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**Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations.**
*(English)*
Zbl 1474.34063

Summary: In this paper, we study the existence, uniqueness and estimate of solutions for nonlinear hybrid implicit Caputo fractional differential equations by using the contraction mapping principle and the generalization of Gronwall’s inequality. After that, we also establish the Ulam stability for the problem at hand. Finally, an example is given to illustrate this work.

### MSC:

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

34A08 | Fractional ordinary differential equations |

34D10 | Perturbations of ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

implicit fractional differential equations; Caputo fractional derivatives; fixed point theorems; existence; uniqueness; Ulam stability
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\textit{A. Lachouri} et al., Math. Morav. 24, No. 1, 109--122 (2020; Zbl 1474.34063)

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### References:

[1] | S. Abbas,Existence of solutions to fractional order ordinary and delay differential equations and applications, Electronic Journal of Differential Equations, 2011(9) (2011), 1-11. · Zbl 1211.34096 |

[2] | K. Aissani, M. Benchohra,Impulsive fractional differential inclusions with statedependent delay, Mathematica Moravica, 23(2) (2019), 97-113. |

[3] | R. P. Agarwal, Y. Zhou, Y. He,Existence of fractional functional differential equations, Computers and Mathematics with Applications, 59 (2010), 1095-1100. · Zbl 1189.34152 |

[4] | B. Ahmad, S. K. Ntouyas,Initial-value problems for hybrid Hadamard fractional differential equations, Electron. J. Differential Equations, 2014(161) (2014), 1-8. · Zbl 1300.34012 |

[5] | B. Ahmad, S. K. Ntouyas,Existence and uniqueness of solutions for CaputoHadamard sequential fractional order neutral functional differential equations, Electronic Journal of Differential Equations, 2017(36) (2017), 1-11. · Zbl 1360.34159 |

[6] | A. Ardjouni,Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions, AIMS Mathematics, 4(4) (2019), 1101-1113. |

[7] | A. Ardjouni, A. Djoudi,Positive solutions for first-order nonlinear Caputo-Hadamard fractional relaxation differential equations, Kragujevac Journal of Mathematics, 45(6) (2021), 897-908. |

[8] | A. Ardjouni, A. Djoudi,Existence of positive periodic solutions for third-order nonlinear delay differential equations with variable coefficients, Mathematica Moravica, 23(2) (2019), 17-28. |

[9] | A. Ardjouni, A. Djoudi,Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via Dhage iteration principle, Ural Mathematical Journal, 5(1) (2019), 3-12. · Zbl 1463.45028 |

[10] | A. Ardjouni, A. Djoudi,Initial-value problems for nonlinear hybrid implicit Caputo fractional differential equations, Malaya Journal of Matematik, 7 (2019), 314-317. |

[11] | A. Ardjouni, A. Djoudi,Stability for nonlinear neutral integro-differential equations with variable delay, Mathematica Moravica, 19(2) (2015), 1-18. · Zbl 1488.45046 |

[12] | A. Ardjouni, A. Djoudi,Stability in nonlinear neutral differential equations with infinite delay, Mathematica Moravica, 18(2) (2014), 91-103. · Zbl 1349.34292 |

[13] | S. Asawasamrit, W. Nithiarayaphaks, S. K. Ntouyas, J. Tariboon,Existence and stability analysis for fractional differential equations with mixed nonlocal conditions, Mathematics 7(117) (2019), 1-11. |

[14] | M. Benchohra, J. E. Lazreg,On stability for nonlinear implicit fractional differential equations, Le Matematiche 70(2) (2015), 49-61. · Zbl 1339.34007 |

[15] | M. Bohner, H. A. El-Morshedy, S. R. Grace, I. Sager,Oscillation of second-order nonlinear difference equations with sublinear neutral term, Mathematica Moravica, 23(1) (2019), 1-10. |

[16] | H. Boulares, A. Ardjouni, Y. Laskri,Positive solutions for nonlinear fractional differential equations, Positivity, 21 (2017), 1201-1212. · Zbl 1377.26006 |

[17] | H. Boulares, A. Ardjouni, Y. Laskri,Stability in delay nonlinear fractional differential equations, Rend. Circ. Mat. Palermo, 65 (2016), 243-253. · Zbl 1373.34114 |

[18] | A. Chidouh, A. Guezane-Lakoud, R. Bebbouchi,Positive solutions of the fractional relaxation equation using lower and upper solutions, Vietnam Jorunal of Mathematics, 44(4) (2016), 739-748. · Zbl 1358.34009 |

[19] | K. Diethelm,The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2010. · Zbl 1215.34001 |

[20] | B. C. Dhage,Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differential Equations & Applications Ele-Math, 5 (2013), 155-184. · Zbl 1279.45005 |

[21] | B. C. Dhage, V. Lakshmikantham,Basic results on hybrid differential equations, Nonlinear Analysis-Hybrid Systems, 4 (2010), 414-424. · Zbl 1206.34020 |

[22] | B. C. Dhage, S. B. Dhage, S. K. Ntouyas,Approximating solutions of nonlinear hybrid differential equations, Applied Mathematics Letters, 34 (2014), 76-80. · Zbl 1314.34036 |

[23] | H. Gabsi, A. Ardjouni, A. Djoudi,Positive periodic solutions of second-order nonlinear neutral differential equations with variable coefficients, Mathematica Moravica, 22(2) (2018), 69-82. · Zbl 1488.34379 |

[24] | F. Ge, C. Kou,Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations, Applied Mathematics and Computation, 257 (2015), 308-316. · Zbl 1338.34103 |

[25] | F. Ge, C. Kou,Asymptotic stability of solutions of nonlinear fractional differential equations of order1< α <2, Journal of Shanghai Normal University, 44(3) (2015), 284-290. |

[26] | A. Guezane Lakoud, R. Khaldi, A. Kılıçman,Existence of solutions for a mixed fractional boundary value problem, Advances in Difference Equations, 2017(164) (2017), 1-9. · Zbl 1422.34041 |

[27] | A. Guezane-Lakoud, S. Ramdane,Existence of solutions for a system of mixed fractional differential equations, Journal of Taibah University for Science, 12(4) (2018), 421-426. |

[28] | M. Haoues, A. Ardjouni, A. Djoudi,Existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo fractional differential equations, Transylvanian Journal of Mathematics and Mechanics, 10(1) (2018), 09-13 · Zbl 1454.34108 |

[29] | D. Henry,Geometric Theory of Semi Linear Parabolic Equations, Springer -Verlag, Berlin, Heidelberge, New York, 1981. |

[30] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo,Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006. · Zbl 1092.45003 |

[31] | C. Kou, H. Zhou, Y. Yan,Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Analysis, 74 (2011), 59755986. · Zbl 1235.34022 |

[32] | K. D. Kucche, S. T. Sutar,On existence and stability results for nonlinear fractional delay differential equations, Boletim da Sociedade Paranaense de Matemática, 36(4) (2018), 55-75. · Zbl 1424.34270 |

[33] | V. Lakshmikantham, A. S. Vatsala,Basic theory of fractional differential equations, Nonlinear Analysis, 69 (2008), 2677-2682. · Zbl 1161.34001 |

[34] | N. Li, C. Wang,New existence results of positive solution for a class of nonlinear fractional differential equations, Acta Mathematica Scientia, 33 (2013), 847-854. · Zbl 1299.34015 |

[35] | I. Podlubny,Fractional Differential Equations, Academic Press, San Diego, 1999. · Zbl 0924.34008 |

[36] | I. A. Rus,Ulam stabilities of ordinary differential equations in a Banach space, Carpathian Journal of Mathematics, 26 (2010), 103-107. · Zbl 1224.34164 |

[37] | D. R. Smart,Fixed Point Theorems, Cambridge Tracts in Mathematics, 66, Cambridge University Press, London-New York, 1974. · Zbl 0297.47042 |

[38] | S. Zhang,The existence of a positive solution for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 252 (2000), 804-812. · Zbl 0972.34004 |

[39] | J. Wang, L. Lv, Y. Zhou,Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Journal of Qualitative Theory of Differential Equations, 2011(63) (2011), 1-10. · Zbl 1340.34034 |

[40] | J. Wang, L. Lv, Y. Zhou,New concepts and results in stability of fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 2530-2538. · Zbl 1252.35276 |

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