Document Type: Research Paper


Department of Mathematics‎, ‎University of Peshawar‎, ‎Peshawar 25000‎, ‎Pakistan


‎In this paper‎, ‎we establish the Hyers--Ulam--Rassias stability and the Hyers--Ulam stability of impulsive Volterra integral equation by using a fixed point method‎.


Main Subjects

[1] M. Akkouchi, Hyers–Ulam–Rassias stability of nonlinear Volterra integral equations via a fixed point approach, Acta. Univ. Apulen. Math. Inform. 26 (2011), 257-266.

[2] K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simulat. (16) (2011), 1970-1977.

[3] M. Benchohra, A. Ouahab, Impulsive neutral functional differential inclusions with variable times, Electron. J. Diff. Equ. 2003 (2003), 1-12.

[4] M. Benchohra, D. Seba, Impulsive fractional differential equations in Banach spaces, Electron. J. Qual. Theo. Diff. Equa. Spec. Edit. (8) (2009), 1-14.

[5] M. Benchohra, B. A. Slimani, Existence and uniqness of solutions to impulsive fractional differential equations, Electron. J. Diff. Equ. (10) (2009), 1-11.

[6] L. Cdariu, V. Radu, On the stability of the Cauchy functional equation a fixed point approach, Grazer. Math. Ber. 346 (2004), 43-52.

[7] L. P. Castro, A. Ramos, Hyers–Ulam–Rassias stability for a class of nonlinear Volterra integral equations, Banach J. Math. Anal. 3 (2009), 36-43.

[8] J. B. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309.

[9] M. Gachpazan, O. Baghani, Hyers–Ulam stability of nonlinear integral equation, Fixed Point Theory Appl. (2010), 2010:927640.

[10] M. Gachpazan, O. Baghani, Hyers–Ulam stability of Volterra integral equation, J. Nonlinear Anal. Appl. 1 (2010), 19-25.

[11] D. Guo, Nonlinear impulsive Volterra integral equations in Banach spaces and applications, J. Appl. Math. Stoch. Anal. 6 (1) (1993), 35-48.

[12] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224.

[13] S. M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. (2007), 2007: 057064.

[14] S. M. Jung, A fixed point approach to the stability of differential equations y′= F(x,y), Bull. Malays. Math. Sci. Soc. 33 (2010), 47-56.

[15] S. M. Jung, S. Sevgin, H. Sevli, On the perturbation of Volterra integro-differential equations, Appl. Math. Lett. 26 (2013), 665-669.

[16] T. Li, A. Zada, Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ. (2016), 2016:153.

[17] T. Li, A. Zada, S. Faisal, Hyers–Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl. 9 (2016), 2070-2075.

[18] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

[19] R. Shah, A. Zada, A fixed point approach to the stability of a nonlinear Volterra integro-differential equation with delay, Hacet. J. Math. Stat. 47 (3) (2018), 615-623.

[20] S. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh, T. Li, Stability of higher–order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl. 9 (2016), 4713-4721.

[21] S. M. Ulam, Problems in modern mathematics, John Wiley & Sons Inc., New York, 1960.

[22] A. Zada, W. Ali, S. Farina, Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Method. Appl. Sci. 40 (2017), 5502-5514.

[23] A. Zada, S. Faisal, Y. Li, On the Hyers–Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces. (2016), 2016:8164978.

[24] A. Zada, S. Faisal, Y. Li, Hyers–Ulam–Rassias stability of non-linear delay differential equations, J. Nonlinear Sci. Appl. 10 (2017), 504-510.

[25] A. Zada, F. U. Khan, U. Riaz, T. Li, Hyers–Ulam stability of linear summation equations, Punjab U. J. Math. 49 (2017), 19-24.

[26] A. Zada, T. Li, S. Ismail, O. Shah, Exponential dichotomy of linear autonomous systems over time scales, Diff. Equa. Appl. 8 (2016), 123-134.

[27] A. Zada, S. O. Shah, Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses, Hacet. J. Math. Stat. 47 (5) (2018), 1196-1205.

[28] A. Zada, S. O. Shah, S. Ismail, T. Li, Hyers–Ulam stability in terms of dichotomy of first order linear dynamic systems, Punjab. Uni. J. Math. 49 (2017), 37-47.

[29] A. Zada, O. Shah, R. Shah, Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comp. 271 (2015), 512-518.

[30] A. Zada, P. Wang, D. Lassoued, T. Li, Connections between Hyers–Ulam stability and uniform exponential stability of 2–periodic linear nonautonomous systems, Adv. Difference Equ. (2017), 2017:192.