Existence and stability results for a nonlinear implicit fractional differential equation with a discrete delay

Abstract

In this paper, we are concerned with a class of nonlinear implicit fractional di?erential equation with a
discrete delay. By means of the contraction mapping principle, we prove the existence of a unique solution.
Then, we investigate the continuous dependence of the solution upon the initial delay data and the Ulam
stability.

Keywords

• Fractional differential equation
• discrete delay
• existence and continuous dependence of solution on initial data
• Ulam stability
• Banach fixed point theorem

Kaynakça

1. [1] R. Agarwal, D. O'Regan and S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60, 6 (2015), 653-676.
2. [2] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373-380.
3. [3] R. Atmania and S. Bouzitouna, Existence and Ulam Stability results for two-orders fractional differential Equation, Acta Math. Univ. Comenianae, Vol. LXXXVIII, 1 (2019), 1-12.
4. [4] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solution of nonlinear fractional pantograph equations, Acta Math. Sci. 33 (3),(2013) 712-720.
5. [5] D. Baleanu, G.C. Wu and S.D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solitons Fractals , 102 (2017), 99-105.
6. [6] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with in?nite delay, J. Math. Anal. Appl., 338(2008), 1340-1350.
7. [7] M. Benchohra, S. Bouriah and J.J. Nieto, Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative, Demonstr. Math. (2019) 52:437-450.
8. [8] Capelas de Oliveira E., Vanterler da C. Sousa J., Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Let., 81, 50-56 (2018).

9. [9] Z. Gao, L. Yang and Z. Luo, Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions, (2013), 2013:43.
10. [10] S. Hristova and C. Tunc, Stability of nonlinear Volterra integro-differential equations with Caputo fractional derivative and bounded delays, EJDE, Vol. 2019, No. 30 (2019), 1-11.
11. [11] D.H. Hyers, On the stability of the linear functional equation,Proceedings of the National Academy of Sciences of the United States of America, vol. 27,(1941) pp. 222-224.
12. [12] D.H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, 1998.
13. [13] R.W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., Vol. 23, No. 5 (2012) 1250056 (9 pages).
14. [14] S.M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Palm Harbor, 2001.
15. [15] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
16. [16] Y. Kuang, Delay differential equations: with applications in population dynamics, Vol. 191 of Mathematics in Science and Engineering, Academic Press, New York, 1993.
17. [17] Y. Li, Y. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45(2009), 1965-1969.
18. [18] I. Podlubny, Fractional Differential Equations, Academic Press, Mathematics in Science and Engineering, vol. 198. Academic Press, New York ,1999.
19. [19] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 2010, 26, 103-107.
20. [20] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Chapter 6, Wiley, New York, NY, USA, 1960.
21. [21] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968.
22. [22] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63, (2011) 1-10.
23. [23] H.Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J.Math. Anal. Appl., (2007), 328, 1075-1081.