Existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative fractional differential equation

Abderrahim Guerfi; Abdelouaheb Ardjouni

doi:10.4067/S0719-06462022000100083

DOI:

https://doi.org/10.4067/S0719-06462022000100083

Abstract

In this work, we study the existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative Caputo fractional differential equation by first inverting it as an integral equation. Then we construct an appropriate mapping and employ the Schauder fixed point theorem to prove our new results. At the end we give an example to illustrate our obtained results.

Keywords

Iterative fractional differential equations , fixed point theorem , existence , uniqueness , continuous dependence , Ulam stability

Abderrahim Guerfi Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, University of Annaba, P.O. Box 12, Annaba 23000, Algeria.
Abdelouaheb Ardjouni Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, University of Annaba, P.O. Box 12, Annaba 23000, Algeria – Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria.

Pages: 83–94
Date Published: 2022-04-04
Vol. 24 No. 1 (2022)

S. Abbas, “Existence of solutions to fractional order ordinary and delay differential equations and applications”, Electron. J. Differential Equations, no. 9, 11 pages, 2011.

A. A. Amer and M. Darus, “An application of univalent solutions to fractional Volterra equation in complex plane”, Transylv. J. Math. Mech., vol. 4, no. 1, pp. 9–14, 2012.

M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay”, J. Math. Anal. Appl., vol. 338, no. 2, pp. 1340–1350, 2008.

H. Boulares, A. Ardjouni and Y. Laskri, “Existence and uniqueness of solutions to fractional order nonlinear neutral differential equations”, Appl. Math. E-Notes, vol. 18, pp. 25–33, 2018.

S. Cheraiet, A. Bouakkaz and R. Khemis, “Bounded positive solutions of an iterative three-point boundary-value problem with integral boundary conditions”, J. Appl. Math. Comput., vol. 65, no. 1-2, pp. 597–610, 2021.

K. Diethelm, “Fractional differential equations, theory and numerical treatment”, TU Braunschweig, Braunschweig, 2003.

A. M. A. El-Sayed, “Fractional order evolution equations”, J. Fract. Calc., vol. 7, pp. 89–100, 1995.

C. Giannantoni, “The problem of the initial conditions and their physical meaning in linear differential equations of fractional order”, Appl. Math. Comput., vol. 141, no. 1, pp. 87–102, 2003.

A. A. Hamoud, “Uniqueness and stability results for Caputo fractional Volterra-Fredholm integro-differential equations”, Zh. Sib. Fed. Univ. Mat. Fiz., vol. 14, no. 3, pp. 313–325, 2021.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Amsterdam: Elsevier Science B. V., 2006.

J. T. Machado, V. Kiryakova and F. Mainardi, “Recent history of fractional calculus”, Commun. Nonlinear Sci. Numer. Simul., vol. 16, no. 3, pp. 1140–1153, 2011.

K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: John Wiley & Sons, Inc., 1993.

K. B. Oldham and J. Spanier, The fractional calculus, Mathematics in Science and Engineering, vol. 111, New York-London: Academic Press, 1974.

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, San Diego, CA: Academic Press, Inc., 1999.

S. Bhalekar, V. Daftardar-Gejji, D. Baleanu and R. Magin, “Generalized fractional order Bloch equation with extended delay”, Internat. J. Bifur. Chaos, vol. 22, no. 4, 1250071, 15 pages, 2012.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Yverdon: Gordon and Breach Science Publishers, 1993.

D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, London-New York: Cambridge University Press, 1974.

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., no. 63, 10 pages, 2011.

H. Y. Zhao and J. Liu, “Periodic solutions of an iterative functional differential equation with variable coefficients”, Math. Methods Appl. Sci., vol. 40, no. 1, pp. 286–292, 2017.