﻿ 一类分数阶非线性微分包含初值问题的可解性
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 浙江大学学报(理学版)  2017, Vol. 44 Issue (3): 287-291  DOI:10.3785/j.issn.1008-9497.2017.03.007 0

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YANG Xiaojuan, HAN Xiaoling. The solvability of Cauchy problem for nonlinear fractional differential inclusions[J]. Journal of Zhejiang University(Science Edition), 2017, 44(3): 287-291. DOI: 10.3785/j.issn.1008-9497.2017.03.007.
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### 文章历史

The solvability of Cauchy problem for nonlinear fractional differential inclusions
YANG Xiaojuan , HAN Xiaoling
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Abstract: In this paper, using Bohnenblust-Karlin's fixed point theorem and combining the upper and lower solution method, we mainly study the solvability of Cauchy problem for nonlinear fractional differential inclusions $\left\{ \begin{gathered} {x^{\left( a \right)}}\left( t \right) \in F\left( {t,x\left( t \right)} \right),\;\;\;t \in J = \left[ {a,b} \right],\;a > 0, \hfill \\ x\left( a \right) = {x_0} \hfill \\ \end{gathered} \right.$ where F:J×R→2R is L1-Carathéodary function, x(α)(t) denotes the conformable fractional derivative of x at t of order α, α∈(0, 1]. By applying this theorem, we arrive at two existence results when the multi-valued nonlinearity F has sub-linear or linear growth about the second variable.
Key words: differential inclusions    fractionl derivatives    existence of solutions    Bohnenblust-Karlin's fixed point theorem
0 引言

 $\left\{ \begin{array}{l} y'\left( t \right) \in F\left( {t,y\left( t \right)} \right),\;\;\;\;\;t \in J = \left[ {0,T} \right],\\ y\left( 0 \right) = y\left( T \right) \end{array} \right.$

 $\left\{ \begin{array}{l} {x^{\left( \alpha \right)}}\left( t \right) = f\left( {t,x\left( t \right)} \right),\;\;\;t \in \left[ {a,b} \right],a > 0,\\ x\left( a \right) = {x_0} \end{array} \right.$

 $\left\{ \begin{array}{l} {x^{\left( \alpha \right)}}\left( t \right) \in F\left( {t,x\left( t \right)} \right),t \in J = \left[ {a,b} \right],a > 0,\\ x\left( a \right) = {x_0} \end{array} \right.$ (1)

1 预备知识

J=[a, b], C(J)为定义在J上的连续实值函数构成的Banach空间, 其范数为‖x=sup{|x(t)|tJ}.设Ck(J)为k次连续可微实值函数构成的Banach空间, 其范数为‖xck=max{‖x, …, ‖x(k)}.设L1(J, R)为定义在J上满足$\int_J {\left| {x\left( t \right)} \right|} {\text{d}}t < + \infty$可测函数构成的Banach空间, 其范数为${\left\| x \right\|_{{L^1}}} = \int_J {\left| {x\left( t \right)} \right|} {\text{d}}t$.设AC(X)表示X的绝对连续函数全体, CC(X)表示X的非空凸的紧子集全体, BCC(X)表示X的有界非空凸的紧子集全体.

 ${T_\alpha }\left( f \right)\left( t \right): = \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{f\left( {t + \varepsilon {t^{1 - \alpha }}} \right) - f\left( t \right)}}{\varepsilon },t > 0,$

 ${f^\alpha }\left( 0 \right): = \mathop {\lim }\limits_{t \to {0^ + }} {f^\alpha }\left( t \right).$

 $I_\alpha ^af\left( t \right): = \int_a^t {\frac{{f\left( \tau \right)}}{{{\tau ^{1 - \alpha }}}}{\rm{d}}\tau } .$

(ⅰ)对任意的xR, 有tF(t, x)是可测的；

(ⅱ)对几乎处处的tJ, 有xF(t, x)是上半连续的；

(ⅲ)对任意的k>0, 存在hkL1(J, R+), 使得‖F(t, x)‖=sup{|v|:vF(t, x)}≤hk(t), 其中|x|≤k, tJ

 $x\left( t \right): = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha J_a^t\left[ {\frac{{g\left( s \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right)$

 $\left\{ \begin{array}{l} {x^\alpha }\left( t \right) + \frac{1}{{{a^\alpha }}}x\left( t \right) = g\left( t \right),\;\;t \in \left[ {a,b} \right],\;\;\;a > 0,\\ x\left( a \right) = {x_0} \end{array} \right.$ (2)

2 主要结果及其证明

 $\left\{ \begin{array}{l} {v_1}\left( t \right) \in F\left( {t,\varphi \left( t \right)} \right),\;\;\;t \in J,a > 0,\\ {\varphi ^\alpha }\left( t \right) \le {v_1}\left( t \right),\;\;\;\;\varphi \left( a \right) \le {x_0}, \end{array} \right.$

 $\left\{ \begin{array}{l} {v_2}\left( t \right) \in F\left( {t,\psi \left( t \right)} \right),\;\;\;t \in J,a > 0,\\ {\psi ^\alpha }\left( t \right) \geqslant {v_2}\left( t \right),\;\;\;\;\psi \left( a \right) \geqslant {x_0}, \end{array} \right.$

 $\left\{ \begin{array}{l} {x^\alpha }\left( t \right) + \frac{1}{{{a^\alpha }}}x\left( t \right) \in F\left( {t,x\left( t \right)} \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x\left( t \right)} \right),\;\;\;t \in J,\\ x\left( a \right) = {x_0},\;\;\;a > 0, \end{array} \right.$ (3)

 $\gamma \left( {t,x\left( t \right)} \right) = \left\{ \begin{array}{l} \varphi \left( t \right),\;\;\;\;\varphi \left( t \right) > x\left( t \right),\\ x\left( t \right),\;\;\;\;\;\varphi \left( t \right) \le x\left( t \right) \le \psi \left( t \right),\\ \psi \left( t \right),\;\;\;\;\;\psi \left( t \right) < x\left( t \right). \end{array} \right.$

F是有非空闭凸值的L1-Carathéodary集值映射, 可得存在ΦL1(J, R+), 使得$\left\| {F\left( {t, x\left( t \right)} \right) + \frac{1}{{{a^a}}}\gamma \left( {t, x\left( t \right)} \right)} \right\| \leqslant \mathit{\Phi} \left( t \right) + \max \frac{1}{{{a^a}}}\left( {\sup \left| {\varphi \left( t \right)} \right|, \sup \left| {\psi \left( t \right)} \right|} \right), t \in J$.定义算子T:C(J, R)→2C(J, R)

 $\begin{array}{l} Tx\left( t \right): = \left\{ {h \in C\left( {J,R} \right):h\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left. {\alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right)} \right\}. \end{array}$

i)对每个xC(J), T(x)是凸的.设h, hT(x), 则存在vSF, xvSF, x, 使得

 $\begin{array}{l} h\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right),\\ \overline {h\left( t \right)} = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{\overline {v\left( s \right)} + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right). \end{array}$

 $\begin{array}{l} \left[ {\lambda h + \left( {1 - \lambda } \right)\bar h} \right]\left( t \right) = \\ \;\;\;\;\;\;\;\lambda {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right) + \\ \;\;\;\;\;\;\;\left( {1 - \lambda } \right){{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}} \times \\ \;\;\;\;\;\;\;\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{\overline {v\left( s \right)} + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right) + \\ \;\;\;\;\;\;\;\alpha \mathscr{J}_a^t\left[ {\frac{{\left( {1 - \lambda } \right)\left( {\overline {v\left( s \right)} + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right] = \\ \;\;\;\;\;\;\;{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{\lambda \left( {v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}} + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {\frac{{\left( {1 - \lambda } \right)\left( {\overline {v\left( s \right)} + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right). \end{array}$

ii)对任意常数r>0, 令Br={xC(J):‖x‖≤r}, 则BrC(J)上的有界闭凸集.下证存在r>0, 使得T(Br)$\subseteq$Br.

 $h\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{\pi }{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right).$

 $\begin{array}{l} \left| {{h_r}\left( t \right)} \right| = \\ \;\;\;\;\left| {{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,y} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right)} \right| \le \\ \;\;\;\;{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}\left| {{x_0}} \right| + {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\alpha \mathscr{J}_a^t \times } \right.\\ \;\;\;\;\left. {\left[ {\frac{{\left| {v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,y} \right)} \right|}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right) \le \frac{K}{C}\left[ {\mathit{\Phi }\left( t \right) } \right.+\\ \;\;\;\;\left. {\max \frac{1}{{{a^\alpha }}}\left( {\sup \left| {\varphi \left( t \right)} \right|,\sup \left| {\psi \left( t \right)} \right|} \right)} \right]\frac{{{b^\alpha } - {a^\alpha }}}{\alpha } + \\ \;\;\;\;K{{\rm{e}}^{\frac{1}{\alpha }}}\left| {{x_0}} \right|. \end{array}$

 $\begin{array}{l} 令\;\;r = K{{\rm{e}}^{\frac{1}{\alpha }}}\left| {{x_0}} \right| + \frac{K}{C}\left[ {\mathit{\Phi }\left( t \right) } \right.+ \\ \;\;\;\;\;\;\;\;\;\left. {\max \frac{1}{{{a^\alpha }}}\left( {\sup \left| {\varphi \left( t \right)} \right|,\sup \left| {\psi \left( t \right)} \right|} \right)} \right]\frac{{{b^\alpha } - {a^\alpha }}}{\alpha }, \end{array}$

iii)T(Br)是等度连续的.设t1, t2J, t1t2, 设xBrhT(x), 则存在vSF, x, 使得对任意tJ, 有

 $h\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right).$

 $\begin{array}{l} \left| {h\left( {{t_2}} \right) - h\left( {{t_1}} \right)} \right| \le \left| {{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_2}}}{a}} \right)}^\alpha }}}{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} - {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_1}}}{a}} \right)}^\alpha }}}{{\rm{e}}^{\frac{1}{\alpha }}}{x_0}} \right| + \\ \;\;\;\;\;\;\;\left| {{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_2}}}{a}} \right)}^\alpha }}}\alpha \mathscr{J}_a^{{t_2}}\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {{t_2},x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right] - } \right.\\ \;\;\;\;\;\;\;\left. {{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_1}}}{a}} \right)}^\alpha }}}\mathscr{J}_a^{{t_1}}\left[ {\frac{{v\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {{t_1},x} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right| \le \\ \;\;\;\;\;\;\;{{\rm{e}}^{\frac{1}{\alpha }}}\left| {{x_0}} \right|\left| {{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_2}}}{a}} \right)}^\alpha }}} - {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_1}}}{a}} \right)}^\alpha }}}} \right| + \\ \frac{K}{C}\left[ {\mathit{\Phi }\left( t \right) + \max \frac{1}{{{a^\alpha }}}\left( {\sup \left| {\varphi \left( t \right)} \right|,\sup \left| {\psi \left( t \right)} \right|} \right)} \right] \times \\ \frac{{\left| {t_2^\alpha - t_1^\alpha } \right|}}{\alpha } \to 0. \end{array}$

iv)T有闭图像.设xnx*, hnT(xn), 且hnh*, 下证h*T(x*).

hnT(xn)可知，存在vnSF, xn, 使得

 $\begin{array}{l} {h_n}\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_1}}}{a}} \right)}^\alpha }}} \times \\ \;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{{v_n}\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,{x_n}} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right). \end{array}$

 $\begin{array}{l} {h_ * }\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{{{t_1}}}{a}} \right)}^\alpha }}} \times \\ \;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{{v_*}\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,{x_*}} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right). \end{array}$

 $\begin{array}{l} \left\| {\left( {{h_n}\left( t \right) - {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{\gamma \left( {t,{x_n}} \right)}}{{{a^\alpha }{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right)} \right) - } \right.\\ \left. {\left( {{h_*}\left( t \right) - {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{\gamma \left( {t,{x_*}} \right)}}{{{a^\alpha }{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right)} \right)} \right\| \to 0. \end{array}$

 $v \to \Gamma \left( v \right)\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( s \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right).$

 $\begin{array}{l} \left( {{h_n}\left( t \right) - {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}}\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{v\left( n \right)}}{{{a^\alpha }{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right)} \right) \in \\ \;\;\;\;\;\;\;\Gamma \left( {{S_{F,{x_n}}}} \right). \end{array}$

 $\begin{array}{l} {h_*}\left( t \right) = {{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{t}{a}} \right)}^\alpha }}} \times \\ \;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{e}}^{\frac{1}{\alpha }}}{x_0} + \alpha \mathscr{J}_a^t\left[ {\frac{{{v_*}\left( s \right) + \frac{1}{{{a^\alpha }}}\gamma \left( {t,{x_*}} \right)}}{{{{\rm{e}}^{ - \frac{1}{\alpha }{{\left( {\frac{s}{a}} \right)}^\alpha }}}}}} \right]} \right). \end{array}$

 $\begin{array}{l} {h^\alpha }\left( {{t_0}} \right) = {\left( {\varphi \left( {{t_0}} \right) - x\left( {{t_0}} \right)} \right)^\alpha } = {\varphi ^\alpha }\left( {{t_0}} \right) - {x^\alpha }\left( {{t_0}} \right) \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;{v_1}\left( t \right) - v\left( t \right) \le 0, \end{array}$

h(a)=φ(a)-x(a)≤x0-x0=0, 即h(a)≤0, 由命题3, 任意tJ, h(t)≤0, 与h(t)>0矛盾, 从而φ(t)≤x(t), tJ.

 $\left\| {F\left( {t,x} \right)} \right\| \le a\left( t \right){\left| x \right|^\mu } + b\left( t \right),\;\;\;\;\left( {t,x} \right) \in J \times R,$

 $\left\| {F\left( {t,x} \right)} \right\| \le a\left( t \right)\left| x \right| + b\left( t \right),\;\;\;\;\left( {t,x} \right) \in J \times R,$

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