Tbilisi Mathematical Journal

A survey and new investigation on $(n,n-k)$-type boundary value problems for higher order impulsive fractional differential equations

Yuji Liu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A survey for studies on boundary value problems of higher order ordinary differential equations is given firstly. Secondly a simple review for studies on solvability of boundary value problems for impulsive fractional differential equations is presented. Thirdly by using a general method for converting an impulsive fractional differential equation with the Riemann-Liouville fractional derivatives to an equivalent integral equation and employing fixed point theorems in Banach space, we establish existence results of solutions for three classes of boundary value problems ($(n,n-k)$ type BVPs) of impulsive higher order fractional differential equations. Some examples are presented to illustrate the efficiency of the results obtained and some mistakes are also corrected at the end of the paper finally. A conclusion section is given at the end of the paper.

Article information

Source
Tbilisi Math. J., Volume 10, Issue 1 (2017), 207-248.

Dates
Received: 10 July 2016
Accepted: 22 August 2016
First available in Project Euclid: 26 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1527300027

Digital Object Identifier
doi:10.1515/tmj-2017-0014

Mathematical Reviews number (MathSciNet)
MR3630173

Zentralblatt MATH identifier
06707546

Subjects
Primary: 92D25: Population dynamics (general)
Secondary: 34A37: Differential equations with impulses 34K15 34K37: Functional-differential equations with fractional derivatives 34K45: Equations with impulses 34B37: Boundary value problems with impulses 34B15: Nonlinear boundary value problems 34B10: Nonlocal and multipoint boundary value problems 92D25: Population dynamics (general) 34A37: Differential equations with impulses 34K15

Keywords
Riemann-Liouville fractional derivative higher order fractional differential system impulse effect fixed point theorem $(n,n-k)$ focal boundary value problem

Citation

Liu, Yuji. A survey and new investigation on $(n,n-k)$-type boundary value problems for higher order impulsive fractional differential equations. Tbilisi Math. J. 10 (2017), no. 1, 207--248. doi:10.1515/tmj-2017-0014. https://projecteuclid.org/euclid.tbilisi/1527300027


Export citation

References

  • B. Ahmad, A. Alsaedi, A. Assolami, Relationship between lower and higher order anti-periodic boundary value problems and existence results, J. Comput. Anal. Appl. 16(2)(2014), 210-219.
  • R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109(2010), 973-1033.
  • R. P. Agarwal, M. Benchohra, B. A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differ. Equ. Math. Phys. 44(2008), 1-21.
  • R.P. Agarwal, M. Bohner, P.J. Y. Wong, Positive solutions and eigenvalues of conjugate boundary value problems, Proc. Edinburgh Math. Soc. (Series 2), 42(2)(1999), 349-374.
  • R. Agarwal, S. Hristova, D. O'Regan, Stability of solutions to impulsive Liouville-Caputo fractional differential equations, Electron. J. Diff. Equ. 58 (2016), 1-22.
  • R.P. Agarwal, D. O'Regan, Positive solutions for $ (p, n-p)$ conjugate boundary value problems, J. Differ. Equ. 150(2)(1998), 462-473.
  • B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems involving fractinal differential equations, Nonlinear Anal. Hybrid Syst. 3(2009), 251-258.
  • B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybrid Syst. 4(2010), 134-141.
  • B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3 (2009), 251-258.
  • B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order. Nonl. Anal. Hybrid Syst. 4(2010), 134-141.
  • C.J. Chyan, J. Henderson, Positive solutions of 2m th-order boundary value problems, Appl. Math. Letters, 15(6)(2002), 767-774.
  • J. M. Davis, P. W. Eloe, J. Henderson, Triple positive solutions and dependence on higher order derivatives, J. Math. Anal. Appl. 237(2)(1999), 710-720.
  • J.M. Davis, J. Henderson, P.J.Y. Wong, General Lidstone problems: multiplicity and symmetry of solutions, J. Math. Anal. Appl. 251(2)(2000), 527-548.
  • J.M. Davis, J. Henderson, Triple positive solutions for $(k, nk) $ conjugate boundary value problems, Math. Slovaca, 51(3)(2001), 313-320.
  • J.M. Davis, J. Henderson, P.K. Rajendra, Eigenvalue intervals for nonlinear right focal problems, Appl. Anal. 74(1-2)(2000), 215-231.
  • P. W. Eloe, J. Henderson, Singular nonlinear $(k, n-k)$ conjugate boundary value problems, J. Differ. Equ. 133(1) (1997), 136-151.
  • P.W. Eloe, J. Henderson, Singular nonlinear $(n�C1, 1)$ conjugate boundary value problems, Georgian Math. J. 4(5)(1997), 401-412.
  • P.W. Eloe, J. Henderson, Positive solutions for $(n- 1, 1)$ conjugate boundary value problems, Nonlinear Anal. TMA, 28(10)(1997), 1669-1680.
  • G. Feltrin, F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, arXiv preprint arXiv:1503.04954, 2015.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, 2000.
  • J. Henderson and A. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl. 59(2010), 1191-1226.
  • J. Henderson, A. Ouahab, Impusive differential inclusions with fractioanl order, Comput. Math. Appl. 59(2010), 1191-1226.
  • J. Henderson, W. Yin, Singular $(k, n- k)$ boundary value problems between conjugate and right focal, J. Comput. Appl. Math. 88(1)(1998), 57-69.
  • V.A. Il'in, E.I. Moiseev, An a priori bound for a solution of the problem conjugate to a nonlocal boundary-value problem of the first kind, Differ. Equ. 24(5)(1988), 795-804.
  • D. Jiang, H. Liu, Existence of positive solutions to $(k, n-k)$ conjugate boundary value problems, Kyushu J. Math. 53(1)(1999), 115-125.
  • N. Kosmatov, On a singular conjugate boundary value problem with infinitely many solutions, Math. Sci. Res. Hot-Line, 4(2000), 9-17.
  • L. Kong, T. Lu, Positive solutions of singular $(n,n-k)$ conjugate boundary value problem, J. Appl. Math. Bioinformatics, 5(1)(2015), 13-24.
  • A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  • Y. Liu, Global existence of solutions for a system of singular fractional differential equations with impulse effects, J. Appl. Math. and Informatics, 33(3-4)(2015), 327-342.
  • Y. Liu, On piecewise continuous solutions of higher order impulsive fractional differential equations and applications, Appl. Math. Comput. 287(2016), 3849.
  • Y. Liu, Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations, J. Nonlinear Sci. Appl. 8 (2015), 340-353.
  • V. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • X. Li, F. Chen, and X. Li, Generalized anti-periodic boundary value problems of impulsive fractional differential equations, Commun. Nonl. Sci. Numer. Simul. 18(1)(2013), 28-41.
  • Y. Liu, W. Ge, Periodic boundary value problems for $n-$th order ordinary differential equations with a $p-$Laplacian, J. Anal. Math. 16(2005), 1-22.
  • X. Lin, D. Jiang, X. Li, Existence and uniqueness of solutions for singular $(k, n-k)$ conjugate boundary value problems, Comput. Math. Appl. 52(3)(2006), 375-382.
  • Z. Liu, X. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonl. Sci. Numer. Simul. 18 (6)(2013), 1362-1373.
  • P. Li, H. Shang, Impulsive problems for fractional differential equations with nonlocal boundary value conditions, Abst. Appl. Anal. 2014 (2014), Article ID 510808, 13 pages.
  • X. Liu, Y. Zhang, H. Shi, Existence and nonexistence results for a fourth-order discrete neumann boundary value problem, Stud. Sci. Math. Hungarica, 51(2)(2014), 186-200.
  • J. Mawhin, Topological degree methods in nonlinear boundary value problems, in: CBMS Regional Conference Series in Mathematics 40, American Math. Soc., Providence, R.I., 1979.
  • R. Ma, Positive solutions for semipositone $(k, n-k)$ conjugate boundary value problems, J. Math. Anal. Appl. 252(1)(2000), 220-229.
  • F. Mainardi, Fractional Calculus: Some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics, 291-348, CISM Courses and Lectures 378, Springer, Vienna, 1997.
  • K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New, York, 1993.
  • I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5(2002), 367-386.
  • I. Podlubny, Fractional Differential Equations. Mathmatics in Science and Engineering, Vol. 198, Academic Press, San Diego, California, USA, 1999.
  • I. Rachunkova, S. Stanek, A singular boundary value problem for odd-order differential equations, J. Math. Anal. Appl. 291(2)(2004), 741-756.
  • P. Shi, L. Dong, Studies on anti-periodic boundary value problems for two classes of special second order impulsive differential equations, Math. Meth. Appl. Sci. 37(1)(2014), 123-135.
  • Y. Tian and Z. Bai, Existence results for three-point impulsive integral boundary value problems involving fractinal differential equations, Comput. Math. Appl. 59(2010), 2601-2609.
  • S. Tian, W. Gao, Positive solutions of singular $(k,n-k)$ conjugate eigenvalue problem, J. Appl. Math. Bioinformatics, 5(2)(2015), 85-97.
  • M. ur-Rehman, P.W. Eloe, Existence and uniqueness of solutions for impulsive fractional differential equations, Appl. Math. Comput. 224 (2013), 422-431.
  • P. J. Y. Wong, Triple positive solutions of conjugate boundary value problems. Comput. Math. Appl. 36(9)(1998), 19-35.
  • X. Wang, Existence of solutions for nonlinear impulsive higher order fractional differential equations, Electron. J. Qual. Theo. Differ. Equ. 80(2011), 1-12.
  • P.J.Y. Wong, R.P. Agarwal, Singular differential equations with (n, p) boundary conditions, Math. Comput. Modelling, 28(1)(1998), 37-44.
  • G. Wang, B. Ahmad, L. Zhang, J.J. Nieto, Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl. 62 (2011), 1389-1397.
  • G. Wang, B. Ahmad, L. Zhang, On impulsive boundary value problems of fractional differential equations with irregular boundary conditions, Abst. Appl. Anal. Volume 2012, Article ID 356132, 18 pages.
  • J. R. Wang, Y. Yang, W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opuscula Math. 30(2010), 361-381.
  • C. Yuan, Multiple positive solutions for $(n-1, 1)$-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual. Theo. Differ. Equ. 2010, 36: 1-12.
  • X. Yang, Existence of solutions for $2n$-order boundary value problem, Appl. Math. Comput., 237(1)(2003), 77-87.
  • A. Yang, J. Henderson, Jr. C. Nelms, Extremal points for a higher-order fractional boundary-value problem, Electron. J. Differ. Equ. 161(2015), 1-12.
  • Y. Yang, J. Zhang, Nontrivial solutions on a kind of fourth-order Neumann boundary value problems, Appl. Math. Comput. 218(13) (2012), 7100-7108.
  • K. Zhao, P. Gong, Positive solutions for impulsive fractional differential equations with generalized periodic boundary value conditions, Adv. Differ. Equ. 225(2014), 1-15.
  • W. Zhou, X. Liu, Existence of solution to a class of boundary value problem for impulsive fractional differential equations. Adv. Differ. Equ. 12(2014), 1-12.
  • W. Zhou, X. Liu, and J. Zhang, Some new existence and uniqueness results of solutions to semilinear impulsive fractional integro-differential equations, Adv. Differ. Equ. 38(2015), 1-16.