武忠文,馬德香
(華北電力大學(xué)數(shù)理學(xué)院,北京,102206)
一類分?jǐn)?shù)階微分方程邊值問題的Lyapunov-type不等式研究
武忠文,馬德香
(華北電力大學(xué)數(shù)理學(xué)院,北京,102206)
一類包含 Caputo分?jǐn)?shù)階導(dǎo)數(shù)的邊值條件情況下的 Caputo分?jǐn)?shù)階微分方程Lyapunov-type不等式被求出.首先,由Caputo分?jǐn)?shù)階導(dǎo)數(shù)的基本概念,把分?jǐn)?shù)階微分方程轉(zhuǎn)化為積分方程,根據(jù)邊值條件,求解出相應(yīng)的格林函數(shù).為了方便研究格林函數(shù)性質(zhì),我們從中提取出函數(shù)F(t,s).運(yùn)用求導(dǎo)的方法,研究函數(shù)F(t,s)的性質(zhì),得到函數(shù)在整個(gè)區(qū)間的上下界.最后,在應(yīng)用方面,對于一類分?jǐn)?shù)階微分方程特征值問題,求解了其特征值的存在區(qū)間;對于一類Mittag-Leffler函數(shù),得到其零解不存在的區(qū)間.
分?jǐn)?shù)階微分方程;Lyapunov-type不等式;格林函數(shù);Mittag-Leffler函數(shù)
考慮下面邊值問題
q(t)是一個(gè)實(shí)連續(xù)函數(shù).如果式(1)存在非奇異解,則
文獻(xiàn)[1]中得到了Lyapunov不等式(2).
文獻(xiàn)[2]中,F(xiàn)erreira研究了一類Caputo分?jǐn)?shù)階邊值問題的Lyapunov型不等式:
q(t)是一個(gè)實(shí)連續(xù)函數(shù).如果式(3)存在非奇異解,則
顯然,令α=2,由式(4)推導(dǎo)出式(2).
文獻(xiàn)[3]中,D.O’Regan和B.Samet研究了一類Riemann-Liouville分?jǐn)?shù)階邊值問題的Lyapunov型不等式:
如果式(5)存在非奇異解,則
文獻(xiàn)[4]中,Jleli和Samet研究了一類Riemann-Liouville分?jǐn)?shù)階邊值問題的Lyapunov型不等式:
如果式(7)存在非奇異解,則
文獻(xiàn)[5]中,Ji Rong和Chuanzhi Bai研究了一類Caputo分?jǐn)?shù)階邊值問題的Lyapunov型不等式:
這里1<α≤2,0<β≤1,并且q:[a,b]→R是一個(gè)連續(xù)函數(shù).若式(9)存在非奇異解,則
在文獻(xiàn)[1-5]的影響下,我們將研究一類包含Caputo分?jǐn)?shù)階導(dǎo)數(shù)的邊值條件情況下的Caputo分?jǐn)?shù)階微分方程:
這里3<α≤4,1<β≤2,q:[a,b]→R是一個(gè)連續(xù)函數(shù).
定義1.1令α>0,Γ(α)是一個(gè)Gamma函數(shù),定義為則函數(shù)y(t)的α階Riemann-Liouville分?jǐn)?shù)階積分定義為
定義1.2令α>0,n=[α]+1,這里[α]是α的整數(shù)部分,則函數(shù)y(t)的α階Caputo分?jǐn)?shù)階微分定義為
引理1.1令γ>α>0,f∈C[a,b],則
引理1.2令y∈C[a,b],且3<α≤4,則
這里c0,c1,c2,c3是實(shí)常數(shù).
引理1.3[6]假設(shè)K(x,t)在區(qū)間[a,b]×[a,b]上是連續(xù)的,且對于任意固定的t∈[a,b],K
引理1.4 y∈C[a,b]是邊值問題式(11)的解,當(dāng)且僅當(dāng)y滿足下面積分不等式
證明:對式(11)兩邊做積分運(yùn)算,則由引理1.2,我們有
根據(jù)條件3<α≤4和y(a)=0,我們有c0=0.
因此
由條件yquot;(a)=0,我們有c3=0.因此,我們有
然后對式(15)兩邊求導(dǎo),由定義1.2,我們有
把上式代入式(14),我們有
證畢.
引理1.5對?(t,s)∈[a,b]×[a,b],我們有
證明:當(dāng)a≤s≤t≤b時(shí),
故對于s∈[a,t],我們有
因此對于s∈[a,t],我們有
當(dāng)a≤t≤s≤b時(shí),F(xiàn)(t,s)≥0顯然成立,且
綜上,對?(t,s)∈[a,b]×[a,b],
成立,證畢.
這個(gè)部分將給出Lyapunov-type不等式及一些推論.
把引理1.5運(yùn)用到式(18),我們有
證畢.
除此之外,在式(11)中令f(y)=y(tǒng),我們得到下面帶線性項(xiàng)的分?jǐn)?shù)階邊值問題.
這里3<α≤4,1<β≤2,q:[a,b]→R是一個(gè)連續(xù)函數(shù).
現(xiàn)在,我們令N=1,并且由定理2.1,我們得到如下推論.
推論2.1如果式(19)有一個(gè)非奇異解y∈[a,b],則
推論2.2若β=2,則由式(17)推導(dǎo)出下面Lyapunov-type不等式:
推論2.3若α=4,1<β≤2,由引理1.4,我們有
則由式(17)推導(dǎo)出下面Lyapunov-type不等式:
在式(19)中令a=0,b=1,然后我們討論下面特征值問題
這里3<α≤4,1<β≤2.
由推論2.1,我們得到下面推論.
證明:假設(shè)y0(t)是式(23)中對應(yīng)特征的一個(gè)特征函數(shù).由推論2.1中的式(20),我們有矛盾.
現(xiàn)在,我們考慮下面含有兩個(gè)參數(shù)的Mittag-Leffler函數(shù)
顯然,對于任意的z≥0,都有Eα,β>0.因此,對于Eα,β的實(shí)零點(diǎn),如果存在,一定是一個(gè)負(fù)實(shí)數(shù).對于Mittag-Leffler函數(shù),參數(shù)1<β≤2,3<α≤4時(shí),我們將用推論2.1去求函數(shù)沒有實(shí)零點(diǎn)的區(qū)間.
證明:由文獻(xiàn)[7]中的定理1,式(23)中通解y(t)為
由式(23)中邊值條件,我們有
對于下面非線性分?jǐn)?shù)階微分方程的邊值問題,我們得到一個(gè)Lyapunov-type不等式
這里3<α≤4,1<β≤2,q:[a,b]→R是一個(gè)連續(xù)函數(shù).
我們假設(shè)非線性項(xiàng)f是可控制的,故由上面定理,得到這些不等式.在證明這些不等式的過程中,得到格林函數(shù)的準(zhǔn)確性質(zhì)是很重要的.
下面特征值問題
這里3<α≤4,1<β≤2.沒有對應(yīng)的特征函數(shù)y(t);另一方面,我們得到對于,Mittag-Leffler函數(shù)沒有實(shí)零點(diǎn).
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Lyapunov-Type Inequality for a Fractional Differential Equation with Boundary Value Problem
WU Zhongwen,MA Dexiang
(Department of Mathematics,North China Electric Power University,Beijing102206,China)
A Lyapunov-type inequality for a Caputo fractional differential equation under boundary condition involving the Caputo fractional derivative is established.Firstly,according to Caputo fractional derivative definitions,a fractional equation is transformed into its equivalent integral equation,according to the boundary value conditions to get the corresponding Green function.In order tofacilitate the studyofthe nature ofthe Green function,the function F(t,s)is extracted.Byusing the method ofderivative in mathematical analysis,the properties offunction F(t,s)is studied toget the upper and lower bounds ofthe function over the whole interval.Finally,as application,for a fractional differential equation eigenvalue problem,a bound of the eigenvalue is obtained.Then,for certain Mittag-Leffler function,an interval where has noreal zeros is gotten.
fractional differential equation;Lyapunov-type inequality;Green function;Mittag-Leffler function
1001-4217(2017)04-0048-08
O 175.1
A
2017-04-03
武忠文(1992—),女(漢族),安徽淮南人,碩士,研究方向:分?jǐn)?shù)階微分方程.E-mail:2672105752@qq.com