Ulam-Hyers stability for a nonlinear Volterra integro-differential equation

Abstract

In this work, the Ulam-Hyers stability and the Ulam-Hyers-Rassias stability for the nonlinear Volterra integro-differential equations are established by employing the method of successive approximation. Some simple examples are given to illustrate the main results.

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Keywords

• Ulam stability
• successive approximation
• nonlinear Volterra integro-differential equation

References

1. [1] M. Gachpazan and O. Baghani, Hyers–Ulam stability of nonlinear integral equation, Fixed Point Theory Appl. 2010, 6 pages, 2010.
2. [2] M. Gachpazan and O. Baghani, Hyers–Ulam stability of volterra integral equation, Inter. J. Nonlinear Anal. Appl. 1 (2), 19–25, 2010.
3. [3] J. Huang and Y. Li, Hyers–Ulam stability of delay differential equations of first order, Math. Nachr. 289 (1), 60–66, 2016.
4. [4] H.V. Jain and H.M. Byrne, Qualitative analysis of an integro-differential equation model of periodic chemotherapy, Appl. Math. Lett. 25 (12), 2132–2136, 2012.
5. [5] M. Janfada and G. Sadeghi, Stability of the Volterra integro-differential equation, Folia Math. 18 (1), 11–20, 2013.
6. [6] C. Jin and J. Luo, Stability of an integro-differential equation, Comput. Math. Appl. 57 (7), 1080–1088, 2009.
7. [7] M. Joshi, An existence theorem for an integro-differential equation, J. Math. Anal. Appl. 62 (1), 114–124, 1978.
8. [8] S.-M. Jung and J. Brzdek, Hyers–Ulam stability of the delay equation $y'(t)=\lambda y(t-\tau)$, Abstr. Appl. Anal. 2010, Art. Id. 372176, 2010.

9. [9] K.D. Kucche and S.T. Sutar, Stability via successive approximation for nonlinear implicit fractional differential equations, Moroccan J. Pure Appl. Anal. 3 (1), 36–55, 2017.
10. [10] K.D. Kucche and S.T. Sutar, On existence and stability results for nonlinear fractional delay differential equations, Bol. Soc. Parana. Mat. 36 (4), 55–75, 2018.
11. [11] V. Lakshmikantham, Theory of Integro-Differential Equations, CRC Press, Boca Raton, Florida, NY, 1995.
12. [12] J.P. Medlock, Integro-differential equation models in ecology and epidemiology, University of Washington, PhD Thesis, 2004.
13. [13] J.A. Oguntuase, On an inequality of Gronwall, J. Inequal. Pure Appl. Math. 2 (1), Art. No. 9, 2001.
14. [14] A.Z. Rahim Shah, A fixed point approach to the stability of a nonlinear volterra integrodifferential equation with delay, Hacet. J. Math. Stat. 47 (3), 615–623, 2018.
15. [15] S. Sevgin and H. Sevli, Stability of a nonlinear volterra integro-differential equation via a fixed point approach, J. Nonlinear Sci. Appl. 9, 200–207, 2016.
16. [16] V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, Dover, New York, NY, 1959.
17. [17] A. Zada and S.O. Shah, Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses, Hacet. J. Math. Stat. 47 (5), 1196–1205, 2018.