Existencia de tres soluciones para el sistema hamiltoniano fraccionario

César Torres Ledesma, Oliverio Pichardo Diestra

Resumen


En este artículo se considera un sistema Hamiltoniano dado por:

(0.1)              −tDα T(0Dα t u(t)) = ∇F(t,u(t)), a.e t ∈ [0,T]

                       u(0) = u(T) = 0.

donde α ∈ (1/2,1), t ∈ [0,T], u ∈ Rn, F : [0,T]×Rn → R es una función dada y ∇F(t,u) es el gradiente de F en u. La novedad de este trabajo es que, usando una versión modificada del teorema del paso de montaña para funcional limitada desde abajo probamos la existencia de por lo menos tres soluciones para (0.1).


Palabras clave


Cálculo fraccionario; derivada fraccionaria; sistema Hamiltoniano fraccionario; problema de valor de contorno

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Referencias


Agarwal R., De Andrade B. and Cuevas C., On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. Difference Equ. ID 179750, 25 pages(2010).

Agarwal R, Benchohra M. and Hamani S., Boundary value problems for fractional differential equations, Georg. Math. J., 16, 3, 401-411(2009).

Agarwal R., Belmekki M. and Benchohra M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ. 9, 47 pages(2009).

Agarwal R, Dos Santos J. and Cuevas C., Analytic resolvent operator and existence results for fractional integro-differential equations, J. Abs. Diff. Equ. 2, 2, 26-47(2012).

Agrawal O., Tenreiro Machado J. and Sabatier J., Fractional derivatives and their application: Nonlinear dynamics, Springer-Verlag, Berlin, 2004.

Anh A. and Mcvinish R., Fractional differential equations driven by L´evy noise, J. Appl. Math. and Stoch. Anal. 16, 2, 97-119(2003).

Atanackovic T. and Stankovic B., On a class of differential equations with left and right fractional derivatives, ZAMM., 87, 537-539(2007).

Bai Z. and Lu¨ H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505(2005).

Baleanu D. and Trujillo J., On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dyn., 52, 331-335(2008).

Benchohra M., Henderson J., Ntouyas S. and Ouahab A., Existence results for fractional order functional differential equations with infinite delay, J. of Math. Anal. Appl., 338, 2, 1340-1350(2008).

Bonanno G., A characterization of the mountain pass geometry for functionals bounded from below, DIE, 25, No 11-12, 1135-1142 (2012).

Bonanno G., Rodríguez-López R. and Tersian S. Existence of solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17, No 3, 717-744 (2014).

Bonanno G. and Riccobono G. Multiplicity results for Sturm-Liouville boundary value problems, Appl. Math. Comput., 210, 294-297 (2009).

Cuevas C., N’Guérékata G. and Sepulveda A., Pseudo almost automorphic solutions to fractional differential and integro-differential equations, CAA, 16, 1, 131-152(2012).

El-Sayed A. Fractional order evolution equations, J. Frac. Cal., 7, 89-100(1995).

Ervin V. and Roop J., Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. Part. Diff. Eqs, 22, 58-76(2006).

Kilbas A., Srivastava H. and Trujillo J., Theory and applications of fractional differential equations, NorthHolland Mathematics Studies, vol 204, Amsterdam, 2006.

Klimek M., Existence and uniqueness result for a certain equation of motion in fractional mechanics, Bull. Polish Acad. Sci. Tech. Sci., 58, No 4, 573-581(2010).

Hilfer R., Applications of fractional calculus in physics, World Scientific, Singapore, 2000.

Jang W., The existence of solutions for boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74, 1987-1994(2011).

Jiao F. and Zhou Y., Existence results for fractional boundary value problem via critical point theory, Intern. Journal of Bif. and Cahos, 22, N 4, 1-17(2012).

Lakshmikantham V., Theory of fractional functional differential equations, Nonl. Anal., 69, 3337-3343(2008).

Lakshmikantham V. and Vatsala A., Basic theory of fractional differential equations, Nonl. Anal., 69, 8, 2677-2682(2008).

Mawhin J. and Willen M., Critical point theory and Hamiltonian systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989.

Metsler R. and Klafter J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37, 161-208(2004).

Miller K. and Ross B., An introduction to the fractional calculus and fractional differential equations, Wiley and Sons, New York, 1993.

N’Guérékata G., A Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Non. Anal., 70, 1873-1876(2009).

Podlubny I., Fractional differential equations, Academic Press, New York, 1999.

Rabinowitz P., Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., No 65, 1986.

Sabatier J., Agrawal O. and Tenreiro Machado J., Advances in fractional calculus. Theoretical developments and applications in physics and engineering, Springer-Verlag, Berlin, 2007.

Samko S., Kilbas A. and Marichev O. Fractional integrals and derivatives: Theory and applications, Gordon and Breach, New York, 1993.

Torres C. Mountain pass solution for fractional boundary value problem, Journal of Fractional Calculus and Applications, 1 (5), 1-10 (2014).

West B., Bologna M. and Grigolini P., Physics of fractal operators, Springer-Verlag, Berlin, 2003.

Zhang S., Existence of a solution for the fractional differential equation with nonlinear boundary conditions, Comput. Math. Appl., 61, 1202-1208(2011).

Received: Mar. 20, 2017.

Accepted: May. 15, 2017.

Corresponding author: etorres@unitru.edu.pe




DOI: http://dx.doi.org/10.17268/sel.mat.2017.01.06

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Short Title: Sel. mat.

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