Abstract
In this paper, by using variational methods and critical point theorems, we prove the existence and multiplicity of solutions for boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Our results extend the second order boundary value problem to the non integer case. Moreover, some conditions to determinate nonnegative solutions are presented and examples are given to illustrate our results.
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References
D. Averna, G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem. Topol. Math. Nonlinear Anal. 2 (2003), 93–103.
G. Bonanno, A. Sciammetta, An existence result of non trivial solution for two points boundary value problems. Bull. Aust. Math. Soc. 84 (2011), 288–299.
G. Bonanno, A critical point theorem via Ekeland variational principle. Nonlinear Anal. 75 (2012), 2992–3007.
G. Bonanno, Relations between the mountain pass theorem and local minima. Adv. Nonlinear Anal. 1, No 3 (2012), 205–220.
G. Bonanno, R. Rodriguez-Lopez, 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://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml.
J. Chen, X.H. Tang., Existence and multiplicity of solutions for some fractional Boundary value problem via critical point theory. Abstract Appl. Anal. 2012 (2012), Article ID 648635, 21 p.; doi:10.1155/2012/648635
G. Cottone, M. Di Paola, M. Zingales, Elastic waves propagation in 1D fractional non local contiuum. Physica E 42 (2009), 95–103.
M. Di Paola, M. Zingales, Long-range cohesive interactions of non local continuum faced by fractional calculus. Internat. J. of Solids and Structures 45 (2008), 5642–5659.
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.
A.A. Kilbas., H.M. Srivastava., J.J. Trujillo., Theory and Applicationa of Fractional Differential equations, North-Holland Mathematics Studies # 204, Elsevier Science B.V., Amsterdam (2006).
I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering # 198, Academic Press, Boston etc. (1999).
P.H. Rabinowitz., Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conf. Ser. in Math. # 65, Amer. Math. Soc., Providence, RI-USA (1986).
R. Rodriguez-Lopez, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17, No 4 (2014), 1016–1038. DOI: 10.2478/s13540- 014-0212-2; http://www.degruyter.com/view/j/fca.2014.17.issue-4/issue-files/fca.2014.17.issue-4.xml.
S.G. Samko., A.A. Kilbas., O.I. Marichev., Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Longhorne, PA–USA (1993).
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Averna, D., Tersian, S. & Tornatore, E. On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations. FCAA 19, 253–266 (2016). https://doi.org/10.1515/fca-2016-0014
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DOI: https://doi.org/10.1515/fca-2016-0014