非线性隐式分数阶微分方程耦合系统初值问题
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摘要
利用不动点定理和向量形式的Gronwall不等式,得到了Caputo分数阶导数定义下的非线性隐式分数阶微分方程耦合系统解的存在性和唯一性,并探讨了解的估值,解对初值的连续依赖性,解对参数和函数的连续依赖性,以及耦合系统的ε-近似解.
By using the fixed point theorem and the Gronwall inequality of vector form,the existence and uniqueness of the solution of the coupled system of nonlinear implicit fractional differential equations under the definition of Caputo fractional derivative are obtained.The estimate on solutions,the continuous dependence on initial values,the continuous dependence on parameters and functions,and e-approximate solutions for coupled systems are also discussed.
By using the fixed point theorem and the Gronwall inequality of vector form,the existence and uniqueness of the solution of the coupled system of nonlinear implicit fractional differential equations under the definition of Caputo fractional derivative are obtained.The estimate on solutions,the continuous dependence on initial values,the continuous dependence on parameters and functions,and e-approximate solutions for coupled systems are also discussed.
引文
[1]Kilbas A A A,Srivastava H M,Trujillo J J.Theory and applications of fractional differential equations.Elsevier Science Limited,2006
[2]Hilfer R,Ed.Applications of Fractional Calculus in Physics.Singapore:World Scientific,2000.
[3]Podlubny I.Fractional Differential Equations.San Diego:Academic Press,1999
[4]苏新卫.分数阶微分方程耦合系统边值问题解的存在性.工程数学学报,2009,26(1):133-137(Su X.The existence of solution to boundary value problems for a coupled system of nonlinear fractional differential equations.Chinese Journal of Engineering Mathematics,2009,26(1):133-137)
[5]Hu Z,Liu W,Chen T.Existence of solutions for a coupled system of fractional differential equations at Resonance.Boundary Value Problem,2012,98:1-13
[6]Aziz Khan,Kamal Shah,Yongjin Li,Tahir Saeed Khan.Ulam type stability for a coupled system of boundary value problems of nonlinear fractional differential equations.Journal of Function Spaces,2017(2017),3046013,8 pages
[7]Martin P.An intrinsic sufficient condition for regular decoupling.Systems and Control Letters,1993,20:383-391
[8]Pritchard F L.On implicit systems of differential equations.Journal of Differential Equations,2003,194:328-363
[9]Rabier P J,Rheinboldt W C.A geometric treatment of implicit differential-algebraic equations.Journal of Differential Equations, 1994,109:110-146
[10]Respondek W.Dynamic input-output linearization and decoupling of nonlinear systems.In:Proceedings of the Second European Control Conference ECC 93,1993,1523-1527
[11]Singh S N.A modified algorithm for invertibility in nonlinear systems.IEEE Transactions on Automatica Control,1981,26:595-598
[12]Kucche K D,Nieto J J,Venktesh V.Theory of nonlinear implicit fractional differential equations.Differential Equations and Dynamical Systems,2016,1-17
[13]胡雷,张淑琴,侍爱玲.分数阶微分方程耦合系统共振边值问题解的存在性.数学物理学报,2014,34A:1313-1326(Hu L,Zhang S,Shi A.Existence of solutions for two-point boundary value problem of a coupled fractional differential equations at resonance.Acta Mathematica Scientia,2014,34A:1313-1326)
[14]钟承奎,范先令,陈文原.非线性泛函分析引论.兰州:兰州大学出版社,1998(Zhong C,Fan X,Chen W.Introduction to Nonlinear Analysis.Lanzhou:Lanzhou University Press,1998)
[15]Feng Y,Wang H.Characterizations of reproducing cone and uniqueness of fixed point.Nonlinear Analysis,2011,74:5759-5765
[2]Hilfer R,Ed.Applications of Fractional Calculus in Physics.Singapore:World Scientific,2000.
[3]Podlubny I.Fractional Differential Equations.San Diego:Academic Press,1999
[4]苏新卫.分数阶微分方程耦合系统边值问题解的存在性.工程数学学报,2009,26(1):133-137(Su X.The existence of solution to boundary value problems for a coupled system of nonlinear fractional differential equations.Chinese Journal of Engineering Mathematics,2009,26(1):133-137)
[5]Hu Z,Liu W,Chen T.Existence of solutions for a coupled system of fractional differential equations at Resonance.Boundary Value Problem,2012,98:1-13
[6]Aziz Khan,Kamal Shah,Yongjin Li,Tahir Saeed Khan.Ulam type stability for a coupled system of boundary value problems of nonlinear fractional differential equations.Journal of Function Spaces,2017(2017),3046013,8 pages
[7]Martin P.An intrinsic sufficient condition for regular decoupling.Systems and Control Letters,1993,20:383-391
[8]Pritchard F L.On implicit systems of differential equations.Journal of Differential Equations,2003,194:328-363
[9]Rabier P J,Rheinboldt W C.A geometric treatment of implicit differential-algebraic equations.Journal of Differential Equations, 1994,109:110-146
[10]Respondek W.Dynamic input-output linearization and decoupling of nonlinear systems.In:Proceedings of the Second European Control Conference ECC 93,1993,1523-1527
[11]Singh S N.A modified algorithm for invertibility in nonlinear systems.IEEE Transactions on Automatica Control,1981,26:595-598
[12]Kucche K D,Nieto J J,Venktesh V.Theory of nonlinear implicit fractional differential equations.Differential Equations and Dynamical Systems,2016,1-17
[13]胡雷,张淑琴,侍爱玲.分数阶微分方程耦合系统共振边值问题解的存在性.数学物理学报,2014,34A:1313-1326(Hu L,Zhang S,Shi A.Existence of solutions for two-point boundary value problem of a coupled fractional differential equations at resonance.Acta Mathematica Scientia,2014,34A:1313-1326)
[14]钟承奎,范先令,陈文原.非线性泛函分析引论.兰州:兰州大学出版社,1998(Zhong C,Fan X,Chen W.Introduction to Nonlinear Analysis.Lanzhou:Lanzhou University Press,1998)
[15]Feng Y,Wang H.Characterizations of reproducing cone and uniqueness of fixed point.Nonlinear Analysis,2011,74:5759-5765