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

Year 2022, Volume: 6 Issue: 2, 246 - 257, 30.06.2022

Rahima Atmania

https://doi.org/10.31197/atnaa.1053966

Cited By: 1

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

References

• [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] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373-380.
• [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] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solution of nonlinear fractional pantograph equations, Acta Math. Sci. 33 (3),(2013) 712-720.
• [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] 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] 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] 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] 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] 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] 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] D.H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, 1998.
• [13] R.W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., Vol. 23, No. 5 (2012) 1250056 (9 pages).
• [14] S.M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Palm Harbor, 2001.
• [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] Y. Kuang, Delay differential equations: with applications in population dynamics, Vol. 191 of Mathematics in Science and Engineering, Academic Press, New York, 1993.
• [17] Y. Li, Y. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45(2009), 1965-1969.
• [18] I. Podlubny, Fractional Differential Equations, Academic Press, Mathematics in Science and Engineering, vol. 198. Academic Press, New York ,1999.
• [19] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 2010, 26, 103-107.
• [20] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Chapter 6, Wiley, New York, NY, USA, 1960.
• [21] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968.
• [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] 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.