Ulam Stability for A Singular Fractional $ 2D$ Nonlinear System

Abstract

In this paper, we study a singular fractional $2D$ nonlinear system. We investigate the existence and uniqueness of solutions in addition to the existence of at least one solution by means of Schauder fixed point theorem, and the contraction mapping principle. Moreover, we define and study the Ulam-Hyers stability and the generalized Ulam-Hyers stability of solutions for such systems. Some applications are presented to illustrate our main results.

Keywords

• Caputo derivative,fixed point,singular fractional differential equation,existence and uniqueness,generalized Ulam-Hyers stability.

References

1. [1] S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit fractional differential and integral equations: existence and stability, Walter de Gruyter GmbH Co KG, Vol. 26, (2018).
2. [2] Z. Dahmani and A. Ta¨ıeb, New existence and uniqueness results for high dimensional fractional differential systems, Facta Nis Ser. Math. Inform. Vol. 30, No. 3, (2015), 281-293.
3. [3] Z. Dahmani and A. Ta¨ıeb, Solvability for high dimensional fractional differential systems with high arbitrary orders, Journal of Advanced Scientific Research In Dynamical And Control Systems. Vol. 7, No. 4, (2015), 51-64.
4. [4] Z. Dahmani and A. Ta¨ıeb, A coupled system of fractional differential equations involing two fractional orders, ROMAI Journal. Vol. 11, No. 2, (2015), 141-177.
5. [5] Z. Dahmani and A. Ta¨ıeb and N. Bedjaoui, Solvability and stability for nonlinear fractional integro-differential systems of hight fractional orders, Facta Nis Ser. Math. Inform. Vol. 31, No. 3 (2016), 629-644.
6. [6] Z. Dahmani and A. Ta¨ıeb, Solvability of a coupled system of fractional differential equations with periodic and antiperiodic boundary conditions, PALM Letters. No. 5, (2015), 29-36.
7. [7] R. Hilfer, Applications of fractional calculus in physics, World Scientific, River Edge, New Jersey. 2000.
8. [8] S. Harikrishnan, R.W. Ibrahim and K. Kanagarajan, On the generalized Ulam-Hyers-Rassias stability for coupled fractional differential equations. Vol. 2018, (2018), 1-13.

9. [9] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier B.V., Amsterdam, The Netherlands, 2006.
10. [10] R. Li, Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition, Advances In Difference Equations. (2014).
11. [11] K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York. 1993.
12. [12] E.C. de Oliveira, J.V.C. Sousa, Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations, Results in Mathematics. Vol. 73, No. 3, (2018).
13. [13] J.V.C. Sousa, K.D. Kuccheb and E.C. de Oliveira, Stability of y-Hilfer impulsive fractional differential equations, Applied Mathematics Letters. Vol. 88, (2019), 73-80.
14. [14] J.V.C. Sousa and E.C. de Oliveira, On the y-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation. Vol. 60, (2018), 72-91.
15. [15] J.V.C. Sousa, D.S. de Oliveira and E.C. de Oliveira, On the existence and stability for noninstantaneous impulsive fractional integro-differential equations, Mathematical Methods in the Applied Sciences. Vol. 42, No. 4, (2018), 1249-1261.
16. [16] A. Ta¨ıeb and Z. Dahmani, A coupled system of nonlinear differential equations involving m nonlinear terms, Georjian Math. Journal. Vol. 23, No. 3, (2016), 447-458.
17. [17] A. Ta¨ıeb and Z. Dahmani, The high order Lane-Emden fractional differential system: Existence, uniqueness and Ulam stabilities, Kragujevac Journal of Mathematics. Vol. 40, No. 2, (2016), 238-259.
18. [18] A. Ta¨ıeb and Z. Dahmani, A new problem of singular fractional differential equations, Journal Of Dynamical Systems And Geometric Theory. Vol. 14, No. 2, (2016), 161-183.
19. [19] A. Ta¨ıeb and Z. Dahmani, On singular fractional differential systems and Ulam-Hyers stabilities, International Journal of Mathematics and Mathematical Sciences. Vol. 14, No. 3, (2016), 262-282.
20. [20] A. Ta¨ıeb and Z. Dahmani, Fractional system of nonlinear integro-differential equations, Journal of Fractional Calculus and Applications. Vol. 10 (1) Jan. 2019, 55-67.
21. [21] A. Ta¨ıeb and Z. Dahmani, Triangular system of higher order singular fractional differential equations, Kragujevac Journal of Math, Accepted 2018.
22. [22] A. Ta¨ıeb, Several results for high dimensional singular fractional systems involving n2-Caputo derivatives, Malaya Journal of Matematik. Vol. 6, No. 3, (2018), 569-581.
23. [23] A. Ta¨ıeb, Stability of singular fractional systems of nonlinear integro-differntial equations, Lobachevskii Journal of Mathematics, Vol. 40, No. 2, (2019), 219-229.
24. [24] A. Ta¨ıeb, Generalized Ulam-Hyers stability of a fractional system of nonlinear integro-differential equations, Int. J. Open Problems Compt. Math. to appear in 2019.
25. [25] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic J Quali TH Diff Equat. No. 63, (2011), 1-10.