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The Applications of Critical-Point Theory to Discontinuous Fractional-Order Differential Equations

  • Yu Tian (a1) and Juan J. Nieto (a2) (a3)


We consider a fractional equation involving the left and right Riemann–Liouville fractional integrals and with Sturm–Liouville boundary-value conditions. We establish the variational structure of the problem and, by using critical-point theory, the existence of an unbounded sequence of solutions is obtained.


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1. Area, I., Losada, J. and Nieto, J. J., On fractional derivatives and primitives of periodic functions, Abstr. Appl. Analysis 2014 (2014), 392598.
2. Belmekki, M., Nieto, J. J. and Rodriguez-Lopez, R., Existence of periodic solution for a nonlinear fractional differential equation, Bound. Value Probl. 2009 (2009), 324561.
3. Benchohra, M., Hamani, S. and Ntouyas, S. K., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlin. Analysis TMA 71 (2009), 23912396.
4. Bonanno, G. and Bisci, G. M., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), 670675.
5. Brezis, H., Functional analysis, Sobolev spaces and partial differential equations (Springer, 2011).
6. Chang, K. C., Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Analysis Applic. 80(1) (1981), 102129.
7. Chang, Y. K. and Nieto, J. J., Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Modelling 49 (2009), 605609.
8. Clarke, F. H., Optimization and nonsmooth analysis (Wiley, 1983).
9. Cresson, J., Inverse problem of fractional calculus of variations for partial differential equations, Commun. Nonlin. Sci. Numer. Simulation 15 (2010), 987996.
10. Ervin, V. J. and Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. PDEs 22 (2006), 5876.
11. Hilfer, R., Applications of fractional calculus in physics (World Scientific, 2000).
12. Jiao, F. and Zhou, Y., Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011), 11811199.
13. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Volume 204 (Elsevier, 2006).
14. Kirchner, J. W., Feng, X. and Neal, C., Fractal streamchemistry and its implications for contaminant transport in catchments, Nature 403 (2000), 524526.
15. Klimek, M., Odzijewicz, T. and Malinowska, A. B., Variational methods for the fractional Sturm–Liouville problem, J. Math. Analysis Applic. 416 (2014), 402426.
16. Lundstrom, B. N., Higgs, M. H., Spain, W. J. and Fairhall, A. L., Fractional differentiation by neocortical pyramidal neurons, Nat. Neurosci. 11 (2008), 13351342.
17. Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and fractional calculus in continuum mechanics (ed. Carpinteri, A. and Mainardi, F.), pp. 291348 (Springer, 1997).
18. Miller, K. S. and Ross, B., An introduction to the fractional calculus and differential equations (John Wiley, 1993).
19. Motreanu, D. and Panagiotopoulos, P. D., Minimax theorems and qualitative properties of the solutions of hemivariational inequalities (Kluwer Academic, Dordrecht, 1999).
20. Ouahab, A., Some results for fractional boundary value problem of differential inclusions, Nonlin. Analysis TMA 69 (2008), 38773896.
21. Podlubny, I., Fractional differential equations (Academic Press, 1999).
22. Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional integral and derivatives: theory and applications (Gordon and Breach, London/New York, 1993).
23. Teng, K. M., Jia, H. E. and Zhang, H. F., Existence and multiplicity results for fractional differential inclusions with Dirichlet boundary conditions, Appl. Math. Computat. 220 (2013), 792801.
24. Wang, J. R. and Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlin. Analysis 12 (2011), 262272.
25. Zhang, Y. H. and Bai, Z. B., Existence of solutions for nonlinear fractional three-point boundary value problem at resonance, J. Appl. Math. Computat. 36 (2011), 417440.


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