仲秋艷 張興秋

(1.濟(jì)寧醫(yī)學(xué)院 信息技術(shù)中心，山東 濟(jì)寧 272067；2.濟(jì)寧醫(yī)學(xué)院 醫(yī)學(xué)信息工程學(xué)院，山東 日照 276826)

含p-Laplacian算子高階分?jǐn)?shù)階微分方程的唯一迭代正解①

仲秋艷1張興秋2

(1.濟(jì)寧醫(yī)學(xué)院 信息技術(shù)中心，山東 濟(jì)寧 272067；2.濟(jì)寧醫(yī)學(xué)院 醫(yī)學(xué)信息工程學(xué)院，山東 日照 276826)

利用單調(diào)迭代技巧,得到了一類含p-Laplacian算子的高階分?jǐn)?shù)階微分方程非局部問題迭代正解的唯一性結(jié)果,同時給出了解的迭代程序和誤差估計.

分?jǐn)?shù)階方程, 單調(diào)迭代,p-Laplacian算子,正解,唯一性

0 引言

近年來,由于在復(fù)雜介質(zhì)的電動力學(xué)、電磁學(xué)、聚合物流變學(xué)、分?jǐn)?shù)控制系統(tǒng)與分?jǐn)?shù)控制器、神經(jīng)的分?jǐn)?shù)模型以及分?jǐn)?shù)回歸模型等諸多方面的應(yīng)用[1-3],分?jǐn)?shù)階微分方程的研究得到了廣泛的關(guān)注和迅速的發(fā)展.文獻(xiàn)[4-16],在不同的邊值條件下,獲得了分?jǐn)?shù)階非局部邊值問題正解以及多個正解的存在性.

如何求出分?jǐn)?shù)階微分方程的正解在應(yīng)用上具有重要的意義.本文主要目的是給出下列具有P-Laplacian算子和積分邊值條件的分?jǐn)?shù)階微分方程(下稱PFDE)

(1)

與文獻(xiàn)[4-7]相比,本文具有以下特征：首先邊值條件更加廣泛;其次,微分方程含有P-Laplacian,這使得考慮的問題更加一般;最后,本文不但給出了解的存在性結(jié)果,而且給出了解的迭代程序及誤差估計.顯然,稍加修改,本文結(jié)果可用于含Riemann-Stieltjes積分的邊值問題.

1 引理

有關(guān)分?jǐn)?shù)階微積分的基本定義和性質(zhì)可在文獻(xiàn)[1-4]找到,從略.

(2)

引理1[17]對y∈C[0,1],y≥0,(2)的唯一解為

其中

G(t,s)=G1(t,s)+G2(t,s),

(3)

(4)

(5)

顯然,G(t,s)在[0,1]×[0,1]上連續(xù).

(a1)G(t,s)≥m1tα-1s(1-s)α-1-i,?t,s∈[0,1];

(a2)G(t,s)≤M1tα-2s(1-s)α-1-i,?t,s∈[0,1];

(a3)G(t,s)>M1tα-1(1-s)α-1-i,?t,s∈[0,1];

(a4)G(t,s)>0,?t,s∈(0,1);

2 主要結(jié)果

記e(t)=tα-1,本文使用以下條件.

(H)f∈C((0,1)×R+,R+),對于固定的t∈(0,1)以及c∈(0,1),f(t,u)關(guān)于u非減且存在常數(shù)0

f(t,cu)≥φp[c(1+η(c))]f(t,u),

(6)

這里η(c)=m(c-r-1).

注1 如果c≥1,那么(6)式等價于

(7)

設(shè)E=C[0,1],令P={x∈C[0,1]∶x(t)≥0,t∈[0,1]}.則P是Banach空間E中正規(guī)常數(shù)為1的正規(guī)錐.定義E的子集D如下

D={u∈P∶存在正數(shù)lu<1

(8)

注意到e(t)∈P,易知D非空.定義算A如

(9)

定理1 設(shè)條件(H)滿足.如果

(10)

那么PFDE(1)有唯一正解w*∈D,且存在常數(shù)

使得ke(t)≤w*(t)≤Ke(t),t∈[0,1],

證明首先證明算子A映D到D.

事實上,對任意u∈D,存在正數(shù)Lu>1>lu滿足

lue(t)≤u(t)≤Lue(t),t∈[0,1].

(11)

由條件(H),引理2,(7),(10),(11)諸式得

(12)

及

(13)

這里,

令

于是

取

(14)

則0<δ<1.由(14)式得

[1+η(δ)]-1w0≤Aw0≤[1+η(δ)]w0.

(15)

un=Aun-1,vn=Avn-1,n=1,2,…,

(16)

注意到f(t,u)關(guān)于u非減,由(6)和(7)兩式得

(17)

(18)

根據(jù)(15)-(18)諸式,我們有

u1=Au0≥δ[1+η(δ)]Aw0≥δw0=u0,

(19)

(20)

由(19)、(20)式,利用數(shù)學(xué)歸納法,我們有

u0≤u1≤…≤un≤…≤vn≤…≤v1≤v0.

(21)

此外,由(17)式得

(22)

注意到u0=δ2v0,我們有

(23)

由于P為正規(guī)常數(shù)為1的正規(guī)錐,故對于任意正整數(shù)P,由un+p-un≤vn-un,我們有

(24)

即{un}是一個Cauchy列.于是un收斂于某w*∈D.顯然

‖vn-w*‖≤‖vn-un‖+‖un-w*‖，

結(jié)合(24)式,我們有vn→w*及un≤w*≤vn，故,un+1=Aun≤Aw*≤Avn=vn+1,n=1,2,…，令n→+∞取極限得w*=Aw*,即,w*∈D是A的一個不動點.

取初值w0=e(t), 則le≤Ae≤Le,這里

u1=Au0≥k[1+η(k)]Ae≥k[1+η(k)]le≥ke=u0,

(25)

v1=Av0≤K[1+η(K-1)]-1Ae≤K[1+η(K-1)]-1Le≤Ke=v0.

(26)

根據(jù)(25)及(26)兩式可知唯一解w*滿足ke(t)≤w*(t)≤Ke(t),t∈[0,1].

(27)

下面在比條件(H)更強(qiáng)的條件下研究PFDE(1).列出以下條件:

(H*)f∈C((0,1)×R+,R+),對于任意固定的t∈(0,1),c∈(0,1),f(t,u)關(guān)于u非減,且存在常數(shù)0<λ*<1使得對于任意(t,u)∈(0,1)×R+,f(t,cu)≥φp(cλ*)f(t,u).

(H0)f∈C((0,1)×R+,R+),對于任意固定的t∈(0,1),c∈(0,1),f(t,u)關(guān)于u非減,且存在常數(shù)0<λ<1使得對于任意(t,u)∈(0,1)×R+,f(t,cu)≥cλf(t,u).

(H1)f∈C(0,1)×R+,R+),存在常數(shù)0<λ1≤λ2<1使得對于任意(t,u)∈(0,1)×R+,cλ2f(t,u)≤f(t,cu)≤cλ1f(t,u),0

證由條件(H*),對任意c∈(0,1),(t,u)∈(0,1)×R+,我們有

f(t,cu)≥φp(cλ*)f(t,u)=φp(cc-1(1-λ*))f(t,u)≥φp[c(1+η(c))]f(t,u),

這里η(c)=m(c-r-1),m=1,r=1-λ*,于是,由定理1可證定理2.

注2 從定理2的證明過程可知,條件(H)弱于(H*).另外,由文獻(xiàn)[12]注 3.4可知(H0)弱于(H1).文獻(xiàn)[12],[14]利用條件(H0)研究微分方程解的存在唯一性,而文獻(xiàn)[18-20]則利用條件(H1)研究某些整數(shù)階微分方程正解存在的充要條件.易知當(dāng)p>2時,(H*)弱于(H0).

3 應(yīng)用舉例

考察下列分?jǐn)?shù)階微分方程

(28)

[1] Samko S G, Kilbas A A, Marichev O I. Fractional Integral and Derivativs[C].//Theory and Applications. Gordon and Breach, Switzerland, 1993.

[2] Podlubny I.Fractional Differential Equations[C].//Mathematics in science and Engineering.NewYork, London, Toronto: Academic Press, 1999.

[3] Kilbas A A,Srivastava H M,TrujilloJ J.Theory and Applications of Fractional Differential Equationss[C]. //North-Holland Mathematics Studies. Elsevier Science B V Amsterdam, 2006.

[4] Wang Yong-qing,Liu Li-shan,Wu Yong-hong. Positive solutions for a nonlocal fractional differential equation[J]. Nonlinear Anal, 2011, 74: 3 599-3 605.

[5] Cabada A, Wang Guotao.Positive solutions of nonlinear fractional differential equations with integral boundary value conditions[J].J Math Anal App,2012,389:403-411.

[6] Feng Mei-qiang, Zhang Xue-mei, Ge Wei-gao.New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions[J].Boundary Value Problems,2011,2011:1-20.

[7] Wang Lin,Zhang Xing-qiu.Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter[J].J Appl Math Comput, 2014, 44: 293-316.

[8] Zhang Xing-qiu, Wang Lin Sun Qian. Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter[J]. Appl Math Comput, 2014, 226: 708-718.

[9] Zhang Xin-guang, Liu Li-shan, Wu Yong-hong, et al. The iterative solutions of nonlinear fractional differential equations[J]. Appl Math Comput, 2013, 219: 4 680-4 691.

[10] Li Shun-jie, Zhang Xin-guang,Wu Yong-hong,et al. Extremal solutions forp-Laplacian differential systems via iterative computation[J]. Appl Math Lett, 2013, 26:1 151-1 158.

[11] Sun Yong-ping, Zhao Min. Positive solutions for a class of fractional differential equations with integral boundary conditions[J]. Appl Math Lett, 2014, 34: 17-21.

[12] Zhang Xin-guang, Han Yue-feng. Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations[J]. Appl Math Lett, 2012, 25: 555-560.

[13] Tian Yuan-sheng, Li Xiao-ping. Existence of positive solution to boundary value problem of fractional differential equations withp-Laplacian operator[J]. J Math Anal Appl, 2015, 47: 237-248.

[14] 仲秋艷，張興秋．一類具有p-Laplacian算子的奇異分?jǐn)?shù)階微分方程無窮點邊值問題的正解[J]. 聊城大學(xué)學(xué)報:自然科學(xué)版，2016, 29(2):25-32.

[15] Zhang Xing-qiu. Positive solutions for singular higher-order fractional differential equations with nonlocal conditions[J]. J Appl Math Comput, 2015, 49: 69-89.

[16] Zhang Xin-guang, Liu Li-shan. A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems withp-Laplacian[J]. Nonlinear Anal, 2008, 68: 3 127-3 137.

[17] 仲秋艷，張興秋.含參數(shù)及p-Laplacian算子的奇異分?jǐn)?shù)階微分方程積分邊值問題的正解[J].山東大學(xué)學(xué)報:理學(xué)版, 2016, 51(6): 78-84.

[18] Wei Zhong-li. A necessary and sufficient condition for the existence of positive solutions of singular super-linearm-point boundary value problems[J]. Appl Math Comput, 2006, 179: 67-78.

[19] Wei Zhong-li, Pang Chang-ci. The method of lower and upper solutions for fourth order singularm-point boundary value problems[J]. J Math Anal Appl, 2006, 322:675-692.

[20] Wei Zhong-li.A class of fourth order singular boundary value problems[J].Appl Math Comput,2004, 153: 865-884.

UniqueIterativePositiveSolutionforHigher-orderFractionalDifferentialEquationswithp-Laplacian

ZHONG Qiu-yan ZHANG Xing-qiu

(1．Department of Information Technology, Jining Medical College, Jining 272067, China; 2．School of Medical Information Engineering, Jining Medical College, Rizhao 276826, China)

By means of monotone iterative technique, uniqueness results of positive solutions for a class of higher nonlocal fractional differential equations withp-Laplacian are obtained. The iterative sequences of solution and error estimation are also considered.

fractional differential equations,monotone iterative technique,p-Laplacian, positive solution, uniqueness

2017-05-12

國家自然科學(xué)基金項目(11571296,11571197);山東省高?？萍及l(fā)展計劃項目(J15LI16);山東省自然科學(xué)基金項目(ZR2015AL002)資助

張興秋,E-mail:zhxq197508@163.com.

O175.8

A

1672-6634(2017)03-0028-07