New uniqueness results for fractional differential equation with dependence on the first order derivative. (English) Zbl 1458.34034
Summary: In this paper, we study the uniqueness of solutions for a fractional differential equation with dependence on the first order derivative. By means of Banach’s contraction mapping principle and a weighted norm in product space, sufficient conditions for the uniqueness of solutions are investigated. An example is given to illustrate the main results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
References:
| [1] | Agarwal, R.P., Benchohra, M., Hamani, S.: A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions. Acta Appl. Math. 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 |
| [2] | Agarwal, R.P., O’Regan, D., Staněk, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57-68 (2010) · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034 |
| [3] | Ahmad, B., Alsaedi, A.: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Appl. 2010, 364560 (2010) · Zbl 1208.34001 · doi:10.1155/2010/364560 |
| [4] | Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916-924 (2010) · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033 |
| [5] | Bai, Z., Lü, H.: Positive solutions for boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052 |
| [6] | Bai, Z., Sun, S., Chen, Y.: The Existence and Uniqueness of a Class of Fractional Differential Equations. Abstr. Appl. Anal. 2014, 486040 (2014) · Zbl 1470.34010 · doi:10.1155/2014/486040 |
| [7] | Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48-54 (2016) · Zbl 1329.34005 · doi:10.1016/j.aml.2015.07.002 |
| [8] | Cui, Y., Liu, L., Zhang, X.: Uniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems. Abstr. Appl. Anal. 2013, 340487 (2013) · Zbl 1295.34034 |
| [9] | Cui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal., Model. Control 23(1), 31-39 (2018). https://doi.org/10.15388/NA.2018.1.3 · Zbl 1420.34009 · doi:10.15388/NA.2018.1.3 |
| [10] | Cui, Y., Ma, W., Wang, X., Su, X.: Uniqueness theorem of differential system with coupled integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2018(9), 1 (2018). https://doi.org/10.14232/ejqtde.2018.1.9 · Zbl 1413.34086 · doi:10.14232/ejqtde.2018.1.9 |
| [11] | Cui, Y., Zou, Y.: An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions. Appl. Math. Comput. 256, 438-444 (2015) · Zbl 1338.34053 |
| [12] | Dong, X., Bai, Z., Zhang, S.: Positive solutions to boundary value problems of p-Laplacian with fractional derivative. Bound. Value Probl. 2017, 5 (2017) · Zbl 1357.34012 · doi:10.1186/s13661-016-0735-z |
| [13] | Graef, J., Kong, L., Kong, Q., Wang, M.: Uniqueness of positive solutions of fractional boundary value problems with nonhomogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 15(3), 509-528 (2012) · Zbl 1279.34007 · doi:10.2478/s13540-012-0036-x |
| [14] | 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 |
| [15] | Hao, X., Sun, H., Liu, L.: Existence results for fractional integral boundary value problem involving fractional derivatives on an infinite interval. Math. Methods Appl. Sci. 41(16), 6984-6996 (2018). https://doi.org/10.1002/mma.5210 · Zbl 1405.34009 · doi:10.1002/mma.5210 |
| [16] | Hao, X., Zuo, M., Liu, L.: Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities. Appl. Math. Lett. 82, 24-31 (2018) · Zbl 1392.34019 · doi:10.1016/j.aml.2018.02.015 |
| [17] | Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0 |
| [18] | Kosmatov, N.: A singular boundary value problem for nonlinear differential equations of fractional order. J. Appl. Math. Comput. 29, 125-135 (2009) · Zbl 1191.34006 · doi:10.1007/s12190-008-0104-x |
| [19] | 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(5), 2943-2958 (2016) · Zbl 1492.47060 · doi:10.22436/jnsa.009.05.87 |
| [20] | Meng, S., Cui, Y.: The Uniqueness Theorem of the Solution for a Class of Differential Systems with Coupled Integral Boundary Conditions. Discrete Dyn. Nat. Soc. 2018, Article ID 9601868 (2018) · Zbl 1417.34056 |
| [21] | Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993) · Zbl 0789.26002 |
| [22] | Min, D., Liu, L., Wu, Y.: Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions. Bound. Value Probl. 2018, Article ID 23 (2018) · Zbl 1409.34015 · doi:10.1186/s13661-018-0941-y |
| [23] | Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999) · Zbl 0924.34008 |
| [24] | Pu, R., Zhang, X., Cui, Y., Li, P., Wang, W.: Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions. J. Funct. Spaces 2017, Article ID 5892616 (2017) · Zbl 1410.34030 |
| [25] | Qi, T., Liu, Y., Cui, Y.: Existence of solutions for a class of coupled fractional differential systems with nonlocal boundary conditions. J. Funct. Spaces 2017, 6703860 (2017) · Zbl 1375.34013 |
| [26] | Qi, T., Liu, Y., Zou, Y.: Existence result for a class of coupled fractional differential systems with integral boundary value conditions. J. Nonlinear Sci. Appl. 10, 4034-4045 (2017) · Zbl 1412.34082 · doi:10.22436/jnsa.010.07.52 |
| [27] | Sun, Q., Ji, H., Cui, Y.: Positive solutions for boundary value problems of fractional differential equation with integral boundary conditions. J. Funct. Spaces 2018, Article ID 6461930 (2018) · Zbl 1391.34018 |
| [28] | Ur Rehman, M., Ali Khan, R.: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 23, 1038-1044 (2010) · Zbl 1214.34007 · doi:10.1016/j.aml.2010.04.033 |
| [29] | Wang, Y., Liu, L.: Uniqueness and existence of positive solutions for the fractional integro-differential equation. Bound. Value Probl. 2017, 12 (2017) · Zbl 1361.45007 · doi:10.1186/s13661-016-0741-1 |
| [30] | Wu, J., Zhang, X., Liu, L., Wu, Y., Cui, Y.: The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Bound. Value Probl. 2018, 82 (2018) · Zbl 1499.34113 · doi:10.1186/s13661-018-1003-1 |
| [31] | Xu, M., Han, Z.: Positive solutions for integral boundary value problem of two-term fractional differential equations. Bound. Value Probl. 2018, 100 (2018) · Zbl 1499.34192 · doi:10.1186/s13661-018-1021-z |
| [32] | Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a singular fractional differential system involving derivatives. Commun. Nonlinear Sci. Numer. Simul. 18, 1400-1409 (2013) · Zbl 1283.34006 · doi:10.1016/j.cnsns.2012.08.033 |
| [33] | Zhang, X., Liu, L., Wu, Y., Cui, Y.: Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a non-square diffusion term. Appl. Math. Lett. 74, 85-93 (2017) · Zbl 1377.35012 · doi:10.1016/j.aml.2017.05.010 |
| [34] | Zhang, X., Liu, L., Wu, Y., Zou, Y.: Existence and uniqueness of solutions for systems of fractional differential equations with Riemann-Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, Article ID 204 (2018). https://doi.org/10.1186/s13662-018-1650-7 · Zbl 1446.34021 · doi:10.1186/s13662-018-1650-7 |
| [35] | Zhang, X., Shao, Z., Zhong, Q.: Positive solutions for semipositone (k,n−k)\((k, n-k)\) conjugate boundary value problems with singularities on space variables. Appl. Math. Lett. 72, 50-57 (2017) · Zbl 1378.34042 · doi:10.1016/j.aml.2017.04.007 |
| [36] | Zhang, X., Wu, Y., Cui, Y.: Existence and nonexistence of blow-up solutions for a Schrodinger equation involving a nonlinear operator. Appl. Math. Lett. 82, 85-91 (2018) · Zbl 1393.35228 · doi:10.1016/j.aml.2018.02.019 |
| [37] | Zhang, X., Zhong, Q.: Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions. Fract. Calc. Appl. Anal. 20(6), 1471-1484 (2017) · Zbl 1395.34010 · doi:10.1515/fca-2017-0077 |
| [38] | Zhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12-19 (2018) · Zbl 1391.34021 · doi:10.1016/j.aml.2017.12.022 |
| [39] | Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68-73 (2017) · Zbl 1376.34014 · doi:10.1016/j.aml.2017.05.011 |
| [40] | Zou, Y., He, G.: A fixed point theorem for systems of nonlinear operator equations and applications to (p1,p2)\((p_1,p_2)\)-Laplacian system. Mediterr. J. Math. 15, 74 (2018) · Zbl 06879914 · doi:10.1007/s00009-018-1119-7 |
| [41] | Zou, Y., Liu, L., Cui, Y.: The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance. Abstr. Appl. Anal. 2014, 314083 (2014) · Zbl 1476.34070 |
| [42] | Zuo, M., Hao, X., Liu, L., Cui, Y.: Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 2017, 161 (2017) · Zbl 1483.34112 · doi:10.1186/s13661-017-0892-8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
