Skip to main content
SpringerLink
Log in

Ulam stabilities of nonlinear iterative integro-differential equations

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we deal with nonlinear iterative integro-functional differential equations (IIFDEs) of first order. We study the uniqueness of solutions and Ulam stabilities regarding that the IIFDEs. The new results in relation to the uniqueness of solutions and Ulam stabilities of the IIFDEs are achieved according to the Banach’s fixed point theorem. The consequences of this paper have scientific novelties and provide new contributions to the subjects of the uniqueness of solutions and the Ulam stabilities of IIFDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Ulam, S.M.: Problems in Modern Mathematics., Science Wiley, New York (1964)

    MATH  Google Scholar 

  2. Abbas, S., Benchohra, M., Petruşel, A.: Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory. Fract. Calc. Appl. Anal. 20(2), 384–398 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akkouchi, M.: On the Hyers-Ulam-Rassias stability of a nonlinear integral equation. Appl. Sci. 21, 1–10 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Biçer, E., Tunç, C.: On the Hyers-Ulam stability of Laguerre and Bessel equations by Laplace transform method. Nonlinear Dyn. Syst. Theory 17(4), 340–346 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Biçer, E., Tunç, C.: On the Hyers-Ulam stability of certain partial differential equations of second order. Nonlinear Dyn. Syst. Theory 17(2), 150–157 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Biçer, E., Tunç, C.: New theorems for Hyers-Ulam stability of Lienard equation with variable time lags. Int. J. Math. Comput. Sci. 13(2), 231–242 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Castro, L.P., Simões, A.M.: Hyers-Ulam-Rassias stability of nonlinear integral equations through the Bielecki metric. Math. Methods Appl. Sci. 41(17), 7367–7383 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castro, L.P., Simões, A.M.: Hyers-Ulam and Hyers-Ulam-Rassias stability for a class of integro-differential equations. In: Taş, K., Baleanu, D., Tenreiro Machado, J.A. (eds.) Mathematical Methods in Engineering, pp. 81–94. Springer, Cham (2019) . (Nonlinear Syst. Complex.)

    Chapter  Google Scholar 

  9. Chauhan, H.V.S., Singh, B., Tunç, C., Tunç, O.: On the existence of solutions of non-linear 2D Volterra integral equations in a Banach space. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116(3), 11 (2022). (Paper No. 101)

    MathSciNet  MATH  Google Scholar 

  10. Deep, A., Deepmala., Tunç, C.: On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications. Arab J. Basic Appl. Sci. 27(1), 279–286 (2020)

    Article  Google Scholar 

  11. Dragičević, D.: Hyers-Ulam stability for a class of perturbed Hill’s equations. Results Math. 76(3), 11 (2021). (Paper No. 129)

    Article  MathSciNet  MATH  Google Scholar 

  12. Egri, E.: Ulam stabilities of a first order iterative functional-differential equation. Fixed Point Theory 12(2), 321–328 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Huang, J., Li, Y.: Hyers-Ulam stability of delay differential equations of first order. Math. Nachr. 289(1), 60–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Janfada, M., Sadeghi, Gh.: Stability of the Volterra integrodifferential equation. Folia Math. 18(1), 11–20 (2013). (561)

    MathSciNet  MATH  Google Scholar 

  15. Jung, S.-M., A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl., p. 9 (2007) (Art. ID 57064)

  16. Jung, S.-M., A fixed point approach to the stability of an integral equation related to the wave equation. In: Abstr. Appl. Anal., p. 4 (2013) (Art. ID 612576)

  17. Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis. In: Springer Optimization and Its Applications, 48. Springer, New York, (2011)

  18. Jung, S.-M., Roh, J.: Hyers-Ulam stability of the time independent Schrödinger equations. Appl. Math. Lett. 74, 147–153 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khan, H., Tunc, C., Chen, W., Khan, A.: Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator. J. Appl. Anal. Comput. 8(4), 1211–1226 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Kucche, K.D., Shikhare, P.U.: Ulam stabilities for nonlinear Volterra-Fredholm delay integro-differential equations. Int. J. Nonlinear Anal. Appl. 9(2), 145–1591 (2018)

    MATH  Google Scholar 

  21. Kucche, K.D., Shikhare, P.U.: Ulam stabilities for nonlinear Volterra delay integro-differential equations. Izv. Nats. Akad. Nauk Armenii Mat. 54(5), 27–43 (2019). (reprinted in J. Contemp. Math. Anal. 54 (2019), no. 5, 276–287)

    MathSciNet  MATH  Google Scholar 

  22. Li, Y., Shen, Y., Hyers-Ulam stability of nonhomogeneous linear differential equations of second order. Int. J. Math. Math. Sci., p. 7 (2009) (Art. ID 576852)

  23. Li, Y., Shen, Y.: Hyers-Ulam stability of linear differential equations of second order. Appl. Math. Lett. 23(3), 306–309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lungu, N., Popa, D.: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 385(1), 86–91 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Miura, T., Hirasawa, G., Takahasi, S.-E.: Note on the Hyers-Ulam-Rassias stability of the first order linear differential equation. Nonlinear Funct. Anal. Appl. 11(5), 851–858 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Obloza, M., Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. No. 13, 259–270 (1993)

  27. Onitsuka, M.: Hyers-Ulam stability of first order linear differential equations of Carathéodory type and its application. Appl. Math. Lett. 90, 61–68 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Otrocol, D., Ilea, V.: Ulam stability for a delay differential equation. Cent. Eur. J. Math. 11(7), 1296–1303 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Petru, T.P., Petruşel, A., Yao, J.-C.: Ulam-Hyers stability for operatorial equations and inclusions via nonself operators. Taiwan. J. Math. 15(5), 2195-2212. (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Petruşel, A., Rus, I.A.: Ulam stability of zero point equations. In: Popa, D., Rassias, T.M., Brzdęk, J. (eds.) Ulam Type Stability, pp. 345–364. Springer, Cham (2019)

    Chapter  MATH  Google Scholar 

  31. Popa, D., Raşa, I.: On the Hyers-Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381(2), 530–537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shah, S.O., Tunç, C., Rizwan, R., Zada, A., Khan, Q.U., Ullah, I., Ullah, I.: Bielecki-Ulam’s types stability analysis of Hammerstein and mixed integro-dynamic systems of non-linear form with instantaneous impulses on time scales. Qual. Theory Dyn. Syst. 21(4), 21 (2022). (Paper No. 107)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, Y., Li, Y.: A general method for the Ulam stability of linear differential equations. Bull. Malays. Math. Sci. Soc. 42(6), 3187–3211 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tunç, C., Biçer, E.: Hyers-Ulam-Rassias stability for a first order functional differential equation. J. Math. Fund. Sci. 47(2), 143–153 (2015)

    Article  MathSciNet  Google Scholar 

  35. Graef, R., Tunç, C.: Continuability and boundedness of multi-delay functional integro-differential equations of the second order. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109(1), 169–173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nieto, J.J., Tunç, O.: An application of Lyapunov-Razumikhin method to behaviors of Volterra integro-differential equations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 197 (2021). https://doi.org/10.1007/s13398-021-01131-2

    Article  MathSciNet  MATH  Google Scholar 

  37. Tunç, C., Tunç, O.: On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation. RACSAM 115, 115 (2021). https://doi.org/10.1007/s13398-021-01058-8

    Article  MathSciNet  MATH  Google Scholar 

  38. Tunç, O., Tunç, C.: Solution estimates to Caputo proportional fractional derivative delay integro-differential equations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 12 (2023). https://doi.org/10.1007/s13398-022-01345-y

    Article  MathSciNet  MATH  Google Scholar 

  39. Tunç, C., Tunç, O., Wen, C. -F., Yao, J.-C. : On the qualitative analyses solutions of new mathematical models of integro-differential equations with infinite delay. Math. Meth. Appl. Sci. (2023), 1–17. https://doi.org/10.1002/mma.9306

Download references

Author information

Authors and Affiliations

  1. Department of Computer, Baskale Vocational School, Van Yuzuncu Yil University, Campus, 65080, Van, Turkey

    Osman Tunç

  2. Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, Campus, 65080, Van, Turkey

    Cemil Tunç

Authors
  1. Osman Tunç
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cemil Tunç
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cemil Tunç.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tunç, O., Tunç, C. Ulam stabilities of nonlinear iterative integro-differential equations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 118 (2023). https://doi.org/10.1007/s13398-023-01450-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-023-01450-6

Keywords

Mathematics Subject Classification

Search

Navigation