Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter September 3, 2014

Multiple solutions to boundary value problem for impulsive fractional differential equations

Rosana Rodríguez-López and Stepan Tersian

Abstract

We study the multiplicity of solutions for fractional differential equations subject to boundary value conditions and impulses. After introducing the notions of classical and weak solutions, we prove the existence of at least three solutions to the impulsive problem considered.

[1] M. Belmekki, J.J. Nieto, R. Rodríguez-López, Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009 (2009), Article ID 324561, 18 pp. 10.1155/2009/324561Search in Google Scholar

[2] G. Bonanno, A. Chinnì, Multiple Solutions for elliptic problems involving the p(x)-Laplacian. Le Matematiche LXVI(I) (2011), 105–113. Search in Google Scholar

[3] G. Bonanno, B. Di Bella, J. Henderson, Existence of solutions to second-order boundary-value problems with small perturbations of impulses. Electr. J. Diff. Eq. 2013 (2013), Article # 126, 1–14. http://dx.doi.org/10.1186/1687-1847-2013-110.1186/1687-1847-2013-1Search in Google Scholar

[4] G. Bonanno, S.A. Marano, On the structure of the critical set of nondifferentiable functionals with a weak compactness condition. Appl. Anal. 89 (2010), 1–10. http://dx.doi.org/10.1080/0003681090339743810.1080/00036810903397438Search in Google Scholar

[5] G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17, No 3 (2014), 717–744; DOI: 10.2478/s13540-014-0196-y; http://link.springer.com/article/10.2478/s13540-014-0195-z. 10.2478/s13540-014-0196-ySearch in Google Scholar

[6] J. Chen, X. H. Tang, Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory. Abstr. Appl. Anal. 2012, Article ID 648635, 21 pp. 10.1155/2012/648635Search in Google Scholar

[7] 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. http://dx.doi.org/10.1016/j.camwa.2011.03.08610.1016/j.camwa.2011.03.086Search in Google Scholar

[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam (2006). http://dx.doi.org/10.1016/S0304-0208(06)80001-010.1016/S0304-0208(06)80001-0Search in Google Scholar

[9] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series, Vol. 301, Longman Sci. & Technical, Harlow and J. Wiley, N. York (1994). Search in Google Scholar

[10] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (Eds.), Springer, Wien (1997), 291–348. 10.1007/978-3-7091-2664-6_7Search in Google Scholar

[11] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Springer, New York (1989). http://dx.doi.org/10.1007/978-1-4757-2061-710.1007/978-1-4757-2061-7Search in Google Scholar

[12] I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego — CA (1999). Search in Google Scholar

[13] W. Rudin, Real and Complex Analysis. Third Ed., Mc-Graw Hill Int. Ed., N. York (1987). Search in Google Scholar

[14] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B. Springer, Berlin-Heidelberg-New York (1990). http://dx.doi.org/10.1007/978-1-4612-0981-210.1007/978-1-4612-0981-2Search in Google Scholar

Published Online: 2014-9-3
Published in Print: 2014-12-1

© 2014 Diogenes Co., Sofia

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Scroll Up Arrow