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Existence results for solutions of nonlinear fractional differential equations. (English) Zbl 1242.34013

Summary: This paper deals with theoretical and constructive existence results for solutions of nonlinear fractional differential equations using the method of upper and lower solutions which generate a closed set. The existence of solutions for nonlinear fractional differential equations involving Riemann-Liouville differential operator in a closed set is obtained by utilizing various types of coupled upper and lower solutions. Furthermore, these results are extended to the finite systems of nonlinear fractional differential equations leading to more general results.

MSC:

34A08 Fractional ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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[1] M. Caputo, “Linear models of dissipation whose Q is almost independent,” Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529-539, 1967.
[2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0924.44003 · doi:10.1080/10652469308819017
[3] R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180-7186, 1995.
[4] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. · Zbl 1059.47501 · doi:10.1142/9789812817747
[5] O. P. Agrawal, “Fractional variational calculus in terms of Riesz fractional derivatives,” Journal of Physics, vol. 40, no. 24, pp. 6287-6303, 2007. · Zbl 1125.26007 · doi:10.1088/1751-8113/40/24/003
[6] S.E. Hamamci, “Stabilization using fractional-order PI and PID controllers,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 329-343, 2008. · Zbl 1170.93023 · doi:10.1007/s11071-007-9214-5
[7] N. Ozdemir, O. P. Agrawal, D. Karadeniz, and B. B. Iskender, “Fractional optimal control problem of an axis-symmetric diffusion-wave propagation,” Physica Scripta, vol. 136, Article ID 014024, pp. 1-5, 2009. · Zbl 1189.26009
[8] S. E. Hamamci and M. Koksal, “Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1621-1629, 2010. · Zbl 1189.93125 · doi:10.1016/j.camwa.2009.08.049
[9] A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232-236, 2009. · Zbl 1175.26004 · doi:10.1016/j.jmaa.2009.04.012
[10] S. D. Lin and H. M. Srivastava, “Some miscellaneous properties and applications of certain operators of fractional calculus,” Taiwanese Journal of Mathematics, vol. 14, no. 6, pp. 2469-2495, 2010. · Zbl 1223.26009
[11] Z. Tomovski, R. Hilfer, and H. M. Srivastava, “Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions,” Integral Transforms and Special Functions, vol. 21, no. 11, pp. 797-814, 2010. · Zbl 1213.26011 · doi:10.1080/10652461003675737
[12] V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511-522, 2004. · Zbl 1058.34002 · doi:10.1016/j.jmaa.2004.01.013
[13] A. A. Kilbas, H. M. Srivatsava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netharland, 2006. · Zbl 1092.45003
[14] V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828-834, 2008. · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[15] V. Lakshmikantham and J. V. Devi, “Theory of fractional differential equations in a Banach space,” European Journal of Pure and Applied Mathematics, vol. 1, no. 1, pp. 38-45, 2008. · Zbl 1146.34042
[16] V. Lakshmikantham, S. Leela, and V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009. · Zbl 1188.37002
[17] V. Lakshmikantham and S. Leela, “A Krasnoselskii-Krein-type uniqueness result for fractional differential equations,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3421-3424, 2009. · Zbl 1177.34004 · doi:10.1016/j.na.2009.02.008
[18] C. Yakar, “Fractional differential equations in terms of comparison results and Lyapunov stability with initial time difference,” Abstract and Applied Analysis, Article ID 762857, 16 pages, 2010. · Zbl 1196.34010 · doi:10.1155/2010/762857
[19] D. Baleanu and J. I. Trujillo, “A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1111-1115, 2010. · Zbl 1221.34008 · doi:10.1016/j.cnsns.2009.05.023
[20] A. Yakar, “Some generalizations of comparison results for fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 8, pp. 3215-3220, 2011. · Zbl 1232.34014 · doi:10.1016/j.camwa.2011.08.035
[21] Y. Zhao, S. Sun, Z. Han, and Q. Li, “Positive solutions to boundary value problems of nonlinear fractional differential equations,” Abstract and Applied Analysis, Article ID 390543, 16 pages, 2011. · Zbl 1210.34009 · doi:10.1155/2011/390543
[22] C. Y. Lee, H. M. Srivastava, and W.-C. Yueh, “Explicit solutions of some linear ordinary and partial fractional differintegral equations,” Applied Mathematics and Computation, vol. 144, no. 1, pp. 11-25, 2003. · Zbl 1033.45006 · doi:10.1016/S0096-3003(02)00389-2
[23] H. M. Srivastava, S. D. Lin, Y. T. Chao, and P.-Y. Wang, “Explicit solutions of a certain class of differential equations by means of fractional calculus,” Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 357-365, 2007. · Zbl 1181.34004 · doi:10.1134/S1061920807030090
[24] P. Y. Wang, S.-D. Lin, and H. M. Srivastava, “Explicit solutions of Jacobi and Gauss differential equations by means of operators of fractional calculus,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 760-769, 2008. · Zbl 1151.34003 · doi:10.1016/j.amc.2007.10.037
[25] A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736-750, 2011. · doi:10.1108/03684921111142287
[26] A. Ashyralyev, F. Dal, and Z. Pınar, “A note on the fractional hyperbolic differential and difference equations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654-4664, 2011. · Zbl 1221.65212 · doi:10.1016/j.amc.2010.11.017
[27] A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations,” in Proceedings of the AIP Conference, vol. 1389, pp. 617-620, 2011.
[28] A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176-1180, 2011. · Zbl 1217.34006 · doi:10.1016/j.aml.2011.02.002
[29] R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1-10, 2007. · Zbl 1123.34302 · doi:10.1016/j.jmaa.2006.12.036
[30] J. Deng and L. Ma, “Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 6, pp. 676-680, 2010. · Zbl 1201.34008 · doi:10.1016/j.aml.2010.02.007
[31] D. B\ualeanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835-1841, 2010. · Zbl 1189.34006 · doi:10.1016/j.camwa.2009.08.028
[32] R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1-10, 2007. · Zbl 1123.34302 · doi:10.1016/j.jmaa.2006.12.036
[33] X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555-560, 2012. · Zbl 1244.34009 · doi:10.1016/j.aml.2011.09.058
[34] W.-X. Zhou and Y.-D. Chu, “Existence of solutions for fractional differential equations with multi-point boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1142-1148, 2012. · Zbl 1245.35153 · doi:10.1016/j.cnsns.2011.07.019
[35] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[36] F. A. McRae, “Monotone iterative technique and existence results for fractional differential equations,” Nonlinear Analysis, vol. 71, no. 12, pp. 6093-6096, 2009. · Zbl 1260.34014 · doi:10.1016/j.na.2009.05.074
[37] J. Vasundhara Devi and Ch. Suseela, “Quasilinearization for fractional differential equations,” Communications in Applied Analysis, vol. 12, no. 4, pp. 407-417, 2008. · Zbl 1184.34015
[38] J. Vasundhara Devi, F. A. McRae, and Z. Drici, “Generalized quasilinearization for fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1057-1062, 2010. · Zbl 1189.34010 · doi:10.1016/j.camwa.2009.05.017
[39] C. Yakar and A. Yakar, “A refinement of quasilinearization method for caputo sense fractional order differential equations,” Abstract and Applied Analysis, vol. 2010, Article ID 704367, 10 pages, 2010. · Zbl 1204.34011 · doi:10.1155/2010/704367
[40] C. Yakar and A. Yakar, “Monotone iterative technique with initial time difference for fractional differential equations,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 2, pp. 331-340, 2011. · Zbl 1236.34009
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