By using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations with periodic boundary conditions. A new result on the existence of solutions of the indicated fractional boundary-value problem is obtained.
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R. Metzler and J. Klafter, “Boundary-value problems for fractional diffusion equations,” Phys. A, 278, 107–125 (2000).
H. Scher and E. Montroll, “Anomalous transit-time dispersion in amorphous solids,” Phys. Rev. B, 12, 2455–2477 (1975).
F. Mainardi, “Fractional diffusive waves in viscoelastic solids,” Nonlinear Waves in Solids, J. L. Wegner and F. R. Norwood (Eds.), Fairfield (1995), pp. 93–97.
K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity,” Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Eds F. Keil, W. Mackens, H. Voss, and J. Werther, Springer-Verlag, Heidelberg (1999), pp. 217–224.
L. Gaul, P. Klein, and S. Kempfle, “Damping description involving fractional operators,” Mech. Syst. Signal Proc., 5, 81–88 (1991).
W. G. Glockle and T. F. Nonnenmacher, “A fractional-calculus approach of self-similar protein dynamics,” Biophys. J., 68, 46–53 (1995).
F. Mainardi, “Fractional calculus: Some basic problems in continuum and statistical mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, Eds A. Carpinteri and F. Mainardi, Springer-Verlag, Wien (1997), pp. 291–348.
F. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: A fractional calculus approach,” J. Chem. Phys., 103, 7180–7186 (1995).
K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York; London (1974).
R. P. Agarwal, D. O’Regan, and S. Stanek, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” J. Math. Anal. Appl., 371, 57–68 (2010).
Z. Bai, “Positive solutions for boundary-value problem of nonlinear fractional differential equation,” J. Math. Anal. Appl., 311, 495–505 (2005).
E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary-value problem for a nonlinear fractional differential equation,” Electron. J. Qual. Theory Different. Equat., 3, 1–11 (2008).
H. Jafari and V. D. Gejji, “Positive solutions of nonlinear fractional boundary-value problems using Adomian decomposition method,” Appl. Math. Comput., 180, 700–706 (2006).
M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary-value problems for differential equations with fractional order and nonlocal conditions,” Nonlin. Anal., 71, 2391–2396 (2009).
S. Liang and J. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equation,” Nonlin. Anal., 71, 5545–5550 (2009).
S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electron. J. Different. Equat., 36, 1–12 (2006).
N. Kosmatov, “A boundary-value problem of fractional order at resonance,” Electron. J. Different. Equat., 135, 1–10 (2010).
Z.Wei, W. Dong, and J. Che, “Periodic boundary-value problems for fractional differential equations involving a Riemann–Liouville fractional derivative,” Nonlin. Anal., 73, 3232–3238 (2010).
W. Jiang, “The existence of solutions to boundary-value problems of fractional differential equations at resonance,” Nonlin. Anal., 74, 1987–1994 (2011).
X. Su, “Boundary-value problem for a coupled system of nonlinear fractional differential equations,” Appl. Math. Lett., 22, 64–69 (2009).
C. Bai and J. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,” Appl. Math. Comput., 150, 611–621 (2004).
B. Ahmad and A. Alsaedi, “Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations,” Fixed Point Theory Appl., Article ID 364560, 17 (2010).
Ahmad Bashir and J. Juan Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions,” Comput. Math. Appl., 58, 1838–1843 (2009).
M. Rehman and R. Khan, “A note on boundary-value problems for a coupled system of fractional differential equations,” Comput. Math. Appl., 61, 2630–2637 (2011).
X. Su, “Existence of solution of boundary-value problem for coupled system of fractional differential equations,” Eng. Math., 26, 134–137 (2009).
W. Yang, “Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions,” Comput. Math. Appl., 63, 288–297 (2012).
J. Mawhin, “Topological degree methods in nonlinear boundary-value problems,” NSFCBMS Reg. Conf. Ser. Math., Amer. Math. Soc., Providence, RI (1979).
V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Acad. Publ., Cambridge (2009).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 11, pp. 1463–1475, November, 2013.
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Hu, Z., Liu, W. Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance. Ukr Math J 65, 1619–1633 (2014). https://doi.org/10.1007/s11253-014-0884-0
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DOI: https://doi.org/10.1007/s11253-014-0884-0