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2016 | 3 | 1 | 42-84
Tytuł artykułu

Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory. Examples are given to illustrate main results. This paper is motivated by [Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Equ. 89(2011), 1-19], [Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance, J, Appl, Math, Comput. 39(2012) 421-443] and [Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl. 2013, 2013: 80].
Wydawca
Rocznik
Tom
3
Numer
1
Strony
42-84
Opis fizyczny
Daty
otrzymano
2015-12-15
zaakceptowano
2016-05-17
online
2016-06-09
Twórcy
autor
  • Department of Mathematics, Guangdong University of Finance and Economics, Guangzhou 510000, P.R.China
Bibliografia
  • [1] B. Ahmad, J.J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58 (2009), 1838-1843. [Crossref]
  • [2] B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal.: H. S. 3 (2009), 251-258.
  • [3] C. Bai, Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance. Electron. J. Qual. Theory Differ. Equ. 89(2011), 1-19.
  • [4] C. Bai, Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance. J. Appl. Math. Comput. 39(2012), 421-443.
  • [5] M. Feckan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(2012), 3050-3060. [Crossref]
  • [6] M. Feckan, Y. Zhou, J. Wang, Response to "Comments on the concept of existence of solution for impulsive fractional differential equations
  • [Commun. Nonlinear Sci. Numer. Simul. 2014;19:401-3.]". 19(2014), 4213-4215.
  • [7] C.S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order. Comput. Math. Appl. 62(2011), 1251-1268. [Crossref]
  • [8] M. Gaber, M. G. Brikaa, Existence results for a coupled system of nonlinear fractional differential equation with four-point boundary conditions. ISRN Math. Anal. 2011, Article ID 468346, 14 pages.
  • [9] L. Hu, S. Zhang, Existence and uniqueness of solutions for a higher-order coupled fractional differential equations at resonance. Adv. Diff. Equ. (2015) 2015: 202. [Crossref]
  • [10] Z. Hu, W. Liu, Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance. Ukrainian Math. J. 65(11) (2014), 1619-1633. [WoS][Crossref]
  • [11] Z. Hu, W. Liu, T. Chen, Existence of solutions for a coupled system of fractional differential equations at resonance. Bound. Value Probl. 2012, 98 (2012). [Crossref]
  • [12] S. Kang, H. Chen, J. Guo, Existence of positive solutions for a system of Caputo fractional difference equations depending on parameters. Adv. Diff. Equ. 2015, 2015: 138. [Crossref]
  • [13] A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.
  • [14] M. Jleli, B. Samet, Existence of positive solutions to a coupled system of fractional differential equations. Math. Methods Appl. Sci. 38(6) (2015), 1014-1031. [Crossref]
  • [15] Y. Liu, T. He, H. Shi, Existence of positive solutions for Sturm-Liouville BVPs of dingular fractional differential equations. U.P.B. Sci. Bull. Series A. 74(2012, )93-108.
  • [16] Y. Liu, X. Yang, Resonant boundary value problems for singular multi-tern fractional differential equations. Diff. Equ. Appl. 5(3) (2013), 409-472.
  • [17] Y. Liu, Global Existence of Solutions for a System of Singular Fractional Differential Equations with Impulse Effects. J. Appl. Math. Informatics. 33(3-4)(2015), 327-342. [Crossref]
  • [18] Y. Liu, Existence of solutions of a class of impulsive periodic type BVPs for singular fractional differential systems. The Korean J. Math. 23(1)(2015), 205-230.
  • [19] Y. Liu, New results on the existence of solutions of boundary value problems for singular fractional differential systems with impulse effects. Tbilisi Math. J. 8(2) (2015), 1-22.
  • [20] Y. Liu, B. Ahmad, A study of impulsive multiterm fractional differential equations with single and multiple base points and applications. The Scientific World J. 2014, Article ID 194346, 28 pages.
  • [21] Y. Liu, P. Yang, IVPs for singular multi-term fractional differential equationswith multiple base points and applications. Appl. Math. 41(4) (2014), 361-384.
  • [22] Y. Li, H. Zhang, Solvability for system of nonlinear singular differential equations with integral boundary conditions. Bound. Value Probl. 2014, 2014: 158. [Crossref]
  • [23] F. Mainardi, Fractional Calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Carpinteri, A. and Mainardi, F. (eds), Springer, New York, 1997.
  • [24] J. Mawhin, Toplogical degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Math., American Math. Soc. Providence, RI, 1979.
  • [25] S.K. Ntouyas and M.Obaid, A coupled system of fractional differential equationswith nonlocal integral boundary conditions. Adv. Differ. Equ. 2012 (2012), 130. [Crossref]
  • [26] I. Podlubny, Geometric and physical interpretation of fractional integration and frac-tional differentiation. Dedicated to the 60th anniversary of Prof. Francesco Mainardi. Fract. Calc. Appl. Anal. 5(2002), 367–386.
  • [27] I. Podlubny, Fractional Differential Equations.Mathmatics in Science and Engineering, Vol. 198, Academic Press, San Diego, California, USA, 1999.
  • [28] I. Podlubny, N. Heymans, Physical interpretation of initial conditions for fractional differential equations with Riemann- Liouville fractional derivative. Rheologica Acta. 45(2006), 765-771. [Crossref]
  • [29] W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Letters. 22(1)(2009), 64-69. [Crossref]
  • [30] K. Shah, R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions. Diff. Equ. Appl. 7(2)(2015), 245-262.
  • [31] S. Sun, Q. Li, Y. Li, Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations. Comput. Math. Appl. 64(2012), 3310-3320. [Crossref]
  • [32] J. Sun, Y. Liu, G. Liu, Existence of solutions for fractional differential systems with anti-periodic boundary conditions. Comput. Math. Appl. 64 (2012), 1557-1566. [Crossref]
  • [33] M. ur Rehman, P.W. Eloe, Existence and uniqueness of solutions for impulsive fractional differential equations. Appl. Math. Comput. 224(2013), 422-431.
  • [34] G. Wang, B. Ahmad, L. Zhang, J. J. Nieto, Comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat. 19 (2014), 401-403. [Crossref]
  • [35] F. Wong, An application of Schauder’s fixed point theorem with respect to higher order BVPs. Proc. American Math. Soc. 126(8)(1998), 2389-2397. [Crossref]
  • [36] J.Wang, M. Feckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations. J.Math. Anal. Appl. 395(2012), [Crossref]
  • [37] J. Wang, Y. Zhou, M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl. 64(2012), 3008-3020. [Crossref]
  • [38] J. Wang, X. Li, W. Wei, On the natural solution of an impulsive fractional differential equation of order q 2 (1, 2). Commun. Nonlinear Sci. Numer. Simul. 17(2012), 4384-4394. [Crossref]
  • [39] Y. Wang, L. Liu, Y. Wu, Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. Adv. Diff. Equ. 2014, 2014: 268. [Crossref]
  • [40] J.Wang, H. Xiang, Z. Liu, Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010 (2010), Article ID 186928, 12 p.
  • [41] A. Yang, W. Ge, Positive solutions for boundary value problems of N-dimension nonlinear fractional differential systems. Bound. Value Probl. 2008, article ID 437453, doi: 10.1155/2008/437453. [Crossref]
  • [42] C. Zhai, M. Hao, Multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. Adv. Diff. Equ. 2015, 2015: 147. [Crossref]
  • [43] K. Zhao, P. Gong, Positive solutions of Riemann-Stieltjes integral boundary problems for the nonlinear coupling system involving fractional-order derivatives. Adv. Diff. Equ. 2014, 2014:254 [Crossref]
  • [44] Y. Zou, L. Liu, Y. Cui, The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance. Abst. Appl. Anal. 2014, Article ID 314083, 8 pages.
  • [45] X. Zhang, C. Zhu, Z. Wu, Solvability for a coupled system of fractional differential equations with impulses at resonance. Bound. Value Probl. 2013, 2013: 80. [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_msds-2016-0004
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