Previous |  Up |  Next


MSC: 34A08, 34B15, 45G15
Riemann-Liouville fractional differential equations, multi-point boundary conditions, positive solutions, existence
We investigate the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with nonnegative nonlinearities which can be nonsingular or singular functions, subject to multi-point boundary conditions that contain fractional derivatives.
[1] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review, 18 (1976), pp. 620–709. DOI 10.1137/1018114 | MR 0415432
[2] Arafa, A. A. M., Rida, S. Z., Khalil, M.: Fractional modeling dynamics of HIV and CD4$^+$ T-cells during primary infection. Nonlinear Biomed. Phys., 6 (1) (2012), pp. 1–7. DOI 10.1186/1753-4631-6-1 | MR 3225564
[3] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J.: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston,2012. MR 2894576
[4] Cole, K.: Electric conductance of biological systems. in: Proc. Cold Spring Harbor Symp. Quant. Biol., Col Springer Harbor Laboratory Press, New York, 1993, pp. 107–116.
[5] Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, New York, 2008. MR 2414740
[6] Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD4$^+$ T-cells. Math. Comp. Model., 50 (2009), pp. 386–392. DOI 10.1016/j.mcm.2009.04.019 | MR 2542785
[7] Djordjevic, V., Jaric, J., Fabry, B., Fredberg, J., Stamenovic, D.: Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng., 31 (2003), pp. 692–699. DOI 10.1114/1.1574026
[8] Ge, Z. M., Ou, C. Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals, 35 (2008), pp. 705–717. DOI 10.1016/j.chaos.2006.05.101
[9] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York, 1988. MR 0959889
[10] Henderson, J., Luca, R.: Existence and multiplicity of positive solutions for a system of fractional boundary value problems. Bound. Value Probl., 2014:60 (2014), pp. 1–17. DOI 10.1186/1687-2770-2014-60 | MR 3352633
[11] Henderson, J., Luca, R.: Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Appl. Math. Comput., 249 (2014), pp. 182–197. MR 3279412
[12] Henderson, J., Luca, R.: Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Bound. Value Probl., 2015:138 (2015), pp. 1–12. DOI 10.1186/s13661-015-0403-8 | MR 3382857
[13] Henderson, J., Luca, R.: Positive solutions for a system of semipositone coupled fractional boundary value problems. Bound. Value Probl., 2016(61) (2016), pp. 1–23. DOI 10.1186/s13661-016-0569-8 | MR 3471318
[14] Henderson, J., Luca, R.: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations. Positive Solutions, Elsevier, Amsterdam, 2016. MR 3430818
[15] Henderson, J., Luca, R.: Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal. Model. Control, 22(1) (2017), pp. 99–114. DOI 10.15388/NA.2017.1.7 | MR 3594626
[16] Henderson, J., Luca, R., Tudorache, A.: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal., 18(2) (2015), pp. 361–386. DOI 10.1515/fca-2015-0024 | MR 3323907
[17] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
[18] Klafter, J., Lim, S. C., (Eds.), R. Metzler: Fractional Dynamics in Physics, Singapore. World Scientific, 2011. MR 2920446
[19] Luca, R., Tudorache, A.: Positive solutions to a system of semipositone fractional boundary value problems. Adv. Difference Equ., 2014(179) (2014), pp. 1–11. DOI 10.1186/1687-1847-2014-179 | MR 3357337
[20] Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339 (2000), pp. 1–77. DOI 10.1016/S0370-1573(00)00070-3 | MR 1809268
[21] Ostoja-Starzewski, M.: Towards thermoelasticity of fractal media. J. Therm. Stress., 30 (2007), pp. 889–896. DOI 10.1080/01495730701495618
[22] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, 1999. MR 1658022
[23] Povstenko, Y.Z.: Fractional Thermoelasticity. New York, Springer, 2015. MR 3287225
[24] Sabatier, J., Agrawal, O. P., (Eds.), J. A. T. Machado: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, 2007. MR 2432163
[25] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993. MR 1347689
[26] Sokolov, I. M., Klafter, J., Blumen, A.: A fractional kinetics. Phys. Today, 55 (2002), pp. 48–54. DOI 10.1063/1.1535007
[27] Zhou, Y., Xu, Y.: Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations. J. Math. Anal. Appl., 320 (2006), pp. 578–590. DOI 10.1016/j.jmaa.2005.07.014 | MR 2225977
Partner of
EuDML logo