**Volume 10, Issue 5, pp 2283--2295**

**Publication Date**: 2017-05-22

http://dx.doi.org/10.22436/jnsa.010.05.01

Peiluan Li - Control science and engineering post-doctoral mobile stations, Henan University of Science and Technology, Luoyang, 471023, China.

Jianwei Ma - College of Information Engineering, Henan University of Science and Technology, Luoyang, 471003, China.

Hui Wang - College of Information Engineering, Henan University of Science and Technology, Luoyang, 471003, China.

Zheqing Li - Network and Information Center, Henan University of Science and Technology, Luoyang, 471003, China.

By the variational methods, the existence criteria of infinitely many nontrivial solutions for fractional differential equations with impulses and perturbation are established. An example is given to illustrate main results. Recent results in the literature are generalized and improved.

Fractional differential equations with impulses and perturbation, infinitely many nontrivial solutions, variational methods.

[1] C.-Z. Bai, Existence of three solutions for a nonlinear fractional boundary value problem via a critical points theorem, Abstr. Appl. Anal., 2012 (2012), 13 pages.

[2] D. Baleanu, H. Khan, H. Jafari, R. A. Khan, M. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Difference Equ., 2015 (2015), 14 pages.

[3] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, (2006).

[4] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75 (2012), 2992–3007.

[5] G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1–10.

[6] G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 717–744.

[7] J. Chen, X. H. Tang, Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory, Abstr. Appl. Anal., 2012 (2012), 21 pages.

[8] J.-N. Corvellec, V. V. Motreanu, C. Saccon, Doubly resonant semilinear elliptic problems via nonsmooth critical point theory, J. Differential Equations, 248 (2010), 2064–2091.

[9] G. D'Aguì, B. Di Bella, S. Tersian, Multiplicity results for superlinear boundary value problems with impulsive effects, Math. Methods Appl. Sci., 39 (2016), 1060–1068.

[10] A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442–1450.

[11] V. J. Erwin, J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22 (2006), 558–576.

[12] M. A. Firoozjaee, S. A. Yousefi, H. Jafari, D. Baleanu, On a numerical approach to solve multi-order fractional differential equations with initialboundary conditions, J. Comput. Nonlinear Dynam., 10 (2015), 6 pages.

[13] B. Ge, Multiple solutions for a class of fractional boundary value problems, Abstr Appl. Anal., 2012 (2012), 16 pages.

[14] Z.-G. Hu, W.-B. Liu, J.-Y. Liu, Ground state solutions for a class of fractional differential equations with Dirichlet boundary value condition, Abstr. Appl. Anal., 2014 (2014), 7 pages.

[15] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181–1199.

[16] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012) , 17 pages.

[17] H. Khalil, R. A. Khan, D. Baleanu, S. H. Saker, Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions, Adv. Difference Equ., 1 (2016), 28 pages.

[18] A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, (2006).

[19] V. Lakshmikantham, D. D. Baınov, P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ, (1989).

[20] V. Lakshmikantham, S. Leela, D. J. Vasundhara, Theory of fractional dynamic systems, Cambridge Scientific Publishers, Cambridge, UK, (2009).

[21] Y.-N. Li, H.-R. Sun, Q.-G. Zhang, Existence of solutions to fractional boundary-value problems with a parameter, Electron. J. Differential Equations, 2013 (2013), 12 pages.

[22] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York, (1989).

[23] N. Nyamoradi, R. Rodríguez-López, On boundary value problems for impulsive fractional differential equatio, Appl. Math. Comput., 271 (2015), 874–892.

[24] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, (1986).

[25] R. Rodríguez-López, S. Tersian, Multiple solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 1016–1038.

[26] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, With a preface by Yu. A. Mitropolskiıand a supplement by S. I. Trofimchuk, Translated from the Russian by Y. Chapovsky, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, World Scientific Publishing Co., Inc., River Edge, NJ, (1995).

[27] H.-R. Sun, Q.-G. Zhang, Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique, Comput. Math. Appl., 64 (2012), 3436–3443.

[28] Y. Tian, W.-G. Ge, Multiple solutions of impulsive Sturm-Liouville boundary value problem via lower and upper solutions and variational methods, J. Math. Anal. Appl., 387 (2012), 475–489.

[29] C. Torres, Mountain pass solution for a fractional boundary value problem, J. Fract. Calc. Appl., 5 (2014), 1–10.

[30] X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, ArXiv, 2016 (2016), 13 pages.

[31] X.-J. Yang, D. Baleanu, Y. Khan, S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian J. Phys., 59 (2014), 36–48.

[32] X.-J. Yang,, F. Gao, J. A. Tenreiro Machado, D. Baleanu, A new fractional derivative involving the normalized sinc function without singular kernel, ArXiv, 2017 (2017), 11 pages.

[33] X.-J. Yang, H. M. Srivastava, J. A. Tenreiro Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753–756.

[34] X.-J. Yang, J. A. Tenreiro Machado, A new fractional operator of variable order: application in the description of anomalous diffusion, ArXiv, 2016 (2016), 13 pages.

[35] X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 5 pages.

[36] Y.-L. Zhao, H.-B. Chen, B. Qin, Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods, Appl. Math. Comput., 257 (2015), 417–427.

[37] Y.-L. Zhao, Y.-L. Zhao, Nontrivial solutions for a class of perturbed fractional differential systems with impulsive effects, Bound. Value Probl., 2016 (2016), 16 pages.

[38] Y. Zhou, Basic theory of fractional differential equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2014).

[39] W.-M. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343–358.