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

Yuji Liu

Nonautonomous Dynamical Systems (2016)

  • Volume: 3, Issue: 1, page 42-84
  • ISSN: 2353-0626

Abstract

top
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].

How to cite

top

Yuji Liu. "Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives." Nonautonomous Dynamical Systems 3.1 (2016): 42-84. <http://eudml.org/doc/281248>.

@article{YujiLiu2016,
abstract = {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].},
author = {Yuji Liu},
journal = {Nonautonomous Dynamical Systems},
keywords = {impulsive fractional differential system; boundary value problem; Schauder’s fixed point theorem; coincidence degree; Schauder's fixed point theorem},
language = {eng},
number = {1},
pages = {42-84},
title = {Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives},
url = {http://eudml.org/doc/281248},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Yuji Liu
TI - Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives
JO - Nonautonomous Dynamical Systems
PY - 2016
VL - 3
IS - 1
SP - 42
EP - 84
AB - 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].
LA - eng
KW - impulsive fractional differential system; boundary value problem; Schauder’s fixed point theorem; coincidence degree; Schauder's fixed point theorem
UR - http://eudml.org/doc/281248
ER -

References

top
  1. [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] Zbl1205.34003
  2. [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.  Zbl1193.34056
  3. [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. [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.  Zbl1302.34048
  5. [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] Zbl1252.35277
  6. [6] M. Feckan, Y. Zhou, J. Wang, Response to "Comments on the concept of existence of solution for impulsive fractional differential equations  Zbl1252.35277
  7. [Commun. Nonlinear Sci. Numer. Simul. 2014;19:401-3.]". 19(2014), 4213-4215.  
  8. [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] Zbl1253.34012
  9. [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.  Zbl1241.34009
  10. [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] 
  11. [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] Zbl1314.34019
  12. [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] 
  13. [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] 
  14. [13] A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.  Zbl0818.26003
  15. [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] Zbl1311.34015
  16. [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.  
  17. [16] Y. Liu, X. Yang, Resonant boundary value problems for singular multi-tern fractional differential equations. Diff. Equ. Appl. 5(3) (2013), 409-472.  Zbl1322.34010
  18. [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] Zbl1356.34015
  19. [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.  
  20. [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.  Zbl1317.34014
  21. [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.  
  22. [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.  Zbl1333.34011
  23. [22] Y. Li, H. Zhang, Solvability for system of nonlinear singular differential equations with integral boundary conditions. Bound. Value Probl. 2014, 2014: 158. [Crossref] 
  24. [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.  Zbl0917.73004
  25. [24] J. Mawhin, Toplogical degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Math., American Math. Soc. Providence, RI, 1979.  
  26. [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] Zbl1350.34010
  27. [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.  Zbl1042.26003
  28. [27] I. Podlubny, Fractional Differential Equations.Mathmatics in Science and Engineering, Vol. 198, Academic Press, San Diego, California, USA, 1999.  
  29. [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] 
  30. [29] W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Letters. 22(1)(2009), 64-69. [Crossref] Zbl1163.34321
  31. [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.  Zbl1336.47071
  32. [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] Zbl1268.34028
  33. [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] Zbl1268.34157
  34. [33] M. ur Rehman, P.W. Eloe, Existence and uniqueness of solutions for impulsive fractional differential equations. Appl. Math. Comput. 224(2013), 422-431.  Zbl1334.34019
  35. [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] 
  36. [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] Zbl0895.34016
  37. [36] J.Wang, M. Feckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations. J.Math. Anal. Appl. 395(2012), [Crossref] 
  38. [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] Zbl1268.34032
  39. [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] Zbl1248.35226
  40. [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] 
  41. [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.  Zbl1207.34012
  42. [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] 
  43. [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] 
  44. [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] Zbl1348.34032
  45. [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.  
  46. [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] 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.