西南大学学报 (自然科学版)  2018, Vol. 40 Issue (12): 120-125.  DOI: 10.13718/j.cnki.xdzk.2018.12.019
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  • 一类新的带有参数δ∈[1, 2)的分数阶可微变分不等式的拓扑处理方法    [PDF全文]
    吴欣锟     
    贵州理工学院 理学院, 贵阳 550003
    摘要:在已有的变分不等式、可微变分不等式、分数阶可微变分不等式的模型的基础上,对一类新的带有参数δ∈[1,2)的分数阶可微变分不等式模型的解的存在性进行了相关的分析和研究.首先,在已有的分数阶可微变分不等式的模型基础上加了一个参数δ∈[1,2),得到了一类新的带有参数δ∈[1,2)的分数阶可微变分不等式模型,对这类新模型给出了详细的阐述;然后证明出该模型的解是非空的.
    关键词可微变分不等式    分数阶可微变分不等式        

    文献[1]首次介绍和研究了一类含有初值的可微变分不等式

    $ \left\{ \begin{array}{l} \dot x\left( t \right) = f\left( {t,x\left( t \right)} \right) + B\left( {t,x\left( t \right)} \right) \cdot u\left( t \right)\\ u\left( t \right) \in {\rm{SOL}}\left( {K,G\left( {t,x\left( t \right)} \right) + S} \right)\\ x\left( 0 \right) = {x_0} \end{array} \right. $ (1)

    其中K$\mathbb{R}^m$的一个非空闭凸子集,Ω≡[0,T$\mathbb{R}^n$,(fBG):Ω$\mathbb{R}^n$×$\mathbb{R}^{n×m}$×$\mathbb{R}^m$S$\mathbb{R}^m$$\mathbb{R}^m$是两个函数.在某些条件下,文献[1]得到了可微变分不等式(1)的一个Caratheodory弱解的存在性.这类可微变分不等式在分析力学、微分纳什游戏和其它相关领域中有着许多重要的应用[1-18].

    简单来讲,分数阶导数0CDtδx(t)是对整数阶导数$ \dot x$(t)的一种拓展.我们一般可以对某个性质较好的可导函数求一阶导数、二阶导数、三阶导数、n阶导数,那么我们是否可以对函数求分数阶导数(如$\frac{1}{2} $阶导数)呢?若某个函数不满足求导条件,我们是否可以使用微积分理论对这个函数进行分析性质的研究呢?根据多方文献的参考得知,答案是肯定的.分数阶微积分的研究热潮产生于20世纪70年代,主要原因是因为研究人员发现分形几何、幂律现象与记忆过程等可以与分数阶微积分建立起密切的联系.分数阶微积分可以作为一种很好的描述与刻画手段.

    如果把式(1)里面的$\dot x\left( t \right)$变成0CDtδx(t),单值函数f(tx(t))变成集值算子F(tx(t)),我们该如何去求解呢?这是本文的主要研究内容.

    首先,研究这个问题是有重要意义的.如果把式(1)里面的$\dot x\left( t \right)$变成0CDtδx(t),单值函数f(tx(t))变成集值算子F(tx(t)),那么式(1)就变成了一个由集值算子、分数阶微分方程和可微变分不等式组成的新式子.众所周知,集值算子F(tx(t))是单值函数f(tx(t))的推广并且比单值函数f(tx(t))更重要,由可微变分不等式和分数阶微分方程的重要性,有理由相信新式子的求解是有重要意义的.

    1 预备知识

    X是度量空间,E是Banach空间,定义:

    $ P\left( E \right) = \left\{ {U \subset E:U \ne \emptyset } \right\} $
    $ B\left( E \right) = \left\{ {U \in P\left( E \right):U\;是有界的} \right\} $
    $ K\left( E \right) = \left\{ {U \in P\left( E \right):U\;是紧的} \right\} $
    $ Kv\left( E \right) = \left\{ {U \in K\left( E \right):U\;是凸的} \right\} $

    我们需要用到集值分析的一些概念和结论:

    定义1  一个集值算子MXP(E)被称为:

    (ⅰ)如果对E的任一闭子集VM-1(V)={xXM(x)∩V}是X的闭子集,则称M是上半连续的;

    (ⅱ)如果对E的任一弱闭子集VM-1(V)={xXM(x)∩V}是X的闭子集,则称M是弱上半连续的;

    (ⅲ)如果图ΓM={(yz):zM(y)}是X×E的一个闭子集,则称M是闭的;

    (ⅳ)如果M是上半连续的,且对X里的每个有界集ΩM(Ω)是E里的相对紧集,则称M是完备上半连续的;

    (ⅴ)如果对X的任一紧子集ΩM(Ω)是E里的相对紧集,则称M是拟紧的.

    引理1[6]  如果MXP(E)是一个闭的、拟紧的集值算子,则M是上半连续的.

    引理2[7]  设E是Banach空间,Ω是另外一个Banach空间的非空子集.如果NΩP(E)是一个映射到弱紧凸集的集值算子,则N是弱上半连续的当且仅当条件{xn}⊂Ωxnx0(x0Ω)和ynN(xn)能够推出{yn}存在子序列弱收敛于y0,其中y0N(x0).

    引理3[6]  设ME的有界闭凸子集,TMKv(M)是完备上半连续的集值映射,则Fix(T)={xxT(x)}是非空紧子集.

    2 主要结果

    考虑下面的这类带有参数δ∈[1,2)的分数阶可微变分不等式:

    $ \begin{array}{*{20}{c}} {{}^CD_t^\delta x\left( t \right) \in F\left( {t,x\left( t \right)} \right) + B\left( {t,x\left( t \right)} \right)u\left( t \right)}&{t \in \left[ {0,h} \right]} \end{array} $ (2)
    $ \begin{array}{*{20}{c}} {\left\langle {v - u\left( t \right),G\left( {t,x\left( t \right)} \right) + Q\left( {u\left( t \right)} \right)} \right\rangle \ge 0}&{\forall v \in K,{\rm{a}}.\;{\rm{e}}.\;t \in \left[ {0,h} \right]} \end{array} $ (3)
    $ \begin{array}{*{20}{c}} {x\left( 0 \right) = a,x\left( h \right) = b}&{a,b \ne 0} \end{array} $ (4)

    其中x(t)∈$\mathbb{R}^{n}$u(t)∈K,1≤δ < 2,CDtδ是分数阶导数的表示符号,FBGQ这4个函数的定义会在下面的定义中给出解释.

    为了处理问题(2)至(4),我们需要下面5个假设成立:

    (F1) FI×$\mathbb{R}^{n}$Kv($\mathbb{R}^{n}$)是上半Caratheodory集值映射,等价于对∀v$\mathbb{R}^{n}$,集值映射F(·,v):IKv($\mathbb{R}^{n}$)确定了一个可测选择,且对a.e. tI,集值映射F(t,·):$\mathbb{R}^{n}$Kv($\mathbb{R}^{n}$)是上半连续的;

    (F2) 对于函数FI×$\mathbb{R}^{n}$Kv($\mathbb{R}^{n}$),存在非减的连续函数ΨF$\mathbb{R}$$\mathbb{R}$和函数ηFLp(I$\mathbb{R}$),使得

    $ \begin{array}{*{20}{c}} {\left\| {F\left( {t,v} \right)} \right\| = \sup \left\{ {\left\| z \right\|:z \in F\left( {t,v} \right)} \right\} \leqslant {\eta _F}\left( t \right){\mathit{\Psi }_F}\left( {\left\| v \right\|} \right)}&{\forall v \in {\mathbb{R}^n},{\text{a}}.\;{\text{e}}.\;t \in I} \end{array} $

    其中p是大于$ \frac{1}{\delta }$的正整数;

    (B) BI×$\mathbb{R}^{n}$$\mathbb{R}^{n×m}$是连续函数,满足

    $ \begin{array}{*{20}{c}} {\left\| {B\left( {t,v} \right)} \right\| \leqslant {\eta _B}}&{\forall v \in {\mathbb{R}^n},\forall t \in I} \end{array} $

    其中ηB是正数;

    (G) 对于连续函数GI×$\mathbb{R}^n$$\mathbb{R}^m$,存在非减的连续函数ΨG$\mathbb{R}$$\mathbb{R}$和函数ηGLp(IR),使得

    $ \begin{array}{*{20}{c}} {\left\| {G\left( {t,v} \right)} \right\| \leqslant {\eta _G}\left( t \right){\mathit{\Psi }_G}\left( {\left\| v \right\|} \right)}&{\forall v \in {\mathbb{R}^n},\forall t \in I} \end{array} $

    (Q) QK$\mathbb{R}^m$是满足下面两个条件的连续函数:

    (Q1) QK上是单调的,也即是说

    $ \begin{array}{*{20}{c}} {\left\langle {u - v,Q\left( u \right) - Q\left( v \right)} \right\rangle \ge 0}&{\forall u,v \in K} \end{array} $

    (Q2) 存在v0K,使得

    $ \mathop {\lim \inf }\limits_{v \in K,\left\| v \right\| \to \infty } \frac{{\left\langle {v - {v_0},Q\left( v \right)} \right\rangle }}{{{{\left\| v \right\|}^2}}} > 0 $

    由上面的条件(F1)和(F2)可知,从C(I$\mathbb{R}^{n}$)映射到P(Lp(I$\mathbb{R}^{n}$))的集值映射

    $ P_F^p\left( x \right) = \left\{ {f \in {L^p}\left( {I,{\mathbb{R}^n}} \right):f\left( t \right) \in F\left( {t,x\left( t \right)} \right),{\text{a}}.\;{\text{e}}.\;t \in I} \right\} $

    是闭的[8],其中P(Lp(I$\mathbb{R}^{n}$))表示Lp(I$\mathbb{R}^{n}$)的所有子集组成的集合.

    定义2[8]  问题(2)至(4)的解xC(I$\mathbb{R}^{n}$)是指存在可积函数uIK和函数fPFp(x),满足:

    $ \begin{array}{*{20}{c}} {x\left( t \right) = a + \frac{1}{h}\left( {b - a} \right)t + \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}\left[ {f\left( s \right) + B\left( {s,x\left( s \right)} \right)x\left( s \right)} \right]{\rm{d}}s} - }\\ {\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {f\left( s \right) + B\left( {s,x\left( s \right)} \right)x\left( s \right)} \right]{\rm{d}}s} \;\;\;\;\;\;\;\;\;t \in I} \end{array} $
    $ \begin{array}{*{20}{c}} {\left\langle {v - u\left( t \right),G\left( {t,x\left( t \right)} \right) + Q\left( {u\left( t \right)} \right)} \right\rangle \ge 0}&{\forall v \in K,{\rm{a}}.\;{\rm{e}}.\;\;t \in I} \end{array} $

    对于函数QK$\mathbb{R}^m$,定义SOL(KQ)为

    $ {\rm{SOL}}\left( {K,Q} \right) = \left\{ {v \in K:\left\langle {w - v,Q\left( v \right)} \right\rangle \ge 0,\forall w \in K} \right\} $ (5)

    依据文献[1]的假设6.2,我们可以得到下面的引理4:

    引理4  如果条件(Q)满足,则对于每个z$\mathbb{R}^m$,解集SOL(Kz+Q(·))是非空的闭凸集,且存在正数ηQ满足

    $ \begin{array}{*{20}{c}} {\left\| v \right\| \le {\eta _Q}\left( {1 + \left\| z \right\|} \right)}&{\forall v \in {\rm{SOL}}\left( {K,z + Q\left( \cdot \right)} \right)} \end{array} $ (6)

    为了解决问题(2)至(4),我们设

    $ U\left( z \right) = {\text{SOL}}\left( {K,z + Q\left( \cdot \right)} \right)\;\;\;\;\;\;\;\;\;\;\forall z \in {\mathbb{R}^m} $

    再定义ΦI×$\mathbb{R}^{n}$P($\mathbb{R}^{n}$)为

    $ \mathit{\Phi }\left( {t,v} \right) = \left\{ {B\left( {t,v} \right)y:y \in U\left( {G\left( {t,v} \right)} \right)} \right\} $ (7)

    则问题(2)至(4)可转化为:

    $ \begin{array}{*{20}{c}} {x\left( t \right) = a + \frac{1}{h}\left( {b - a} \right)t + \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}\left[ {f\left( s \right) + B\left( {s,x\left( s \right)} \right)x\left( s \right)} \right]{\rm{d}}s} - }\\ {\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {f\left( s \right) + B\left( {s,x\left( s \right)} \right)x\left( s \right)} \right]{\rm{d}}s} \;\;\;\;\;\;\;\;\;t \in I} \end{array} $ (8)
    $ f \in P_F^p\left( x \right),g \in P_\mathit{\Phi }^p\left( x \right) $ (9)

    为了解决问题(8)至(9),我们引入集值映射ΣC(I$\mathbb{R}^{n}$)→P(C(I$\mathbb{R}^{n}$))为

    $ \begin{array}{l} \mathit{\Sigma }\left( x \right) = \left\{ {a + \frac{1}{h}\left( {b - a} \right)t + \frac{1}{{\mathit{\Gamma }\left( \alpha \right)}}\int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}\left[ {f\left( s \right) + g\left( s \right)} \right]{\rm{d}}s} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {\left[ {f\left( s \right) + g\left( s \right)} \right]{\rm{d}}s} :f \in P_F^p\left( x \right),g \in P_\mathit{\Phi }^p\left( x \right)} \right\} \end{array} $ (10)

    xC(I$\mathbb{R}^{n}$)是问题(8)到(9)的解等价于x是集值映射Σ的不动点.

    引理5[8]  在(F1),(F2),(B),(G)和(Q)的假设下,PFpPΦp是弱上半连续的.

    定理1  设从Lp(I$\mathbb{R}^{n}$)到C(I$\mathbb{R}^{n}$)的映射W

    $ W\left( f \right)\left( t \right) = a + \frac{1}{h}\left( {b - a} \right) + \frac{1}{{\mathit{\Gamma }\left( \sigma \right)}}\int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}f\left( s \right){\rm{d}}s} - \frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\alpha - 1}}f\left( s \right){\rm{d}}s} $

    W是完备连续的.

      设映射

    $ {W_1}\left( f \right)\left( t \right) = \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}f\left( s \right){\rm{d}}s} $

    依据文献[8]可知W1是完备连续的.

    再设映射

    $ {W_2}\left( f \right)\left( t \right) = \frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}f\left( s \right){\rm{d}}s} $

    下面证W2是完备连续的.

    W2是完备连续的等价于W2是连续的,且W2把有界集映射到相对紧集. W2连续是显然的.下面证明对于任一有界集ΩLp(I$\mathbb{R}^{n}$),W2(Ω)是C(I$\mathbb{R}^{n}$)里的相对紧集.因为W2(Ω)是C(I$\mathbb{R}^{n}$)里的相对紧集等价于对∀tIW2(Ω)(t)有界且W2(Ω)是C(I$\mathbb{R}^{n}$)里的等度连续集.对∀tIW2(Ω)(t)有界是显然的.设Ω的上界为M,下面证明W2(Ω)是C(I$\mathbb{R}^{n}$)里的等度连续集.

    因为对∀ε>0,∀tTI,只要

    $ \left| {t - T} \right| < \frac{{h\mathit{\Gamma }\left( \delta \right)}}{{M\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}{\rm{d}}s} }}\varepsilon $

    就有

    $ \begin{array}{l} \left| {\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}f\left( s \right){\rm{d}}s} - \frac{1}{h}\frac{T}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}f\left( s \right){\rm{d}}s} } \right| = \\ \left| {\frac{1}{{\mathit{h\Gamma }\left( \delta \right)}} \cdot \int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}f\left( s \right){\rm{d}}s} \cdot \left( {t - T} \right)} \right| \le \\ \frac{1}{{\mathit{h\Gamma }\left( \delta \right)}} \cdot \int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left| {f\left( s \right)} \right|{\rm{d}}s} \cdot \left| {t - T} \right| \le \\ \frac{M}{{\mathit{h\Gamma }\left( \delta \right)}} \cdot \int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}{\rm{d}}s} \cdot \left| {t - T} \right| < \varepsilon \end{array} $

    所以W2(Ω)是C(I$\mathbb{R}^{n}$)里的等度连续集.所以W2是完备连续的.所以映射W是完备连续的.

    定理2  映射Σ是完备上半连续的.

      证明与文献[8]中的证明类似.

    定理3  假设(F1),(F2),(B),(G),(Q)这5个条件成立.如果

    $ \begin{array}{l} \mathop {\lim \inf }\limits_{k \to \infty } \left[ {\frac{{{\mathit{\Psi }_F}\left( k \right)}}{{\mathit{h\Gamma }\left( \delta \right)}}\mathop {\sup }\limits_{t \in J} \int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}{\eta _F}\left( s \right){\rm{d}}s} + \frac{{{\mathit{\Psi }_G}\left( k \right)}}{{\mathit{h\Gamma }\left( \delta \right)}}{\eta _Q}{\eta _B}\mathop {\sup }\limits_{t \in J} \int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}{\eta _G}\left( s \right){\rm{d}}s} + } \right.\\ \left. {\frac{1}{{\mathit{h\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}{\eta _G}\left( s \right){\mathit{\Psi }_G}\left( k \right){\rm{d}}s} + \frac{1}{{\mathit{h\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}{\mathit{\Psi }_F}\left( k \right){\eta _F}\left( s \right){\rm{d}}s} } \right] < 1 \end{array} $ (11)

    则问题(8)和(9)至少有一个解.

      根据定理2,Σ是完备上半连续的.为了利用引理3,我们还需证明集合C(I$\mathbb{R}^{n}$)里的任意一个以原点为圆心、$\mathbb{R}$($\mathbb{R}$>0)为半径的球$B_{\mathbb{R}}$都满足Σ($B_{\mathbb{R}}$)⊂$B_{\mathbb{R}}$.利用反证法,假设在C(I$\mathbb{R}^{n}$)里存在一个序列{xk}满足‖xkCkykΣ(xk),‖ykCk对所有的k$\mathbb{N}$+都成立,此时

    $ \mathop {\lim \inf }\limits_{k \to \infty } \frac{{{{\left\| {{y_k}} \right\|}_C}}}{k} \ge 1 $

    根据集值映射Σ的定义,可知存在fkPFp(xk)和gkPΦp(xk),满足

    $ \begin{array}{*{20}{c}} {{y_k}\left( t \right) = a + \frac{1}{h}\left( {b - a} \right)t + W\left( {{f_k} + {g_k}} \right)\left( t \right)}&{t \in I} \end{array} $

    对∀tI,因为

    $ \begin{array}{l} W\left( {{f_k} + {g_k}} \right)\left( t \right) = \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}\left[ {{f_k}\left( s \right) + {g_k}\left( s \right)} \right]{\rm{d}}s} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {{f_k}\left( s \right) + {g_k}\left( s \right)} \right]{\rm{d}}s} \end{array} $

    所以

    $ \begin{array}{l} \left\| {{y_k}\left( t \right)} \right\| = \left\| {a + \frac{1}{h}\left( {b - a} \right)t + \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}\left[ {{f_k}\left( s \right) + {g_k}\left( s \right)} \right]{\rm{d}}s} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {{f_k}\left( s \right) + {g_k}\left( s \right)} \right]{\rm{d}}s} } \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;b + \left\| {\frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}\left[ {{f_k}\left( s \right) + {g_k}\left( s \right)} \right]{\rm{d}}s} } \right\| + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left\| {\frac{1}{h}\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {{f_k}\left( s \right) + {g_k}\left( s \right)} \right]{\rm{d}}s} } \right\| \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;b + \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\mathop {\sup }\limits_{t \in J} \left\{ {\int_0^t {{{\left( {t - s} \right)}^{\delta - 1}}\left[ {\left\| {{f_k}\left( s \right)} \right\| + \left\| {{g_k}\left( s \right)} \right\|} \right]{\rm{d}}s} } \right\} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{h}\mathop {\sup }\limits_{t \in J} \left\{ {\frac{t}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {\left\| {{f_k}\left( s \right)} \right\| + \left\| {{g_k}\left( s \right)} \right\|} \right]{\rm{d}}s} } \right\} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;b + \frac{{{\mathit{\Psi }_F}\left( k \right)}}{{\mathit{\Gamma }\left( \alpha \right)}}\mathop {\sup }\limits_{t \in J} \int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}{\eta _F}\left( t \right){\rm{d}}s} + \frac{{{\eta _B}{\eta _Q}}}{{\mathit{\Gamma }\left( \alpha \right)}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\sup }\limits_{t \in J} \left\{ {\int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}\left[ {1 - {\eta _G}\left( s \right){\mathit{\Psi }_G}\left( k \right)} \right]{\rm{d}}s} } \right\} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}{\mathit{\Psi }_F}\left( k \right){\eta _r}\left( s \right){\rm{d}}s} + \frac{1}{{\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {1 - {\eta _G}\left( s \right){\mathit{\Psi }_G}\left( k \right)} \right]{\rm{d}}s} \end{array} $

    所以

    $ \begin{array}{l} \mathop {\lim \inf }\limits_{k \to \infty } \frac{{{{\left\| {{y_k}} \right\|}_C}}}{k} \le \mathop {\lim \inf }\limits_{k \to \infty } \left\{ {\frac{{{\mathit{\Psi }_F}\left( k \right)}}{{\mathit{\Gamma }\left( \alpha \right)}}\mathop {\sup }\limits_{t \in J} \int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}{\eta _F}\left( t \right){\rm{d}}s} } \right. + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{\eta _B}{\eta _Q}}}{{k\mathit{\Gamma }\left( \alpha \right)}}\mathop {\sup }\limits_{t \in J} \left\{ {\int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}\left[ {1 + {\eta _G}\left( s \right){\mathit{\Psi }_G}\left( k \right)} \right]{\rm{d}}s} } \right\} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{{k\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}{\mathit{\Psi }_F}\left( k \right){\eta _F}\left( s \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{1}{{k\mathit{\Gamma }\left( \delta \right)}}\int_0^h {{{\left( {h - s} \right)}^{\delta - 1}}\left[ {1 + {\eta _G}\left( s \right){\mathit{\Psi }_G}\left( k \right)} \right.} } \right\}{\rm{d}}s \end{array} $

    所以依据(11)式,我们有

    $ \mathop {\lim \inf }\limits_{k \to \infty } \frac{{{{\left\| {{y_k}} \right\|}_C}}}{k} < 1 $

    这与前面的假设是矛盾的.定理3得证.

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    Topological Methods for a New Class of Fractional Differential Variational Inequalities with δ∈[1, 2)
    WU Xin-kun     
    College of Science, Guizhou Institute of Technology, Guiyang 550003, China
    Abstract: In this paper, a new class of fractional differential variational inequalities with δ∈[1, 2) are introduced and studied. Firstly, a parameter, δ∈[1, 2), is added to fractional differential variational inequalities, and a new class of fractional differential variational inequalities with δ∈[1, 2) is obtained. Finally, some lemmas and theorems are used to prove that the set of solutions of these differential variational inequalities are non-blank.
    Key words: differential variational inequality    fractional differential variational inequality    solutions    
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