一类带积分边界条件的分数阶微分方程的正解
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摘要
分数阶微积分理论在空气动力学、复杂介质电动力学、控制理论、信号与图像处理、流变学等诸多问题上显示出独特优势,其理论和应用的研究已成为一个热点,研究分数阶微分方程及其边值问题为上述问题提供了重要的理论依据;考虑一类带有积分边界条件的分数阶微分方程的边值问题,首先应用分数阶微积分的有关结论得到了线性分数阶微分方程边值问题解的表达式,获得了相应的格林函数及其性质,给出格林函数的一个新的上界的估计;再利用Schauder不动点定理,得到了此边值问题的正解存在性结果.
Fractional calculus theory shows unique advantage in aerodynamics,electrodynamics in complex medium,control theory,signal and image processing,rheology and many other issues. The study of such kind of problems has received considerable attention both in theory and applications. The investigation of fractional differential equations and their boundary value problems provide an important theoretical basis for the above problems. We consider the fractional differential equations with integral boundary conditions. First,we give the expression of the solution of the boundary value problem of the linear fractional differential equation,analyze the properties of the Green function,and give new estimate of a new upper bound of the Green function. Then by the Schauder fixed point theorem,we get the existence results of positive solution for the boundary value problems.
Fractional calculus theory shows unique advantage in aerodynamics,electrodynamics in complex medium,control theory,signal and image processing,rheology and many other issues. The study of such kind of problems has received considerable attention both in theory and applications. The investigation of fractional differential equations and their boundary value problems provide an important theoretical basis for the above problems. We consider the fractional differential equations with integral boundary conditions. First,we give the expression of the solution of the boundary value problem of the linear fractional differential equation,analyze the properties of the Green function,and give new estimate of a new upper bound of the Green function. Then by the Schauder fixed point theorem,we get the existence results of positive solution for the boundary value problems.
引文
[1]ANATOLY A,HARI M S,JUAN J T.Theory and Applications of Fractional Differential Equations[J].Elsevier,2006,204(49-52):2453-2461
[2]ONNENMACHERT F,NMETZLER R.On the RiemannLiouville Fractional Calculus and Some Recent Applications[J].Fractals,1995(3):557-566
[3]ZHENG Z.On the Developments and Applications of Fractional Differential Equations[J].Journal of Xuzhou Normal University,2008(2):26
[4]JIANG F,XU X,CAO Z.The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applicatio[J].Abstract and Applied Analysis,2013(2):710-719
[5]BAI Z,LU H.Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation[J].Journal of Mathematical Analysis and Applications,2005(311):495-505
[6]LI X Y,LIU S.Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation[J].Applied Mathematics and Compution,2011(217):9278-9285
[7]JOLIO C.On Existence and Uniqueness of Positive Solutions to a Class of Frational Boundary Value Problems[J].Spring,2011(1):1-9
[8]DELBOSCO J,LUI R.Existence and Uniqueness for a Nonlinear Fractional Differential Equation[J].Journal of Mathematical Analysis and Applications,1996(204):609-625
[9]CABALLERO J.Positive Solutions for a Class of Singular Fractional Boundary Value Problems Comput[J].Computers and Mathematics with Applications,2011(62):1325-1332
[10]CHRISTOPHER S G.Existence of a Positive Solution to a Class of Fractional Differential Equations[J].Applied Mathematics Letters,2010(23):1050-1055
[11]JOHN R G.Positive Solutions for a Fractional Boundary Value Problem[J].Applied Mathematics Letters,2010(56):49-55
[12]PODLUBNY I.Fractional Differential Equations[M].San Diego:Academic Press,1999
[13]SUN Y,ZHAO M.Positive Solutions for a Class of Fractional Differential Equations with Integral Boundary Conditions[J].Applied Mathematics Letters,2014(34):17-21
[14]XI L,JIA M,WU B.Existence and Uniqueness of Solution for Fracional Differential Equations with Integral Boundary Conditions[J].Electron Joual Theo,2009(69):1-10
[15]NEAMPREM K N.Positive Solutions for Fractional Differential Systems with Nonlocal Riemann-Liouville Fractional Integral Boundary Conditions[J].Spring,2017(21):825-845
[16]UONG S.Positive Solutions of Singular Fractional Differential Equations with Integral Boundary Conditons[J].Elsevier,2013(27):1053-1059
[17]YANG A,WANG H.Positive Solutions for Higher-order Nonlinear Fractional Differential Equation with Integral Boundary Condition[J].Electron Jqual Theo,2011(1):1-15
[18]FENG H,ZHAI C.Existence and Uniqueness of Positive Solution for a Class of Fractional Differential Equation with Integral Boundary Conditions[J].Nonlinear Anal,2017(22):160-172
[19]ZHOU C.Existence and Uniquencess of Positive Solutions to Higher-order Nonlinear Fractional Differential Equational with Integral Boundary Conditions[J].Appl Math Mech,2012(234):3670-3674
[20]YANG W.Positive Solutions for Singular Coupled Integral Boundary Value Problems of Nonlinear Hadmard Fractional Differential Equations[J].J Nonlinearsci Appl,2015(8):110-129
[2]ONNENMACHERT F,NMETZLER R.On the RiemannLiouville Fractional Calculus and Some Recent Applications[J].Fractals,1995(3):557-566
[3]ZHENG Z.On the Developments and Applications of Fractional Differential Equations[J].Journal of Xuzhou Normal University,2008(2):26
[4]JIANG F,XU X,CAO Z.The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applicatio[J].Abstract and Applied Analysis,2013(2):710-719
[5]BAI Z,LU H.Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation[J].Journal of Mathematical Analysis and Applications,2005(311):495-505
[6]LI X Y,LIU S.Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation[J].Applied Mathematics and Compution,2011(217):9278-9285
[7]JOLIO C.On Existence and Uniqueness of Positive Solutions to a Class of Frational Boundary Value Problems[J].Spring,2011(1):1-9
[8]DELBOSCO J,LUI R.Existence and Uniqueness for a Nonlinear Fractional Differential Equation[J].Journal of Mathematical Analysis and Applications,1996(204):609-625
[9]CABALLERO J.Positive Solutions for a Class of Singular Fractional Boundary Value Problems Comput[J].Computers and Mathematics with Applications,2011(62):1325-1332
[10]CHRISTOPHER S G.Existence of a Positive Solution to a Class of Fractional Differential Equations[J].Applied Mathematics Letters,2010(23):1050-1055
[11]JOHN R G.Positive Solutions for a Fractional Boundary Value Problem[J].Applied Mathematics Letters,2010(56):49-55
[12]PODLUBNY I.Fractional Differential Equations[M].San Diego:Academic Press,1999
[13]SUN Y,ZHAO M.Positive Solutions for a Class of Fractional Differential Equations with Integral Boundary Conditions[J].Applied Mathematics Letters,2014(34):17-21
[14]XI L,JIA M,WU B.Existence and Uniqueness of Solution for Fracional Differential Equations with Integral Boundary Conditions[J].Electron Joual Theo,2009(69):1-10
[15]NEAMPREM K N.Positive Solutions for Fractional Differential Systems with Nonlocal Riemann-Liouville Fractional Integral Boundary Conditions[J].Spring,2017(21):825-845
[16]UONG S.Positive Solutions of Singular Fractional Differential Equations with Integral Boundary Conditons[J].Elsevier,2013(27):1053-1059
[17]YANG A,WANG H.Positive Solutions for Higher-order Nonlinear Fractional Differential Equation with Integral Boundary Condition[J].Electron Jqual Theo,2011(1):1-15
[18]FENG H,ZHAI C.Existence and Uniqueness of Positive Solution for a Class of Fractional Differential Equation with Integral Boundary Conditions[J].Nonlinear Anal,2017(22):160-172
[19]ZHOU C.Existence and Uniquencess of Positive Solutions to Higher-order Nonlinear Fractional Differential Equational with Integral Boundary Conditions[J].Appl Math Mech,2012(234):3670-3674
[20]YANG W.Positive Solutions for Singular Coupled Integral Boundary Value Problems of Nonlinear Hadmard Fractional Differential Equations[J].J Nonlinearsci Appl,2015(8):110-129