Mathematica Bohemica, online first, 40 pp.

# Existence of solutions of impulsive boundary value problems for singular fractional differential systems

## Yuji Liu

#### Received March 28, 2014.   First published April 28, 2017.

Yuji Liu, Dept. of Mathematics, Guangdong University of Finance and Economics, No. 21 Chisha Road, Guangzhou, Haizhu District, 510000, P. R. China, e-mail: liuyuji888@sohu.com

Abstract: A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point theorem: Leray-Schauder Nonlinear Alternative (Lemma 2.1). An example is given to illustrate the efficiency of the main theorems, see Example 5.1.

Keywords: singular fractional differential system; impulsive boundary value problem; fixed point theorem

Classification (MSC 2010): 34A08, 26A33, 39B99, 45G10, 34B37, 34B15, 34B16

DOI: 10.21136/MB.2017.0029-14

Full text available as PDF.

References:
[1] A. Arara, M. Benchohra, N. Hamidi, J. J. Nieto: Fractional order differential equations on an unbounded domain. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 580-586. DOI 10.1016/j.na.2009.06.106 | MR 2579326 | Zbl 1179.26015
[2] M. Belmekki, J. J. Nieto, R. Rodriguez-López: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. (electronic only) 2009 (2009), Article ID 324561, 18 pages. DOI 10.1155/2009/324561 | MR 2525590 | Zbl 1181.34006
[3] M. Belmekki, J. J. Nieto, R. Rodriguez-López: Existence of solution to a periodic boundary value problem for a nonlinear impulsive fractional differential equation. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2014 (2014), Article No. 16, 27 pages. DOI 10.14232/ejqtde.2014.1.16 | MR 3199694 | Zbl 06439060
[4] R. Dehghani, K. Ghanbari: Triple positive solutions for boundary value problem of a nonlinear fractional differential equation. Bull. Iran. Math. Soc. 33 (2007), 1-14. MR 2374531 | Zbl 1148.34008
[5] G. L. Karakostas: Positive solutions for the {\$\Phi\$}-Laplacian when {\$\Phi\$} is a sup-multiplicative- like function. Electron. J. Differ. Equ. (electronic only) 2004 (2004), Article No. 68, 12 pages. MR 2057655 | Zbl 1057.34009
[6] E. R. Kaufmann, E. Mboumi: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2008 (2008), Article No. 3, 11 pages. DOI 10.14232/ejqtde.2008.1.3 | MR 2369417 | Zbl 1183.34007
[7] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). DOI 10.1016/S0304-0208(06)80001-0 | MR 2218073 | Zbl 1092.45003
[8] A. A. Kilbas, J. J. Trujillo: Differential equations of fractional order: methods, results and problems I. Appl. Anal. 78 (2001), 153-192. DOI 10.1080/00036810108840931 | MR 1887959 | Zbl 1031.34002
[9] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6. World Scientific, Singapore (1989). MR 1082551 | Zbl 0719.34002
[10] Y. Liu: Positive solutions for singular FDEs. Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 73 (2011), 89-100. MR 2799434 | Zbl 1240.34113
[11] Y. Liu: New results on the existence of solutions of boundary value problems for singular fractional differential systems with impulse effects. Tbil. Math. J. (electronic only) 8 (2015), 1-22. DOI 10.1515/tmj-2015-0003 | MR 3323916 | Zbl 06418062
[12] F. Mainardi: Fractional Calculus: Some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics, Udine, 1996 CISM Courses and Lectures 378. Springer, New York (1997), 291-348. MR 1611587
[13] J. Mawhin: Topological Degree Methods in Nonlinear Boundary Value Problems. Expository lectures held at Harvey Mudd College, Claremont, Calif., 1977 CBMS Regional Conference Series in Mathematics 40. American Mathematical Society, Providence (1979). MR 0525202 | Zbl 0414.34025
[14] K. S. Miller, B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience Publication. John Wiley & Sons, New York (1993). MR 1219954 | Zbl 0789.26002
[15] K. S. Miller, S. G. Samko: Completely monotonic functions. Integral Transforms Spec. Funct. 12 (2001), 389-402. DOI 10.1080/10652460108819360 | MR 1872377 | Zbl 1035.26012
[16] A. M. Nahušev: The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Dokl. Akad. Nauk SSSR 234 308-311 (1977), Russian English translation in Sov. Math., Dokl. 18 (1977), 666-670. MR 0454145 | Zbl 0376.34015
[17] J. J. Nieto: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23 (2010), 1248-1251. DOI 10.1016/j.aml.2010.06.007 | MR 2665605 | Zbl 1202.34019
[18] J. J. Nieto: Comparison results for periodic boundary value problem of fractional differential equations. Fract. Differ. Calc. 1 (2011), 99-104. DOI dx.doi.org/10.7153/fdc-01-05 | MR 2995577
[19] I. Podlubny: Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications Mathematics in Science and Engineering 198. Academic Press, San Diego (1999). MR 1658022 | Zbl 0924.34008
[20] I. Podlubny: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5 (2002), 367-386; correction ibid 6 (2003), 109-110. MR 1967839 | Zbl 1042.26003
[21] S. Z. Rida, H. M. El-Sherbiny, A. A. M. Arafa: On the solution of the fractional nonlinear Schrödinger equation. Phys. Lett., A 372 (2008), 553-558. DOI 10.1016/j.physleta.2007.06.071 | MR 2378723 | Zbl 1217.81068
[22] X. Su: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22 (2009), 64-69. DOI 10.1016/j.aml.2008.03.001 | MR 2483163 | Zbl 1163.34321
[23] X. Wang, C. Bai: Periodic boundary value problems for nonlinear impulsive fractional differential equation. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2011 (2011), Article No. 3, 15 pages. DOI 10.14232/ejqtde.2011.1.3 | MR 2756028 | Zbl 1207.35029
[24] Z. Wei, W. Dong: Periodic boundary value problems for Riemann-Liouville sequential fractional differential equations. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2011 (2011), Article No. 87, 13 pages. DOI 10.14232/ejqtde.2011.1.87 | MR 2854026 | Zbl 06528091
[25] Z. Wei, W. Dong, J. Che: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 3232-3238. DOI 10.1016/j.na.2010.07.003 | MR 2680017 | Zbl 1202.26017
[26] C. Yuan: Multiple positive solutions for {\$(n-1,1)\$}-type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2011 (2011), Article No. 13, 12 pages. MR 2771149 | Zbl 1261.34014
[27] C. Yuan, D. Jiang, D. O'Regan, R. P. Agarwal: Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2012 (2012), Article No. 13, 17 pages. DOI 10.14232/ejqtde.2012.1.13 | MR 2889755 | Zbl 06476163
[28] S. Zhang: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. (electronic only) 2006 (2006), Article No. 36, 12 pages. MR 2213580 | Zbl 1096.34016
[29] Y. Zhao, S. Sun, Z. Han, M. Zhang: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 217 (2011), 6950-6958. DOI 10.1016/j.amc.2011.01.103 | MR 2775685 | Zbl 1227.34011