DOI: https://doi.org/10.11568/kjm.2015.23.1.205

### Existence of solutions of a class of impulsive periodic type BVPs for singular fractional differential systems

#### Abstract

#### Keywords

#### Subject classification

92D25, 34A37, 34K15#### Sponsor(s)

This work was supported by the National Natural Science Foundation of China (No: 11401111), the Natural Science Foundation of Guangdong province (No:S2011010001900) and the Foundation for High-level talents in Guangdong Higher Education Project.#### Full Text:

PDF#### References

A. Arara, M. Benchohra, N. Hamidi and J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Analysis 72 (2010), 580–586. (Google Scholar)

R. P. Agarwal, M. Benchohra and B. A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differential Equations Math. Phys. 44 (2008), 1–21. (Google Scholar)

B. Ahmad and J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topological Methods in Nonlinear Analysis 35 (2010), 295–304. (Google Scholar)

B. Ahmad and J. J. Nieto, Existence of solutions for impulsive anti-periodic boundary value problems of fractional order, Taiwanese Journal of Mathematics 15 (3) (2011), 981–993. (Google Scholar)

B. Ahmad and S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems 3 (2009), 251–258. (Google Scholar)

M. Benchohra, J. Graef and S. Hamani, Existence results for boundary value problems with nonlinear frational differential equations, Applicable Analysis 87 (2008), 851–863. (Google Scholar)

M. Belmekki, Juan J. Nieto and Rosana Rodriguez-Lopez, Existence of peri- odic solution for a nonlinear fractional differential equation, Boundary Value Problems 2009 (2009), Article ID 324561, doi:10.1155/2009/324561. (Google Scholar)

M. Belmekki, Juan J. Nieto and Rosana Rodriguez-Lopez, Existence of solu- tion to a periodic boundary value problem for a nonlinear impulsive fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations 16 (2014), 1–27. (Google Scholar)

M. Benchohra and B. A. Slimani, Impulsive fractional differential equations, Electron. J. Differential Equations 10 (2009), 1–11. (Google Scholar)

R. Caponetto, G. Dongola and L. Fortuna, Frational order systems Modeling and control applications, World Scientific Series on nonlinear science, Ser. A, Vol. 72, World Scientific, Publishing Co. Pte. Ltd. Singapore, 2010. (Google Scholar)

K. Diethelm, Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution, J. Eur. Syst. Autom. 42 (2008), 665–676. (Google Scholar)

W. H. Deng and C. P. Li, Chaos synchronization of the fractional Lu system, Physica A 353 (2005), 61–72. (Google Scholar)

J. Dabas, A. Chauhan, and M. Kumar, Existence of the Mild Solutions for Im- pulsive Fractional Equations with Infinite Delay, International Journal of Dif- ferential Equations 2011 (2011), Article ID 793023, 20 pages. (Google Scholar)

R. Dehghant and K. Ghanbari, Triple positive solutions for boundar, Bulletin of the Iranian Mathematical Society 33 (2007), 1–14. (Google Scholar)

S. Das and P.K.Gupta, A mathematical model on fractional Lotka-Volterra equa- tions, Journal of Theoretical Biology 277 (2011), 1–6. (Google Scholar)

H. Ergoren and A. Kilicman, Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions, Abstract and Applied Analysis 2012 (2012), Article ID 387629, 15 pages. (Google Scholar)

M. Feckan, Y. Zhou and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun Nonlinear Sci Numer Simulat 17 (2012), 3050–3060. (Google Scholar)

L. J. Guo, Chaotic dynamics and synchronization of fractional-order Genesio- Tesi systems, Chinese Physics 14 (2005), 1517–1521. (Google Scholar)

G. L. Karakostas, Positive solutions for the Φ−Laplacian when Φ is a sup- multiplicative-like function, Electron. J. Diff. Eqns. 68 (2004), 1–12. (Google Scholar)

E. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 3 (2008), 1–11. (Google Scholar)

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Frational Differential Equations, Elsevier Science B. V. Amsterdam, 2006. (Google Scholar)

Y. Liu, Positive solutions for singular FDES, U.P.B. Sci. Series A, 73 (2011), 89–100. (Google Scholar)

Y. Liu, Solvability of multi-point boundary value problems for multiple term Riemann-Liouville fractional differential equations, Comput. Math. Appl. 64 (4) (2012), 413–431. (Google Scholar)

C. Li and G. Chen, Chaos and hyperchaos in the fractional-order Rossler equations, Physica A 341 (2004), 55–61. (Google Scholar)

Z. Liu and X. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 18 (6) (2013), 1362–1373. (Google Scholar)

Z. Liu, L. Lu and I. Szanto, Existence of solutions for fractional impulsive differential equations with p-Laplacian operator, Acta Mathematica Hungarica 141 (3) (2013), 203–219. (Google Scholar)

J. Mawhin, Topological degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Math., American Math. Soc. Providence, RI, 1979. (Google Scholar)

G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Anal. 72 (2010), 1604–1615. (Google Scholar)

K. S. Miller and S. G. Samko, Completely monotonic functions, Integr. Transf. Spec. Funct. 12 (2001), 389–402. (Google Scholar)

A. M. Nakhushev, The Sturm-Liouville Problem for a Second Order Ordinary Differential equations with fractional derivatives in the lower terms, Dokl. Akad. Nauk SSSR 234 (1977), 308–311. (Google Scholar)

J. J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Applied Mathematics Letters 23 (2010), 1248–1251. (Google Scholar)

J. J. Nieto, Comparison results for periodic boundary value problems of fractional differential equations, Fractional Differential Equations 1 (2011), 99–104. (Google Scholar)

N Ozalp and I Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Advances in Difference Equations, (2012) 2012, 189. (Google Scholar)

I. Petras, Chaos in the fractional-order Volta’s system: modeling and simulation, Nonlinear Dyn. 57 (2009), 157–170. (Google Scholar)

I. Petras, Fractional-Order Feedback Control of a DC Motor, J. of Electrical Engineering 60 (2009), 117–128. (Google Scholar)

I. Podlubny, Frational Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, USA, 1999. (Google Scholar)

S. Z. Rida, H.M. El-Sherbiny and A. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Physics Letters A 372 (2008), 553–558. (Google Scholar)

M. Rehman and R. Khan, A note on boundaryvalueproblems for a coupled system of fractional differential equations, Computers and Mathematics with Applica- tions 61 (2011), 2630–2637. (Google Scholar)

G. Wang, B. Ahmad and L. Zhang, Impulsive anti-periodic boundary value prob- lem for nonlinear differential equations of fractional order, Nonlinear Analysis 74 (2011), 792–804. (Google Scholar)

X. Wang, C. Bai, Periodic boundary value problems for nonlinear impulsive fractional differential equations, Electronic Journal of Qualitative Theory and Differential Equations, 3 (2011), 1-15. (Google Scholar)

X. Wang and H. Chen, Nonlocal Boundary Value Problem for Impulsive Differ- ential Equations of Fractional Order, Advances in Difference Equations, (2011) 2011, 404917. (Google Scholar)

Z. Wei, W. Dong and J. Che, Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative, Non- linear Analysis: Theory, Methods and Applications 73 (2010), 3232–3238. (Google Scholar)

Z. Wei and W. Dong, Periodic boundary value problems for Riemann-Liouville fractional differential equations, Electronic Journal of Qualitative Theory of Dif- ferential Equations, 87 (2011), 113. (Google Scholar)

J. Wang, H. Xiang and Z. Liu, Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations, International Journal of Differential Equations 2010 (2010), Article ID 186928, 12 pages, doi:10.1155/2010/186928. (Google Scholar)

P. K. Singh and T Som, Fractional Ecosystem Model and Its Solution by Ho- motopy Perturbation Method, International Journal of Ecosystem 2 (5) (2012), 140–149. (Google Scholar)

Y. Tian and Z. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Computers and Mathematics with Applications 59 (2010), 2601–2609. (Google Scholar)

M. S. Tavazoei and M. Haeri, Chaotic attractors in incommensurate fractional order systems, Physica D 327 (2008), 2628–2637. (Google Scholar)

M. S. Tavazoei and M. Haeri, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Analysis 69 (2008), 1299–1320. (Google Scholar)

A. Yang and W. Ge, Positive solutions for boundary value problems of N- dimension nonlinear fractional differential systems, Boundary Value Problems, 2008, article ID 437453, doi: 10.1155/2008/437453. (Google Scholar)

S. Zhang, The existence of a positive solution for a nonlinear fractional differ- ential equation, J. Math. Anal. Appl. 252 (2000), 804–812. (Google Scholar)

S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equation, Electron. J. Diff. Eqns. 36 (2006), 1–12. (Google Scholar)

X. Zhao and W. Ge, Some results for fractional impulsive boundary value prob- lems on infinite intervals, Applications of Mathematics 56 (4) (2011), 371–387. (Google Scholar)

X. Zhang, X. Huang and Z. Liu, The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Analysis: Hybrid Systems 4 (2010), 775–781. (Google Scholar)

Y. Zhao, S. Sun, Z. Han, M. Zhang, Positive solutions for boundary value prob- lems of nonlinear fractional differential equations, Applied Mathematics and Computation, 217 (2011), 6950–6958. (Google Scholar)

Z. Liu, L. Lu and I. Szanto, Existence of solutions for fractional impulsive dif- ferential equations with p-Laplacian operator, Acta Math. Hungar. 141 (2013), 203–219. (Google Scholar)

### Refbacks

- There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr