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Abstract: In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator.By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained.Some examples are also presented to illustrate the application of our main results.
Key words: fractional differential equations; Riemann-Liouville fractional derivative; fixed point theorem; fractional order linear derivative operator
CLC number: O 175.8 Document code: A Article ID: 1000-5137(2014)05-0496-10
Foundation item: This work is supported by Natural Science Foundation of China (No.11171220); Support Projects of University of Shanghai for Science and Technology (No.14XPM01).
*Corresponding author: Xiping Liu(1962-),Professor,E-mail:xipingliu@163.com;1 Introduction
Fractional differential equations have been widely applied in various modern scientific fields,such as physics,chemistry,electrodynamics of complex medium,electrical circuits and biology,etc (see ［1-5］).Recently,there are many papers discussing the existence of solutions of fractional differential equations (see ［6-10］ and the references therein) or fractional differential equations with the fractional order linear derivative operator (see ［11-13］ and the references therein).
Existence of positive solutions for integral boundary value problem of fractional differential equations 2 Preliminaries
In this section,we introduce some notations,definitions and basic lemmas about the Riemann-Liouville fractional derivative,which are used to prove our main results.
Definition 1 (see ［1］) The Riemann-Liouville fractional integral of order α> 0 of a function x:［a,b］→
References:
［1］ I. Podlubny.Fractional differential equations ［M］.San Diego:Academic Press,1999.
［2］ K. B. Miller,B.Ross.An introduction to the fractional calculus and fractional differential equations ［M］.New York:Wiely,1993.
［3］ A. A. Kilbas,H. M. Stivastava,J. J. Trujillo.Theory and applications of fractional differential equations,in:North-Holland Mathematics Studies ［J］.Elsevier Science B V Amsterdam,2006,204.
［4］ V. Lakshmikantham,S. Leela,J. Vasundhara,et al.Theory of fractional dynamic system ［M］.Cambridge:Cambridge Academic Publishers,2009.
［5］ M. Javidi,N. Nyamoradi.Dynamic analysis of a fractional order phytoplankton model ［J］.J. Appl. Anal. Comput.,2013,3:343-355. ［6］ M. Jia,X. Liu.Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions ［J］.Appl. Math. Comput.,2014,232:313-323.
［7］ X. Liu,F. Li,M. Jia,E. Zhi.Existence and uniqueness of the solutions for fractional differential equations with nonlinear boundary conditions ［J］.Abstr. Appl. Anal.,2014,Article ID 758390,11,pp.1-11.
［8］ Z. Bai,H. Lu.Positive solutions for boundary value problem of nonlinear fractional differential equation ［J］.J. Math. Anal. Appl.,2005,311:495-505.
［9］ X. Liu,L. Lin,H. Fang.Existence of positive solutions for nonlocal boundary value problem of fractional differential equation ［J］.Cent. Eur. J. Phys.,2013,11:1423-1432.
［10］ M. Jia,X. Liu.The existence of positive solutions for fractional differential equations with integral and disturbance parameter in boundary conditions ［J］.Abstr. Appl. Anal.,2014,Article ID 131548,14,pp.1-14.
［11］ D. Baleanu,O. G. Mustafa.On the global existence of solutions to a class of fractional differential equations ［J］.Math. Comput. Appl.,2010,59:1835-1841.
［12］ A. Babakhani.Existence and uniqueness of solution for class of fractional order differential equations on an unbounded domain ［J］.Adv. Difference Equ.,2012,41:1-8.
［13］ J. R. Graef,L. Kong,Q. Kong,M. Wang.Uniqueness and parameter dependence of positive solutions to higher order boundary value problems with fractional q-derivatives ［J］.J. Appl. Anal. Comput.,2013,3:21-35.
［14］ M. A. Krasnosel′skii.Positive solutions of operator equations ［M］.Noordhoff.Groningen.1964.
［15］ R. W. Leggett,L. R. Williams.Multiple positive fixed points of nonlinear operators on ordered Banach spaces ［J］.J.Math Indiana Univ,1979,28:673-688.
(Zhenzhen Feng,Zhenyu Bao)
Key words: fractional differential equations; Riemann-Liouville fractional derivative; fixed point theorem; fractional order linear derivative operator
CLC number: O 175.8 Document code: A Article ID: 1000-5137(2014)05-0496-10
Foundation item: This work is supported by Natural Science Foundation of China (No.11171220); Support Projects of University of Shanghai for Science and Technology (No.14XPM01).
*Corresponding author: Xiping Liu(1962-),Professor,E-mail:xipingliu@163.com;1 Introduction
Fractional differential equations have been widely applied in various modern scientific fields,such as physics,chemistry,electrodynamics of complex medium,electrical circuits and biology,etc (see ［1-5］).Recently,there are many papers discussing the existence of solutions of fractional differential equations (see ［6-10］ and the references therein) or fractional differential equations with the fractional order linear derivative operator (see ［11-13］ and the references therein).
Existence of positive solutions for integral boundary value problem of fractional differential equations 2 Preliminaries
In this section,we introduce some notations,definitions and basic lemmas about the Riemann-Liouville fractional derivative,which are used to prove our main results.
Definition 1 (see ［1］) The Riemann-Liouville fractional integral of order α> 0 of a function x:［a,b］→
References:
［1］ I. Podlubny.Fractional differential equations ［M］.San Diego:Academic Press,1999.
［2］ K. B. Miller,B.Ross.An introduction to the fractional calculus and fractional differential equations ［M］.New York:Wiely,1993.
［3］ A. A. Kilbas,H. M. Stivastava,J. J. Trujillo.Theory and applications of fractional differential equations,in:North-Holland Mathematics Studies ［J］.Elsevier Science B V Amsterdam,2006,204.
［4］ V. Lakshmikantham,S. Leela,J. Vasundhara,et al.Theory of fractional dynamic system ［M］.Cambridge:Cambridge Academic Publishers,2009.
［5］ M. Javidi,N. Nyamoradi.Dynamic analysis of a fractional order phytoplankton model ［J］.J. Appl. Anal. Comput.,2013,3:343-355. ［6］ M. Jia,X. Liu.Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions ［J］.Appl. Math. Comput.,2014,232:313-323.
［7］ X. Liu,F. Li,M. Jia,E. Zhi.Existence and uniqueness of the solutions for fractional differential equations with nonlinear boundary conditions ［J］.Abstr. Appl. Anal.,2014,Article ID 758390,11,pp.1-11.
［8］ Z. Bai,H. Lu.Positive solutions for boundary value problem of nonlinear fractional differential equation ［J］.J. Math. Anal. Appl.,2005,311:495-505.
［9］ X. Liu,L. Lin,H. Fang.Existence of positive solutions for nonlocal boundary value problem of fractional differential equation ［J］.Cent. Eur. J. Phys.,2013,11:1423-1432.
［10］ M. Jia,X. Liu.The existence of positive solutions for fractional differential equations with integral and disturbance parameter in boundary conditions ［J］.Abstr. Appl. Anal.,2014,Article ID 131548,14,pp.1-14.
［11］ D. Baleanu,O. G. Mustafa.On the global existence of solutions to a class of fractional differential equations ［J］.Math. Comput. Appl.,2010,59:1835-1841.
［12］ A. Babakhani.Existence and uniqueness of solution for class of fractional order differential equations on an unbounded domain ［J］.Adv. Difference Equ.,2012,41:1-8.
［13］ J. R. Graef,L. Kong,Q. Kong,M. Wang.Uniqueness and parameter dependence of positive solutions to higher order boundary value problems with fractional q-derivatives ［J］.J. Appl. Anal. Comput.,2013,3:21-35.
［14］ M. A. Krasnosel′skii.Positive solutions of operator equations ［M］.Noordhoff.Groningen.1964.
［15］ R. W. Leggett,L. R. Williams.Multiple positive fixed points of nonlinear operators on ordered Banach spaces ［J］.J.Math Indiana Univ,1979,28:673-688.
(Zhenzhen Feng,Zhenyu Bao)