具非线性项变号的分数阶微分方程非齐次边值问题正解的存在性
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摘要
考虑一类具非线性项变号的分数阶微分方程非齐次积分边值问题正解的存在性,用锥拉伸与锥压缩不动点定理,建立并证明该边值问题正解的存在性定理,并给出实例说明所得结论的合理性.
We considered the existence of positive solutions for a class of nonhomogeneous integral boundary value problems of fractional differential equations with sign changing nonlinearities.By using fixed point theorems of cone expansion and cone compression,we established and proved the existence theorems of positive solutions for the boundary value problem,and gave some examples to illustrate the rationality of the conclusions.
We considered the existence of positive solutions for a class of nonhomogeneous integral boundary value problems of fractional differential equations with sign changing nonlinearities.By using fixed point theorems of cone expansion and cone compression,we established and proved the existence theorems of positive solutions for the boundary value problem,and gave some examples to illustrate the rationality of the conclusions.
引文
[1]PODLUBNY I.Fractional Differential Equations[M].New York:Academic Press,1999.
[2]DIETHELM K.The Analysis of Fractional Differential Equations[M].Berlin:Springer,2010.
[3]SAMKO S G,KILBAS A A,MARICHEV O I.Fractional Integrals and Derivatives:Theory and Applications[M].Philadelphia,PA:Gordon and Breach Science Publishers,1993.
[4]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of Fractional Differential Equations[M].Amsterdam:Elsevier,2006.
[5]白占兵.分数阶微分方程边值问题理论及应用[M].北京:中国科学技术出版社,2012:12.(BAI Zhanbing.Theory and Application of Boundary Value Problems for Fractional Differential Equations[M].Beijing:China Science and Technology Press,2012:12.)
[6]LIU Yuji.Positive Solutions of Boundary Value Problems for Fractional Order Differential Equations by Monotonic Iteration Method[J].Rendiconti del Circolo Matematico di Palermo Series 2,2017,66(3):457-469.
[7]刘帅,贾梅,秦小娜.带积分边值条件的分数阶微分方程解的存在性和唯一性[J].上海理工大学学报,2014,36(5):409-415.(LIU Shuai,JIA Mei,QIN Xiaona.Existence and Uniqueness of Solutions of the Fractional Differential Equation with Integral Boundary Value Conditions[J].Journal of University of Shanghai for Science and Technology,2014,36(5):409-415.)
[8]李晓晨,刘锡平,李燕,等.无穷区间上含参数分数阶微分方程积分边值问题正解的存在性[J].吉林大学学报(理学版),2017,55(1):13-21.(LI Xiaochen,LIU Xiping,LI Yan,et al.Existence of Positive Solutions for Integral Boundary Value Problems of Fractional Differential Equations with Parameters in Infinite Interval[J].Journal of Jilin University(Science Edition),2017,55(1):13-21.)
[9]LIU Suli,LI Huilai,DAI Qun,et al.Existence and Uniqueness Results for Nonlocal Integral Boundary Value Problems for Fractional Differential Equations[J/OL].Advances in Difference Equations,2016-05-03.doi:10.1186/s13662-016-0847-x.
[10]LIU Xiping,JIA Mei,GE Weigao.The Method of Lower and Upper Solutions for Mixed Fractional Four-Point Boundary Value Problem with p-Laplacian Operator[J].Applied Mathematics Letters,2017,65:56-62.
[11]SU Xiaofeng,JIA Mei,LI Mengmeng.The Existence and Nonexistence of Positive Solutions for Fractional Differential Equations with Nonhomogeneous Boundary Conditions[J/OL].Advances in Difference Equations,2016-01-28.doi:10.1186/s13662-016-0750-5.
[12]LIU Yang,XIE Dapeng,BAI Chuanzhi,et al.Multiple Positive Solutions for a Coupled System of Fractional Multi-point BVP with p-Laplacian Operator[J/OL].Advances in Difference Equations,2017-06-15.doi:10.1186/s13662-017-1221-3.
[13]FENG Meiqiang,ZHANG Xuemei,GE Weigao.New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions[J/OL].Boundary Value Problems,2011-07-20.doi:10.1155/2011/720702.
[14]王淑,贾梅,祁卫杰.非线性项变号的分数阶微分方程边值问题正解的存在性[J].上海理工大学学报,2013,35(1):1-6.(WANG Shu,JIA Mei,QI Weijie.Existence of Positive Solutions for a Class of Fractional Differential Equations with Sign Changing Nonlinearities[J].Journal of University of Shanghai for Science and Technology,2013,35(1):1-6.)
[15]WANG Yongqing,LIU Lishan,WU Yonghong.Positive Solutions for a Class of Fractional Boundary Value Problem with Changing Sign Nonlinearity[J].Nonlinear Analysis:Theory Methods&Applications,2011,74(17):6434-6441.
[16]HENDERSON J,LUCA R.Positive Solutions for a System of Semipositone Coupled Fractional Boundary Value Problems[J/OL].Boundary Value Problems,2016-03-08.doi:10.1186/s13661-016-0569-8.
[17]WANG Ying,LIU Lishan,ZHANG Xinguang,et al.Positive Solutions of an Abstract Fractional Semipositone Differential System Model for Bioprocesses of HIV Infection[J].Applied Mathematics&Computation,2015,258:312-324.
[18]HE Jiankun,JIA Mei,LIU Xiping,et al.Existence of Positive Solutions for a High Order Fractional Differential Equation Integral Boundary Value Problem with Changing Sign Nonlinearity[J/OL].Advances in Difference Equations,2018-02-05.https://doi.org/10.1186/s13662-018-1465-6.
[19]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法[M].2版.济南:山东科学技术出版社,2006:190-218.(GUO Dajun,SUN Jingxian,LIU Zhaoli.Functional Methods for Nonlinear Ordinary Differential Equations[M].2nd ed.Jinan:Shandong Science and Technology Press,2006:190-218.)
[2]DIETHELM K.The Analysis of Fractional Differential Equations[M].Berlin:Springer,2010.
[3]SAMKO S G,KILBAS A A,MARICHEV O I.Fractional Integrals and Derivatives:Theory and Applications[M].Philadelphia,PA:Gordon and Breach Science Publishers,1993.
[4]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of Fractional Differential Equations[M].Amsterdam:Elsevier,2006.
[5]白占兵.分数阶微分方程边值问题理论及应用[M].北京:中国科学技术出版社,2012:12.(BAI Zhanbing.Theory and Application of Boundary Value Problems for Fractional Differential Equations[M].Beijing:China Science and Technology Press,2012:12.)
[6]LIU Yuji.Positive Solutions of Boundary Value Problems for Fractional Order Differential Equations by Monotonic Iteration Method[J].Rendiconti del Circolo Matematico di Palermo Series 2,2017,66(3):457-469.
[7]刘帅,贾梅,秦小娜.带积分边值条件的分数阶微分方程解的存在性和唯一性[J].上海理工大学学报,2014,36(5):409-415.(LIU Shuai,JIA Mei,QIN Xiaona.Existence and Uniqueness of Solutions of the Fractional Differential Equation with Integral Boundary Value Conditions[J].Journal of University of Shanghai for Science and Technology,2014,36(5):409-415.)
[8]李晓晨,刘锡平,李燕,等.无穷区间上含参数分数阶微分方程积分边值问题正解的存在性[J].吉林大学学报(理学版),2017,55(1):13-21.(LI Xiaochen,LIU Xiping,LI Yan,et al.Existence of Positive Solutions for Integral Boundary Value Problems of Fractional Differential Equations with Parameters in Infinite Interval[J].Journal of Jilin University(Science Edition),2017,55(1):13-21.)
[9]LIU Suli,LI Huilai,DAI Qun,et al.Existence and Uniqueness Results for Nonlocal Integral Boundary Value Problems for Fractional Differential Equations[J/OL].Advances in Difference Equations,2016-05-03.doi:10.1186/s13662-016-0847-x.
[10]LIU Xiping,JIA Mei,GE Weigao.The Method of Lower and Upper Solutions for Mixed Fractional Four-Point Boundary Value Problem with p-Laplacian Operator[J].Applied Mathematics Letters,2017,65:56-62.
[11]SU Xiaofeng,JIA Mei,LI Mengmeng.The Existence and Nonexistence of Positive Solutions for Fractional Differential Equations with Nonhomogeneous Boundary Conditions[J/OL].Advances in Difference Equations,2016-01-28.doi:10.1186/s13662-016-0750-5.
[12]LIU Yang,XIE Dapeng,BAI Chuanzhi,et al.Multiple Positive Solutions for a Coupled System of Fractional Multi-point BVP with p-Laplacian Operator[J/OL].Advances in Difference Equations,2017-06-15.doi:10.1186/s13662-017-1221-3.
[13]FENG Meiqiang,ZHANG Xuemei,GE Weigao.New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions[J/OL].Boundary Value Problems,2011-07-20.doi:10.1155/2011/720702.
[14]王淑,贾梅,祁卫杰.非线性项变号的分数阶微分方程边值问题正解的存在性[J].上海理工大学学报,2013,35(1):1-6.(WANG Shu,JIA Mei,QI Weijie.Existence of Positive Solutions for a Class of Fractional Differential Equations with Sign Changing Nonlinearities[J].Journal of University of Shanghai for Science and Technology,2013,35(1):1-6.)
[15]WANG Yongqing,LIU Lishan,WU Yonghong.Positive Solutions for a Class of Fractional Boundary Value Problem with Changing Sign Nonlinearity[J].Nonlinear Analysis:Theory Methods&Applications,2011,74(17):6434-6441.
[16]HENDERSON J,LUCA R.Positive Solutions for a System of Semipositone Coupled Fractional Boundary Value Problems[J/OL].Boundary Value Problems,2016-03-08.doi:10.1186/s13661-016-0569-8.
[17]WANG Ying,LIU Lishan,ZHANG Xinguang,et al.Positive Solutions of an Abstract Fractional Semipositone Differential System Model for Bioprocesses of HIV Infection[J].Applied Mathematics&Computation,2015,258:312-324.
[18]HE Jiankun,JIA Mei,LIU Xiping,et al.Existence of Positive Solutions for a High Order Fractional Differential Equation Integral Boundary Value Problem with Changing Sign Nonlinearity[J/OL].Advances in Difference Equations,2018-02-05.https://doi.org/10.1186/s13662-018-1465-6.
[19]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法[M].2版.济南:山东科学技术出版社,2006:190-218.(GUO Dajun,SUN Jingxian,LIU Zhaoli.Functional Methods for Nonlinear Ordinary Differential Equations[M].2nd ed.Jinan:Shandong Science and Technology Press,2006:190-218.)