Opuscula Math. 37, no. 2 (2017), 281-301
http://dx.doi.org/10.7494/OpMath.2017.37.2.281

Opuscula Mathematica

# Existence of three solutions for impulsive nonlinear fractional boundary value problems

Shapour Heidarkhani
Massimiliano Ferrara
Giuseppe Caristi

Abstract. In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.

Keywords: fractional differential equation, impulsive condition, classical solution, variational methods, critical point theory.

Mathematics Subject Classification: 34A08, 34B37, 58E05, 58E30, 26A33.

Full text (pdf)

1. C. Bai, Existence of solutions for a nonlinear fractional boundary value problem via a local minimum theorem, Electron. J. Differ. Equ. 2012 (2012) 176, 1-9.
2. C. Bai, Infinitely many solutions for a perturbed nonlinear fractional boundary value-problem, Electron. J. Differ. Equ. 2013 (2013) 136, 1-12.
3. C. Bai, Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Eqs. 2011 (2011) 89, 1-19.
4. D. Bainov, P. Simeonov, Systems with Impulse Effect, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, 1989.
5. M. Belmekki, J.J. Nieto, R. Rodríguez-López, Existence of periodic solution for a nonlinear fractional differential equation, Bound. Value Probl. 2009 (2009), Art. ID 324561, 18 pp.
6. M. Benchohra, A. Cabada, D. Seba, An existence result for nonlinear fractional differential equations on Banach spaces, Bound. Value Probl. 2009 (2009), Art. ID 628916, 11 pp.
7. M. Benchohra, J. Henderson, S.K. Ntouyas, Theory of Impulsive Differential Equations, Contemporary Mathematics and Its Applications, vol. 2, Hindawi Publishing Corporation, New York, 2006.
8. G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary-value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal. 3 (2014), 717-744.
9. H. Chen, Z. He, New results for perturbed Hamiltonian systems with impulses, Appl. Math. Comput. 1218 (2012), 9489-9497.
10. 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), 1-21.
11. J.-N. Corvellec, V.V. Motreanu, C. Saccon, Doubly resonant semilinear elliptic problems via nonsmooth critical point theory, J. Differ. Equ. 248 (2010), 2064-2091.
12. M. De la Sen, On Riemann-Liouville and Caputo impulsive fractional calculus, [in:] Proc. of the World Congress on Engineering 2011, vol. 1, London, UK, 2011.
13. K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, [in:] F. Keil, W. Mackens, H. Voss, J. Werther (eds), Computing in Chemical Engineering, II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, 217-224.
14. M. Ferrara, S. Heidarkhani, Multiple solutions for perturbed $$p$$-Laplacian boundary value problem with impulsive effects, Electron. J. Differ. Equ. 2014 (2014) 106, 1-14.
15. M. Ferrara, G. Molica Bisci, Remarks for one-dimensional fractional equations, Opuscula Math. 34 (2014) 4, 691-698.
16. M. Galewski, G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci. 39 (2016), 1480-1492.
17. L. Gaul, P. Klien, S. Kempfle, Damping description involving fractional operators, Mech. Syst. Signal Pr. 5 (1991), 81-88.
18. W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophysical J. 68 (1995), 46-53.
19. J.R. Graef, L. Kong, Q. Kong, Multiple solutions of systems of fractional boundary value problems, Appl. Anal. 94 (2015), 1288-1304.
20. J.R. Graef, L. Kong, Q. Kong, M. Wang, Fractional boundary value problems with integral boundary conditions, Appl. Anal. 92 (2013), 2008-2020.
21. S. Heidarkhani, Multiple solutions for a nonlinear perturbed fractional boundary value problem, Dynam. Sys. Appl. 23 (2014), 317-332.
22. S. Heidarkhani, Infinitely many solutions for nonlinear perturbed fractional boundary value problems, Annals of the University of Craiova, Math. Comput. Sci. Ser. 41 (2014) 1, 88-103.
23. S. Heidarkhani, M. Ferrara, A. Salari, Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses, Acta. Appl. Math. 139 (2015), 81-94.
24. S. Heidarkhani, A. Salari, Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type, Stud. Sci. Math. Hungarica, to appear.
25. S. Heidarkhani, A. Salari, Nontrivial Solutions for impulsive fractional differential systems through variational methods, Comput. Math. Appl. (2016), http://dx.doi.org/10.1016/j.camwa.2016.04.016
26. S. Heidarkhani, Y. Zhao, G. Caristi, G.A. Afrouzi, S. Moradi, Infinitely many solutions for perturbed impulsive fractional differential systems, Appl. Anal. (2016), http://dx.doi.org/10.1080/00036811.2016.1192147
27. S. Heidarkhani, Y. Zhou, G. Caristi, G.A. Afrouzi, S. Moradi, Existence results for fractional differential systems through a local minimization principle, Comput. Math. Appl. (2016), http://dx.doi.org/10.1016/j.camwa.2016.04.012
28. R. Hilferm, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
29. 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.
30. T.D. Ke, D. Lan, Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal. 17 (2014) 1, 96-121.
31. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
32. L. Kong, Existence of solutions to boundary value problems arising from the fractional advection dispersion equation, Electron. J. Diff. Equ. 2013 (2013) 106, 1-15.
33. V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677-2682.
34. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, vol. 6, World Scientific, Teaneck, NJ, 1989.
35. F. Li, Z. Liang, Q. Zhang, Existence of solutions to a class of nonlinear second order two-point boundary value problems, J. Math. Anal. Appl. 312 (2005), 357-373.
36. X. Liu, A.R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. Probl. Eng. 2 (1996), 277-299.
37. F. Mainardi, Fractional Calculus: some basic problems in continuum and statistical mechanics, [in:] A. Carpinteri, F. Maniardi (eds), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, 291-348.
38. J. Mawhin, M. Willem, Critical Point Theorey and Hamiltonian Systems, Springer, New York, 1989.
39. F. Metzler, W. Schick, H.G. Kilan, T.F. Nonnenmacher, Relaxation in filled polymers: A Fractional Calculus approach, J. Chemical Phys. 103 (1995), 7180-7186.
40. G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett. 27 (2014), 53-58.
41. G. Molica Bisci, V. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differ. Equ. 54 (2015), 2985-3008.
42. G. Molica Bisci, V. Rădulescu, Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl. 22 (2015), 721-739.
43. G. Molica Bisci, R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl. 13 (2015), 371-394.
44. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
45. P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, [in:] CBMS, vol. 65, American Mathematical Society, 1986.
46. B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (2009), 4151-4157.
47. R. Rodríguez-López, S. Tersian, Multiple solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal. 17 (2014), 1016-1038.
48. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, PA, 1993.
49. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
50. J. Sun, H. Chen, J.J. Nieto, M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal. TMA 72 (2010), 4575-4586.
51. Y. Tian, Z. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Comput. Math. Appl. 59 (2010), 2601-2609.
52. J. Wang, M. Feckan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynam. Part. Differ. Eqs. 58 (2011), 345-361.
53. E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. II, Springer, Berlin-Heidelberg-New York, 1985.
• Shapour Heidarkhani
• Razi University, Faculty of Sciences, Department of Mathematics, 67149 Kermanshah, Iran
• Massimiliano Ferrara
• University Mediterranea of Reggio Calabria, Department of Law and Economics, Via dei Bianchi, 2 - 89131 Reggio Calabria, Italy
• Giuseppe Caristi
• University of Messina, Department of Economics, Via dei Verdi, 75, Messina, Italy
• Razi University, Faculty of Sciences, Department of Mathematics, 67149 Kermanshah, Iran
• Communicated by Marek Galewski.
• Revised: 2016-06-20.
• Accepted: 2016-06-22.
• Published online: 2017-01-03.