Existence and stability results for fractional differential equations with two Caputo fractional derivatives. (English) Zbl 1474.34031
Summary: In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our 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 |
| 34D10 | Perturbations of ordinary differential equations |
References:
| [1] | S. Belarbi and Z. Dahmani:Some applications of Banach fixed point and Leray Schauder theorems for fractional boundary value problems. Journal of Dynamical Systems and Geometric Theories. 11(1-2), 2013, 53-79. · Zbl 1317.34007 |
| [2] | M. Benchohra, S. Hamani and S.K. Ntouyas:Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2009, 2391- 2396. · Zbl 1198.26007 |
| [3] | R. Ben Taher, M. EL Fetnassi and M. Rachidi:On the stability of some rational difference equations and Ostrowski conditions. Journal of Interdisciplinary Mathematics. 16(1), 2013, pp. 19-36. |
| [4] | A. V. Bitsadze:On the theory of nonlocal boundary value problems. Dokl. Akad. Nauk SSSR. 277, 1984, 17-19. · Zbl 0586.30036 |
| [5] | L. Byszewski:Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 1991, 494-505. · Zbl 0748.34040 |
| [6] | M. Houas:Existence of solutions for fractional differential equations involving two Riemann-Liouville fractional orders. Anal. Theory Appl. 34 (3)2018, 253-274. · Zbl 1438.34036 |
| [7] | M. Houas, Z. Dahmani and M. Benbachir:New results for a boundary value problem for differential equations of arbitrary order. International Journal of Modern Mathematical Sciences. 7(2), 2013, 195-211. |
| [8] | M. Houas and Z. Dahmani:New results for multi-point boundary value problems involving a sequence of Caputo fractional derivatives. Electronic Journal of Mathematics and its Applications. 1 (2), 2015, 48-62. |
| [9] | M. Houas and Z. Dahmani:On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions. Lobachevskii Journal of Mathematics. 37(2), 2016, 120-127. · Zbl 1344.34015 |
| [10] | A.A. Kilbas and S.A. Marzan:Nonlinear differential equation with the Caputo fraction derivative in the space of continuously differentiable functions. Differential Equations. 41(1), 2005, 84-89. · Zbl 1160.34301 |
| [11] | V. Lakshmikanthan and A.S. Vatsala:Basic theory of fractional differential equations. Nonlinear Anal. 69(8), 2008, 2677-2682. · Zbl 1161.34001 |
| [12] | N. Lungu and D. Popa:Hyers-Ulam stability of a first order partial di erential equation. J. Math. Anal. Appl. 385, 2012, 86-91. · Zbl 1236.39030 |
| [13] | F. Mainardi:Fractional calculus, Some basic problem in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics. Springer, Vienna. 1997. · Zbl 0917.73004 |
| [14] | S K. Ntouyas:boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opuscula Math. 33(1), 2013, 117-138. · Zbl 1277.34008 |
| [15] | D. O’Regan:Fixed-point theory for the sum of two operators. Appl. Math. Lett. 9, 1996 1-8. · Zbl 0858.34049 |
| [16] | I. A. Rus:Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26, 2010, 103-107. · Zbl 1224.34164 |
| [17] | S.M. Ulam:A collection of mathematical problems. Interscience, New York. 1968. |
| [18] | Z. Wei, C. Pang and Y. Ding:Positive solutions of singular Caputo fractional differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 17, 2012, 3148-3160. · Zbl 1246.35205 |
| [19] | J.R Wang and Z. Lin:Ulam’s type stability of Hadamard type fractional integral equations. Filomat. 28(7), 2014, 1323-1331. · Zbl 1463.45023 |
| [20] | J.R. Wang, M. Feckan and Y. Zhou:Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 2012, 258-264. · Zbl 1254.34022 |
| [21] | J.R. Wang, L. Lv and Y. Zhou:Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electronic Journal of Qualitative Theory of Differential Equations. 63, 2011, 1-10. · Zbl 1340.34034 |
| [22] | J.R. Wang and X. Li:Ulam-Hyers stability of fractional Langevin equations. Applied Mathematics and Computation. 258, 2015, 72-83. · Zbl 1338.39047 |
| [23] | R.Yan, S. Sun, Y. Sun and Z. Han:Boundary value problems for fractional differential equations with nonlocal boundary conditions. Advances in Difference Equations. 2013:176, 2013, 1-12. · Zbl 1390.34057 |
| [24] | W. Zhon and W. Lin:Nonlocal and multiple-point boundary value problem for fractional differential equations. Computers and Mathematics with Applications. 59(3), 2010, 1345-1351. · Zbl 1189.34036 |
| [25] | J. Zhao, P. Wang and W. Ge:Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 2011, 402-413 · Zbl 1221.34053 |
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.
