# 具有左右分数阶导数和时滞的非瞬时脉冲微分方程非线性边值问题Nonlinear Boundary Value Problems for Non-Instantaneous Pulse Differential Equations with Left-Right Fractional Derivatives and Delays

DOI: 10.12677/AAM.2021.104136, PDF, HTML, XML, 下载: 7  浏览: 26

Abstract: In this paper, we study a class of special non-instantaneous impulsive differential equations with left and right fractional derivatives and delays. The equations have cross delays and nonlinear boundary conditions. Based on the upper and lower solution method, we obtain the existence theorems of multiple positive solutions.

1. 引言

$\left\{\begin{array}{l}{}^{c}D{}_{{b}^{_}}^{\alpha }{}^{c}D{}_{{a}^{+}}^{\alpha }T\left(t\right)+\lambda T\left(t\right)=0,\\ T\left(a\right)={T}_{0},T\left(b\right)={T}_{1},\end{array}$

$\left\{\begin{array}{l}-\left({}_{{0}^{+}}D{}_{t}^{\alpha }u\left(t\right)\right)=f\left(t,v\left(t\right)\right),\text{}t\in \left(0,T\right),\\ -\left({}_{t}D{}_{{T}^{-}}^{\beta }v\left(t\right)\right)=g\left(t,u\left(t\right)\right),\text{}t\in \left(0,T\right),\\ u\left(0\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{}_{{0}^{+}}D{}_{t}^{\alpha -1}u\left(T\right)={r}_{1}{}_{t}D{}_{{T}^{-}}^{\beta -1}v\left(\xi \right),\\ v\left(T\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{}_{t}D{}_{{T}^{-}}^{\beta -1}v\left(0\right)={r}_{2}{}_{{0}^{+}}D{}_{t}^{\alpha -1}u\left(\xi \right),\end{array}$

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }u\left(t\right)={f}_{1}\left(t,u\left(t\right),u\left(t+{\tau }_{1}\right)\right),\text{}t\in \left[0,\xi \right],\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }u\left(t\right)={f}_{2}\left(t,u\left(t\right),u\left(t-{\tau }_{2}\right)\right),\text{}t\in \left(\xi ,1\right],\\ \Delta u\left(\xi \right)=I\left(\xi ,u\left(\xi \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {u}^{\prime }\left(\xi \right)=Q\left(\xi ,u\left(\xi \right)\right),\\ {h}_{0}\left(u\left(0\right),u\left(1\right)\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}u\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}u\left(1\right)\right)=0\end{array}$ (1)

2. 线性边值问题

${}_{{a}^{+}}I{}_{t}^{\alpha }{}_{{}^{{a}^{+}}}{}^{c}D{}_{t}^{\alpha }u\left(t\right)=u\left(t\right)+{c}_{1}+{c}_{2}\left(t-a\right)+{c}_{3}{\left(t-a\right)}^{2}+\cdots +{c}_{n}{\left(t-a\right)}^{n-1},$

${}_{t}I{}_{{b}^{-}}^{\alpha }{}_{t}{}^{c}D{}_{{b}^{-}}^{\alpha }u\left(t\right)=u\left(t\right)+{d}_{1}+{d}_{2}\left(b-t\right)+{d}_{3}{\left(b-t\right)}^{2}+\cdots +{d}_{n}{\left(b-t\right)}^{n-1},$

${\alpha }_{1}\prec {\beta }_{1}\prec {\alpha }_{2}\prec {\beta }_{2}$,

$A:\left[{\alpha }_{1},{\beta }_{2}\right]\to E$ 是全连续算子，且为强增算子，使

${\alpha }_{1}\prec A{\alpha }_{1}$, $A{\beta }_{1}\prec {\beta }_{1}$, ${\alpha }_{2}\prec A{\alpha }_{2}$, $A{\beta }_{2}\underset{_}{\prec }{\beta }_{2}$.

${\alpha }_{1}\underset{_}{\prec }{x}_{1}\prec \prec {\beta }_{1}$, ${\alpha }_{2}\prec \prec {x}_{2}\underset{_}{\prec }{\beta }_{2}$, ${\alpha }_{2}\overline{)\underset{_}{\prec }}{x}_{2}\overline{)\underset{_}{\prec }}{\beta }_{2}$.

$J=\left[0,1\right]$${J}_{0}=J\\xi$$E=PC\left[J,ℝ\right]=\left\{u:J\to ℝ:u在\text{ }{J}_{0}上是连续的,u\left({\xi }^{+}\right)与u\left({\xi }^{-}\right)存在且u\left({\xi }^{-}\right)=u\left(\xi \right)\right\}$。显然E是Banach空间且定义其范数为

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

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

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }u\left(t\right)=h\left(t\right),\text{}t\in \left(0,\xi \right),\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }u\left(t\right)=y\left(t\right),t\in \left(\xi ,1\right),\\ \Delta u\left(\xi \right)=I,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {u}^{\prime }\left(\xi \right)=Q,\\ {m}_{1}u\left(0\right)+{n}_{1}u\left(1\right)={\gamma }_{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}u\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}u\left(1\right)={\gamma }_{1}\end{array}$ (2)

E中存在唯一解

$u\left(t\right)=\left\{\begin{array}{l}{\int }_{0}^{\xi }{G}_{1}\left(t,s\right)h\left(s\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right)y\left(s\right)\text{d}s+{\Delta }_{2}+\frac{t}{{\Delta }_{1}}\left(-\frac{Q{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\right),\text{\hspace{0.17em}}t\in \left[0,\xi \right],\\ {\int }_{\xi }^{1}{G}_{2}\left(t,s\right)y\left(s\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(t,s\right)h\left(s\right)\text{d}s+{\Delta }_{3}+\frac{t}{{\Delta }_{1}}\left(\frac{Q{m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\right),\text{\hspace{0.17em}}t\in \left(\xi ,1\right].\end{array}$ (3)

${G}_{1}\left(t,s\right)=\left\{\begin{array}{l}{g}_{1}\left(t,s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le s\le t\le \xi ,\\ {g}_{1}\left(t,s\right)+\frac{1}{\Gamma \left(\alpha \right)}{\left(s-t\right)}^{\alpha -1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le s\le \xi ;\end{array}$ (4)

${G}_{2}\left(t,s\right)=\left\{\begin{array}{l}{g}_{2}\left(t,s\right)+\frac{1}{\Gamma \left(\beta \right)}{\left(t-s\right)}^{\beta -1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\xi \le s\le t\le 1,\\ {g}_{2}\left(t,s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\xi \le t\le s\le 1;\end{array}$ (5)

${\Delta }_{1}=\frac{{m}_{2}{\xi }^{2-\alpha }}{\Gamma \left(3-\alpha \right)}-\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)};{\Delta }_{2}=-\frac{1}{{m}_{1}+{n}_{1}}\left({n}_{1}I+{n}_{1}\left(1-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}-{\gamma }_{0}\right);$

${\Delta }_{3}=-\frac{1}{{m}_{1}+{n}_{1}}\left(-{m}_{1}I+\left({m}_{1}\xi +{n}_{1}+\frac{{n}_{1}{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}-{\gamma }_{0}\right);$

${g}_{1}\left(t,s\right)=-\frac{1}{{m}_{1}+{n}_{1}}\left(\frac{{m}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)+\frac{t{m}_{2}}{{\Delta }_{1}};{g}_{2}\left(t,s\right)=-\frac{{n}_{1}}{{m}_{1}+{n}_{1}}\left(\frac{1}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}+\frac{{n}_{2}}{{\Delta }_{1}}\right)-\frac{t{n}_{2}}{{\Delta }_{1}}.$

$u\left(t\right)={}_{t}I{}_{\xi -}^{\alpha }h\left(t\right)+{c}_{0}+{c}_{1}t=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{t}^{\xi }{\left(s-t\right)}^{\alpha -1}h\left(s\right)\text{d}s+{c}_{0}+{c}_{1}t,$

${u}^{\prime }\left(t\right)=-\frac{1}{\Gamma \left(\alpha -1\right)}{\int }_{t}^{\xi }{\left(s-t\right)}^{\alpha -2}h\left(s\right)\text{d}s+{c}_{1},$

${}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}u\left(t\right)={\int }_{t}^{\xi }h\left(s\right)\text{d}s-\frac{{c}_{1}}{\Gamma \left(3-\alpha \right)}{\left(\xi -t\right)}^{2-\alpha }$

${}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }u\left(t\right)=y\left(t\right)$ 的解为：

$u\left(t\right)={}_{{\xi }^{+}}I{}_{t}^{\beta }y\left(t\right)+{c}_{2}+{c}_{3}t=\frac{1}{\Gamma \left(\beta \right)}{\int }_{\xi }^{t}{\left(t-s\right)}^{\beta -1}y\left(s\right)\text{d}s+{c}_{2}+{c}_{3}t,$

${u}^{\prime }\left(t\right)=\frac{1}{\Gamma \left(\beta -1\right)}{\int }_{\xi }^{t}{\left(t-s\right)}^{\beta -2}y\left(s\right)\text{d}s+{c}_{3},$

${}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}u\left(t\right)={\int }_{\xi }^{t}y\left(s\right)\text{d}s+\frac{{c}_{3}}{\Gamma \left(3-\beta \right)}{\left(t-\xi \right)}^{2-\beta }$

$\left\{\begin{array}{l}{c}_{2}-{c}_{0}=I-Q\xi ,\\ {c}_{3}-{c}_{1}=Q.\end{array}$

$\left\{\begin{array}{l}{c}_{0}=-\frac{1}{{m}_{1}+{n}_{1}}\left({\int }_{0}^{\xi }\left(\frac{{m}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)h\left(s\right)\text{d}s+{n}_{1}{\int }_{\xi }^{1}\left(\frac{1}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}-\frac{{n}_{2}}{{\Delta }_{1}}\right)y\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}I+{n}_{1}\left(1-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}-{\gamma }_{0}\right),\\ {c}_{1}=\frac{1}{{\Delta }_{1}}\left({m}_{2}{\int }_{0}^{\xi }h\left(s\right)\text{d}s+{n}_{2}{\int }_{\xi }^{1}y\left(s\right)\text{d}s+\frac{Q{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\right),\\ {c}_{2}=-\frac{1}{{m}_{1}+{n}_{1}}\left({\int }_{0}^{\xi }\left(\frac{{m}_{1}{s}^{\alpha -1}}{\Gamma \left(\alpha \right)}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)h\left(s\right)\text{d}s+{n}_{1}{\int }_{\xi }^{1}\left(\frac{{\left(1-s\right)}^{\beta -1}}{\Gamma \left(\beta \right)}-\frac{{n}_{2}}{{\Delta }_{1}}\right)y\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{m}_{1}I+\left({m}_{1}\xi +{n}_{1}+\frac{{n}_{1}{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}-{\gamma }_{0}\right),\\ {c}_{3}=\frac{1}{{\Delta }_{1}}\left({m}_{2}{\int }_{0}^{\xi }h\left(s\right)\text{d}s+{n}_{2}{\int }_{\xi }^{1}y\left(s\right)\text{d}s+\frac{Q{m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\right).\end{array}$

$\begin{array}{c}u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{t}^{\xi }{\left(s-t\right)}^{\alpha -1}h\left(s\right)\text{d}s-\frac{1}{{m}_{1}+{n}_{1}}\left({\int }_{0}^{\xi }\left(\frac{{m}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)h\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}+{n}_{1}{\int }_{\xi }^{1}\left(\frac{1}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}-\frac{{n}_{2}}{{\Delta }_{1}}\right)y\left(s\right)\text{d}s+{n}_{1}I+{n}_{1}\left(1-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}-{\gamma }_{0}\right)\\ \text{\hspace{0.17em}}+\frac{t}{{\Delta }_{1}}\left({m}_{2}{\int }_{0}^{\xi }h\left(s\right)\text{d}s+{n}_{2}{\int }_{\xi }^{1}y\left(s\right)\text{d}s+\frac{Q{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\right)\\ ={\int }_{0}^{\xi }{G}_{1}\left(t,s\right)h\left(s\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right)y\left(s\right)\text{d}s+{\Delta }_{2}+\frac{t}{{\Delta }_{1}}\left(\frac{Q{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\right).\end{array}$

$t\in \left(\xi ,1\right]$ 时，

$\begin{array}{c}u\left(t\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_{\xi }^{t}{\left(t-s\right)}^{\beta -1}y\left(s\right)\text{d}s-\frac{1}{{m}_{1}+{n}_{1}}\left({\int }_{0}^{\xi }\left(\frac{{m}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)h\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}+{n}_{1}{\int }_{\xi }^{1}\left(\frac{1}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}+\frac{{n}_{2}}{{\Delta }_{1}}\right)y\left(s\right)\text{d}s-{m}_{1}I+\left({m}_{1}\xi +{n}_{1}+\frac{{n}_{1}{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}-{\gamma }_{0}\right)\\ \text{\hspace{0.17em}}+\frac{t}{{\Delta }_{1}}\left({m}_{2}{\int }_{0}^{\xi }h\left(s\right)\text{d}s+{n}_{2}{\int }_{\xi }^{1}y\left(s\right)\text{d}s+\frac{Q{m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\right)\\ ={\int }_{\xi }^{1}{G}_{2}\left(t,s\right)y\left(s\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(t,s\right)h\left(s\right)\text{d}s+{\Delta }_{3}+\frac{t}{{\Delta }_{1}}\left(\frac{Q{m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\right).\end{array}$

(H1) ${m}_{i},{n}_{i}\in ℝ\left(i=1,2\right)$, ${n}_{1}>0,\text{\hspace{0.17em}}{n}_{2}>0,-{n}_{1}>{m}_{1}>-\frac{{n}_{1}}{\xi },{\gamma }_{1}\le 0,{\gamma }_{0}\le 0$

${m}_{2}>\mathrm{max}\left\{\frac{\Gamma \left(3-\alpha \right){\xi }^{\alpha -1}{n}_{2}}{\Gamma \left(3-\beta \right)\left(\Gamma \left(3-\alpha \right)\Gamma \left(\alpha \right)-\xi \right)},-\frac{\Gamma \left(3-\alpha \right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right){\xi }^{2-\alpha }}\right\}.$

1) $0<{G}_{1}\left(0,s\right)\le {G}_{1}\left(t,s\right)\le {G}_{1}\left(\xi ,s\right)$，对任意 $\left(t,s\right)\in \left[0,\xi \right]×\left[0,\xi \right]$

2) $0<{G}_{2}\left(\xi ,s\right)\le {G}_{2}\left(t,s\right)\le {G}_{2}\left(1,s\right)$，对任意 $\left(t,s\right)\in \left[\xi ,1\right]×\left[\xi ,1\right]$

${G}_{1}\left(t,s\right)={g}_{1}\left(t,s\right)=-\frac{1}{{m}_{1}+{n}_{1}}\left(\frac{{m}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)+\frac{t{m}_{2}}{{\Delta }_{1}},\frac{\partial {g}_{1}\left(t,s\right)}{\partial t}=\frac{{m}_{2}}{{\Delta }_{1}}>0;$

$0\le s\le t\le \xi$ 时，由于 ${G}_{1}\left(t,s\right)={g}_{1}\left(t,s\right)+\frac{1}{\Gamma \left(\alpha \right)}{\left(s-t\right)}^{\alpha -1}$

$\frac{\partial {G}_{1}\left(t,s\right)}{\partial t}=-\frac{1}{\Gamma \left(\alpha -1\right)}{\left(s-t\right)}^{\alpha -2}+\frac{{m}_{2}}{{\Delta }_{1}},\frac{{\partial }^{2}{G}_{1}\left(t,s\right)}{\partial {t}^{2}}=\frac{\alpha -2}{\Gamma \left(\alpha -1\right)}{\left(s-t\right)}^{\alpha -3}<0,$

$\frac{\partial {G}_{1}\left(t,s\right)}{\partial t}\ge \frac{\partial {G}_{1}\left(s,s\right)}{\partial t}=\frac{{m}_{2}}{{\Delta }_{1}}>0$。因此， ${G}_{1}\left(t,s\right)$ 是关于t的单调递增函数，且

${G}_{1}\left(0,s\right)\le {G}_{1}\left(t,s\right)\le {G}_{1}\left(\xi ,s\right).$

$\begin{array}{l}{G}_{1}\left(0,s\right)={g}_{1}\left(0,s\right)+\frac{1}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}=-\frac{1}{{m}_{1}+{n}_{1}}\left(\frac{{m}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)+\frac{1}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-\frac{1}{{m}_{1}+{n}_{1}}\left(\frac{-{n}_{1}}{\Gamma \left(\alpha \right)}{s}^{\alpha -1}+\frac{{n}_{1}{m}_{2}}{{\Delta }_{1}}\right)>-\frac{{n}_{1}}{{m}_{1}+{n}_{1}}\left(\frac{-1}{\Gamma \left(\alpha \right)}{\xi }^{\alpha -1}+\frac{{m}_{2}}{{\Delta }_{1}}\right)>0,\end{array}$

$0<{G}_{1}\left(0,s\right)\le {G}_{1}\left(t,s\right)\le {G}_{1}\left(\xi ,s\right)$ 成立。

2) 对于 $t\in \left[\xi ,1\right]$，当 $\xi \le t\le s\le 1$ 时，

${G}_{2}\left(t,s\right)={g}_{2}\left(t,s\right)=-\frac{{n}_{1}}{{m}_{1}+{n}_{1}}\left(\frac{1}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}+\frac{{n}_{2}}{{\Delta }_{1}}\right)+\frac{t{n}_{2}}{{\Delta }_{1}},\frac{\partial {g}_{2}\left(t,s\right)}{\partial t}=\frac{{n}_{2}}{{\Delta }_{1}}>0;$

$\xi \le s\le t\le 1$ 时，

${G}_{2}\left(t,s\right)={g}_{2}\left(t,s\right)+\frac{1}{\Gamma \left(\beta \right)}{\left(t-s\right)}^{\beta -1},\frac{\partial {G}_{2}\left(t,s\right)}{\partial t}=\frac{{n}_{2}}{{\Delta }_{1}}+\frac{1}{\Gamma \left(\beta -1\right)}{\left(t-s\right)}^{\beta -2}>0,$

${G}_{2}\left(t,s\right)$ 是关于t的单调递增函数，那么

${G}_{2}\left(\xi ,s\right)\le {G}_{2}\left(t,s\right)\le {G}_{2}\left(1,s\right).$

${G}_{2}\left(\xi ,s\right)=-\frac{{n}_{1}}{{m}_{1}+{n}_{1}}\left(\frac{1}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}+\frac{{n}_{2}}{{\Delta }_{1}}\right)+\frac{{n}_{2}}{{\Delta }_{1}}=\frac{1}{{m}_{1}+{n}_{1}}\left(\frac{{n}_{1}}{\Gamma \left(\beta \right)}{\left(1-s\right)}^{\beta -1}+\frac{{n}_{2}\left(1-\left({m}_{1}+{n}_{1}\right)\xi \right)}{{\Delta }_{1}}\right)>0,$

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }u\left(t\right)\ge 0,\text{}t\in \left(0,\xi \right),\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }u\left(t\right)\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(\xi ,1\right),\\ \Delta u\left(\xi \right)=I,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {u}^{\prime }\left(\xi \right)=Q,\\ {m}_{1}u\left(0\right)+{n}_{1}u\left(1\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}u\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}u\left(1\right)\ge 0,\end{array}$ (4)

$u\left(t\right)\ge 0$$t\in \left[0,1\right]$

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }u\left(t\right)=h\left(t\right),\text{}t\in \left[0,\xi \right],\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }u\left(t\right)=y\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(\xi ,1\right],\\ \Delta u\left(\xi \right)=I,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {u}^{\prime }\left(\xi \right)=Q,\text{}\\ {m}_{1}u\left(0\right)+{n}_{1}u\left(1\right)={\gamma }_{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}u\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}u\left(1\right)={\gamma }_{1}.\end{array}$

$u\left(t\right)=\left\{\begin{array}{l}{\int }_{0}^{\xi }{G}_{1}\left(t,s\right)h\left(s\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right)y\left(s\right)\text{d}s+{\Delta }_{2}+\frac{t}{{\Delta }_{1}}\left(\frac{Q{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\right),\text{\hspace{0.17em}}t\in \left[0,\xi \right],\\ {\int }_{\xi }^{1}{G}_{2}\left(t,s\right)y\left(s\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(t,s\right)h\left(s\right)\text{d}s+{\Delta }_{2}+\frac{t}{{\Delta }_{1}}\left(\frac{Q{m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\right),\text{\hspace{0.17em}}t\in \left(\xi ,1\right].\end{array}$

3. 分数阶微分方程的上下解方法

(H2) 对任意 ${u}_{1}\le {u}_{2},{v}_{1}\le {v}_{2}$$f\left(t,{u}_{1},{v}_{1}\right)\le f\left(t,{u}_{2},{v}_{2}\right),t\in \left[0,\xi \right]$

$g\left(t,{u}_{1},{v}_{1}\right)\le g\left(t,{u}_{2},{v}_{2}\right),t\in \left(\xi ,1\right]$, $I\left({u}_{1}\right)\le I\left({u}_{2}\right),Q\left({u}_{1}\right)\le Q\left({u}_{2}\right),t\in \left[0,1\right]$.

$g\left(t,{u}_{1},{v}_{1}\right), $I\left({u}_{1}\right).

(H3) 若(H1)成立，且 ${x}_{1},{x}_{2},{y}_{1},{y}_{2}\in ℝ$，当 ${x}_{1}\le {x}_{2},{y}_{1}\le {y}_{2}$ 时，

${h}_{0}\left({x}_{2},{y}_{2}\right)-{h}_{0}\left({x}_{1},{y}_{1}\right)\le -{m}_{1}\left({x}_{2}-{x}_{1}\right)-{n}_{1}\left({y}_{2}-{y}_{1}\right)$,

${h}_{1}\left({x}_{2},{y}_{2}\right)-{h}_{0}\left({x}_{1},{y}_{1}\right)\ge -{m}_{2}\left({x}_{2}-{x}_{1}\right)-{n}_{2}\left({y}_{2}-{y}_{1}\right)$.

$P=\left\{u\in E:u\left(t\right)\ge 0,t\in \left[0,1\right]\right\}$，显然P为E中的正规体锥。且若 $u\left(t\right)\le v\left(t\right)$, $t\in \left[0,1\right]$，则 $u\underset{_}{\prec }v\in P$

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }u\left(t\right)={f}_{1}\left(t,x\left(t\right),x\left(t+{\tau }_{\text{1}}\right)\right),\text{}t\in \left(\text{0,}\xi \right)\text{,}\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }u\left(t\right)={f}_{2}\left(t,x\left(t\right),x\left(t-{\tau }_{2}\right)\right),\text{}t\in \left(\xi \text{,1}\right)\text{,}\\ \Delta u\left(\xi \right)=I\left(\xi ,x\left(\xi \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {u}^{\prime }\left(\xi \right)=Q\left(\xi ,x\left(\xi \right)\right),\\ {m}_{1}u\left(0\right)+{n}_{1}u\left(1\right)={h}_{0}\left(x\left(0\right),x\left(1\right)\right)+{m}_{1}x\left(0\right)+{n}_{1}x\left(1\right):={\gamma }_{0}\left(x\right),\\ {m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}u\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}u\left(1\right)\\ ={h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}x\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}x\left(1\right)\right)+{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}x\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}x\left(1\right):={\gamma }_{1}\left(x\right).\end{array}$ (5)

$u\left(t\right)=\left\{\begin{array}{l}{\int }_{0}^{\xi }{G}_{1}\left(t,s\right){f}_{1}\left(s,x\left(s\right),x\left(s+{\tau }_{1}\right)\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right){f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\Delta }_{2}^{x}+\frac{t}{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\left(x\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,\xi \right],\\ {\int }_{\xi }^{1}{G}_{2}\left(t,s\right){f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(t,s\right){f}_{1}\left(s,x\left(s\right),x\left(s+{\tau }_{1}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\Delta }_{3}^{x}+\frac{t}{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\left(x\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(\xi ,1\right].\end{array}$

${\Delta }_{2}^{x}=-\frac{1}{{m}_{1}+{n}_{1}}\left({n}_{1}I\left(\xi ,x\left(\xi \right)\right)+{n}_{1}\left(1-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q\left(\xi ,x\left(\xi \right)\right)-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}\left(x\right)-{\gamma }_{0}\left(x\right)\right);$

${\Delta }_{3}^{x}=-\frac{1}{{m}_{1}+{n}_{1}}\left(-{m}_{1}I\left(\xi ,x\left(\xi \right)\right)+\left({m}_{1}\xi +{n}_{1}+\frac{{n}_{1}{n}_{2}}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)Q\left(\xi ,x\left(\xi \right)\right)-\frac{{n}_{1}}{{\Delta }_{1}}{\gamma }_{1}\left(x\right)-{\gamma }_{0}\left(x\right)\right).$

$Tx\left(t\right)=\left\{\begin{array}{l}{\int }_{0}^{\xi }{G}_{1}\left(t,s\right){f}_{1}\left(s,x\left(s\right),x\left(s+{\tau }_{1}\right)\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right){f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\Delta }_{2}^{x}+\frac{t}{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\left(x\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,\xi \right],\\ {\int }_{\xi }^{1}{G}_{2}\left(t,s\right){f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(t,s\right){f}_{1}\left(s,x\left(s\right),x\left(s+{\tau }_{1}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\Delta }_{3}^{x}+\frac{t}{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){m}_{2}{\xi }^{2-\alpha }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\left(x\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(\xi ,1\right].\end{array}$

$\underset{n\to \infty }{\mathrm{lim}}\left({f}_{1}\left(t,{x}_{n}\left(t\right),{x}_{n}\left(t+{\tau }_{1}\right)\right)-{f}_{1}\left(t,x\left(t\right),x\left(t+{\tau }_{1}\right)\right)\right)=0,$

$\underset{n\to \infty }{\mathrm{lim}}\left({f}_{2}\left(t,{x}_{n}\left(t\right),{x}_{n}\left(t-{\tau }_{2}\right)\right)-{f}_{2}\left(t,x\left(t\right),x\left(t-{\tau }_{2}\right)\right)\right)=0,$

$\underset{n\to \infty }{\mathrm{lim}}|I\left({x}_{n}\left(t\right)\right)-I\left(x\left(t\right)\right)|=0,\text{\hspace{0.17em}}\underset{n\to \infty }{\mathrm{lim}}|Q\left({x}_{n}\left(t\right)\right)-Q\left(x\left(t\right)\right)|=0,$

$\underset{n\to \infty }{\mathrm{lim}}\left({\gamma }_{0}\left({x}_{n}\right)-{\gamma }_{0}\left(x\right)\right)=0,\text{\hspace{0.17em}}\underset{n\to \infty }{\mathrm{lim}}\left({\gamma }_{1}\left({x}_{n}\right)-{\gamma }_{1}\left(x\right)\right)=0,$

$A=\left[0,\xi \right]×\left[-{\stackrel{¯}{M}}_{0},{\stackrel{¯}{M}}_{0}\right]×\left[-{\stackrel{¯}{M}}_{0},{\stackrel{¯}{M}}_{0}\right],\text{\hspace{0.17em}}B=\left[\xi ,1\right]×\left[-{\stackrel{¯}{M}}_{0},{\stackrel{¯}{M}}_{0}\right]×\left[-{\stackrel{¯}{M}}_{0},{\stackrel{¯}{M}}_{0}\right].$

$\begin{array}{l}|T\left({x}_{n}\right)-T\left(x\right)|\\ =|{\int }_{0}^{\xi }{G}_{1}\left(t,s\right)\left({f}_{1}\left(s,{x}_{n}\left(s\right),{x}_{n}\left(s+{\tau }_{1}\right)\right)-{f}_{1}\left(s,x\left(t\right),x\left(s+{\tau }_{1}\right)\right)\right)\text{d}s\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right)\left({f}_{2}\left(s,{x}_{n}\left(s\right),{x}_{n}\left(s-{\tau }_{2}\right)\right)-{f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left({\Delta }_{2}^{{x}_{n}}-{\Delta }_{2}^{x}\right)+\frac{t}{{\Delta }_{1}}\left(\frac{\left(Q\left(\xi ,{x}_{n}\left(\xi \right)\right)-Q\left(\xi ,x\left(\xi \right)\right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-\left({\gamma }_{1}\left({x}_{n}\right)-{\gamma }_{1}\left(x\right)\right)\right)|\end{array}$

$\begin{array}{l}|T\left({x}_{n}\right)-T\left(x\right)|\\ =|{\int }_{0}^{\xi }{G}_{1}\left(t,s\right)\left({f}_{1}\left(s,{x}_{n}\left(s\right),{x}_{n}\left(s+{\tau }_{1}\right)\right)-{f}_{1}\left(s,x\left(t\right),x\left(s+{\tau }_{1}\right)\right)\right)\text{d}s\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right)\left({f}_{2}\left(s,{x}_{n}\left(s\right),{x}_{n}\left(s-{\tau }_{2}\right)\right)-{f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left({\Delta }_{2}^{{x}_{n}}-{\Delta }_{2}^{x}\right)+\frac{t}{{\Delta }_{1}}\left(\frac{\left(Q\left(\xi ,{x}_{n}\left(\xi \right)\right)-Q\left(\xi ,x\left(\xi \right)\right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-\left({\gamma }_{1}\left({x}_{n}\right)-{\gamma }_{1}\left(x\right)\right)\right)|\end{array}$

$\Omega \subset P$ 为有界集，由 ${f}_{1}$${f}_{2}$，I，Q的连续性得，存在 ${\stackrel{¯}{M}}_{2}>0$，使得对任意 $t\in \left[0,\xi \right]$, $u,v\in \Omega$，有 $|{f}_{1}\left(t,u,v\right)|\le {\stackrel{¯}{M}}_{2}$ ；对任意 $t\in \left(\xi ,1\right]$$u,v\in \Omega$$|{f}_{2}\left(t,u,v\right)|\le {\stackrel{¯}{M}}_{2}$, $|I|\le {\stackrel{¯}{M}}_{2}$, $|Q|\le {\stackrel{¯}{M}}_{2}$, $|{\gamma }_{0}\left(x\right)|\le {\stackrel{¯}{M}}_{2}$, $|{\gamma }_{1}\left(x\right)|\le {\stackrel{¯}{M}}_{2}$

$|{\Delta }_{2}^{x}|=-\frac{1}{{m}_{1}+{n}_{1}}\left({n}_{1}\left(2-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }+\Gamma \left(3-\beta \right)}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)-1\right){\stackrel{¯}{M}}_{2};$

$|{\Delta }_{3}^{x}|=-\frac{1}{{m}_{1}+{n}_{1}}\left({m}_{1}\left(\xi -1\right)+{n}_{1}\left(1+\frac{{n}_{2}-\Gamma \left(3-\beta \right)}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)-1\right){\stackrel{¯}{M}}_{2}$

$\begin{array}{c}\underset{t\in \left[0,\xi \right]}{\mathrm{sup}}|Tu\left(t\right)|\le |{\int }_{0}^{\xi }{G}_{1}\left(\xi ,s\right){f}_{1}\left(s,x\left(s\right),x\left(s+{\tau }_{1}\right)\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(1,s\right){f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\text{d}s\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \text{\hspace{0.17em}}+{\Delta }_{2}^{x}+\frac{\xi }{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\left(x\right)\right)|\\ \le \left({\int }_{0}^{\xi }{G}_{1}\left(\xi ,s\right)\text{d}s+{\int }_{\xi }^{1}{g}_{2}\left(1,s\right)\text{d}s-\frac{1}{{m}_{1}+{n}_{1}}\left({n}_{1}\left(2-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }-\Gamma \left(3-\beta \right)}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)-1\right)\\ \text{\hspace{0.17em}}+\frac{\xi }{{\Delta }_{1}}\left(\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-1\right)\right){\stackrel{¯}{M}}_{2},\end{array}$

$\begin{array}{c}\underset{t\in \left(\xi ,1\right]}{\mathrm{sup}}|Tu\left(t\right)|\le |{\int }_{\xi }^{1}{G}_{2}\left(1,s\right){f}_{2}\left(s,x\left(s\right),x\left(s-{\tau }_{2}\right)\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(\xi ,s\right){f}_{1}\left(s,x\left(s\right),x\left(s+{\tau }_{1}\right)\right)\text{d}s\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \text{\hspace{0.17em}}+{\Delta }_{3}^{x}+\frac{\xi }{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\left(x\right)\right)|\\ \le \left({\int }_{\xi }^{1}{G}_{2}\left(1,s\right)\text{d}s+{\int }_{0}^{\xi }{g}_{1}\left(\xi ,s\right)\text{d}s-\frac{1}{{m}_{1}+{n}_{1}}\left({m}_{1}\left(\xi -1\right)+{n}_{1}\left(1+\frac{{n}_{2}-\Gamma \left(3-\beta \right)}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)-1\right)\\ \text{\hspace{0.17em}}+\frac{1}{{\Delta }_{1}}\left(\frac{{m}_{2}{\xi }^{2-\alpha }}{\Gamma \left(3-\beta \right)}-1\right)\right){\stackrel{¯}{M}}_{2}.\end{array}$

$\begin{array}{c}|Tu\left({t}_{2}\right)-Tu\left({t}_{1}\right)|=|{\int }_{0}^{\xi }\left({G}_{1}\left({t}_{1},s\right)-{G}_{1}\left({t}_{2},s\right)\right){f}_{1}\left(s,u\left(s\right),u\left(s+{\tau }_{1}\right)\right)\text{d}s\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \text{\hspace{0.17em}}+{\int }_{\xi }^{1}\left({g}_{2}\left({t}_{1},s\right)-{g}_{2}\left({t}_{2},s\right)\right){f}_{2}\left(s,u\left(s\right),u\left(s-{\tau }_{2}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}+\frac{{t}_{1}-{t}_{2}}{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\left(x\right)\right)|\\ \le {\stackrel{¯}{M}}_{2}\left({\int }_{0}^{\xi }|{G}_{1}\left({t}_{1},s\right)-{G}_{1}\left({t}_{2},s\right)|\text{d}s+{\int }_{\xi }^{1}|{g}_{2}\left({t}_{1},s\right)-{g}_{2}\left({t}_{2},s\right)|\text{d}s\right)\\ \text{\hspace{0.17em}}+|{t}_{1}-{t}_{2}|\frac{|{n}_{2}{\left(1-\xi \right)}^{2-\beta }-\Gamma \left(3-\beta \right)\left(1-{n}_{2}\right){\stackrel{¯}{M}}_{2}}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\\ <\epsilon .\end{array}$

$\begin{array}{c}|Tu\left({t}_{3}\right)-Tu\left({t}_{4}\right)|=|{\int }_{\xi }^{1}\left({G}_{2}\left({t}_{3},s\right)-{G}_{1}\left({t}_{4},s\right)\right){f}_{2}\left(s,u\left(s\right),u\left(s-{\tau }_{2}\right)\right)\text{d}s\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \text{\hspace{0.17em}}+{\int }_{0}^{\xi }\left({g}_{1}\left({t}_{3},s\right)-{g}_{1}\left({t}_{4},s\right)\right){f}_{1}\left(s,u\left(s\right),u\left(s+{\tau }_{1}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}+\frac{{t}_{3}-{t}_{4}}{{\Delta }_{1}}\left(\frac{Q\left(\xi ,x\left(\xi \right)\right){m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\left(x\right)\right)|\\ \le {\stackrel{¯}{M}}_{2}\left({\int }_{\xi }^{1}|{G}_{2}\left({t}_{3},s\right)-{G}_{1}\left({t}_{4},s\right)|\text{d}s+{\int }_{0}^{\xi }|{g}_{1}\left({t}_{3},s\right)-{g}_{1}\left({t}_{4},s\right)|\text{d}s\right)\\ \text{\hspace{0.17em}}+|{t}_{3}-{t}_{4}|\frac{|{m}_{2}{\xi }^{2-\alpha }-\left({m}_{2}+1\right)\Gamma \left(3-\beta \right)|}{{\Delta }_{1}\Gamma \left(3-\beta \right)}{\stackrel{¯}{M}}_{2}\\ <\epsilon .\end{array}$

${f}_{1}\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t+{\tau }_{1}\right)\right)-{f}_{1}\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t+{\tau }_{1}\right)\right)\ge 0,\text{\hspace{0.17em}}t\in \left[0,\xi \right],$

${f}_{2}\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right)\right)-{f}_{2}\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\tau }_{2}\right)\right)\ge 0,\text{\hspace{0.17em}}t\in \left(\xi ,1\right],$

$\left(I\left(\xi ,{x}_{2}\left(\xi \right)\right)-I\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)\ge 0,\text{\hspace{0.17em}}\left(Q\left(\xi ,{x}_{2}\left(\xi \right)\right)-Q\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,1\right].$

${f}_{1}\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t+{\tau }_{1}\right)\right)-{f}_{1}\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t+{\tau }_{1}\right)\right)>0,$

$\left(I\left(\xi ,{x}_{2}\left(\xi \right)\right)-I\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)>0,\text{\hspace{0.17em}}\left(Q\left(\xi ,{x}_{2}\left(\xi \right)\right)-Q\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)>0,$

$\begin{array}{c}{\gamma }_{0}\left({x}_{2}\right)-{\gamma }_{0}\left({x}_{1}\right)={h}_{0}\left({x}_{2}\left(0\right),{x}_{2}\left(1\right)\right)-{h}_{0}\left({x}_{1}\left(0\right),{x}_{1}\left(1\right)\right)\\ \text{\hspace{0.17em}}+\left({m}_{1}{x}_{2}\left(0\right)+{n}_{1}{x}_{2}\left(1\right)\right)-\left(\begin{array}{c}{m}_{1}{x}_{1}\left(0\right)+{n}_{1}{x}_{1}\left(1\right)\\ \le 0,\end{array}\right)\end{array}$

$\begin{array}{c}{\gamma }_{1}\left({x}_{2}\right)-{\gamma }_{1}\left({x}_{1}\right)={h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{x}_{2}\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{x}_{2}\left(1\right)\right)-{h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{x}_{2}\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{x}_{2}\left(1\right)\right)\\ \text{\hspace{0.17em}}+{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{x}_{2}\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{x}_{2}\left(1\right)-\left({m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{x}_{1}\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{x}_{1}\left(1\right)\right)\\ \le 0,\end{array}$

$\begin{array}{c}{\Delta }_{2}^{{x}_{2}}-{\Delta }_{2}^{{x}_{1}}=-\frac{1}{{m}_{1}+{n}_{1}}\left({n}_{1}\left(I\left(\xi ,{x}_{2}\left(\xi \right)\right)-I\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)\right)\\ \text{\hspace{0.17em}}+{n}_{1}\left(1-\xi +\frac{{n}_{2}{\left(1-\xi \right)}^{2-\beta }}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)\left(Q\left(\xi ,{x}_{2}\left(\xi \right)\right)-Q\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)\end{array}$

$-\frac{{n}_{1}}{{\Delta }_{1}}\left({\gamma }_{1}\left({x}_{2}\right)-{\gamma }_{1}\left({x}_{1}\right)\right)-\left({\gamma }_{0}\left({x}_{2}\right)-{\gamma }_{0}\left({x}_{1}\right)\right)\ge 0,$

$\begin{array}{l}T{x}_{2}\left(t\right)-T{x}_{1}\left(t\right)\\ ={\int }_{0}^{\xi }{G}_{1}\left(t,s\right)\left({f}_{1}\left(s,{x}_{2}\left(s\right),{x}_{2}\left(s+{\tau }_{1}\right)\right)-{f}_{1}\left(s,{x}_{1}\left(s\right),{x}_{1}\left(s+{\tau }_{1}\right)\right)\right)\text{d}s+{\Delta }_{2}^{{x}_{2}}-{\Delta }_{2}^{{x}_{1}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{\xi }^{1}{g}_{2}\left(t,s\right)\left({f}_{2}\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right)\right)-{f}_{2}\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\tau }_{2}\right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{t}{{\Delta }_{1}}\left(\frac{\left(Q\left(\xi ,{x}_{2}\left(\xi \right)\right)-Q\left(\xi ,{x}_{1}\left(\xi \right)\right)\right){n}_{2}{\left(1-\xi \right)}^{2-\beta }}{\Gamma \left(3-\beta \right)}-{\gamma }_{1}\left({x}_{2}\right)-{\gamma }_{1}\left({x}_{1}\right)\right)\\ >{\int }_{0}^{\xi }{G}_{1}\left(t,s\right)\left({f}_{1}\left(s,{x}_{2}\left(s\right),{x}_{2}\left(s+{\tau }_{1}\right)\right)-{f}_{1}\left(s,{x}_{1}\left(s\right),{x}_{1}\left(s+{\tau }_{1}\right)\right)\right)\text{d}s\\ >0.\end{array}$

$\left({f}_{2}\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right)\right)-{f}_{2}\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\tau }_{2}\right)\right)\right)>0,$

$\begin{array}{c}{\Delta }_{3}^{{x}_{2}}-{\Delta }_{3}^{{x}_{1}}=-\frac{1}{{m}_{1}+{n}_{1}}\left(-{m}_{1}\left(I\left(\xi ,{x}_{2}\left(\xi \right)\right)-I\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)\right)\\ \text{\hspace{0.17em}}+\left({m}_{1}\xi +{n}_{1}+\frac{{n}_{1}{n}_{2}}{{\Delta }_{1}\Gamma \left(3-\beta \right)}\right)\left(Q\left(\xi ,{x}_{2}\left(\xi \right)\right)-Q\left(\xi ,{x}_{1}\left(\xi \right)\right)\right)\\ \text{\hspace{0.17em}}-\frac{{n}_{1}}{{\Delta }_{1}}\left({\gamma }_{1}\left({x}_{2}\right)-{\gamma }_{1}\left({x}_{1}\right)-\left({\gamma }_{0}\left({x}_{2}\right)-{\gamma }_{0}\left({x}_{1}\right)\right)\right)\\ \ge 0,\end{array}$

$\begin{array}{l}T{x}_{2}\left(t\right)-T{x}_{1}\left(t\right)\\ ={\int }_{\xi }^{1}{G}_{2}\left(t,s\right)\left({f}_{2}\left(s,{x}_{2}\left(s\right),{x}_{2}\left(s-{\tau }_{2}\right)\right)-{f}_{2}\left(s,{x}_{1}\left(s\right),{x}_{1}\left(s-{\tau }_{2}\right)\right)\right)\text{d}s+{\Delta }_{3}^{{x}_{2}}-{\Delta }_{3}^{{x}_{1}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{\xi }{g}_{1}\left(t,s\right)\left({f}_{1}\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t+{\tau }_{1}\right)\right)-{f}_{1}\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t+{\tau }_{1}\right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{t}{{\Delta }_{1}}\left(\frac{\left(Q\left(\xi ,{x}_{2}\left(\xi \right)\right)-Q\left(\xi ,{x}_{1}\left(\xi \right)\right)\right){m}_{2}}{\Gamma \left(3-\beta \right)}{\xi }^{2-\alpha }-{\gamma }_{1}\left({x}_{2}\right)-{\gamma }_{1}\left({x}_{1}\right)\right)\\ >{\int }_{\xi }^{1}{G}_{2}\left(t,s\right)\left({f}_{2}\left(s,{x}_{2}\left(s\right),{x}_{2}\left(s-{\tau }_{2}\right)\right)-{f}_{2}\left(s,{x}_{1}\left(s\right),{x}_{1}\left(s-{\tau }_{2}\right)\right)\right)\text{d}s\\ >0.\end{array}$

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }\alpha \left(t\right)\le {f}_{1}\left(t,\alpha \left(t\right),\alpha \left(t+{\tau }_{1}\right)\right),\text{}t\in \left[0,\xi \right],\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }\alpha \left(t\right)\le {f}_{2}\left(t,\alpha \left(t\right),\alpha \left(t-{\tau }_{2}\right)\right),\text{}t\in \left(\xi ,1\right],\\ \Delta \alpha \left(\xi \right)\le I\left(\xi ,u\left(\xi \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {\alpha }^{\prime }\left(\xi \right)\le Q\left(\xi ,\alpha \left(\xi \right)\right),\\ {h}_{0}\left(\alpha \left(0\right),\alpha \left(1\right)\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}\alpha \left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}\alpha \left(1\right)\right)\ge 0.\end{array}$

$\beta$ 为边值问题(1)的一个下解，若 $\beta$ 满足

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }\beta \left(t\right)\ge {f}_{1}\left(t,\beta \left(t\right),\beta \left(t+{\tau }_{1}\right)\right),\text{}t\in \left[0,\xi \right],\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }\beta \left(t\right)\ge {f}_{2}\left(t,\beta \left(t\right),\beta \left(t-{\tau }_{2}\right)\right),\text{}t\in \left(\xi ,1\right],\\ \Delta \beta \left(\xi \right)\ge I\left(\xi ,\beta \left(\xi \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {\beta }^{\prime }\left(\xi \right)\ge Q\left(\xi ,\beta \left(\xi \right)\right),\\ {h}_{0}\left(\beta \left(0\right),\beta \left(1\right)\right)\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}\beta \left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}\beta \left(1\right)\right)\le 0.\end{array}$

4. 主要结论

${\alpha }_{1}\prec {\beta }_{1}\prec {\alpha }_{2}\prec {\beta }_{2}.$

${\alpha }_{1}\underset{_}{\prec }{u}_{1}\prec \prec {\beta }_{1},\text{\hspace{0.17em}}{\alpha }_{2}\prec \prec {u}_{2}\underset{_}{\prec }{\beta }_{2},\text{\hspace{0.17em}}{\alpha }_{2}\overline{)\underset{_}{\prec }}{u}_{3}\overline{)\underset{_}{\prec }}{\beta }_{1}.$

$\left\{\begin{array}{l}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }\left(T{\alpha }_{1}\right)\left(t\right)={f}_{1}\left(t,{\alpha }_{1}\left(t\right),{\alpha }_{1}\left(t+{\tau }_{1}\right)\right),\text{}t\in \left(0,\xi \right),\\ {}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }\left(T{\alpha }_{1}\right)\left(t\right)={f}_{2}\left(t,{\alpha }_{1}\left(t\right),{\alpha }_{1}\left(t-{\tau }_{2}\right)\right),\text{}t\in \left(\xi ,1\right),\\ \Delta \left(T{\alpha }_{1}\right)\left(\xi \right)=I\left(\xi ,{\alpha }_{1}\left(\xi \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta {\left(T{\alpha }_{1}\right)}^{\prime }\left(\xi \right)=Q\left(\xi ,{\alpha }_{1}\left(\xi \right)\right),\\ {m}_{1}\left(T{\alpha }_{1}\right)\left(0\right)+{n}_{1}\left(T{\alpha }_{1}\right)\left(1\right)={h}_{0}\left({\alpha }_{1}\left(0\right),{\alpha }_{1}\left(1\right)\right)+{m}_{1}{\alpha }_{1}\left(0\right)+{n}_{1}{\alpha }_{1}\left(1\right),\\ {m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}\left(T{\alpha }_{1}\right)\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}\left(T{\alpha }_{1}\right)\left(1\right)\\ ={h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left(1\right)\right)+{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left(1\right).\end{array}$

$\alpha \left(t\right)=\left(T{\alpha }_{1}\right)\left(t\right)-{\alpha }_{1}\left(t\right)$。由于 ${\alpha }_{1}$ 是边值问题(1)的一个下解，则

${}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }\alpha \left(t\right)={}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }\left(T{\alpha }_{1}\right)\left(t\right)-{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }{\alpha }_{1}\left(t\right)={f}_{1}\left(t,{\alpha }_{1}\left(t\right),{\alpha }_{1}\left(t+{\tau }_{1}\right)\right)-{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha }{\alpha }_{1}\left(t\right)\ge 0,\text{}t\in \left(0,\xi \right),$

${}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }\alpha \left(t\right)={}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }\left(T{\alpha }_{1}\right)\left(t\right)-{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }{\alpha }_{1}\left(t\right)={f}_{2}\left(t,{\alpha }_{1}\left(t\right),{\alpha }_{1}\left(t-{\tau }_{2}\right)\right)-{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta }{\alpha }_{1}\left(t\right)\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(\xi ,1\right),$

$\begin{array}{c}{m}_{1}\alpha \left(0\right)+{n}_{1}\alpha \left(1\right)={m}_{1}\left(\left(T{\alpha }_{1}\right)\left(0\right)-{\alpha }_{1}\left(0\right)\right)+{n}_{1}\left(\left(T{\alpha }_{1}\right)\left(1\right)-{\alpha }_{1}\left(1\right)\right)\\ =\left({m}_{1}\left(T{\alpha }_{1}\right)\left(0\right)+{n}_{1}\left(T{\alpha }_{1}\right)\left(1\right)\right)-\left({m}_{1}{\alpha }_{1}\left(0\right)+{n}_{1}{\alpha }_{1}\left(1\right)\right)\\ ={h}_{0}\left({\alpha }_{1}\left(0\right),{\alpha }_{1}\left(1\right)\right)+{m}_{1}{\alpha }_{1}\left(0\right)+{n}_{1}{\alpha }_{1}\left(1\right)-\left({m}_{1}{\alpha }_{1}\left(0\right)+{n}_{1}{\alpha }_{1}\left(1\right)\right)\\ ={h}_{0}\left({\alpha }_{1}\left(0\right),{\alpha }_{1}\left(1\right)\right)\\ \le 0.\end{array}$

$\begin{array}{l}{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}\alpha \left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}\alpha \left(1\right)\\ ={m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}\left(\left(T{\alpha }_{1}\right)\left(0\right)-{\alpha }_{1}\left(0\right)\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}\left(\left(T{\alpha }_{1}\right)\left(1\right)-{\alpha }_{1}\left(1\right)\right)\\ ={m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}\left(T{\alpha }_{1}\right)\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}\left(T{\alpha }_{1}\right)\left(1\right)-\left({m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left(1\right)\right)\\ ={h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left(1\right)\right)+{m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left( 1 \right)\end{array}$

$-\left({m}_{2}{}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right)+{n}_{2}{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left(1\right)\right)={h}_{1}\left({}_{t}{}^{c}D{}_{{\xi }^{-}}^{\alpha -1}{\alpha }_{1}\left(0\right),{}_{{\xi }^{+}}{}^{c}D{}_{t}^{\beta -1}{\alpha }_{1}\left(1\right)\right)\ge 0.$

$\Delta \alpha \left(\xi \right)=\Delta \left(T{\alpha }_{1}\right)\left(\xi \right)-\Delta {\alpha }_{1}\left(\xi \right)=I\left(\xi ,{\alpha }_{1}\left(\xi \right)\right)-\Delta {\alpha }_{1}\left(\xi \right)\ge 0,$

$\Delta {\alpha }^{\prime }\left(\xi \right)=\Delta {\left(T{\alpha }_{1}\right)}^{\prime }\left(\xi \right)-\Delta {{\alpha }^{\prime }}_{1}\left(\xi \right)=Q\left(\xi ,{\alpha }_{1}\left(\xi \right)\right)-\Delta {{\alpha }^{\prime }}_{1}\left(\xi \right)\ge 0.$

${\alpha }_{2}\prec T{\alpha }_{2}.$

$T{\beta }_{1}\prec {\beta }_{1},\text{\hspace{0.17em}}T{\beta }_{2}\underset{_}{\prec }{\beta }_{2}.$

${\alpha }_{1}\underset{_}{\prec }{x}_{1}\prec \prec {\beta }_{1},\text{\hspace{0.17em}}{\alpha }_{2}\prec \prec {x}_{2}\underset{_}{\prec }{\beta }_{2},\text{\hspace{0.17em}}{\alpha }_{2}\overline{)\underset{_}{\prec }}{x}_{2}\overline{)\underset{_}{\prec }}{\beta }_{2}.$

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