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Year 2020, Volume: 49 Issue: 4, 1261 - 1269, 06.08.2020
https://doi.org/10.15672/hujms.483606

Abstract

References

  • [1] M. Gachpazan and O. Baghani, Hyers–Ulam stability of nonlinear integral equation, Fixed Point Theory Appl. 2010, 6 pages, 2010.
  • [2] M. Gachpazan and O. Baghani, Hyers–Ulam stability of volterra integral equation, Inter. J. Nonlinear Anal. Appl. 1 (2), 19–25, 2010.
  • [3] J. Huang and Y. Li, Hyers–Ulam stability of delay differential equations of first order, Math. Nachr. 289 (1), 60–66, 2016.
  • [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] M. Janfada and G. Sadeghi, Stability of the Volterra integro-differential equation, Folia Math. 18 (1), 11–20, 2013.
  • [6] C. Jin and J. Luo, Stability of an integro-differential equation, Comput. Math. Appl. 57 (7), 1080–1088, 2009.
  • [7] M. Joshi, An existence theorem for an integro-differential equation, J. Math. Anal. Appl. 62 (1), 114–124, 1978.
  • [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] 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] 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] V. Lakshmikantham, Theory of Integro-Differential Equations, CRC Press, Boca Raton, Florida, NY, 1995.
  • [12] J.P. Medlock, Integro-differential equation models in ecology and epidemiology, University of Washington, PhD Thesis, 2004.
  • [13] J.A. Oguntuase, On an inequality of Gronwall, J. Inequal. Pure Appl. Math. 2 (1), Art. No. 9, 2001.
  • [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] 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] V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, Dover, New York, NY, 1959.
  • [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.

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

Year 2020, Volume: 49 Issue: 4, 1261 - 1269, 06.08.2020
https://doi.org/10.15672/hujms.483606

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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References

  • [1] M. Gachpazan and O. Baghani, Hyers–Ulam stability of nonlinear integral equation, Fixed Point Theory Appl. 2010, 6 pages, 2010.
  • [2] M. Gachpazan and O. Baghani, Hyers–Ulam stability of volterra integral equation, Inter. J. Nonlinear Anal. Appl. 1 (2), 19–25, 2010.
  • [3] J. Huang and Y. Li, Hyers–Ulam stability of delay differential equations of first order, Math. Nachr. 289 (1), 60–66, 2016.
  • [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] M. Janfada and G. Sadeghi, Stability of the Volterra integro-differential equation, Folia Math. 18 (1), 11–20, 2013.
  • [6] C. Jin and J. Luo, Stability of an integro-differential equation, Comput. Math. Appl. 57 (7), 1080–1088, 2009.
  • [7] M. Joshi, An existence theorem for an integro-differential equation, J. Math. Anal. Appl. 62 (1), 114–124, 1978.
  • [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] 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] 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] V. Lakshmikantham, Theory of Integro-Differential Equations, CRC Press, Boca Raton, Florida, NY, 1995.
  • [12] J.P. Medlock, Integro-differential equation models in ecology and epidemiology, University of Washington, PhD Thesis, 2004.
  • [13] J.A. Oguntuase, On an inequality of Gronwall, J. Inequal. Pure Appl. Math. 2 (1), Art. No. 9, 2001.
  • [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] 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] V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, Dover, New York, NY, 1959.
  • [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.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Vu Ho 0000-0001-7274-6096

Ngo Van Hoa This is me 0000-0002-4603-4682

Publication Date August 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 4

Cite

APA Ho, V., & Hoa, N. V. (2020). Ulam-Hyers stability for a nonlinear Volterra integro-differential equation. Hacettepe Journal of Mathematics and Statistics, 49(4), 1261-1269. https://doi.org/10.15672/hujms.483606
AMA Ho V, Hoa NV. Ulam-Hyers stability for a nonlinear Volterra integro-differential equation. Hacettepe Journal of Mathematics and Statistics. August 2020;49(4):1261-1269. doi:10.15672/hujms.483606
Chicago Ho, Vu, and Ngo Van Hoa. “Ulam-Hyers Stability for a Nonlinear Volterra Integro-Differential Equation”. Hacettepe Journal of Mathematics and Statistics 49, no. 4 (August 2020): 1261-69. https://doi.org/10.15672/hujms.483606.
EndNote Ho V, Hoa NV (August 1, 2020) Ulam-Hyers stability for a nonlinear Volterra integro-differential equation. Hacettepe Journal of Mathematics and Statistics 49 4 1261–1269.
IEEE V. Ho and N. V. Hoa, “Ulam-Hyers stability for a nonlinear Volterra integro-differential equation”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, pp. 1261–1269, 2020, doi: 10.15672/hujms.483606.
ISNAD Ho, Vu - Hoa, Ngo Van. “Ulam-Hyers Stability for a Nonlinear Volterra Integro-Differential Equation”. Hacettepe Journal of Mathematics and Statistics 49/4 (August 2020), 1261-1269. https://doi.org/10.15672/hujms.483606.
JAMA Ho V, Hoa NV. Ulam-Hyers stability for a nonlinear Volterra integro-differential equation. Hacettepe Journal of Mathematics and Statistics. 2020;49:1261–1269.
MLA Ho, Vu and Ngo Van Hoa. “Ulam-Hyers Stability for a Nonlinear Volterra Integro-Differential Equation”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, 2020, pp. 1261-9, doi:10.15672/hujms.483606.
Vancouver Ho V, Hoa NV. Ulam-Hyers stability for a nonlinear Volterra integro-differential equation. Hacettepe Journal of Mathematics and Statistics. 2020;49(4):1261-9.