西南大学学报 (自然科学版)  2019, Vol. 41 Issue (12): 69-73.  DOI: 10.13718/j.cnki.xdzk.2019.12.010
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  • 一类新的混合分数阶可微变分不等式的拓扑处理方法    [PDF全文]
    吴欣锟    
    黔南民族师范学院 数学与统计学院, 贵州 都匀 558000
    摘要:在可微变分不等式和分数阶可微变分不等式的基础上首次引进和研究了一类新的混合分数阶可微变分不等式.给出了这类新的混合分数阶可微变分不等式的模型,并详细地说明了模型中的符号所代表的数学意义,证明了该模型的解集是非空的.
    关键词分数阶可微变分不等式    混合分数阶可微变分不等式    解集    

    文献[1-8]介绍和研究了一类含有初值的可微变分不等式

    $ \left\{ {\begin{array}{*{20}{l}} {\dot x(t) = f(t,x(t)) + B(t,x(t)) \cdot u(t)}\\ {u(t) \in {\rm{SOL}}(K,G(t,x(t)) + S)}\\ {x(0) = {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} $mS$ \mathbb{R} $m$ \mathbb{R} $m是两个函数.在某些条件下,文献[1]得到了不等式(1)的Caratheodory弱解的存在性.

    文献[9-10]将不等式(1)推广到了分数阶的情形,其数学表达式为

    $ \left\{ \begin{array}{l} {}^CD_t^\delta x(t) \in F(t,x(t)) + B(t,x(t))u(t)\;\;\;\;\;\;\;t \in {I_1}\\ \langle v - u(t),G(t,x(t)) + Q(u(t))\rangle \ge 0\;\;\;\;\;\;\;\;\;\;\forall v \in K,{\rm{ a}}{\rm{.}}\;{\rm{e}}{\rm{. }}t \in {I_1}\\ x(0) = {x_0} + h(x) \end{array} \right. $ (2)

    其中x(t)∈$ \mathbb{R} $nu(t)∈K,0≤δ<1,I1=[0,T],CDtδ是分数阶导数的表示符号,F是一个从I1×$ \mathbb{R} $nKv($ \mathbb{R} $n)的满足一定条件的映射,Kv($ \mathbb{R} $n)在文中有定义,B是一个从I1×$ \mathbb{R} $n$ \mathbb{R} $n×m的满足一定条件的映射,G是从I1×$ \mathbb{R} $n$ \mathbb{R} $m的满足一定条件的映射,Q是从K$ \mathbb{R} $m的满足一定条件的映射.

    本文把δ的研究范围改成(1,2],再把(2)式中的变分不等式推广到更一般的混合变分不等式,得到的新的这类带有参数δ∈[1,2)的分数阶混合可微变分不等式为

    $ \left\{ \begin{array}{l} {}^CD_t^\delta x(t) \in F(t,x(t)) + B(t,x(t))u(t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t \in I\\ \langle v - u(t),G(t,x(t)) + Q(u(t))\rangle + \varphi \left( v \right) - \varphi \left( {u\left( t \right)} \right) \ge 0\;\;\;\;\;\;\;\;\;\;\forall v \in K,{\rm{ a}}{\rm{.}}\;{\rm{e}}{\rm{. }}t \in I\\ x(0) = a,x\left( h \right) = b\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;a,b \ne 0 \end{array} \right. $ (3)

    其中x(t)∈$ \mathbb{R} $nu(t)∈K,1≤δ<2,I=[0,h],CDtδ是分数阶导数的表示符号,FBGQ这4个映射的定义与(2)式中的定义是相同的,φ是一个从$ \mathbb{R} $m到(-∞,+∞]的真凸下半连续函数.

    1 预备知识

    X是度量空间,E是Banach空间.

    $ P(E) = \{ U \subset E:U \ne \emptyset \} \;\;\;\;B(E) = \left\{ {U \in P(E):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\} $

    定义1  对于一个集值算子MXP(E),

    (a) 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;

    (b) 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;

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

    (d) 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;

    (e) 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[3]  如果MXP(E)是一个闭的和拟紧的集值算子,则M是上半连续的.

    引理2  设E是Banach空间,Ω是另外一个Banach空间的非空子集,如果NΩP(E)是一个映射到弱紧且凸集的集值算子,则N是弱上半连续的当且仅当条件{xn}⊂Ωxnx0ΩynN(xn)能够推出{yn}存在一个子序列弱收敛于y0,其中y0N(x0).

    引理3[3]  设ΜE的有界闭凸子集,再设ΤMKv(M)是一个完备的上半连续集值映射,则Fix(T)={xxT(x)}是非空的紧子集.

    2 主要结果

    在这一部分,我们来分析和研究(3)式的解的存在性.

    定义2[7]  对于一个函数x:[0,+∞)→$ \mathbb{R} $n,它的Caputo导数CDtδx(t)被定义成

    $ {}^CD_t^\delta x(t) = \frac{1}{{\mathit{\Gamma }(2 - \delta )}}\int_0^t {\frac{{x''(s)}}{{{{(t - s)}^{\delta - 1}}}}} {\rm{d}}s\;\;\;\;1 \le \delta < 2 $

    其中$ \mathit{\Gamma}(2-\delta)=\int_{0}^{+\infty} \mathrm{e}^{-t} t^{1-\delta} \mathrm{d} t$,符号Γ表示伽玛函数.

    定义3[9]  对于函数φ$ \mathbb{R} $m→(-∞,+∞],如果φ满足下面两个条件:

    (a) $\forall x\in {{\mathbb{R}}^{m}}, \varphi (x)\le \underset{y\to x}{\mathop{\lim \inf }}\, \varphi (y)$

    (b) $\forall r\in \mathbb{R}, {{V}_{r}}=\left\{ x\in {{\mathbb{R}}^{m}}:\varphi (x)>r \right\}$$ \mathbb{R} $m中的一个开子集.

    则称φ是下半连续的.

    为了得到(3)式解的存在性,我们需要下面6个假设成立:

    (F1) FI×$ \mathbb{R} $nKv($ \mathbb{R} $n)是上半Carathéodory集值映射,等价于说对∀v$ \mathbb{R} $n,集值映射F(·,v):IKv($ \mathbb{R} $n)确定了一个可测选择,且对几乎处处tI,集值映射F(t,·):$ \mathbb{R} $nKv($ \mathbb{R} $n)是上半连续的;

    (F2) 对于函数FI×$ \mathbb{R} $nKv($ \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

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

    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

    $ \left\| {B(t,v)} \right\| \leqslant {\eta _B}\;\;\;\;\forall v \in {\mathbb{R}^n},\forall t \in I $

    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} $和函数ηGLp(I$ \mathbb{R} $),使得

    $ \left\| {G(t,v)} \right\| \leqslant {\eta _G}(t){\mathit{\Psi }_G}(\left\| v \right\|)\;\;\;\;\forall v \in {\mathbb{R}^n},\;\;\forall t \in I $

    (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

    $ \langle u - v,Q(u) - Q(v)\rangle \geqslant 0\;\;\;\;\forall u,v \in K $

    (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(v)} \right\rangle }}{{{{\left\| v \right\|}^2}}} > 0 $

    (Φ)函数φ$ \mathbb{R} $m→(-∞,+∞]是真凸下半连续函数.

    从条件(F1)和(F2),我们可以推出从C(I$ \mathbb{R} $n)映射到P(Lp(I$ \mathbb{R} $n))的集值映射

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

    是闭的,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4  (3)式的一个解xC(I$ \mathbb{R} $n)是指存在一个可积函数uIK和函数fPFp(x),满足

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

    对于一个函数QK$ \mathbb{R} $m,我们定义SOL(KQφ)为

    $ {\rm{SOL}}(K,Q,\varphi ) = \{ v \in K:\langle w - v,Q(v)\rangle + \varphi (w) - \varphi (v) \ge 0,\quad \forall w \in K\} $ (4)

    引理4[9]  如果条件(Q)和条件(Φ)满足,则对于每一个z$ \mathbb{R} $m,解集SOL(Kz+Q(·),φ)是非空的闭凸集,且存在正数ηQ满足

    $ \left\| v \right\| \le {\eta _Q}(1 + \left\| z \right\|)\;\;\;\;\forall v \in \text{SOL} (K,z + Q( \bullet ),\varphi ) $ (5)

    为了解决(3)式,设

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

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

    $ \mathit{\Phi }(t,v) = \{ B(t,v)y:y \in U(G(t,v)) $ (6)

    则可以把上面的(3)式转化为

    $ \begin{array}{l} x(t) = a + \frac{1}{h}(b - a)t + \frac{1}{{\mathit{\Gamma }(\delta )}}\int_0^t {{{(t - s)}^{\delta - 1}}} [f(s) + g(s)]{\rm{d}}s - \\ \;\;\;\;\;\;\;\;\;\frac{1}{h}\frac{t}{{\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} [f(s) + g(s)]{\rm{d}}s\;\;\;\;\;\;t \in I,f \in P_F^p(x),g \in P_\mathit{\Phi }^p(x) \end{array} $ (7)

    为了解决(7)式,引入集值映射ΣC(I$ \mathbb{R} $n)→P(C(I$ \mathbb{R} $n))为

    $ \begin{array}{l} \Sigma (x) = \left\{ {a + \frac{1}{h}(b - a)t + \frac{1}{{\mathit{\Gamma }(\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}} [f(s) + g(s)]{\rm{d}}s - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\frac{1}{h}\frac{t}{{\mathit{\Gamma }(\delta )}}\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} $ (8)

    xC(I$ \mathbb{R} $n)是(7)式的解等价于说x是集值映射Σ的不动点.

    引理5  在条件(F1),(F2),(B),(G),(Q)和(Φ)的假设下,PFpPΦp是弱上半连续的.

      当条件(Q)和条件(Φ)满足时,不等式(5)成立,剩下的证明过程与文献[2]中引理3.5的证明过程是一样的.

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

    $ W(f)(t) = a + \frac{1}{h}(b - a)t + \frac{1}{{\mathit{\Gamma }(\delta )}}\int_0^t {{{(t - s)}^{\delta - 1}}} f(s){\rm{d}}s - \frac{1}{h}\frac{t}{{\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} f(s){\rm{d}}s $ (9)

    定理1  映射W是完备连续的.

      证明与文献[2]中相应结论的证明过程类似.

    定理2  (8)式的算子Σ是完备上半连续的.

      证明与文献[2]中相应结论的证明过程类似.

    定理3  假设(F1),(F2),(B),(G),(Q)和(Φ)这6个条件成立.如果

    $ \begin{array}{l} \mathop {\lim \inf }\limits_{k \to \infty } \left[ {\frac{{{\mathit{\Psi }_F}(k)}}{{k\mathit{\Gamma }(\delta )}}\mathop {\sup }\limits_{t \in I} \int_0^t {{{(t - s)}^{\delta - 1}}} {\eta _F}(s){\rm{d}}s + \frac{{{\mathit{\Psi }_G}(k)}}{{k\mathit{\Gamma }(\delta )}}{\eta _Q}{\eta _B}\mathop {\sup }\limits_{t \in I} \int_0^t {{{(t - s)}^{\delta - 1}}} {\eta _G}(s){\rm{d}}s + } \right.\\ \left. {\frac{{{\Psi _F}(k)}}{{k\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} {\eta _F}(s){\rm{d}}s + \frac{{{\mathit{\Psi }_G}(k)}}{{k\mathit{\Gamma }(\delta )}}{\eta _Q}{\eta _B}\int_0^h {{{(h - s)}^{\delta - 1}}} {\eta _G}(s){\rm{d}}s} \right] < 1 \end{array} $ (10)

    其中I=[0,h],则(7)式至少有一个解.

      依据定理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} $n)中任意一个以原点为圆心,$ \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} $n)里存在一个序列{xk}满足‖xkCkykΣ(xk),‖ykCk对∀k$ \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

    $ {y_k}(t) = a + \frac{1}{h}(b - a)t + W\left( {{f_k} + {g_k}} \right)(t)\;\;\;\;t \in I $

    对∀tI,因为

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

    所以

    $ \begin{array}{l} \left\| {{y_k}(t)} \right\| = ||a+\frac{1}{h}(b-a)t+\frac{1}{\mathit{\Gamma} (\delta )}\int_{0}^{t}{{{(t-s)}^{\delta -1}}} \\ \left[ {{f}_{k}}(s)+{{g}_{k}}(s) \right]\text{d}s-\frac{1}{h}\frac{t}{\mathit{\Gamma} (\delta )}\int_{0}^{h}{{{(h-s)}^{\delta -1}}}\left[ {{f}_{k}}(s)+{{g}_{k}}(s) \right]\text{d}s|| \le \\ b + \left\| {\frac{1}{{\mathit{\Gamma }(\delta )}}\int_0^t {{{(t - s)}^{\delta - 1}}} \left[ {{f_k}(s) + {g_k}(s)} \right]{\rm{d}}s} \right\| + \left\| {\frac{1}{h}\frac{t}{{\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} \left[ {{f_k}(s) + {g_k}(s)} \right]{\rm{d}}s} \right\| \le \\ b + \frac{1}{{\mathit{\Gamma }(\delta )}}\mathop {\sup }\limits_{t \in I} \left\{ {\int_0^t {{{(t - s)}^{\delta - 1}}} \left[ {\left\| {{f_k}(s)} \right\| + \left\| {{g_k}(s)} \right\|} \right]{\rm{d}}s} \right\} + \\ \frac{1}{h}\mathop {\sup }\limits_{t \in I} \left\{ {\frac{t}{{\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} \left[ {\left\| {{f_k}(s)} \right\| + \left\| {{g_k}(s)} \right\|} \right]{\rm{d}}s} \right\} \le \\ b + \frac{{{\mathit{\Psi }_F}(k)}}{{\mathit{\Gamma }(\alpha )}}\mathop {\sup }\limits_{t \in I} \int_0^t {{{(t - s)}^{\delta - 1}}} {\eta _F}(s){\rm{d}}s + \frac{{{\eta _Q}{\eta _B}}}{{\mathit{\Gamma }(\delta )}}\mathop {\sup }\limits_{t \in I} \left\{ {\int_0^t {{{(t - s)}^{\delta - 1}}} \left[ {1 + {\eta _G}(s){\mathit{\Psi }_G}(k)} \right]{\rm{d}}s} \right\} + \\ \frac{{{\mathit{\Psi }_F}(k)}}{{\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} {\eta _F}(s){\rm{d}}s + \frac{{{\eta _Q}{\eta _B}}}{{\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} \left[ {1 + {\eta _G}(s){\mathit{\Psi }_G}(k)} \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}(k)}}{{k\mathit{\Gamma }(\delta )}}\mathop {\sup }\limits_{t \in I} \int_0^t {{{(t - s)}^{\delta - 1}}} {\eta _F}(s){\rm{d}}s + } \right.\\ \frac{{{\eta _Q}{\eta _B}}}{{k\mathit{\Gamma }(\delta )}}\mathop {\sup }\limits_{t \in I} \left\{ {\int_0^t {{{(t - s)}^{\delta - 1}}} \left( {1 + {\eta _G}(s){\mathit{\Psi }_G}(k)} \right){\rm{d}}s} \right\} + \frac{{{\mathit{\Psi }_F}(k)}}{{k\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} {\eta _F}(s){\rm{d}}s + \\ \left. {\frac{{{\eta _Q}{\eta _B}}}{{k\mathit{\Gamma }(\delta )}}\int_0^h {{{(h - s)}^{\delta - 1}}} \left( {1 + {\eta _G}(s){\mathit{\Psi }_G}(k)} \right){\rm{d}}s} \right] \end{array} $

    所以依据(10)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 Mixed Differential Variational Inequalities
    WU Xin-kun    
    College of Mathematics and Statistics, Qiannan Normal University for Nationalities, Duyun Guizhou 558000, China
    Abstract: Based on fractional differential variational inequality and fractional mixed differential variational inequality, a new class of fractional mixed differential variational inequalities are introduced and studied in this paper. First, a model of this class of fractional mixed differential variational inequalities is given. Then, a detailed description is given of the mathematical meanings of the symbols in this model. Finally, it is proved that the set of solutions of this model is non-empty.
    Key words: fractional differential variational inequality    fractional mixed differential variational inequality    set of solutions    
    X