A study of Caputo-Hadamard-type fractional differential equations with nonlocal boundary conditions. (English) Zbl 1339.34014
Summary: Existence and uniqueness results of positive solutions to nonlinear boundary value problems for Caputo-Hadamard fractional differential equations by using some fixed point theorems are presented. The related Green’s function for the boundary value problem is given, and some useful properties of Green’s function are obtained. Example is presented to illustrate the main results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
References:
| [1] | Gambo, Y. Y.; Jarad, F.; Baleanu, D.; Abdeljawad, T., On Caputo modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2014, article 10 (2014) · Zbl 1343.26002 · doi:10.1186/1687-1847-2014-10 |
| [2] | Ahmad, B.; Ntouyas, S. K.; Alsaedi, A., New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions, Boundary Value Problems, 275, 14 (2013) · Zbl 1291.34037 |
| [3] | Tariboon, J.; Ntouyas, S. K.; Sudsutad, W., Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations, Boundary Value Problems, 2014, article 253 (2014) · Zbl 1307.34017 · doi:10.1186/s13661-014-0253-9 |
| [4] | Thiramanus, P.; Ntouyas, S. K.; Tariboon, J., Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.34054 · doi:10.1155/2014/902054 |
| [5] | Ahmad, B.; Ntouyas, S. K., An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.34094 · doi:10.1155/2014/705809 |
| [6] | Ahmad, B.; Ntouyas, S. K., A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fractional Calculus and Applied Analysis. An International Journal for Theory and Applications, 17, 2, 348-360 (2014) · Zbl 1312.34005 · doi:10.2478/s13540-014-0173-5 |
| [7] | Jarad, F.; Abdeljawad, T.; Baleanu, D., Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012, 1, article 142 (2012) · Zbl 1346.26002 · doi:10.1186/1687-1847-2012-142 |
| [8] | Kilbas, A. A., Hadamard-type fractional calculus, Journal of the Korean Mathematical Society, 38, 6, 1191-1204 (2001) · Zbl 1018.26003 |
| [9] | Podlubny, I., Fractional Differential Equations (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010 |
| [10] | Pooseh, S.; Almeida, R.; Torres, D. F. M., Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numerical Functional Analysis and Optimization, 33, 3, 301-319 (2012) · Zbl 1248.26013 · doi:10.1080/01630563.2011.647197 |
| [11] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, 204 (2006), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1092.45003 |
| [12] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003 |
| [13] | Ahmad, B.; Ntouyas, S. K.; Tariboon, J., Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential equations, Advances in Difference Equations, 2015, 1, article 293 (2015) · Zbl 1422.34015 · doi:10.1186/s13662-015-0625-1 |
| [14] | Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications, 141 (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0960.54027 · doi:10.1017/cbo9780511543005 |
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