研究了如下一类同时具有瞬时和非瞬时脉冲的带p-Laplacian算子的分数阶微分方程边值问题

\left\{\begin{array}{ll}\frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\mathrm{1}}{\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\alpha }{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right)\right)\right]+\lambda \nabla F\left(t,u\left(t\right)\right)=\mathrm{0},& t\in \left.\left(s_j,t_{j+\mathrm{1}}\right.\right],j=\mathrm{0,1},\cdots ,m,\\ \mathrm{\Delta }\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right)\right)\left(t_j\right)=\mu I_{j}u\left(t_j\right),& j=\mathrm{1,2},\cdots ,m,\\ \begin{array}{l}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right)\right)\left(t\right)=\\ \left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right)\right)\left(t_{{}_j}^{+}\right),\end{array}& t\in \left.\left(t_j,s_j\right.\right],j=\mathrm{1,2},\cdots ,m,\\ \begin{array}{l}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right)\right)\left(t_{{}_j}^{-}\right)=\\ \left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right)\right)\left(t_{{}_j}^{+}\right),\end{array}& j=\mathrm{1,2},\cdots ,m;u\left(\mathrm{0}\right)=u\left(T\right)=\mathrm{0}\end{array}\right.（3）

解的存在性。与问题（2）相比，我们在更一般的p>\mathrm{1}的情况下寻找问题（3）的解，并将线性微分算子{}_{\mathrm{0}}{}^c{}_t^{\alpha }{}_{\mathrm{0}}{}^{}D{}_t^{\alpha }和{}_t{}^c{}_T^{\alpha }{}_t{}^{}D{}_T^{\alpha }推广到非线性微分算子{}_{\mathrm{0}}{}^c{}_t^{\alpha }{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }\right)和{}_t{}^c{}_T^{\alpha }{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }\right)，这意味着我们的问题（3）包含了问题（2）在p=\mathrm{2}和\mu =\lambda =\mathrm{1}时的特例。其中，，\lambda和\mu是两个正实参数，{\Phi }_p\left(s\right)={\left|s\right|}^{p-\mathrm{2}}s\hspace{1em}\left(s\ne \mathrm{0}\right),\hspace{1em}{\Phi }_p\left(\mathrm{0}\right)=\mathrm{0}，{}_{\mathrm{0}}{}^{}{}_t^{\alpha }和{}_t{}^{}{}_T^{\alpha }分别是左、右Riemann-Liouville分数阶导数，{}_{{}_{\mathrm{0}}{}^{}}{}^c{}_t^{\alpha }和{}_t{}^c{}_T^{\alpha }分别是左、右Caputo分数阶导数，\alpha \in \left.\left(\frac{\mathrm{1}}{p},\mathrm{1}\right.\right]。F:\left[\mathrm{0},T\right]\times {\mathbb{R}}^N\to \mathbb{R}是一个连续函数，\nabla F\left(t,u\left(t\right)\right)是F在u\left(t\right)处的梯度。对于任意的j=\mathrm{1,2},\dots ,m，都有，瞬时脉冲I_j\in C\left(\mathbb{R},\mathbb{R}\right)在点t_j处开始突变，非瞬时脉冲在有限区间\left.\left(t_j,s_j\right.\right]内持续，且有

{}_{\mathrm{0}}{}^c{}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\left(s_j^{\pm }\right)=\underset{t\to s_j^{\pm }}{\mathrm{l}\mathrm{i}\mathrm{m}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\left(t\right)\right)，

{}_{\mathrm{0}}{}^c{}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\left(t_j^{\pm }\right)=\underset{t\to t_j^{\pm }}{\mathrm{l}\mathrm{i}\mathrm{m}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\left(t\right)\right)，

\mathrm{\Delta }\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\right)\left(t_j\right)=\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\right)\left(t_j^{+}\right)-

\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha -\mathrm{1}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\right)-{}_t{}^{c}D{D}_T^{\alpha -\mathrm{1}}{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\right)\right)\left(t_{{}_j}^{-}\right)。

1 预备知识

设X是一个实Banach空间。用{\mathrm{\Gamma }}_X表示所有满足如下性质的泛函\phi :X\to \mathbb{R}的类：如果序列{\left\{x_n\right\}}_{n\in N}弱收敛于X且\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\phi \left(x_n\right)\le \phi \left(x\right)，那么，存在一个{\left\{x_n\right\}}_{n\in N}的子序列强收敛于x。

定理1^［14］ 设X是一个可分自反的实Banach空间，\phi :X\to \mathbb{R}是一个强制的、弱下半连续的C^{\mathrm{1}}函数，在X的每个有界子集上有界，与{\mathrm{\Gamma }}_X有关，其导数在X^{\mathrm{*}}上有一个连续的逆。\omega :X\to \mathbb{R}是一个具有紧导数的C^{\mathrm{1}}函数。假设\phi存在一个严格的局部最小值u_{\mathrm{0}}，使得\phi \left(u_{\mathrm{0}}\right)=\omega \left(u_{\mathrm{0}}\right)=\mathrm{0}。最后，令

{\rho }_{\mathrm{1}}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{0},\underset{‖u‖\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\omega \left(u\right)}{\phi \left(u\right)},\underset{u\to u_{\mathrm{0}}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\omega \left(u\right)}{\phi \left(u\right)}\right\}， {\rho }_{\mathrm{2}}=\underset{u\in {\phi }^{-\mathrm{1}}\left(\mathrm{0},+\mathrm{\infty }\right)}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\omega \left(u\right)}{\phi \left(u\right)}，（4）

假设，则对于每个紧区间\left[{\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\right]\in \left(\frac{\mathrm{1}}{{\rho }_{\mathrm{2}}},\frac{\mathrm{1}}{{\rho }_{\mathrm{1}}}\right)（规定\frac{\mathrm{1}}{\mathrm{0}}=+\mathrm{\infty },\frac{\mathrm{1}}{+\mathrm{\infty }}=\mathrm{0}），存在R>\mathrm{0}满足以下性质：对于每个\lambda \in \left[{\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\right]以及任意具有紧导数的C^{\mathrm{1}}函数\phi :X\to \mathbb{R}，存在\xi >\mathrm{0}，使得对于每一个\mu \in \left[\mathrm{0},\xi \right]，方程{\phi }^{\text{'}}\left(u\right)-\mu {\phi }^{\text{'}}\left(u\right)-\lambda {\omega }^{\text{'}}\left(u\right)=\mathrm{0}在X中至少有三个范数小于R的解。

定义1^［15-16］ 设，n\in \mathbb{N}，t\in \left[\mathrm{0},T\right]。分别用{}_{\mathrm{0}}{}^{}{}_t^{\alpha }\left(u\left(t\right)\right)和{}_t{}^{}{}_T^{\alpha }\left(u\left(t\right)\right)表示u\left(t\right)的左右Riemann-Liouville分数阶导数，且定义如下

{}_{\mathrm{0}}{}^{}{}_t^{\alpha }u\left(t\right)=\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}{}_{\mathrm{0}}{}^{}D{}_t^{\alpha -n}u\left(t\right)=\frac{\mathrm{1}}{\mathrm{\Gamma }\left(n-\alpha \right)}\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}\left({\int }_{\mathrm{0}}^t{\left(t-\tau \right)}^{n-\alpha -\mathrm{1}}u\left(\tau \right)\mathrm{d}\tau \right)，

{}_t{}^{}{}_T^{\alpha }u\left(t\right)={\left(-\mathrm{1}\right)}^n\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}{}_t{}^{}D{}_T^{\alpha -n}u\left(t\right)=\frac{{\left(-\mathrm{1}\right)}^n}{\mathrm{\Gamma }\left(n-\alpha \right)}\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}\left({\int }_t^T{\left(\tau -t\right)}^{n-\alpha -\mathrm{1}}u\left(\tau \right)\mathrm{d}\tau \right)。

定义2^［15-16］ 设，n\in \mathbb{N}，u\left(t\right)\in A{C}^n\left(\left[\mathrm{0},T\right],\mathbb{R}\right)。分别用{}_{{}_{\mathrm{0}}{}^{}}{}^c{}_t^{\alpha }\left(u\left(t\right)\right)和{}_{{}_t{}^{}}{}^c{}_T^{\alpha }\left(u\left(t\right)\right)表示u\left(t\right)的左右Caputo分数阶导数，且定义如下

{}_{{}_{\mathrm{0}}{}^{}}{}^c{}_t^{\alpha }u\left(t\right)=\frac{\mathrm{1}}{\mathrm{\Gamma }\left(n-\alpha \right)}\left({\int }_{\mathrm{0}}^t{\left(t-\tau \right)}^{n-\alpha -\mathrm{1}}{u}^n\left(\tau \right)\mathrm{d}\tau \right)，

{}_t{}^c{}_T^{\alpha }u\left(t\right)=\frac{{\left(-\mathrm{1}\right)}^n}{\mathrm{\Gamma }\left(n-\alpha \right)}\left({\int }_t^T{\left(\tau -t\right)}^{n-\alpha -\mathrm{1}}{u}^n\left(\tau \right)\mathrm{d}\tau \right)，

且有

{}_{\mathrm{0}}{}^c{}_t^{\alpha }u\left(t\right)={}_{\mathrm{0}}{}^{}D{}_t^{\alpha }u\left(t\right)-{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}\frac{u^i\left(\mathrm{0}\right)}{\mathrm{\Gamma }\left(i-\alpha +\mathrm{1}\right)}{\left(t\right)}^{i-\alpha }， {}_t{}^c{}_T^{\alpha }u\left(t\right)={}_t{}^{}D{}_T^{\alpha }u\left(t\right)-{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}\frac{u^i\left(T\right)}{\mathrm{\Gamma }\left(i-\alpha +\mathrm{1}\right)}{\left(T-t\right)}^{i-\alpha }。（5）

现在，我们定义一个赋范空间\left(X_{\alpha }^p,{‖\cdot ‖}_{\alpha }^p\right)

X_{\alpha }^p=\left\{u\left(t\right)\in L^p\left(\left[\mathrm{0},T\right],\mathbb{R}\right)|{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }u\left(t\right)\in L^p\left(\left[\mathrm{0},T\right],\mathbb{R}\right),u\left(\mathrm{0}\right)=u\left(T\right)=\mathrm{0}\right\}，（6）

且对于任意的u\left(t\right)\in X_{\alpha }^p有范数

{‖u‖}_{\alpha }^p={\left({\int }_{\mathrm{0}}^T{\left|u\left(t\right)\right|}^p\mathrm{d}t+{\int }_{\mathrm{0}}^T{\left|{}_{\mathrm{0}}{}^{}D{D}_t^{p}u\left(t\right)\right|}^p\mathrm{d}t\right)}^{\frac{\mathrm{1}}{p}}。（7）

如文献［16］所示，X_{\alpha }^p是一个自反的Banach空间。特别是，鉴于式（5），对于任意的u\left(t\right)\in X_{\alpha }^p我们可以推断出

{}_{\mathrm{0}}{}^{}{}_t^{\alpha }u\left(t\right)={}_{\mathrm{0}}{}^{c}D{}_t^{\alpha }u\left(t\right)， {}_t{}^{}{}_T^{\alpha }u\left(t\right)={}_t{}^{c}D{}_T^{\alpha }u\left(t\right)。（8）

引理1 （文献［17］命题3.3） 设，。对于所有的u\in X_{\alpha }^p，有

{‖u‖}_{L^p}\le \frac{T^{\alpha }}{\mathrm{\Gamma }\left(\alpha +\mathrm{1}\right)}{‖{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u‖}_{L^p}。

此外，如果\alpha >\frac{\mathrm{1}}{\mathrm{p}}和\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}=\mathrm{1}，那么

{‖u‖}_{\mathrm{\infty }}\le \frac{T^{\alpha -\mathrm{1}/P}}{\mathrm{\Gamma }\left(\alpha \right){\left(\left(\alpha -\mathrm{1}\right)q+\mathrm{1}\right)}^{\mathrm{1}/q}}{‖{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u‖}_{L^P}。

其中{‖u‖}_{L^p}={\left({\int }_{\mathrm{0}}^T{\left|u\left(t\right)\right|}^p\mathrm{d}t\right)}^{\mathrm{1}/p}和{‖u‖}_{\mathrm{\infty }}=\underset{t\in \left[\mathrm{0},T\right]}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|u\left(t\right)\right|。令U=\frac{T^{\alpha }}{\mathrm{\Gamma }\left(\alpha +\mathrm{1}\right)}，B=\frac{T^{\alpha -\frac{\mathrm{1}}{p}}}{\mathrm{\Gamma }\left(\alpha \right){\left(\left(\alpha -\mathrm{1}\right)q+\mathrm{1}\right)}^{\frac{\mathrm{1}}{q}}}。

注1^［17］ 根据引理1，范数（7）式与{‖u‖}_{\alpha }^p={‖{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u‖}_{L^p}相同。

为了在进一步的讨论中简化符号，在本文的其余部分中，用\left(X,‖\cdot ‖\right)代替\left(X_{\alpha }^p,{‖\cdot ‖}_{\alpha }^p\right)来表示赋范空间。

引理2 （文献［17］命题3.4） 设\alpha \in \left.\left(\frac{\mathrm{1}}{p},\mathrm{1}\right.\right]，。假设任意序列{\left\{u_n\right\}}_{n\in N}在X中弱收敛于u。 那么在C\left(\left[\mathrm{0},T\right],\mathbb{R}\right)中当n\to \mathrm{\infty }时有u_k\to u， 即n\to \mathrm{\infty }时，{‖u_n-u‖}_{\mathrm{\infty }}\to \mathrm{0}。

引理3^［16］ 设\alpha >\mathrm{0}，p\ge \mathrm{1}，q\ge \mathrm{1}，\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}\le \mathrm{1}+\alpha（当p,q\ne \mathrm{1}时\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}=\mathrm{1}+\alpha）。如果u\in L^p\left(a,b\right)，v\in L^q\left(a,b\right)，则有{\int }_a^b\left({}_a{}^{}{D}_t^{\alpha }u\left(t\right)\right)v\left(t\right)\mathrm{d}t={\int }_a^{b}u\left(t\right)\left({}_t{}^{}{D}_b^{\alpha }v\left(t\right)\right)\mathrm{d}t。

引理4 假设u\left(t\right)\in X_{\alpha }^p是边值问题（Boundary Value Problems， BVP）（3）式的一个弱解，如果满足下面定义的关系，即对于任意的v\left(t\right)\in X有

\frac{\mathrm{1}}{\mathrm{2}}\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right){}_t{}^{}D{D}_T^{\alpha }v\left(t\right)+{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right){}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }v\left(t\right)\right]\mathrm{d}t-\mu {\displaystyle \sum _{j=\mathrm{1}}^m}{I}_j\left(u\left(t_j\right)\right)v\left(t_j\right)=\lambda {\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_j+\mathrm{1}}\nabla F_j\left(t,u\left(t\right)\right)v\left(t\right)\mathrm{d}t\mathrm{。}

证明 由引理3，{}_t{}^{}{}_T^{\alpha }u\left(t\right)=-\frac{\mathrm{d}}{\mathrm{d}t}{}_t{}^{}D{}_T^{\alpha -\mathrm{1}}u\left(t\right)和（8）式，我们得到对于任意的u,v\in X，有

\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[\left({\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right),{}_t{}^{}D{D}_T^{\alpha }v\left(t\right)\right)+\left({\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right),{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }v\left(t\right)\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[\left({\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right),{}_t{}^{c}D{D}_T^{\alpha }v\left(t\right)\right)+\left({\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right),{}_{\mathrm{0}}{}^{c}D{D}_t^{\alpha }v\left(t\right)\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v^{\text{'}}\left(t\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v^{\text{'}}\left(t\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}{}_{\mathrm{0}}{}^{c}D{D}_t^{\alpha }u\left(t\right){}_t{}^{c}D{D}_T^{\alpha }v\left(t\right)+{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}{}_t{}^{c}D{D}_T^{\alpha }u\left(t\right){}_{\mathrm{0}}{}^{c}D{D}_t^{\alpha }v\left(t\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}\left[{}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v^{\text{'}}\left(t\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v^{\text{'}}\left(t\right)\right]\mathrm{d}t+\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{1}}^m}{\int }_{t_j}^{s_j}\left[{}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v^{\text{'}}\left(t\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v^{\text{'}}\left(t\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{0}}^m}{}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v\left(t\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v\left(t\right)\left|{}_{s_j^{+}}^{t_{j+\mathrm{1}}^{-}}+\right.\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}\left[{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v\left(t\right)-{}_t{}^{}D{D}_T^{\alpha }{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v\left(t\right)\right]\mathrm{d}t+\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{1}}^m}{}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v\left(t\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v\left(t\right)\left|{}_{t_j^{+}}^{s_j^{-}}\right.+\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{1}}^m}{\int }_{t_j}^{s_j}\left[\frac{\mathrm{d}}{\mathrm{d}t}\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)\right)v\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}\left({}_t{}^{}{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)\right)v\left(t\right)\right]\mathrm{d}t,\end{array}（9）

然后，问题（3）上的给定条件，将（9）式变化

\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}{}_{\mathrm{0}}{}^{c}D{D}_t^{\alpha }u\left(t\right){}_t{}^{c}D{D}_T^{\alpha }v\left(t\right)+{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}{}_t{}^{c}D{D}_T^{\alpha }u\left(t\right){}_{\mathrm{0}}{}^{c}D{D}_t^{\alpha }v\left(t\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}\left[{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v\left(t\right)-{}_t{}^{}D{D}_T^{\alpha }{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v\left(t\right)\right]\mathrm{d}t-\\ {\displaystyle \sum _{j=\mathrm{1}}^m}\mathrm{\Delta }\left[\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t_j\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t_j\right)\right)\right)v\left(t_j\right)\right.\left.-\left({}_t{}^{}{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t_j\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t_j\right)\right)\right)v\left(t_j\right)\right]+\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{1}}^m}{}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_j^{-}\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_j^{-}\right)\right)v\left(s_j\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_j^{-}\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(s_j^{-}\right)\right)v\left(s_j\right)-\\ \left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_j^{+}\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_j^{+}\right)\right)v\left(s_j\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(s_j^{+}\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(s_j^{+}\right)\right)v\left(s_j\right)\right)+\\ {}_{\mathrm{0}}{}^{}{}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(T\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(T\right)\right)v\left(T\right)-{}_t{}^{}D{}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(T\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(T\right)\right)v\left(T\right)-\\ \left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(\mathrm{0}\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(\mathrm{0}\right)\right)v\left(\mathrm{0}\right)-{}_t{}^{}D{D}_T^{\alpha -\mathrm{1}}{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(\mathrm{0}\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(\mathrm{0}\right)\right)v\left(\mathrm{0}\right)\right)=\\ \frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}\left[{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right)v\left(t\right)-{}_t{}^{}D{D}_T^{\alpha }{\left|{}_{\mathrm{0}}{}^c{D}_t^{\alpha }u\left(t\right)\right|}^{p-\mathrm{2}}\left({}_t{}^c{D}_T^{\alpha }u\left(t\right)\right)v\left(t\right)\right]\mathrm{d}t-\mu {\displaystyle \sum _{j=\mathrm{1}}^m}{I}_j\left(u\left(t_j\right)\right)v\left(t_j\right)\mathrm{。}\end{array}（10）

将v\left(t\right)代入（3）式的方程，先从s_j到t_{j+\mathrm{1}}两边积分，再从j=\mathrm{0}到j=m相加，然后从（10）式中推导得

\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right){}_t{}^{}D{}_T^{\alpha }v\left(t\right)+{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right){}_{\mathrm{0}}{}^{}D{}_t^{\alpha }v\left(t\right)\mathrm{d}t-\mu {\displaystyle \sum _{j=\mathrm{1}}^m}{I}_j\left(u\left(t_j\right)\right)v\left(t_j\right)=\lambda {\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_j+\mathrm{1}}\nabla F_j\left(t,u\left(t\right)\right)v\left(t\right)\mathrm{d}t\mathrm{。}

证毕。

接下来，我们定义一些函数\phi ,\varphi ,\omega :X\to \mathbb{R}如下

\phi \left(u\right)=-\frac{\mathrm{1}}{p}{\int }_{\mathrm{0}}^T\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right),{}_t{}^{}D{D}_T^{\alpha }u\left(t\right)\right)\mathrm{d}t，（11）

\varphi \left(u\right)=-{\displaystyle \sum _{j=\mathrm{1}}^m}{\int }_{\mathrm{0}}^{\left(u\left(t_j\right)\right)}{I}_j\left(s\right)\mathrm{d}s， \omega \left(u\right)={\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}{F}_j\left(t,u\left(t\right)\right)\mathrm{d}t。（12）

通过标准论证表明，\phi \in C^{\mathrm{1}}\left(X,\mathbb{R}\right)和\varphi和\omega具有连续的Gâteaux导数。对于任何v\left(t\right)\in X，它们的Gâteaux导数如下

{\phi }^{\text{'}}\left(u\right)\left(v\right)=-\frac{\mathrm{1}}{p}\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u\left(t\right)\right){}_t{}^{}D{D}_T^{\alpha }v\left(t\right)+{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }u\left(t\right)\right){}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }v\left(t\right)\right]\mathrm{d}t，（13）

{\varphi }^{\text{'}}\left(u\right)\left(v\right)=-{\displaystyle \sum _{j=\mathrm{1}}^m}{I}_j\left(u\left(t_j\right)\right)v\left(t_j\right)， {\omega }^{\text{'}}\left(u\right)\left(v\right)={\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}\nabla F_j\left(t,u\left(t\right)\right)v\left(t\right)\mathrm{d}t。（14）

显然，我们寻找问题（3）的弱解的一种方法是求函数\phi -\mu \varphi -\lambda \omega的临界点。此外，与文献［13］中的引理3类似，BVP（3）的弱解也是一个经典解。

2 主要结果

在本节中，我们利用定理1，建立了BVP（3）解的多重性。

定理2 假设存在非负常数k_j，j=\mathrm{0,1},\cdots ,m和一个函数\eta \left(t\right)\in X\backslash \left\{\mathrm{0}\right\}，使得

\mathrm{m}\mathrm{a}\mathrm{x}\left\{\underset{\left|x\right|\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{F_j\left(t,x\right)}{{\left|x\right|}^p},\underset{\left|x\right|\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{F_j\left(t,x\right)}{{\left|x\right|}^p}\right\}\le k_j， j=\mathrm{0,1},\cdots ,m，（15）

和

\frac{{\sum }_{j=\mathrm{0}}^m{\int }_{s_j}^{t_{j+\mathrm{1}}}{F}_j\left(t,\eta \left(t\right)\right)\mathrm{d}t}{{‖\eta ‖}^p}>T{B}^p\underset{\mathrm{0}\le j\le m}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{k_j\right\}。（16）

则对于每个紧区间\left[{\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\right]\subset \left(\frac{\mathrm{1}}{{\rho }_{\mathrm{2}}},\frac{\mathrm{1}}{{\rho }_{\mathrm{1}}}\right)（{\rho }_{\mathrm{1}},{\rho }_{\mathrm{2}}的定义见定理1），存在R>\mathrm{0}，使得对于每一个\lambda \in \left[{\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\right]，存在\xi >\mathrm{0}，对于每一个\mu \in \left[\mathrm{0},\xi \right]，BVP（3）在X中至少有三个解x_i，其中，i=\mathrm{1,2},\mathrm{3}。

证明 首先，我们考虑泛函\phi。假设M是X的一个子集的界，即‖x‖\le M在X的一个子集上。那么，由（11）式，我们有\left|\phi \left(x\right)\right|\le \frac{\mathrm{1}}{p}{‖x‖}^p\le \frac{M^p}{p}，这表明\phi是X的每个有界子集上的有界泛函。进一步，由（13）式，我们得到

\begin{array}{c}\left({\phi }^{\text{'}}\left(x\right)-{\phi }^{\text{'}}\left(y\right)\right)\left(x-y\right)=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)\right){}_t{}^{}D{D}_T^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)+{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }x\left(t\right)\right){}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }y\left(t\right)\right){}_t{}^{}D{D}_T^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)+{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }y\left(t\right)\right){}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\right]\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)\right)-{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }y\left(t\right)\right)\right]{}_t{}^{}D{D}_T^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\mathrm{d}t=\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }x\left(t\right)\right)-{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }y\left(t\right)\right)\right]{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\mathrm{d}t\mathrm{。}\end{array}（17）

考虑文献［18］中引入的不等式，即

（18）

并且利用Hölder不等式，当时，我们得到

\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T{\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)-{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }y\left(t\right)\right|}^p\mathrm{d}t\le \\ \frac{\mathrm{1}}{\mathrm{2}}{\left({\int }_{\mathrm{0}}^T{\left(\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)\right|+\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }y\left(t\right)\right|\right)}^p\mathrm{d}t\right)}^{\frac{\mathrm{2}-p}{\mathrm{2}}}{\left({\int }_{\mathrm{0}}^T\frac{{\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)-{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }y\left(t\right)\right|}^{\mathrm{2}}}{{\left(\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)\right|+\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }y\left(t\right)\right|\right)}^{\mathrm{2}-p}}\mathrm{d}t\right)}^{{}^{\frac{P}{\mathrm{2}}}}\le \end{array}

\frac{K}{\mathrm{2}}{\left({‖x‖}^p+{‖y‖}^p\right)}^{\frac{\mathrm{2}-p}{\mathrm{2}}}{\left({\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)\right)-{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }y\left(t\right)\right)\right]{}_t{}^{}D{D}_T^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\mathrm{d}t\right)}^{\frac{p}{\mathrm{2}}}。

同理可得

\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T{\left|{}_t{}^{}{D}_T^{\alpha }x\left(t\right)-{}_t{}^{}D{D}_T^{\alpha }y\left(t\right)\right|}^p\mathrm{d}t\le \frac{K}{\mathrm{2}}{\left({‖x‖}^p+{‖y‖}^p\right)}^{\frac{\mathrm{2}-p}{\mathrm{2}}}{\left({\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }x\left(t\right)\right)-{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }y\left(t\right)\right)\right]{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\mathrm{d}t\right)}^{\frac{p}{\mathrm{2}}}，

其中K={\mathrm{2}}^{\frac{\left(p-\mathrm{1}\right)\left(\mathrm{2}-p\right)}{\mathrm{2}}}，因此对于任意的x,y\in X,\hspace{1em}x\ne y，有

\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)\right)-{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }y\left(t\right)\right)\right]{}_t{}^{}D{D}_T^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\mathrm{d}t+\\ \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T\left[{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }x\left(t\right)\right)-{\Phi }_p\left({}_t{}^{}{D}_T^{\alpha }y\left(t\right)\right)\right]{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }\left(x\left(t\right)-y\left(t\right)\right)\mathrm{d}t\ge K^{-\frac{\mathrm{2}}{P}}{\left({‖x‖}^p+{‖y‖}^p\right)}^{\frac{P-\mathrm{2}}{P}}{‖x-y‖}^{\mathrm{2}}>\mathrm{0}\end{array}，

当p\ge \mathrm{2}，对于任意的x,y\in X,\hspace{1em}x\ne y时，结合（17）式和（18）式，我们有

\left({\phi }^{\text{'}}\left(x\right)-{\phi }^{\text{'}}\left(y\right)\right)\left(x-y\right)\ge \frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T{\left|{}_{\mathrm{0}}{}^{}{D}_t^{\alpha }x\left(t\right)-{}_{\mathrm{0}}{}^{}D{D}_t^{\alpha }y\left(t\right)\right|}^p\mathrm{d}t+\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathrm{0}}^T{\left|{}_t{}^{}{D}_T^{\alpha }x\left(t\right)-{}_t{}^{}D{D}_T^{\alpha }y\left(t\right)\right|}^p\mathrm{d}t>\mathrm{0}。

因此，对任意的p>\mathrm{1}，在上述讨论的基础上，我们得到\left({\phi }^{\text{'}}\left(x\right)-{\phi }^{\text{'}}\left(y\right)\right)\left(x-y\right)>\mathrm{0}，即{\phi }^{\text{'}}是严格单调算子。由文献［19］中的定理26可知，在X^{\mathrm{*}}上存在{\phi }^{\text{'}}的逆，且该逆是连续的。由（11）式，我们有\phi \left(u\right)=-\frac{\mathrm{1}}{p}{\int }_{\mathrm{0}}^T\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }u(t),{}_t{}^{}D{D}_T^{\alpha }u(t)\right)\mathrm{d}t\ge \frac{\mathrm{1}}{p\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\pi }\alpha \right)\right|}{‖u‖}_{\alpha }^p，这表明当‖u‖\to \mathrm{\infty }时，\phi \left(u\right)\to \mathrm{\infty }。因此，\phi是强制的。类似于文献［11］，我们可以得到\phi是序列弱下半连续的且属于{\mathrm{\Gamma }}_X。

另一方面，由于F_j和I_j的连续性，很容易证明\phi和\omega的导数是紧的。另外，存在泛函\phi的严格局部极小值0，使得\phi \left(\mathrm{0}\right)=\omega \left(\mathrm{0}\right)=\mathrm{0}。

根据式（15）知，存在{\epsilon }_{\mathrm{1}},{\epsilon }_{\mathrm{2}}>\mathrm{0}，使得对于任意的t\in \left[\mathrm{0},T\right],\hspace{1em}\left|x\right|\in \left(\mathrm{0},{\epsilon }_{\mathrm{1}}\right)\bigcup \left({\epsilon }_{\mathrm{2}},\mathrm{\infty }\right)，有

F_j\left(t,x\left(t\right)\right)\le k_j{\left|x\left(t\right)\right|}^p，（19）

由于F_j具有连续性，故存在r>\mathrm{0},\sigma >p，使得对于任意的t\in \left[\mathrm{0},T\right],x\in \mathbb{R}，有

F_j\left(t,x\left(t\right)\right)\le k_j{\left|x\left(t\right)\right|}^p+r{\left|x\left(t\right)\right|}^{\sigma }。（20）

记k^{\mathrm{*}}=\underset{\mathrm{0}\le j\le m}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{k_j\right\}，则基于（20）式和引理1，我们有

\omega \left(x\right)={\displaystyle \sum _{j=\mathrm{0}}^m}{\int }_{s_j}^{t_{j+\mathrm{1}}}{F}_j\left(t,x\left(t\right)\right)\mathrm{d}t\le k^{\mathrm{*}}T{B}^p{‖x‖}^p+rT{B}^{\sigma }{‖x‖}^{\sigma }，

因此对于任意的\sigma >p，有

\underset{x\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\omega \left(x\right)}{\phi \left(x\right)}\le \underset{x\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{k^{\mathrm{*}}T{B}^p{‖x‖}^p+rT{B}^{\sigma }{‖x‖}^{\sigma }}{\frac{\mathrm{1}}{p\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\pi }\alpha \right)\right|}{‖u‖}_{\alpha }^p}=p\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\pi }\alpha \right)\right|k^{\mathrm{*}}T{B}^p\le p{k}^{\mathrm{*}}T{B}^p，（21）

此外，由（19）式可得

\begin{array}{c}\underset{x\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\omega \left(x\right)}{\phi \left(x\right)}\le \\ \underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{{\sum }_{j=\mathrm{0}}^m{\int }_{\left|x\right|\le {\epsilon }_{\mathrm{2}}}{F}_j\left(t,x\left(t\right)\right)\mathrm{d}t}{\frac{\mathrm{1}}{p\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\pi }\alpha \right)\right|}{‖u‖}_{\alpha }^p}+\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{{\sum }_{j=\mathrm{0}}^m{\int }_{\left|x\right|>{\epsilon }_{\mathrm{2}}}{F}_j\left(t,x\left(t\right)\right)\mathrm{d}t}{\frac{\mathrm{1}}{p\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\pi }\alpha \right)\right|}{‖u‖}_{\alpha }^p}\le \\ \underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{{\sum }_{j=\mathrm{0}}^m{\int }_{\left|x\right|>{\epsilon }_{\mathrm{2}}}{F}_j\left(t,x\left(t\right)\right)\mathrm{d}t}{\frac{\mathrm{1}}{p\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\pi }\alpha \right)\right|}{‖u‖}_{\alpha }^p}\le p{k}^{\mathrm{*}}T{B}^P\mathrm{。}\end{array}（22）

因此，结合（21）式和（22）式，我们有{\rho }_{\mathrm{1}}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{0},\underset{‖u‖\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\omega \left(u\right)}{\phi \left(u\right)},\underset{u\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\omega \left(u\right)}{\phi \left(u\right)}\right\}\le p{k}^{\mathrm{*}}T{B}^p。

此外，利用（16）式，我们有

{\rho }_{\mathrm{2}}=\underset{x\in {\phi }^{-\mathrm{1}}\left(\left(\mathrm{0},+\mathrm{\infty }\right)\right)}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\omega \left(x\right)}{\phi \left(x\right)}=\underset{x\in X\backslash \left\{\mathrm{0}\right\}}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\omega \left(x\right)}{\phi \left(x\right)}\ge \frac{{\sum }_{j=\mathrm{0}}^m{\int }_{s_j}^{t_{j+\mathrm{1}}}{F}_j\left(t,\eta \left(t\right)\right)\mathrm{d}t}{\frac{\mathrm{1}}{p}{\int }_{\mathrm{0}}^T\left({}_{\mathrm{0}}{}^{}{D}_t^{\alpha }\eta \left(t\right),{}_t{}^{}D{D}_T^{\alpha }\eta \left(t\right)\right)\mathrm{d}t}>p{k}^{\mathrm{*}}T{B}^P\ge {\rho }_{\mathrm{1}}。

由定理1可知，定理2成立。

例 证明下列边值问题解的存在性

\left\{\begin{array}{ll}\frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\mathrm{1}}{\mathrm{2}}\left({}_{\mathrm{0}}{}^c{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}{\Phi }_p\left({}_t{}^{}{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)\right)\right]+\lambda \nabla F\left(t,u\left(t\right)\right)=\mathrm{0},& t\in \left.\left(s_j,t_{j+\mathrm{1}}\right.\right],j=\mathrm{0,1},\cdots ,m,\\ \mathrm{\Delta }\left({}_{\mathrm{0}}{}^c{D}_t^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_t{}^{}{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)\right)\left(t_j\right)=\mu I_{j}u\left(t_j\right),& j=\mathrm{1,2},\cdots ,m,\\ \begin{array}{l}\left({}_{\mathrm{0}}{}^c{D}_t^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_t{}^{}{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)\right)\left(t\right)=\\ \left({}_{\mathrm{0}}{}^c{D}_t^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_t{}^{}{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)\right)\left(t_{{}_j}^{+}\right),\end{array}& t\in \left.\left(t_j,s_j\right.\right],j=\mathrm{1,2},\cdots ,m,\\ \begin{array}{l}\left({}_{\mathrm{0}}{}^c{D}_t^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_t{}^{}{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)\right)\left(t_{{}_j}^{-}\right)=\\ \left({}_{\mathrm{0}}{}^c{D}_t^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_{\mathrm{0}}{}^{}{D}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)-{}_t{}^{c}D{D}_T^{-\frac{\mathrm{1}}{\mathrm{4}}}{\Phi }_p\left({}_t{}^{}{D}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)\right)\right)\left(t_{{}_j}^{+}\right),\end{array}& j=\mathrm{1,2},\cdots ,m;u\left(\mathrm{0}\right)=u\left(T\right)=\mathrm{0},\end{array}\right.（23）

其中。

证明 取F_j(t,u(t))={\mathrm{e}}^{-\left|u(t)\right|}{u}^{\mathrm{4}}(t)且有

u\left(t\right)=\left\{\begin{array}{ll}\mathrm{3}t,& t\in \left.\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{3}}\right.\right),\\ \mathrm{1},& t\in \left[\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}}\right],\\ \mathrm{3}\left(\mathrm{1}-t\right),& t\in \left.\left(\frac{\mathrm{2}}{\mathrm{3}},\mathrm{1}\right.\right]\mathrm{。}\end{array}\right.

显然，F_j是关于u的C^{\mathrm{1}}泛函且有F_j\left(t,\mathrm{0}\right)=\mathrm{0}，满足\underset{\left|u(t)\right|\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{F_j\left(t,u(t)\right)}{{\left|u(t)\right|}^{\mathrm{3}}}=\underset{\left|u(t)\right|\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{F_j\left(t,u(t)\right)}{{\left|u(t)\right|}^{\mathrm{3}}}=\mathrm{0}。通过直接计算，我们得到

{}_t{}^c{}_T^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)={}_{\mathrm{0}}{}^{c}D{}_t^{\frac{\mathrm{3}}{\mathrm{4}}}u\left(t\right)=\frac{\mathrm{1}}{\mathrm{\Gamma }\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\left\{\begin{array}{ll}\mathrm{12}{t}^{\frac{\mathrm{3}}{\mathrm{4}}},& t\in \left.\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{3}}\right.\right),\\ \mathrm{4}\sqrt[\mathrm{4}]{\mathrm{27}},& t\in \left[\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}}\right],\\ \mathrm{4}\sqrt[\mathrm{4}]{\mathrm{27}}-\mathrm{12}{\left(t-\frac{\mathrm{2}}{\mathrm{3}}\right)}^{\frac{\mathrm{1}}{\mathrm{4}}},& t\in \left.\left(\frac{\mathrm{2}}{\mathrm{3}},\mathrm{1}\right.\right],\end{array}\right.

且‖u(t)‖\approx \mathrm{8.483},\hspace{1em}{B}^{\mathrm{3}}=\mathrm{1.391}\mathrm{2}。选取k^{\mathrm{*}}={\mathrm{10}}^{-\mathrm{5}}，则有

\frac{{\int }_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{3}}}{F}_j\left(t,u\left(t\right)\right)\mathrm{d}t+{\int }_{\frac{\mathrm{1}}{\mathrm{3}}}^{\frac{\mathrm{2}}{\mathrm{3}}}{F}_j\left(t,u\left(t\right)\right)\mathrm{d}t+{\int }_{\frac{\mathrm{2}}{\mathrm{3}}}^{\mathrm{1}}{F}_j\left(t,u\left(t\right)\right)\mathrm{d}t}{{‖u(t)‖}^{\mathrm{3}}}\approx \mathrm{2.8}\times {\mathrm{10}}^{-\mathrm{4}}>\mathrm{3}{k}^{\mathrm{*}}T{B}^{\mathrm{3}}\approx \mathrm{4.12}\times {\mathrm{10}}^{-\mathrm{5}}。

因此，利用定理2，我们得到：对于每一个紧区间\left[{\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\right]\subset \left(\mathrm{3.6}\times {\mathrm{10}}^{\mathrm{3}},\mathrm{2.4}\times {\mathrm{10}}^{\mathrm{4}}\right)，存在R>\mathrm{0}，且满足下列性质：对于每一个\lambda \in \left[{\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\right]，存在\xi >\mathrm{0}，使得对于每一个\mu \in \left[\mathrm{0},\xi \right]，问题（23）至少有三个范数小于R的解。

3 结论

在本文研究中，我们主要利用变分法建立问题（3）解的存在性。首先，从问题（2）出发，我们将线性分数阶微分方程推广到区间\left.\left(s_j,t_{j+\mathrm{1}}\right.\right]上的非线性分数阶微分方程。其次，我们给出问题（3）的弱解的定义，且使其解的多重性依赖于两个参数\lambda和\mu。最后，我们得到当p=\mathrm{2}和\mu =\lambda =\mathrm{1}时，问题（3）退化为问题（2）。因此，我们补充和推广了现有文献［13］的结果。

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基金资助

国家自然科学基金(11961039)

甘肃省基础研究创新群体项目(25JRRA805)