﻿ 非瞬时脉冲分数阶微分方程边值问题解的存在性与唯一性
 上海理工大学学报  2020, Vol. 42 Issue (5): 430-435 PDF

Existence and uniqueness of solutions for boundary value problems of fractional differential equations with non-instantaneous impulses
ZHENG Wenjing, JIA Mei, LI Tingle
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The mutation described by non-instantaneous pulses will stay in a limited time interval. This phenomenon is common in clinical medicine, bioengineering, chemistry, physics and other fields. In order to reflect the change law of things more profoundly and accurately, the existence and uniqueness of solutions for a class of boundary value problems of fractional differential equations with non-instantaneous impulses were studied. The operator was defined by establishing an integral equation equivalent to the boundary value problem, and its complete continuity was proved. By using Schauder fixed point theorem, the sufficient conditions for the existence of solutions of the boundary value problems were obtained. The uniqueness theorem of solutions was obtained by using the contraction mapping principle.
Key words: non-instantaneous impulses     Caputo derivative     Schauder fixed point theorem     contraction mapping principle
1 问题的提出

 $\left\{ \begin{split} & {}^c{\rm D}_{{s_k} + }^\alpha u(t)\!\! =\!\! f(t,u(t)),{\rm{ }}t \in ({s_k},{t_{k + 1}}],{\rm{ }}k\!\! =\!\! 0,1,2, \cdots ,m \\ & {}^c{\rm D}_{{t_k} + }^\beta u(t) = g(t,u(t)),{\rm{ }}t \in ({t_k},{s_k}],{\rm{ }}k = 1,2, \cdots ,m \\ &\!\! \Delta u{|_{t={t_k}}}\!\!\!=\!\!\!{Q_k}({t_k},u({t_k})),\Delta u{|_{t={s_k}}}\!\!\!=\!\!0,u'(s_k^+)\!\!=\!\!u'(s_k^-)\!\!=\!\!0, \\ & \quad\quad k = 1,2, \cdots ,m, \\ & u'(0) = u(1) = 0 \end{split} \!\!\!\!\!\!\!\!\!\!\! \right.$ (1)

2 预备知识与引理

 $\begin{split} & PC(J,\mathbb{R}): = \{ u:J \to \mathbb{R}|u \in C(J',\mathbb{R}),\;u(t_k^{}) = u(t_k^ - ), \\ & \;\;\;\;\;\;\;\;\;\;u(t_k^ + ){\text{存在}}\;,\;k = 1,2,3, \cdots ,m\} \end{split}$

 $\left\{ \begin{split} & {}^c{\rm D}_{{s_k} + }^\alpha u(t) = {y_k}(t),{\rm{ }}t \in ({s_k},{t_{k + 1}}],{\rm{ }}k = 0,\;1,\;2,\; \cdots ,\;m\; \\ & {}^c{\rm D}_{{t_k} + }^\beta u(t) = {h_k}(t),{\rm{ }}t \in ({t_k},{s_k}], k = 1,\;2,\; \cdots ,\;m \\ & \Delta u{|_{t = {t_k}}} = {q_k}, \Delta u{|_{t = {s_k}}} = 0,\;u'(s_k^ + ) = u'(s_k^ - ) = 0,\\ &\qquad k = 1,\;2,\; \cdots ,\;m \\ & u'(0) = u(1) = 0 \end{split} \right. \!\!\!\!\!\!\!\!\!\!\!\!\!$ (2)

 $u(t) \!\!=\!\! \left\{ \begin{split} & I_{0 + }^\alpha {y_0}(t) - \sum\limits_{i = 1}^{m + 1} {I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i})} - \sum\limits_{i = 1}^m {(I_{{t_i} + }^\beta {h_i}({s_i}) -} \\ & \;\;\;\; { I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i}))} - \sum\limits_{i = 1}^m {{q_i}} ,\;t \in [0,{t_1}] \\ & I_{{s_k} + }^\alpha {y_k}(t) - \sum\limits_{i = k + 1}^{m + 1} {I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i})} - \\ &\; \sum\limits_{i = k + 1}^m {(I_{{t_i} + }^\beta {h_i}({s_i}) - I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i}))} \!\! -\!\! \sum\limits_{i = k + 1}^m {{q_i}} ,\!\!\!\!\!\! \\ & \;\;\; t \in ({s_k},{t_{k + 1}}],k = 1,2, \cdots ,m - 1\\ & \!\!-\!\! \sum\limits_{i = k + 1}^{m + 1} {I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i})} + I_{{t_k} + }^\beta {h_k}(t) \!\!- \!\!I_{{t_k} + }^{\beta - 1}{h_k}({s_k})(t - {t_k})\! -\!\! \\ & \;\;\; \sum\limits_{i = k}^m {I_{{t_i} + }^\beta {h_i}({s_i})} + \sum\limits_{i = k}^m {I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i})} - \sum\limits_{i = k}^m {{q_i}} + {q_k}, \\ & \;\;\; \;\;t \in ({t_k},{s_k}],k = 1,2, \cdots ,m\\ & I_{{s_m} + }^\alpha {y_m}(t) - I_{{s_m} + }^\alpha {y_m}(1),\;\;\;t \in ({s_m},1]。\end{split} \right. \!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$ (3)

 $u(t) = I_{0 + }^\alpha {y_0}(t) + {c_0},u(t_1^{}) = u(t_1^ - ) = I_{0 + }^\alpha {y_0}({t_1}) + {c_0}$ (4)

$t \in ({t_1},{s_1}]$ 时，考虑边值问题

 $\left\{ \begin{split} {}^c{\rm D}_{{t_1} + }^\beta u(t) = {h_1}(t),{\rm{ }}t \in ({t_1},{s_1}]{\rm{ }}\\ u(t_1^ + ) = u(t_1^ - ) + {q_1},u'(s_1^ - ) = 0 \end{split} \right.$

${\rm D}_{{t_1}^ + }^\beta (u(t)) = {h_1}(t)$ $u'(s_1^ - ) = 0$ ，可得

 $u(t) = I_{{t_1} + }^\beta {h_1}(t) - I_{{t_1} + }^{\beta - 1}{h_1}({s_1})(t - {t_1}) + {d_1}$

$u(t_1^ + ) = u(t_1^ - ) + {q_1}$ ，所以

 ${d_1} = u(t_1^ - ) + {q_1} = I_{0 + }^\alpha {y_0}({t_1}) + {c_0} + {q_1}$ (5)

 $\begin{split} u(t) =& I_{{t_1} + }^\beta {h_1}(t) - I_{{t_1} + }^{\beta - 1}{h_1}({s_1})(t - {t_1}) + \\ &I_{0 + }^\alpha {y_0}({t_1}) + {q_1} + {c_0} \end{split}$ (6)
 $\begin{split} & u(s_1^ - ) = I_{{t_1} + }^\beta {h_1}({s_1}) - I_{{t_1} + }^{\beta - 1}{h_1}({s_1})({s_1} - {t_1}) + \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\; I_{0 + }^\alpha {y_0}({t_1}) + {q_1} + {c_0} \end{split}$ (7)

$t \in ({s_1},{t_2}]$ 时，考虑Cauchy问题

 $\left\{ \begin{split} & {}^c{\rm D}_{{s_1} + }^\alpha u(t) = {y_1}(t) \\ & u(s_1^ + ) = u(s_1^ - ),\;u'(s_1^ + ) = 0\; \\ \end{split} \right.$

${}^c{\rm D}_{{s_1} + }^\alpha u(t) = {y_1}(t),\;u'(s_1^ + ) = 0$ 可得， $u(t) = I_{{s_1} + }^\alpha {c_1},$ 所以

 $\begin{split} &\;\;\;\; u(s_1^ + ) = {c_1} = u(s_1^ - ) = I_{{t_1} + }^\beta {h_1}({s_1}) - \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\; I_{{t_1} + }^{\beta - 1}{h_1}({s_1})({s_1} - {t_1}) + I_{0 + }^\alpha {y_0}({t_1}) + {q_1} + {c_0} \end{split}$ (8)

 $\begin{split} & u(t) = I_{{s_1} + }^\alpha {y_1}(t) + I_{{t_1} + }^\beta {h_1}({s_1}) - I_{{t_1} + }^{\beta - 1}{h_1}({s_1})({s_1} - {t_1}) + \\ & \;\;\;\;\;\;\;\;\;\;\; I_{0 + }^\alpha {y_0}({t_1}) + {q_1} + {c_0} \end{split}$ (9)

$t \in ({t_k},{s_k}]$ 时，由边值问题

 $\left\{ \begin{split} &{}^c{\rm D}_{{t_k} + }^\beta u(t) = {h_k}(t) \\ & u(t_k^ + ) = u(t_k^ - ) + {q_k},\;u'(s_k^ - ) = 0 \end{split} \right.$

 $\left\{ \begin{split} & u(t) = I_{{t_k} + }^\beta {h_k}(t) - I_{{t_k} + }^{\beta - 1}{h_k}({s_k})(t - {t_k}) + {d_k} \\ & {d_k} = u(t_k^ - ) + {q_k} = I_{{s_{k - 1}} + }^\alpha {y_{k - 1}}({t_k}) + {c_{k - 1}} + {q_k} \end{split} \right.$ (10)

 $\begin{split} & u(t) = I_{{t_k} + }^\beta {h_k}(t) - I_{{t_k} + }^{\beta - 1}{h_k}({s_k})(t - {t_k}) + \\ &\;\;\;\;\;\;\;\;\;\;\;I_{{s_{k - 1}} + }^\alpha {y_{k - 1}}({t_k}) + {c_{k - 1}} + {q_k} \end{split}$ (11)
 $\begin{split} & u(s_k^ - ) = I_{{t_k} + }^\beta {h_k}({s_k}) - I_{{t_k} + }^{\beta - 1}{h_k}({s_k})({s_k} - {t_k}) + \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;I_{{s_{k - 1}} + }^\alpha {y_{k - 1}}({t_k}) + {c_{k - 1}} + {q_k} \end{split}$ (12)

$t \in ({s_k},{t_{k + 1}}]$ 时，根据Cauchy问题

 $\left\{ {\begin{split} & {{}^c{\rm D}_{{s_k} + }^\alpha u(t) = {y_k}(t)} \\ & {u(s_k^ + ) = u(s_k^ - ),\;u'(s_k^ + ) = 0} \end{split}} \right.$

 $\left\{ \begin{split} & u(t) = I_{{s_k} + }^\alpha {y_k}(t) + {c_k}\\ & u(s_k^ + ) = {c_k} = u(s_k^ - ) = I_{{t_k} + }^\beta {h_k}({s_k}) - \\ &\;\; I_{{t_k} + }^{\beta - 1}{h_k}({s_k})({s_k} - {t_k}) + I_{{s_{k - 1}} + }^\alpha {y_{k - 1}}({t_k}) + {c_{k - 1}} + {q_k} \end{split} \right.$ (13)

 \begin{aligned} & u(t) = I_{{s_k} + }^\alpha {y_k}(t) + I_{{t_k} + }^\beta {h_k}({s_k}) - \\ &\;\;I_{{t_k} + }^{\beta - 1}{h_k}({s_k})({s_k} - {t_k}) + I_{{s_{k - 1}} + }^\alpha {y_{k - 1}}({t_k}) + {c_{k - 1}} + {q_k}\end{aligned} (14)

 $\begin{split} & {c_k} - {c_{k - 1}} = I_{{t_k} + }^\beta {h_k}({s_k}) - I_{{t_k} + }^{\beta - 1}{h_k}({s_k})({s_k} - {t_k}) + \\ & \;\;\;\;\;\;\;\;\;\;\;\; I_{{s_{k - 1}} + }^\alpha {y_{k - 1}}({t_k}) + {q_k},\;\;k = 1,2,3, \cdots ,m\; \end{split}$

 $\begin{split} & {c_k} = \sum\limits_{i = 1}^k (I_{{t_i} + }^\beta {h_i}({s_i}) - I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i}) + \\ &\;\;\;\;\;\;\;\;\; I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i}) + {q_i}) + {c_0} \end{split}$ (15)

 $\begin{split} & 0 = u(1) = I_{{s_m} + }^\alpha {y_m}(1) + {c_m}= I_{{s_m} + }^\alpha {y_m}(1) +\\ & \;\;\;\;\; \sum\limits_{i = 1}^m (I_{{t_i} + }^\beta {h_i}({s_i}) - I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i}) + \\ & \;\;\;\;\; I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i}) + {q_i}) + {c_0} \end{split}$

 $\begin{split} & {c_m} = - I_{{s_m} + }^\alpha {y_m}(1),{c_0} = - \sum\limits_{i = 1}^{m + 1} {I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i})} - \\ &\;\;\;\;\;\;\;\; \sum\limits_{i = 1}^m {(I_{{t_i} + }^\beta {h_i}({s_i}) - I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i}) + {q_i}} ) \end{split}$ (16)

 $\begin{split} & {c_k} = - \sum\limits_{i = k + 1}^m {(I_{{t_i} + }^\beta {h_i}({s_i}) - I_{{t_i} + }^{\beta - 1}{h_i}({s_i})({s_i} - {t_i}) + {q_i}} )\; - \\ & \;\;\;\;\;\;\;\; \sum\limits_{i = k + 1}^{m + 1} {I_{{s_{i - 1}} + }^\alpha {y_{i - 1}}({t_i})} ,\;k = 1,2, \cdots ,m - 1 \end{split}$ (17)

 $\chi (x,y,z) = \left\{ \begin{split} & 1,\;\;\;\;x \leqslant z \leqslant y\\ & 0,\;{\rm{ }} \;\;\; {\text{其他}} \end{split} \right.$
 \begin{aligned} & \!\!\!{{W_{\rm{1}}}(t,s) = \frac{1}{{\Gamma (\alpha )}}} \cdot \\ &\;\;\;\; {\left\{ \begin{split} &\chi (0,t,s){(t - s)^{\alpha - 1}} - \sum\limits_{i = 1}^{m + 1} {\chi ({s_{i - 1}},{t_i},s){{({t_i} - s)}^{\alpha - 1}}} ,\\ & \;\;\;\;\;0 \leqslant t \leqslant {t_1} 0 \leqslant s \leqslant 1\\ & \chi ({s_k},t,s){(t - s)^{\alpha - 1}} \!- \!\!\sum\limits_{i = k + 1}^{m + 1} {\chi ({s_{i - 1}},{t_i},s){{({t_i}\!\! -\!\! s)}^{\alpha - 1}},} \\ & \;\;\;\;\;{s_k} < t \leqslant {t_{k + 1}},\;0 \leqslant s \leqslant 1,\;k = 1,2, \cdots ,m - 1\\ &- \sum\limits_{i = k + 1}^{m + 1} \chi ({s_{i - 1}},{t_i},s){{({t_i} - s)}^{\alpha - 1}},\;{t_k} < t \leqslant {s_k}, 0 \leqslant s \leqslant 1,\,\\ & \;\;\;\;\;k = 1,2, \cdots ,m \\ & \chi ({s_m},t,s){(t - s)^{\alpha - 1}} - \chi ({s_m},1,s){(1 - s)^{\alpha - 1}},\\ & \;\;\;\;\;{s_m} < t \leqslant 1,\;0 \leqslant s \leqslant 1 \end{split} \right.} \end{aligned} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (18)
 \begin{aligned} &\!\!\! {{W_2}(t,s) = \frac{1}{{\Gamma (\beta )}}}\cdot \\ & {\left\{ \begin{split} & - \sum\limits_{i = 1}^m {\chi ({t_i},{s_i},s){{({s_i} - s)}^{\beta - 2}}((2 - \beta ){s_i} + (\beta - 1){t_i} - s} ),\;\; \\ & \;\;\;\;\;\;\;\; 0 \leqslant t \leqslant {t_1},\;0 \leqslant s \leqslant 1,\\ & - \sum\limits_{i = k + 1}^m {\chi ({t_i},{s_i},s){{({s_i} - s)}^{\beta - 2}}((2 - \beta ){s_i} + (\beta - 1){t_i} - s} ),\\ &\;\;\;\;\;\;\;\;{s_k} < t \leqslant {t_{k + 1}},\;0 \leqslant s \leqslant 1,\;k = 1,2, \cdots ,m - 1\\ &\chi ({t_k},t,s){(t - s)^{\beta - 1}} - \sum\limits_{i = k}^m \chi ({t_i},{s_i},s){{({s_i} - s)}^{\beta - 2}}\text{·} \\ & \;\;\;\;\;\;\;\; ((2 - \beta ){s_i} + (\beta - 1){t_i} - s )- (\beta - 1)\chi ({t_k},{s_k},s)\text{·} \\ &\;\;\;\;\;\;\;\; \;\; (t - {t_k}){({s_k} - s)^{\beta - 2}},\;\;{t_k} < t \leqslant {s_k}, \\ &\;\;\;\;\;\;\;\; 0 \leqslant s \leqslant 1,\;k = 1,2,3, \cdots ,m\\ & 0,\;{s_m} < t \leqslant 1,\;\,0 \leqslant s \leqslant 1 \end{split} \right.} \end{aligned} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (19)

 $Au(t) = \int_0^1 {{W_1}(t,s)f(s,u(s)){\rm{d}}s}$
 $Bu(t) = \int_0^1 {{W_2}(t,s)} g(s,u(s)){\rm{d}}s$
 $Gu(t) = \left\{ \begin{split} & - \sum\limits_{i = 1}^m {{Q_i}({t_i},u({t_i})),\;\;\;\;\;t \in [0,{t_1}]}\\ & - \sum\limits_{i = k + 1}^m {{Q_i}({t_i},u({t_i})),\;\;} t \in ({t_k},{t_{k + 1}}],\\ & \;\;\;\;k = 1,2, \cdots ,m - 1\; \\ & 0,\;t \in ({t_m},1] \end{split} \right.$
 $Tu(t) = Au(t) + Bu(t) + Gu(t)$

$T:PC(J,\mathbb{R}) \to PC(J,\mathbb{R})$

 $\left| {{W_1}(t,s)} \right| \leqslant \frac{{m + 2}}{{\Gamma (\alpha )}},\;\;\left| {{W_2}(t,s)} \right| \leqslant \frac{{2(m + 1)}}{{\Gamma (\beta )}}$

 $\begin{split} & \left| {{W_1}(t,s)} \right| \leqslant \frac{1}{{\Gamma (\alpha )}}\Bigg(\left| {\chi (0,t,s){{(t - s)}^{\alpha - 1}}} \right| + \\ & \;\;\;\;\;\;\; \left| {\sum\limits_{i = 1}^{m + 1} {\chi ({s_{i - 1}},{t_i},s){{({t_i} - s)}^{\alpha - 1}}} } \right|\Bigg) \leqslant \frac{{m + 2}}{{\Gamma (\alpha )}} \end{split}$

${s_k} < t \leqslant {t_{k + 1}},\;0 \leqslant s \leqslant 1,\;k = 1,2, \cdots ,m - 1$ 时，

 $\begin{split} & \left| {{W_{\rm{1}}}(t,s)} \right| \leqslant \frac{1}{{\Gamma (\alpha )}}\Bigg(\left| {\chi ({s_k},t,s){{(t - s)}^{\alpha - 1}}} \right| + \\ & \;\;\;\;\;\;\; \left| { - \sum\limits_{i = k + 1}^{m + 1} {\chi ({s_{i - 1}},{t_i},s){{({t_i} - s)}^{\alpha - 1}}\;} } \right|\Bigg) < \frac{{m + 2}}{{\Gamma (\alpha )}} \end{split}$

${t_k} < t \leqslant {s_k},{\rm{ }}0 \leqslant s \leqslant 1,\;k = 1,2, \cdots ,m$ 时，

 $\begin{split} & \left| {{W_1}(t,s)} \right| \leqslant \frac{1}{{\Gamma (\alpha )}}\Bigg(\left| {\sum\limits_{i = k + 1}^m {\chi ({s_{i - 1}},{t_i},s){{({t_i} - s)}^{\alpha - 1}}} } \right| + \\ &\;\;\;\;\;\;\; \left| {\chi ({s_m},1,s){{(1 - s)}^{\alpha - 1}}} \right|\Bigg) \leqslant \frac{{m + 1}}{{\Gamma (\alpha )}} < \frac{{m + 2}}{{\Gamma (\alpha )}} \end{split}$

${s_m} < t \leqslant 1,\;0 \leqslant s \leqslant 1$ 时，

 $\begin{split} & \left| {{W_1}(t,s)} \right| = \frac{1}{{\Gamma (\alpha )}}| {\chi ({s_m},t,s){(t - s)}^{\alpha - 1} -} \\ &\;\;\;\;\;\;\; {\chi ({s_m},1,s){{(1 - s)}^{\alpha - 1}}} | \leqslant \frac{2}{\Gamma (\alpha )} \end{split}$

 $\begin{array}{l} f(s,{u_n}(s)) - f(s,u(s)) \to 0,g(s,{u_n}(s)) - \\ \;\;\;\;\;g(s,u(s)) \to 0,n \to \infty\\ {Q_k}(s,{u_n}(s)) - {Q_k}(s,u(s)) \to 0,\;k = 1,2, \cdots ,m,n \to \infty \end{array}$

$r > 0,{\rm{ }}{B_r} = \{ u \in PC(J,\mathbb{R}):\left\| u \right\| \leqslant r\}$ ，记 ${M_f} =$ $\mathop {\max }\limits_{(t,u) \in [0,1] \times [ - r,r]} \left| {f(t,u)} \right|$ ${M_g} = \mathop {\max }\limits_{(t,u) \in [0,1] \times [ - r,r]} \left| {g(t,u)} \right|$ ${M_Q} =$ $\mathop {\max }\limits_{1 \leqslant k \leqslant m,u \in [ - r,r]}$ $\left| {{Q_k}({t_k},u)} \right|$ 。对任意 $u \in {B_r}$ ，根据引理2可得

 $\begin{split} & \left| {Tu(t)} \right| = \left| {Au(t) + Bu(t) + Gu(t)} \right| \leqslant \\ & \;\;\;\; \int_0^1 {\left| {{W_1}(t,s)f(s,u(s))} \right|\rm{d}} s +\\ & \;\;\;\; \;\int_0^1 {{{\left| W \right.}_2}(t,s)} g(s,u(s)\left. ) \right| {\rm{d}} s + \left| {Gu(t)} \right| \leqslant \\ & \;\;\;\; \frac{{(m + 2){M_f}}}{{\Gamma (\alpha )}} + \frac{{2(m + 1){M_g}}}{{\Gamma (\beta )}} + (m + 1){M_Q} \end{split}$

 $\begin{split} & |{ ({t'_1} - s)^{\beta - 1}} - {({t'_2} - s)^{\beta - 1}}| < \frac{{\Gamma (\beta )\varepsilon }}{{3{M_g}{\rm{ + }}1}}\;\; \\ & |{({t'_1} - s)^{\alpha - 1}} - {({t'_2} - s)^{\alpha - 1}}| < \frac{{\Gamma (\alpha )\varepsilon }}{{2{M_f}{\rm{ + }}1}} \end{split}$

 $\begin{split} & \left| {Tu({{t'_1}}) - Tu({{t'_2}})} \right|\leqslant | Au({{t'_1}}) - Au({{t'_2}})| + \\ & \quad|Bu({{t'_1}}) - Bu({{t'_2}})| + |Gu({{t'_1}}) - Gu({{t'_2}}) | \leqslant \\ &\quad \frac{{{M_f}}}{{\Gamma (\alpha )}} \Bigg( \int_0^{{{t'_1}}} {({{({{t'_2}} - s)}^{\alpha - 1}} - {{({{t'_1}} - s)}^{\alpha - 1}})\rm{d}} s + \\ &\quad \int_{{{t'_1}}}^{{{t'_2}}} {{{({{t'_2}} - s)}^{\alpha - 1}}\rm{d}} s \Bigg) < \frac{{{M_f}}}{{\Gamma (\alpha )}} \frac{{\Gamma (\alpha )\varepsilon }}{{2{M_f} + 1}} + \frac{{{M_f \delta }}}{{\Gamma (\alpha )}} < \varepsilon \quad\quad \end{split}$

${t_1}^\prime < {t_2}^\prime \in ({t_k},{s_k}],\;|{t_1}^\prime - {t_2}^\prime |\; < \delta ,\;\;k = 1,2, \cdots ,m$ 时，

 $\begin{split} & | {Tu({{t'_1}}) - Tu({{t'_2}})} | \leqslant | Au({{t'_1}}) - Au({{t'_2}})| +\\ & \;\;\;\; |Bu({{t'_1}}) - Bu({{t'_2}})| + |Gu({{t'_1}}) - Gu({{t'_2}}) |\leqslant \\ & \;\;\;\;\frac{{{M_g}}}{{\Gamma (\beta )}} \Bigg(\int_{{t_k}}^{{{t'}_1}} | {{{({{t'_1}} - s)}^{\beta - 1}} - {{({{t'_2}} - s)}^{\beta - 1}}} | {\rm{d} }s + \\ &\;\;\;\; \int_{{{t'_1}}}^{{{t'_2}}} {{{({{t'_2}} - s)}^{\beta - 1}}\rm{d}} s + \int_0^1 | (\beta - 1)\chi ({t_k},{s_k},s)({{t'_2}} - {{t'_1}}) \cdot \\ &\;\;\;\;{{({s_k} - s)}^{\beta - 2}} | {\rm{d}} s \Bigg) \leqslant \dfrac{{{M_g}}}{{\Gamma (\beta )}} \dfrac{{\Gamma (\beta )}}{{3{M_g} + 1}} + \dfrac{{{M_g}}}{{\Gamma (\beta )}} 2 \delta < \varepsilon \end{split}$

${t_1}^\prime < {t_2}^\prime \in ({s_m},1],\;|{t_1}^\prime - {t_2}^\prime |\; < \delta$ 时，

 $\begin{split} & \left| {Tu({{t'_1}}) - Tu({{t'_2}})} \right|\leqslant \\ & \;\;\;\;\;\dfrac{{{M_f}}}{{\Gamma (\alpha )}}\Bigg(\int_{{s_m}}^{{{t'_1}}} {\left| {{{({{t'_1}} - s)}^{\alpha - 1}} - {{({{t'_2}} - s)}^{\alpha - 1}}} \right|\rm{d}} s + \\ & \;\;\;\;\;\int_{{{t'_1}}}^{{{t'_2}}} {{{({{t'_2}} - s)}^{\alpha - 1}}\rm{d}} s \Bigg) \leqslant \dfrac{{{M_f}}}{{\Gamma (\alpha )}} \dfrac{{\Gamma (\alpha )\varepsilon }}{{2{M_f} + 1}} + \dfrac{{{M_f}}}{{\Gamma (\alpha )}} \delta < \varepsilon \quad\quad \end{split}$

3 边值问题解的存在性与唯一性

 ${N_0} = \frac{{m + 2}}{{\Gamma (\alpha )}},{N_1} = \frac{{2(m + 2)}}{{\Gamma (\beta )}},{N_2} = m + 1$

H1 存在非负实数 ${a_0},{a_1},{b_0},{b_1},{c_0},{c_1}$ ，常数 $\sigma ,\theta ,\gamma > 0,$ 使得对任意的 $t \in J$ 及任意的 $u \in \mathbb{R}$

 $\begin{split} & |f(t,x)| \leqslant {a_0} + {a_1}|u{|^\sigma },|g(t,u)| \leqslant {b_0} + {b_1}{\left| u \right|^\theta }, \\ & |{Q_k}(t,u)| \leqslant {l_0} + {l_1}{\left| u \right|^\gamma },k = 1,2, \cdots ,m \end{split}$

H2 存在常数 ${L_1},{L_2},{L_3} \geqslant 0$ ，使得对任意的 $t \in J$ 及任意的 ${u_1},{u_2} \in \mathbb{R}$

 $\begin{split} & \left| {f(t,{u_1}) - f(t,{u_2})} \right| \leqslant {L_1}\left| {{u_1} - {u_2}} \right| \\ & \left| {g(t,{u_1}) - g(t,{u_2})} \right| \leqslant {L_2}\left| {{u_1} - {u_2}} \right| \end{split}$

 $\begin{split} & {r_1} \geqslant \max \{ 1,4({N_0}{a_0} + {N_1}{b_0} + {N_2}{l_0}), \\ & \;\;\;\;\;\;\; {(4{N_0}{a_1})^{\frac{1}{{1 - \sigma }}}},{(4{N_1}{b_1})^{\frac{1}{{1 - \theta }}}},{(4{N_2}{l_1})^{\frac{1}{{1 - \gamma }}}}\} \\ & D = \{ u \in PC(J,\mathbb{R}):{\left\| u \right\|_{PC}} \leqslant {r_1}\} \end{split}$

D $PC(J,\mathbb{R})$ 中非空有界闭凸集。

 $\begin{split} & \left| {Tu(t)} \right| = \left| {Au(t) + Bu(t) + Gu(t)} \right| \leqslant \\ & \;\;\;\;\;\int_0^1 {\left| {{W_1}(t,s)f(s,u(s))} \right|\rm{d} }s + \\ & \;\;\;\;\;\int_0^1 {{{\left| W \right.}_2}(t,s)} g(s,u(s)\left. ) \right|{\rm{d}} s + \left| {Gu(t)} \right| < \\ & \;\;\;\;\; \frac{{(m + 2){a_0}}}{{\Gamma (\alpha )}} + \frac{{2(m + 1){b_0}}}{{\Gamma (\beta )}} + (m + 1){l_0} + \\ &\;\;\;\;\;\frac{{(m + 2){a_1}{r_1}^\sigma }}{{\Gamma (\alpha )}} + \;\frac{{2(m + 1){b_1}{r_1}^\theta }}{{\Gamma (\beta )}} + (m + 1){l_1}{r_1}^\gamma < \\ &\;\;\;\;\; {N_0}{a_0} + {N_1}{b_0} + {N_2}{l_0} + \;{N_0}{a_1}{r_1}^\sigma + \; \\ & \;\;\;\;\; {N_1}{b_1}{r_1}^\theta+ {N_2}{l_1}{r_1}^\gamma \leqslant {r_1} \end{split}$

 $\begin{split} & \left| {Tu(t)} \right| = \left| {Au(t) + Bu(t) + Gu(t)} \right| \leqslant\\ & \;\;\;\;\;\; \int_0^1 {\left| {{W_1}(t,s)f(s,u(s))} \right|\rm{d}} s + \\ & \;\;\;\;\;\; \int_0^1 {\left| {{W_2}} \right.(t,s)} g(s,u(s)\left. ) \right|{\rm{d}} s + \left| {Gu(t)} \right| < \\ & \;\;\;\;\;\; \frac{{(m + 2){a_0}}}{{\Gamma (\alpha )}} + \frac{{2(m + 1){b_0}}}{{\Gamma (\beta )}} + (m + 1){l_0} + \\ & \;\;\;\;\;\; \Bigg(\frac{{(m + 2){a_1}}}{{\Gamma (\alpha )}} + \;\frac{{2(m + 1){b_1}}}{{\Gamma (\beta )}} + (m + 1){l_1}\Bigg){r_2} <\\ & \;\;\;\;\;\; {N_0}{a_0} + {N_1}{b_0} + {N_2}{l_0} + \;{N_0}{a_1}{r_2} + \;{N_1}{b_1}{r_2} +\\ & \;\;\;\;\;\; {N_2}{l_1}{r_2} \leqslant {r_2} \end{split}$ (20)

 $\begin{split} & \left| {T{u_1}(t) - T{u_2}(t)} \right| = \left| {{A{u_1}(t) - A{u_2}(t)+ }} \right. \\ & \;\;\;\;\; \left. { B{u_1}(t) - B{u_2}(t) + G{u_1}(t) - G{u_2}(t)} \right|\leqslant \\ & \;\;\;\;\; \int_0^1 {\left| {{W_1}(t,s\left. ) \right|\left| f \right.(s,{u_1}(s)) - f(s,{u_{\rm{2}}}(s))} \right|\rm{d} }s +\\ & \;\;\;\;\; \int_0^1 {\left| {{W_2}} \right.(t,s\left. ) \right|} \left| g \right.(s,{u_{\rm{1}}}(s)\left. {) - g(s,{u_2}(s)} \right|{\rm{d}} s + \\ &\;\;\;\;\; \left| {G{u_1}(t) - G{u_2}(t)} \right| <\\ &\;\;\;\;\; \left(\frac{{(m + 2){L_1}}}{{\Gamma (\alpha )}} + \frac{{2(m + 1){L_2}}}{{\Gamma (\beta )}} + (m + 2){L_3}\right){\left\| {{u_1} - {u_2}} \right\|_{PC}} = \\ &\;\;\;\;\; ({N_0}{L_1} + {N_1}{L_2} + {N_2}{L_3}){\left\| {{u_1} - {u_2}} \right\|_{PC}} \end{split}$

 [1] PODLUBNY I. Fractional differential equations[M]. San Diego: Academic Press, 1999. [2] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier, 2006. [3] 白占兵. 分数阶微分方程边值问题理论及应用[M]. 北京: 中国科学技术出版社, 2013. [4] 李秀红, 寇春海, 刘晓波. 分数阶微分方程边值问题解的存在性[J]. 东华大学学报(自然科学版), 2011, 37(2): 240-245. [5] 李燕, 刘锡平, 李晓晨, 等. 具逐项分数阶导数的积分边值问题正解的存在性[J]. 上海理工大学学报, 2016, 38(6): 511-516. [6] YANG L, SHEN C F, XIE D P. Multiple positive solutions for nonlinear boundary value problem of fractional order differential equation with the Riemann-Liouville derivative[J]. Advances in Difference Equations, 2014, 2014: 284. DOI:10.1186/1687-1847-2014-284 [7] AHMAD B, SIVASUNDARAM S. Existence of solutions for impulsive integral boundary value problems of fractional order[J]. Nonlinear Analysis: Hybrid Systems, 2010, 4(1): 134-141. DOI:10.1016/j.nahs.2009.09.002 [8] 智二涛, 刘锡平, 李凡凡. 分数阶脉冲微分方程边值问题正解的存在性[J]. 吉林大学学报(理学版), 2014, 52(3): 482-488. [9] LIU X P, JIA M. Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations[J]. Mathematical Methods in the Applied Sciences, 2016, 39(3): 475-487. DOI:10.1002/mma.3495 [10] MYSHKIS A D, SAMOILENKO A M. Systems with impulses in prescribed moments of the time[J]. Mat. Sb, 1976, 74(1): 202-208. [11] 张莎, 贾梅, 李燕, 等. 分数阶脉冲微分方程三点边值问题解的存在性和唯一性[J]. 山东大学学报(理学版), 2017, 52(2): 66-72. [12] WANG J R, ZHOU Y, FE KAN M. On recent developments in the theory of boundary value problems for impulsive fractional differential equations[J]. Computers and Mathematics with Applications, 2012, 64(10): 3008-3020. DOI:10.1016/j.camwa.2011.12.064 [13] 陈辉, 贾梅, 何健堃. 一类具有非瞬时脉冲的分数阶微分方程积分边值问题解的存在性[J]. 上海理工大学学报, 2017, 39(6): 521-527. [14] AGARWAL R, HRISTOVA S, O’REGAN D. Iterative techniques for the initial value problem for Caputo fractional differential equations with non-instantaneous impulses[J]. Applied Mathematics and Computation, 2018, 334: 407-421. DOI:10.1016/j.amc.2018.04.004