一類具有初邊值條件的非線性分?jǐn)?shù)階微分方程組解的存在性與唯一性

李雪梅,代 群,李輝來

(1.吉林大學(xué) 數(shù)學(xué)學(xué)院,長春 130012；2.長春理工大學(xué) 理學(xué)院,長春 130022)

?

一類具有初邊值條件的非線性分?jǐn)?shù)階微分方程組解的存在性與唯一性

李雪梅1,代 群2,李輝來1

(1.吉林大學(xué) 數(shù)學(xué)學(xué)院,長春 130012；2.長春理工大學(xué) 理學(xué)院,長春 130022)

考慮一類具有初邊值條件的耦合非線性分?jǐn)?shù)階微分方程組解的存在性與唯一性問題,應(yīng)用Schauder和Banach不動點定理得到了此類方程組解的存在性與唯一性條件.

耦合方程組；分?jǐn)?shù)階微分方程；存在性；唯一性；Schauder不動點定理；Banach不動點定理

分?jǐn)?shù)階微分方程在工程技術(shù)以及各種材料和工藝的遺傳特性描述中應(yīng)用廣泛[1-6].文獻(xiàn)[7]討論了一個具有邊值條件的耦合分?jǐn)?shù)階微分方程;文獻(xiàn)[8]分析了一種分?jǐn)?shù)階微分方程系統(tǒng).本文考慮具有兩點邊值條件的分?jǐn)?shù)階微分方程組:

(1)

其中:0<αi≤1;01;Dαi表示Riemann-Liouville分?jǐn)?shù)階微分.

定義2[10]令n-1<α≤n,n∈,則稱為α階Riemann-Liouville微分.

由分?jǐn)?shù)階積分和微分的定義可知IαIβf(t)=Iα+βf(t),DαIαf(t)=f(t).

引理1初邊值問題(1)等價于如下方程組:

證明：對式(1)兩邊分別進(jìn)行α1和α2次積分,得

由于u(0)=0,v(0)=0,從而C1=C2=0,證畢.

定理1(存在性) 若fi:Wi→(i=1,2)連續(xù),則初邊值條件(1)有解(u(t),v(t)):[0,a*]→2.其中:

Wi=[0,a]×[Dmiu(0)-l1i,Dmiu(0)+l1i]×[Dniv(0)-l2i,Dniv(0)+l2i];

證明：令F(u(t),v(t))=(F1(u(t),v(t)),F2(u(t),v(t))),其中

由于函數(shù)Dm1u(s),Dn1v(s),(t-s)αi-m1-1在[0,a*]上連續(xù),則DmiF1(t)在[0,a*]連續(xù).同理DniF2(t)在[0,a*]上連續(xù).于是,有

同理|DniF2(t)-DniF2(0)|

從而F(u(t),v(t))是有界且逐點定義的.

且

因此,如果|t2-t1|<δ,則

同理,有

從而得F(U)是一個等度連續(xù)集,于是F是一個全連續(xù)算子,再由Schauder不動點定理知,F有一個不動點(u(t),v(t)):[0,a*]→為方程(1)的解.

定理2(唯一性)w=[0,a*]×[Dmiu(0)-l,Dmiu(0)+l]×[Dniu(0)-l,Dniu(0)+l],若fi(s,Dmiu(s),Dniv(s))(i=1,2)在w上連續(xù),并存在兩個正函數(shù)m(t),n(t)滿足:

且如果

成立,則方程(1)有唯一正解.其中:

證明：由定理1可知F(U)?U,

而

類似地,有

則

‖(Dm1u1(t),Dn1v1(t))-(Dm1u2(t),Dn1v2(t))‖∞≤k1‖(u1(t),v1(t))-(u2(t),v2(t))‖∞,

從而

同理有

|F2(u1(t),v1(t))-F2(u2(t),v2(t))|≤θ‖(u1(t),v1(t))-(u2(t),v2(t))‖∞,

則

|F(u1(t),v1(t))-F(u2(t),v2(t))|≤max{ρ,θ}‖(u1(t),v1(t))-(u2(t),v2(t))‖∞,

由定理1知F是全連續(xù)算子,又由Banach不動點定理可知,算子F在U中有唯一不動點,其即為方程組(1)的唯一正解.

[1] BAI Zhanbing,Lü Haishen.Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations [J].Math Anal Appl,2005,311(2):495-505.

[2] CHANG Yongkai,Nieto J J.Some New Existence Results for Fractional Differential Inclusions with Boundary Conditions [J].Math Comput Modelling,2009,49(3/4):605-609.

[3] DENG Weihua.Numerical Algorithm for the Time Fractional Fokker-Planck Equation [J].J Comput Phys,2007,227(2):1510-1522.

[4] BAI Chuanzhi,FANG Jinxuan.The Existence of a Positive Solution for a Singular Coupled System of Nonlinear Fractional Differential Equations [J].Appl Math Comput,2004,150(3):611-621.

[5] CHEN Yong,AN Hongli.Numerical Solutions of Coupled Burgers Equations with Time- and Space-Fractional Derivatives [J].Appl Math Comput,2008,200(1):87-95.

[6] Daftardar-Gejji V.Positive Solutions of a System of Non-autonomous Fractional Differential Equations [J].J Math Anal Appl,2005,302(1):56-64.

[7] SU Xinwei.Boundary Value Problem for a Coupled System of Nonlinear Fractional Differential Equations [J].Appl Math Lett,2009,22(1):64-69.

[8] Daftardar-Gejji V,Babakhani A.Analysis of a System of Fractional Differential Equations [J].Math Anal Appl,2004,293(2):511-522.

[9] Kilbas A A,Srivastava H M,Trujillo J J.Theory and Applications of Fractional Differential Equations [M].Amsterdam:Elsevier,2006.

[10] Podlubny I.Fractional Differential Equations [M].San Diego:Academic Press,1999.

(責(zé)任編輯：趙立芹)

ExistenceandUniquenessforaCoupledSystemofNonlinearFractionalDifferentialEquationswithInitialValueConditions

LI Xuemei1,DAI Qun2,LI Huilai1

(1.CollegeofMathematics,JilinUniversity,Changchun130012,China；2.CollegeofScience,ChangchunUniversityofScienceandTechnology,Changchun130022,China)

We studied a coupled system of nonlinear fractional equations with initial value condition and got the existence and uniqueness of it by means of the Schauder fixed point theorem and Banach fixed point theorem.

coupled system;fractional differential equations;existence;uniqueness;Schauder fixed point theorem;Banach fixed point theorem

10.13413/j.cnki.jdxblxb.2015.03.03

2014-10-24.

李雪梅(1989—),女,漢族,碩士,從事偏微分方程的研究,E-mail：1304833191@qq.com.通信作者：李輝來(1962—),男,漢族,博士,教授,博士生導(dǎo)師,從事偏微分方程的研究,E-mail：lihuilai@mail.jlu.edu.cn.

國家自然科學(xué)基金(批準(zhǔn)號：11271154).

O175.08

：A

：1671-5489(2015)03-0363-04