具p-Laplacian算子的分数阶脉冲微分方程边值问题解的存在性

Existence of Solutions for Fractional Impulsive Differential Equations with p-Laplacian Operator

摘要: 本文讨论了一类分数阶脉冲微分方程的边值问题解的存在性，应用一些不动点定理给出了脉冲微分方程解的存在性的充分条件。

Abstract: In this paper, we discuss the existence of solutions of boundary value problems for a class of fractional impulsive differential equations. Some fixed point theorems are used to obtain sufficient conditions for the existence of solutions of Impulsive Differential Equations.

1. 引言

分数阶微分方程在科技、经济和工程等领域都得到了广泛的应用，引起了广大数学学者的关注和研究，并在分数阶微分系统方面得到了很多的研究成果 [1] - [9] ，含有p-Laplacian算子的微分方程也逐渐热门 [1] [2] [3] ，而对非线性分数阶脉冲微分方程脉冲边值问题的研究相对而言比较少，在文献 [4] - [9] 中也研究了含有脉冲项的分数阶微分方程的边值问题，因此本文主要研究含有p-Laplacian算子的分数阶脉冲微分方程边值问题解的存在性。

在文献 [6] 中，作者运用Banach压缩映射原理和Leray-Schauder不动点定理研究了分数阶差分方程边值问题

$\{\begin{array}{l}{}^C{D}_{0+}^{q}u\left(t\right)=f\left(t,u\left(t\right)\right),1<q\le 2,t\in J^{\prime },\\ \Delta \left(u\left(t_k\right)\right)=I_k\left(u\left(t_k\right)\right),\Delta \left(u^{\prime }\left(t_k\right)\right)=Q_k\left(u\left(t_k\right)\right),\\ au\left(0\right)-b{u}^{\prime }\left(0\right)=x_0,cu\left(1\right)+d{u}^{\prime }\left(1\right)=x_1,\end{array}$ (1)

解的存在性，这里是标准的Caputo分数阶导数，其中 $f\in \left(C\times R,R\right)$ ， $I_k,Q_k\in C\left(R,R\right)$ ，是连续函数。

文献 [2] 中，作者运用不动点定理研究了如下边值问题

$\{\begin{array}{l}{}^C{D}_{0+}^{\beta }{\phi }_p\left({}^C{D}_{0+}^{\beta }x\left(t\right)\right)=f\left(t,x\left(t\right)\right),1<\alpha \le 2,t\in J^{\prime },\\ x\left(0\right)+x\left(1\right)=0,{}^{C}D{}_{0+}^{\alpha }x\left(0\right)+{}^{C}D{}_{0+}^{\alpha }x\left(1\right)=0,\end{array}$ (2)

解的存在性，这里 ${}^C{D}_{0+}^{\beta }$ 是标准的Caputo分数阶导数。

受上述文献启发，本文应用不动点定理研究含p-Laplacian算子的分数阶脉冲微分方程边值问题

$\{\begin{array}{l}{\phi }_p\left({}^C{D}_{0+}^{\alpha }u\left(t\right)\right)=f\left(t,u\left(t\right)\right),t\in J^{\prime }\\ \Delta \left(u\left(t_k\right)\right)=I_k\left(u\left(t_k\right)\right),\Delta \left(u^{\prime }\left(t_k\right)\right)=Q_k\left(u\left(t_k\right)\right),\\ au\left(0\right)-b{u}^{\prime }\left(0\right)=x_0,cu\left(0\right)+d{}^{C}D{}_{0+}^{\beta }u\left(1\right)=x_1\end{array}$ (3)

解的存在性，其中 ${}^C{D}_{0+}^{\alpha }$ 是标准的Caputo分数阶导数，

$1<\alpha \le 2$ ， $\alpha -1<\beta \le 1$ ， $a\ge 0$ ， $b>0$ ， $c\ge 0$ ， $d>0$ ，

$\left(\delta =bc+a\left(c+\frac{d}{\Gamma \left(2-\beta \right)}\right)\right)$ ， $x_0,x_1\in R$ ， $f\in C\left(J\times R,R\right)$ ， $I_k,J_k\in R$ ，

$J=\left[0,1\right]$ ， $0=t_0<t_1<\cdots <t_k<\cdots <t_m<t_{m+1}=1$ ， $J^{\prime }=J\backslash \left\{t_1,t_2,\cdots ,t_m\right\}$

其脉冲项为

$\Delta u\left(t_k\right)=u\left(t_k^{+}\right)-u\left(t_k^{-}\right)$ ， $\Delta u^{\prime }\left(t_k\right)=u^{\prime }\left(t_k^{+}\right)-u^{\prime }\left(t_k^{-}\right)$

这里 $u\left(t_k^{+}\right)$ 与 $u\left(t_k^{-}\right)$ 分别为 $u\left(t_k\right)$ 在 $t=t_k$ 处的右左极限， $k=1,2,\cdots ,m$ ，其中 ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$ 为p-Laplacian算子， $p>1$ ，并且 ${\phi }_p\left(s\right)$ 的逆算子 ${\phi }_q\left(s\right)$ ， $\frac{1}{p}+\frac{1}{q}=1$ 。

2. 相关定义及引理

在这一节，为了后面的讨论，我们给出一些定义及相关引理。

令 $J=\left[0,1\right]$ ， $0=t_0<t_1<\cdots <t_k<\cdots <t_m<t_{m+1}=1$ ， $J_0=\left[0,t_1\right]$ ， $J_1=\left(t_1,t_2\right]$ ， $\cdots$ ， $J_m=\left(t_m,1\right]$ ， $PC\left(J,R\right)=\left\{u:J\to R|u\in C\left(J_k\right),k=0,1,\cdots ,m,并且u\left(t_k^{+}\right)存在\right\}$ ，其空间上范数为 $\Vert u\Vert =\underset{t\in J}{\sup }\left|u\left(t\right)\right|$ ，有

$PC\left(J,R\right)=\left\{u:J\to R|u\in C\left(J_k\right),k=0,1,\cdots ,m,并且u\left(t_k^{+}\right),u^{\prime }\left(t_k^{+}\right)存在\right\}$ ，与其范数 ${\Vert u\Vert }_{P{C}^{\prime }}=\underset{t\in J}{\max }\left\{\Vert u\Vert ,\Vert u^{\prime }\Vert \right\}$ ，显然 $PC$ 与 $P{C}^{\prime }$ 为Banach空间。

定理1：设 $E$ 为Banach空间， $\Omega$ 为 $E$ 上的非空有界闭凸子集，算子 $T:\bar{\Omega }\to E$ 为全连续算子，使得 $\Vert Tu\Vert \le \Vert u\Vert$ ， $\forall u\in \bar{\Omega }$ ，则 $T$ 在 $\bar{\Omega }$ 存在不动点。

定理2：设 $E$ 为Banach空间， $T:E\to E$ 为全连续算子， $V=\left\{u\in E|u=\mu Tu,0<\mu <1\right\}$ ，为有界集，则 $T$ 在 $\bar{\Omega }$ 存在不动点。

引理1： [1] 令 $\alpha >0$ ，则分数阶微分方程 ${}^C{D}_{0+}^{\alpha }u\left(t\right)=0$ 有唯一解

$u\left(t\right)=c_0+c_{1}t+\cdots +c_{n-1}{t}^{n-1},c_i\in R,n=\left[\alpha \right]+1$

引理2： [1] 令 $\alpha >0$ ，则有

$I_{0+}^{\alpha }{}^{C}D{}_{0+}^{\alpha }u\left(t\right)=u\left(t\right)+c_0+c_{1}t+\cdots +c_{n-1}{t}^{n-1},c_i\in R,i=0,1,\cdots ,n-1,n=\left[\alpha \right]+1$

对于边值问题(3)，可通过 ${\phi }_p\left(s\right)$ 的逆算子 ${\phi }_q\left(s\right)$ 将其转化为等价的边值问题

$\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }u\left(t\right)={\phi }_q\left(f\left(t,u\left(t\right)\right)\right),t\in J^{\prime }\\ \Delta \left(u\left(t_k\right)\right)=I_k\left(u\left(t_k\right)\right),\Delta \left(u^{\prime }\left(t_k\right)\right)=Q_k\left(u\left(t_k\right)\right),\\ au\left(0\right)-b{u}^{\prime }\left(0\right)=x_0,cu\left(0\right)+d^C{D}_{0+}^{\beta }u\left(1\right)=x_1\end{array}$ (4)

引理3：对于给定函数 $y\in C\left[0,1\right]$ ， $u\left(t\right)$ 为如下边值问题的解

$\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }u\left(t\right)={\phi }_q\left(y\left(t\right)\right),t\in J^{\prime }\\ \Delta \left(u\left(t_k\right)\right)=I_k\left(u\left(t_k\right)\right),\Delta \left(u^{\prime }\left(t_k\right)\right)=Q_k\left(u\left(t_k\right)\right),\\ au\left(0\right)-b{u}^{\prime }\left(0\right)=x_0,cu\left(0\right)+d^C{D}_{0+}^{\beta }u\left(1\right)=x_1\end{array}$ (5)

当且仅当

$u\left(t\right)=\{\begin{array}{l}\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}t-c_1-c_{2}t}\text{,}t\in J_0\\ \frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t_k}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}t}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\frac{t-t_k}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}t}}\\ +{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\frac{t_k-t_i}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}t}}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\left(t-t_k\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\left(t_k-t_i\right)Q_i\left(u\left(t_i\right)\right)}-c_1-c_{2}t,t\in J_k\end{array}$

其中

$\begin{array}{l}{c}_1=\frac{1}{\delta }(\frac{bc}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+\frac{bc\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{bc\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc\left(1-t_m\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}bc\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{bd}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +\frac{bd}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-b{x}_1-c{x}_0-\frac{d{x}_0}{\Gamma \left(2-\beta \right)})\end{array}$

$\begin{array}{l}{c}_2=\frac{1}{\delta }(\frac{ac}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+\frac{ac\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{ac\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac\left(1-t_m\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}ac\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{ad}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +\frac{ad}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-a{x}_1-c{x}_0)\end{array}$

证明：设 $u\left(t\right)$ 为(5)的解，由(2)可得

$u\left(t\right)=I_{0+}^{\alpha }y\left(t\right)-c_1-c_{2}t=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}t}-c_1-c_{2}t,t\in J_0,c_1,c_2\in R$

则有

$u^{\prime }\left(t\right)=\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}t}-c_2$ (6)

若 $t\in J_1$ ，有

$u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t_1}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}-d_1-d_2\left(t-t_1\right)$

$u^{\prime }\left(t\right)=\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_1}^t{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}-d_2$

其中 $d_1,d_2\in R$ ，可得

$u\left(t^{-}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t_1}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}-c_1-c_2{t}_1,u\left(t^{+}\right)=-d_1$

$u^{\prime }\left(t^{-}\right)=\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_1}^t{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}-c_2,u^{\prime }\left(t^{+}\right)=-d_2$

由上述定义 $\Delta u\left(t_k\right)=u\left(t_k^{+}\right)-u\left(t_k^{-}\right)$ ， $\Delta u^{\prime }\left(t_k\right)=u^{\prime }\left(t_k^{+}\right)-u^{\prime }\left(t_k^{-}\right)$ ，则

$-d_1=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{t_1}{\left(t_1-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}-c_1-c_2{t}_1+I_1\left(u\left(t_1\right)\right)$

$-d_2=\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_0^{t_1}{\left(t_1-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}-c_2+Q_1\left(u\left(t_1\right)\right)$

因此

$\begin{array}{l}u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t_k}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}t}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\frac{t-t_k}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}t}}\\ +{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\frac{t_k-t_i}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}t}}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\left(t-t_k\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\left(t_k-t_i\right)Q_i\left(u\left(t_i\right)\right)}-c_1-c_{2}t\end{array}$

由边值条件 $au\left(0\right)-b{u}^{\prime }\left(0\right)=x_0$ ， $cu\left(0\right)+d{}^{C}D{}_{0+}^{\beta }u\left(1\right)=x_1$ ，

$\begin{array}{l}-a{c}_1+b{c}_2=x_0\\ c{c}_1+\left(c+\frac{1}{\Gamma \left(2-\beta \right)}\right)\\ =\frac{c}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\frac{c\left(1-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}c{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}c\left(1-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}bc\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}\frac{d}{\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+\frac{d}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-x_1\end{array}$

可得

$\begin{array}{l}{c}_1=\frac{1}{\delta }(\frac{bc}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+\frac{bc\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{bc\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc\left(1-t_m\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}bc\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{bd}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +\frac{bd}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-b{x}_1-c{x}_0-\frac{d{x}_0}{\Gamma \left(2-\beta \right)})\end{array}$

$\begin{array}{l}{c}_2=\frac{1}{\delta }(\frac{ac}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+\frac{ac\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{ac\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac\left(1-t_m\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}ac\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{ad}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +\frac{ad}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-a{x}_1-c{x}_0)\end{array}$

命题得证。

3. 主要结果

定义算子 $T:PC\left(J,R\right)\to PC\left(J,R\right)$ 如下

$\begin{array}{l}Tu\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t_k}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}}\\ +{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\frac{t-t_k}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}}+{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\frac{t_k-t_i}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\left(t-t_k\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\left(t_k-t_i\right)Q_i\left(u\left(t_i\right)\right)}-c_1-c_{2}t\end{array}$

其中

$\begin{array}{l}{c}_1=\frac{1}{\delta }(\frac{bc}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}+\frac{bc\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{bc\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc\left(1-t_m\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}bc\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{bd}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(y\left(s\right)\right)\text{d}s}}\\ +\frac{bd}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-b{x}_1-c{x}_0-\frac{d{x}_0}{\Gamma \left(2-\beta \right)})\end{array}$

$\begin{array}{l}{c}_2=\frac{1}{\delta }(\frac{ac}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}+\frac{ac\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{ac\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac{I}_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac\left(1-t_m\right)Q_i\left(u\left(t_i\right)\right)}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}ac\left(t_m-t_i\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{ad}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}s}}\\ +\frac{ad}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{Q}_i\left(u\left(t_i\right)\right)}-a{x}_1-c{x}_0)\end{array}$

由引理3，令 $y\left(t\right)=f\left(t,u\left(t\right)\right)$ 则边值问题(3)有解等价存在不动点，即 $u=Tu$ ，于是(3)有解当且仅当 $u=Tu$ 。

定理3：令 $\underset{u\to 0}{\lim }\frac{{\phi }_q\left(f\left(t,u\right)\right)}{u}=0$ ， $\underset{u\to 0}{\lim }\frac{I_k\left(u\right)}{u}=0$ ， $\underset{u\to 0}{\lim }\frac{Q_k\left(u\right)}{u}=0$ ，并且有

$\begin{array}{l}\Lambda =\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(m+1\right)\left(c\left(a+b\right)+\delta \right)+\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{\left(m+1\right)\left(a+b\right)d}{\Gamma \left(2-\beta \right)}\right)\\ +\frac{m\left(c\left(a+b\right)+\delta \right)}{\delta }{\delta }_2+\frac{1}{\delta }\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{md\left(a+b\right)}{\Gamma \left(2-\beta \right)}\right){\delta }_3\\ +\frac{1}{\delta }\left(\left(a+b\right)\left|x_1\right|+\frac{2c+d}{\Gamma \left(2-\beta \right)}\left|x_0\right|\right)\le 1\end{array}$

证明：首先，证明 $T:PC\left(J,R\right)\to PC\left(J,R\right)$ 是全连续算子， $T$ 作用于 $f,I_k,Q_k$ 均连续，

设 $\Omega$ 为 $T:PC\left(J,R\right)\to PC\left(J,R\right)$ 的有界子集，则存在常数 $L_i>0\left(i=1,2,3,4\right)$ 使得 $\left|f\left(t,u\right)\right|\le L_1$ ， $\left|I_k\left(u\right)\right|\le L_2$ ， $\left|Q_k\left(u\right)\right|\le L_3$ ， $\left|{\phi }_q\left(f\left(t,u\right)\right)\right|\le L_4$ ，又

$\begin{array}{l}\left|c_1\right|\le \frac{1}{\delta }(\frac{bc}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}+\frac{bc\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{bc\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc\left|I_i\left(u\left(t_i\right)\right)\right|}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc\left(1-t_m\right)\left|Q_i\left(u\left(t_i\right)\right)\right|}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}bc\left(t_m-t_i\right)\left|Q_i\left(u\left(t_i\right)\right)\right|}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{bd}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}\\ +\frac{bd}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}\left|Q_i\left(u\left(t_i\right)\right)\right|}+b\left|x_1\right|+c\left|x_0\right|+\frac{d\left|x_0\right|}{\Gamma \left(2-\beta \right)})\end{array}$

$\begin{array}{l}\le \frac{1}{\delta }(\frac{bc{L}_4}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}\text{d}s}}+\frac{bc{L}_4}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{bc{L}_4}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc{L}_2}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}bc\left(1-t_m\right)L_3}+{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}bc\left(t_m-t_i\right)L_3}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{bd{L}_4}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\text{d}s}}\\ +\frac{bd}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{L}_3}+b\left|x_1\right|+c\left|x_0\right|+\frac{d\left|x_0\right|}{\Gamma \left(2-\beta \right)})\\ \le \frac{1}{\delta }\frac{bc\left(m+1\right)L_4}{\Gamma \left(\alpha +1\right)}+\frac{1}{\delta }\frac{bcm{L}_4}{\Gamma \left(\alpha \right)}+\frac{1}{\delta }{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{bcm{L}_4}{\Gamma \left(\alpha \right)}}+\frac{bcm{L}_2}{\delta }+\frac{bcm{L}_3}{\delta }+\frac{bc\left(m-1\right)L_3}{\delta }\\ +\frac{1}{\delta }\frac{bd(m+1)L_4}{\Gamma \left(\alpha \right)\Gamma \left(2-\beta \right)}+\frac{1}{\delta }\frac{bdm{L}_3}{\Gamma \left(2-\beta \right)}+\frac{1}{\delta }b\left|x_1\right|+\frac{1}{\delta }c\left|x_0\right|+\frac{1}{\delta }\frac{d\left|x_0\right|}{\Gamma \left(2-\beta \right)}\end{array}$

$\begin{array}{l}\left|c_2\right|\le \frac{1}{\delta }(\frac{ac}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}+\frac{ac\left(1-t_m\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{ac\left(t_m-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac\left|I_i\left(u\left(t_i\right)\right)\right|}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac\left(1-t_m\right)\left|Q_i\left(u\left(t_i\right)\right)\right|}\\ +{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}ac\left(t_m-t_i\right)\left|Q_i\left(u\left(t_i\right)\right)\right|}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{ad}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\left|{\phi }_q\left(f\left(s,u\right)\right)\right|\text{d}s}}\\ +\frac{ad}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}\left|Q_i\left(u\left(t_i\right)\right)\right|}-a\left|x_1\right|-c\left|x_0\right|)\end{array}$

$\begin{array}{l}\le \frac{1}{\delta }(\frac{ac{L}_4}{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -1}\text{d}s}}+\frac{ac{L}_4}{\Gamma \left(\alpha -1\right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\text{d}s}}+{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}\frac{ac{L}_4}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac{L}_2}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}ac\left(1-t_m\right)L_3}+{\displaystyle \underset{i=1}{\overset{m-1}{\sum }}ac\left(t_m-t_i\right)L_3}+{\displaystyle \underset{i=1}{\overset{m+1}{\sum }}\frac{ad{L}_4}{\Gamma \left(\alpha -1\right)\Gamma \left(2-\beta \right)}{\displaystyle {\int }_{t_{i-1}}^{t_i}{\left(t_i-s\right)}^{\alpha -2}\text{d}s}}\\ +\frac{ad}{\Gamma \left(2-\beta \right)}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{L}_3}-a\left|x_1\right|-c\left|x_0\right|)\\ \le \frac{1}{\delta }\frac{ac\left(m+1\right)L_4}{\Gamma \left(\alpha +1\right)}+\frac{1}{\delta }\frac{acm{L}_4}{\Gamma \left(\alpha \right)}+\frac{1}{\delta }\frac{ac(m-1)L_4}{\Gamma \left(\alpha \right)}+\frac{acm{L}_2}{\delta }+\frac{acm{L}_3}{\delta }\\ +\frac{ac\left(m-1\right)L_3}{\delta }+\frac{1}{\delta }\frac{ad\left(m+1\right)L_4}{\Gamma \left(\alpha \right)\Gamma \left(2-\beta \right)}+\frac{1}{\delta }\frac{adm{L}_3}{\Gamma \left(2-\beta \right)}+\frac{1}{\delta }a\left|x_1\right|+\frac{1}{\delta }c\left|x_0\right|\end{array}$

$\begin{array}{l}\left|Tu\left(t\right)\right|\le \frac{1}{\delta }\frac{L_4}{\Gamma \left(\alpha +1\right)}\left(m+1\right)\left(c\left(a+b\right)+\delta \right)+\frac{1}{\delta }\frac{L_4}{\Gamma \left(\alpha +1\right)}\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{\left(m+1\right)\left(a+b\right)d}{\Gamma \left(2-\beta \right)}\right)\\ +\frac{m\left(c\left(a+b\right)+\delta \right)}{\delta }{L}_2+\frac{m}{\delta }\left(\left(2p-1\right)\left(\delta +c\left(a+b\right)\right)+\frac{md\left(a+b\right)}{\Gamma \left(2-\beta \right)}\right)L_3+\frac{1}{\delta }\left(\left(a+b\right)\left|x_1\right|+\frac{2c+d}{\Gamma \left(2-\beta \right)}\left|x_0\right|\right):=L\end{array}$

因此 $\Vert Tu\Vert \le L$ 。

另一方面

$\left|T{u}^{\prime }\left(t\right)\right|\le \frac{1}{\delta }\frac{ac\left(m+1\right)L_4}{\Gamma \left(\alpha +1\right)}+\frac{acm{L}_2}{\delta }+\frac{L_3}{\delta }\left(\left(m+1\right)\left(\delta +\frac{ad}{\Gamma \left(2-\beta \right)}\right)+ac\left(2m-1\right)\right)+\frac{\left(a\left|x_1\right|+c\left|x_0\right|\right)}{\delta }:=\bar{L}$

因此对于 $t_1,t_2\in J_k,t_1\le t_2,0\le k\le m$ 有

$\left|Tu\left(t_2\right)-Tu\left(t_1\right)\right|\le {\displaystyle {\int }_{t_1}^{t_2}{\left(Tu\right)}^{\prime }\left(s\right)\text{d}s}\le \bar{L}\left(t_2-t_1\right)$

则 $t$ 在每一个子空间等度连续，由Arzela-Ascoli定理， $T$ 为全连续算子，由条件 $\underset{u\to 0}{\lim }\frac{{\phi }_q\left(f\left(t,u\right)\right)}{u}=0$ ， $\underset{u\to 0}{\lim }\frac{I_k\left(u\right)}{u}=0$ ， $\underset{u\to 0}{\lim }\frac{Q_k\left(u\right)}{u}=0$ ，存在常数 $r>0,{\delta }_i>0\left(i=1,2,3\right)$ 有 $\left|{\phi }_q\left(f\left(t,u\right)\right)\right|\le {\delta }_1\left|u\right|$ ， $\left|I_i\left(u\right)\right|\le {\delta }_2\left|u\right|$ ， $\left|Q_i\left(u\right)\right|\le {\delta }_3\left|u\right|$ ，由

$\begin{array}{l}\Lambda =\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(m+1\right)\left(c\left(a+b\right)+\delta \right)+\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{\left(m+1\right)\left(a+b\right)d}{\Gamma \left(2-\beta \right)}\right)\\ +\frac{m\left(c\left(a+b\right)+\delta \right)}{\delta }{\delta }_2+\frac{1}{\delta }\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{md\left(a+b\right)}{\Gamma \left(2-\beta \right)}\right){\delta }_3+\frac{1}{\delta }\left(\left(a+b\right)\left|x_1\right|+\frac{2c+d}{\Gamma \left(2-\beta \right)}\left|x_0\right|\right)\le 1\end{array}$

对于 $u\in PC\left(J,R\right)$ ，定义 $\Omega =\left\{u\in PC\left(J,R\right)|\Vert u\Vert <r\right\}$ ， $u\in \partial \Omega$ 时， $\Vert u\Vert =r$ ，有

$\begin{array}{l}\left|Tu\left(t\right)\right|\le \{\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(m+1\right)\left(c\left(a+b\right)+\delta \right)+\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{\left(m+1\right)\left(a+b\right)d}{\Gamma \left(2-\beta \right)}\right)\\ +\frac{m\left(c\left(a+b\right)+\delta \right)}{\delta }{\delta }_2+\frac{1}{\delta }\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{md\left(a+b\right)}{\Gamma \left(2-\beta \right)}\right){\delta }_3+\frac{1}{\delta }\left(\left(a+b\right)\left|x_1\right|+\frac{2c+d}{\Gamma \left(2-\beta \right)}\left|x_0\right|\right)\}{\Vert u\Vert }_{P{C}^{\prime }}\end{array}$

有 ${\Vert Tu\Vert }_{P{C}^{\prime }}\le {\Vert u\Vert }_{P{C}^{\prime }}$ ，由定理1算子 $T$ 至少存在一个不动点，则(3)至少存在一个解。

定理4：假设存在正常数 $L_i\ge 0\left(i=1,2,3,4\right)$ 使得 $\left|f\left(t,u\right)\right|\le L_1$ ， $\left|I_k\left(u\right)\right|\le L_2$ ， $\left|Q_k\left(u\right)\right|\le L_3$ ， $\left|{\phi }_q\left(f\left(t,u\right)\right)\right|\le L_4$ ，则(3)至少存在一个解。

证明：由定理3，算子 $T$ 为全连续算子，此时假设 $V=\left\{u\in E|u=\mu Tu,0<\mu <u\right\}$ 为有界集，令 $u\in V$ ，我们有

$\begin{array}{l}u\left(t\right)=\frac{\mu }{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t_k}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}+\frac{\mu }{\Gamma \left(\alpha \right)}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle {\int }_{t_k}^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}\frac{\mu \left(t-t_k\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}}+{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\frac{\mu \left(t_k-t_i\right)}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_{t_{-1k}}^{t_k}{\left(t-s\right)}^{\alpha -2}{\phi }_q\left(f\left(s,u\right)\right)\text{d}t}}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}\mu I_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}\mu \left(t-t_k\right)Q_i\left(u\left(t_i\right)\right)}+{\displaystyle \underset{i=1}{\overset{k-1}{\sum }}\mu \left(t_k-t_i\right)Q_i\left(u\left(t_i\right)\right)}-\mu c_1-\mu c_{2}t\end{array}$

由条件，有

$\begin{array}{l}\left|u\left(t\right)\right|=\mu \left|Tu\left(t\right)\right|\\ \le \frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(m+1\right)\left(c\left(a+b\right)+\delta \right)+\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{\left(m+1\right)\left(a+b\right)d}{\Gamma \left(2-\beta \right)}\right)\\ +\frac{m\left(c\left(a+b\right)+\delta \right)}{\delta }{\delta }_2+\frac{1}{\delta }\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{md\left(a+b\right)}{\Gamma \left(2-\beta \right)}\right){\delta }_3+\frac{1}{\delta }\left(\left(a+b\right)\left|x_1\right|+\frac{2c+d}{\Gamma \left(2-\beta \right)}\left|x_0\right|\right)\end{array}$

则对 $t\in J$ ，

$\begin{array}{l}{\Vert u\left(t\right)\Vert }_{P{C}^{\prime }}\le \frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(m+1\right)\left(c\left(a+b\right)+\delta \right)+\frac{1}{\delta }\frac{{\delta }_1}{\Gamma \left(\alpha +1\right)}\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{\left(m+1\right)\left(a+b\right)d}{\Gamma \left(2-\beta \right)}\right)\\ +\frac{m\left(c\left(a+b\right)+\delta \right)}{\delta }{\delta }_2+\frac{1}{\delta }\left(\left(2m-1\right)\left(c\left(a+b\right)+\delta \right)+\frac{md\left(a+b\right)}{\Gamma \left(2-\beta \right)}\right){\delta }_3+\frac{1}{\delta }\left(\left(a+b\right)\left|x_1\right|+\frac{2c+d}{\Gamma \left(2-\beta \right)}\left|x_0\right|\right)\end{array}$

又由 $V$ 为有界集，故由定理2算子 $T$ 至少存在一个不动点，则(3)至少存在一个解。

4. 例子

对于如下具p-Laplacian算子的非线性分数阶脉冲微分方程边值问题

$\{\begin{array}{l}{\phi }_{5/3}\left({}^C{D}_{0+}^{\frac{5}{4}}u\left(t\right)\right)=\frac{\cos t}{{\left(t+5\sqrt{2}\right)}^2}\frac{u\left(t\right)}{1+u\left(t\right)},\\ \Delta \left(u\left(\frac{1}{2}\right)\right)=\frac{\left|u\left(\frac{1}{2}\right)\right|}{20+\left|u\left(\frac{1}{2}\right)\right|},\Delta \left(u^{\prime }\left(\frac{1}{2}\right)\right)=Q_k\left(\frac{\left|u\left(\frac{1}{2}\right)\right|}{20+\left|u\left(\frac{1}{2}\right)\right|}\right),\\ u\left(0\right)-u^{\prime }\left(0\right)=0,u\left(0\right)+{}^{C}D{}_{0+}^{\frac{1}{2}}u\left(1\right)=0\end{array}$

这里 $p=\frac{5}{3},\alpha =\frac{5}{4},a=b=c=d=m=1,\beta =\frac{1}{2},x_0=x_1=0$ ，取 $L_1=\frac{1}{50},L_2=\frac{1}{20},L_3=\frac{1}{40}$ 有 $\Lambda \le 1$ 满足条件，由定理3和定理4，边值问题(3)至少存在一个解。

基金项目

新疆维吾尔自治区研究生科研创新项目(XJGRI2016136)，新疆高校科研计划重点项目(XJEDU2014I040)。