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

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.

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  • M. Akkouchi, Hyers-Ulam-Rassias stability of nonlinear Volterra integral equations via a xed point approach, Acta Univ. Apulensis Math. Inform., 26 (2011), 257266.
  • L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a xed point approach, Grazer Math. Ber., 346 (2004), 4352.
  • 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.
  • 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.
  • 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.
  • M. Gachpazan and O. Baghani, Hyers-Ulam stability of Volterra integral equation, J. Nonl. Anal. Appl., 1 (2010), 1925.
  • M. Gachpazan and O. Baghani, Hyers-Ulam stability of nonlinear integral equation, Fix. P. Theo. Appl., 2010 (2010), 6 pages.
  • D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A, 27 (1941), 222224.
  • S. M. Jung, A xed point approach to the stability of a Volterra integral equation. Fix. P. Theo. Appl., 2007 (2007), 9 pages.
  • 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.
  • S. M. Jung, S. Sevgin and H. Sevli, On the perturbation of Volterra integro-differential equations, Appl. Math. Lett., 26 (2013), 665669.
  • 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.
  • T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297300.
  • 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.
  • S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, New York, (1960).
  • 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.