A fixed point approach to the stability of a nonlinear volterra integrodifferential equation with delay

Abstract

By using a fixed point method, we prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of a nonlinear Volterra integrodifferential equation with delay. Two examples are presented to support the usability of our results.

Keywords

• Hyers-Ulam-Rassias stability,Hyers-Ulam stability,Volterra integrodifferential equation with delay,fixed point approach

References

1. M. Akkouchi, Hyers-Ulam-Rassias stability of nonlinear Volterra integral equations via a xed point approach, Acta Univ. Apulensis Math. Inform., 26 (2011), 257266.
2. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a xed point approach, Grazer Math. Ber., 346 (2004), 4352.
3. L. P. Castro and A. Ramos, Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations, Banach J. Math. Anal., 3 (2009), 3643.
4. L. P. Castro, A. Ramos, Hyers-Ulam and Hyers-Ulam-Rassias stability of Volterra integral equations with delay, Integral methods in science and engineering, Birkhauser Boston, Inc., Boston, MA, 1 (2010), 8594.
5. J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305309.
6. M. Gachpazan and O. Baghani, Hyers-Ulam stability of Volterra integral equation, J. Nonl. Anal. Appl., 1 (2010), 1925.
7. M. Gachpazan and O. Baghani, Hyers-Ulam stability of nonlinear integral equation, Fix. P. Theo. Appl., 2010 (2010), 6 pages.
8. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A, 27 (1941), 222224.

9. S. M. Jung, A xed point approach to the stability of a Volterra integral equation. Fix. P. Theo. Appl., 2007 (2007), 9 pages.
10. S. M. Jung, A fixed point approach to the stability of differential equations y0 = F (x, y), Bull. Malays. Math. Sci. Soc., 33 (2010), 4756.
11. S. M. Jung, S. Sevgin and H. Sevli, On the perturbation of Volterra integro-differential equations, Appl. Math. Lett., 26 (2013), 665669.
12. J. R. Morales and E. M. Rojas, Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear integral equations with delay, Int. J. Nonl. Anal. Appl., 2 (2011), 16.
13. T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297300.
14. J. M. Rassias and M. Eslamian, Fixed points and stability of nonic functional equation in quasi--normed spaces, Cont. Anal. Appl. Math., 3 (2015), 293309.
15. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, New York, (1960).
16. A. Zada, R. Shah and T. Li, Integral type contraction and coupled coincidence xed point theorems for two pairs in G-metric spaces, Hacet. J. Math. Stat., 45 (2016), 14751484.