Uniqueness and existence of positive solutions for the fractional integro-differential equation. (English) Zbl 1361.45007
Summary: In this paper, we study the uniqueness and existence of positive solutions for the fractional integro-differential equation with the integral boundary value problem. By means of the Banach contraction principle and the Krasnoselskii fixed point theorem, the sufficient conditions on the uniqueness and existence of positive solutions are investigated. An example is given to illustrate the main results.
MSC:
| 45J05 | Integro-ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 45M20 | Positive solutions of integral equations |
| 45G10 | Other nonlinear integral equations |
Keywords:
nonlinear fractional integro-differential equation; integral conditions; uniqueness; existence; positive solutions; boundary value problemReferences:
| [1] | Atanackovic, TM, Stankovic, B: On a class of differential equations with left and right fractional derivatives. Z. Angew. Math. Mech. 87, 537-546 (2009) · Zbl 1131.34003 · doi:10.1002/zamm.200710335 |
| [2] | Debnath, L: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413-3442 (2003) · Zbl 1036.26004 · doi:10.1155/S0161171203301486 |
| [3] | Xu, H: Analytical approximations for a population growth model with fractional order. Commun. Nonlinear Sci. Numer. Simul. 14, 1978-1983 (2009) · Zbl 1221.65210 · doi:10.1016/j.cnsns.2008.07.006 |
| [4] | Kempfle, S, Schäfer, I, Beyer, H: Fractional calculus via functional calculus: theory and applications. Nonlinear Dyn. 29, 99-127 (2002) · Zbl 1026.47010 · doi:10.1023/A:1016595107471 |
| [5] | Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003 |
| [6] | Scudo, FM: Vito Volterra and theoretical ecology. Theor. Popul. Biol. 2, 1-23 (1971) · Zbl 0241.92001 · doi:10.1016/0040-5809(71)90002-5 |
| [7] | TeBeest, KG: Numerical and analytical solutions of Volterra’s population model. SIAM Rev. 39, 484-493 (1997) · Zbl 0892.92020 · doi:10.1137/S0036144595294850 |
| [8] | He, J., Nonlinear oscillation with fractional derivative and its applications, Dalian, China |
| [9] | He, J: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86-90 (1999) |
| [10] | Bai, Z, Qiu, T: Existence of positive solution for singular fractional differential equation. Appl. Math. Comput. 215, 2761-2767 (2009) · Zbl 1185.34004 |
| [11] | Li, H, Liu, L, Wu, Y: Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2015, 232 (2015) · Zbl 1332.34010 · doi:10.1186/s13661-015-0493-3 |
| [12] | Graef, JR, Kong, L: Existence of positive solutions to a higher order singular boundary value problem with fractional Q-derivatives. Fract. Calc. Appl. Anal. 16(3), 695-708 (2013) · Zbl 1312.39014 · doi:10.2478/s13540-013-0044-5 |
| [13] | Wang, Y, Liu, L, Wu, Y: Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. Adv. Differ. Equ. 2014, 268 (2014) · Zbl 1417.34062 · doi:10.1186/1687-1847-2014-268 |
| [14] | Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403-411 (2012) · Zbl 1232.34010 · doi:10.1016/j.jmaa.2011.11.065 |
| [15] | Guo, L, Liu, L, Wu, Y: Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions. Bound. Value Probl. 2016, 114 (2016) · Zbl 1343.34013 · doi:10.1186/s13661-016-0621-8 |
| [16] | Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26-33 (2014) · Zbl 1320.35007 · doi:10.1016/j.aml.2014.05.002 |
| [17] | Xu, Y, He, Z: Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition. J. Appl. Math. Comput. 44, 417-435 (2014) · Zbl 1296.34042 · doi:10.1007/s12190-013-0700-2 |
| [18] | Wang, Y, Liu, L, Zhang, X, Wu, Y: Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection. Appl. Math. Comput. 258, 312-324 (2015) · Zbl 1338.92143 |
| [19] | Wang, Y, Zhang, J: Positive solutions for higher-order singular fractional differential system with coupled integral boundary conditions. Adv. Differ. Equ. 2016, 117 (2016) · Zbl 1419.34042 · doi:10.1186/s13662-016-0844-0 |
| [20] | Liu, L, Zhang, X, Jiang, J, Wu, Y: The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems. J. Nonlinear Sci. Appl. 9, 2943-2958 (2016) · Zbl 1492.47060 |
| [21] | Sun, F, Liu, L, Zhang, X, Wu, Y: Spectral analysis for a singular differential system with integral boundary conditions. Mediterr. J. Math. 13, 4763-4782 (2016) · Zbl 1349.34080 · doi:10.1007/s00009-016-0774-9 |
| [22] | Hao, X, Liu, L, Wu, Y: Positive solutions for nonlinear fractional semipositone differential equation with nonlocal boundary conditions. J. Nonlinear Sci. Appl. 9, 3992-4002 (2016) · Zbl 1355.34015 |
| [23] | Guo, L, Liu, L, Wu, Y: Uniqueness of iterative positive solutions for the singular fractional differential equations with integral boundary conditions. Bound. Value Probl. 2016, 147 (2016) · Zbl 1348.34013 · doi:10.1186/s13661-016-0652-1 |
| [24] | Guo, L, Liu, L, Wu, Y: Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions. Nonlinear Anal., Model. Control 21(5), 635-650 (2016) · Zbl 1420.34015 · doi:10.15388/NA.2016.5.5 |
| [25] | Jiang, J, Liu, L: Existence of solutions for a sequential fractional differential system with coupled boundary conditions. Bound. Value Probl. 2016, 159 (2016) · Zbl 1347.34013 · doi:10.1186/s13661-016-0666-8 |
| [26] | Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002 |
| [27] | Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999) · Zbl 0924.34008 |
| [28] | Smart, DR: Fixed Point Theorems. Cambridge University Press, Cambridge (1980) · Zbl 0427.47036 |
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