On unique solutions of integral equations by progressive contractions

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.

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

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

Abstract

 The authors consider Hammerstein-type integral equations for the purpose of obtaining new results on the uniqueness of solutions on an infinite interval. The approach used in the proofs is based on the technique called progressive contractions due to T. A. Burton. Here the authors apply the Burton’s method to a general Hammerstein type integral equation that also yields the existence of solutions. In most of the existing literature, investigators prove uniqueness of solutions of integral equations by applying some type of fixed point theorem which can be tedious and challenging, often patching together solutions on short intervals after making complicated translations. In this article, using the progressive contractions throughout three simple short steps, each of the three steps is an elementary contraction mapping on a short interval, we improve the technique due to T. A. Burton for a general Hammerstein type integral equation and obtain the uniqueness of solutions on an infinite interval. These are advantages of the used method to prove the uniqueness of solutions.

Keywords

Main Subjects

References

  • [1] S. Abbas, M. Benchohra, J. R. Graef, and J. Henderson, Implicit Fractional Differential and Integral Equations, Existence and Stability, De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 26 (2018).
  • [2] Z. Arab and C. Tun¸c, Well-posedness and regularity of some stochastic time-fractional integral equations in Hilbert space, J. Taibah Univ. Sci., 16 (2022), 788–798.
  • [3] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-differential Equations, Math. in Sci. Engin., Academic Press, Orlando, 178 (1985).
  • [4] T. A. Burton, Existence and uniqueness results by progressive contractions for integro-differential equations, Nonlinear Dyn. Syst. Theory, 16 (2016), 366–371.
  • [5] T. A. Burton, Integral equations, transformations, and a Krasnosel’skii-Schaefer type fixed point theorem, Electron. J. Qual. Theory Differ. Equ., (66) (2016), 1–13.
  • [6] T. A. Burton, A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions, Fixed Point Theory, 20 (2019), 107–111.
  • [7] T. A. Burton, An existence theorem for a fractional differential equation using progressive contractions, J. Fract. Calc. Appl., J. Fract. Calc. Appl., 8(1) (2017), 168–172.
  • [8] T. A. Burton and I. K. Purnaras, Global existence and uniqueness of solutions of integral equations with delay: progressive contractions. Electron. J. Qual. Theory Differ. Equ., (49) (2017), 6.
  • [9] H. V. S. Chauhan, B. S. Singh, C. Tun¸c, and O. Tun¸c, On the existence of solutions of non-linear 2D Volterra integral equations in a Banach space, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116(101) (2022), 11.
  • [10] A. Deep, Deepmala, and C. Tun¸c, On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications, Arab J. Basic Appl. Sci., 27 (2020), 279–286.
  • [11] B. C. Dhage, A. V. Deshmukh, and J. R. Graef, On asymptotic behavior of a nonlinear functional integral equation, Comm. Appl. Nonlinear Anal., 15 (2008), 55–67.
  • [12] L. E. El’sgol’ts,` Introduction to the Theory of Differential Equations with Deviating Arguments, Translated from the Russian by R. J. McLaughlin, Holden-Day, San Francisco, 1966.
  • [13] J. R. Graef, S. R. Grace, and E. Tunc¸, On the oscillation of certain integral equations, Publ. Math. Debrecen, 90 (2017), 195–204.
  • [14] J. R. Graef, S. R. Grace, and E. Tun¸c, Asymptotic behavior of solutions of certain integro-differential equations, PanAmer. Math. J., 29 (2019), 45–60.
  • [15] J. R. Graef, S. R. Grace, and E. Tun¸c, On the asymptotic behavior of solutions of certain integro–differential equations, J. Appl. Anal. Comput., 9 (2019), 1305–1318.
  • [16] J. R. Graef, S. R. Grace, S. Panigrahi, and E. Tun¸c, On the oscillatory behavior of Volterra integral equations on time-scales, PanAmer. Math. J., 23 (2013), 35–41.
  • [17] J. K. Hale, Theory of Functional Differential Equations, Second edition, Appl. Math. Sci., Vol. 3, Springer-Verlag, New York, 1977.
  • [18] V. Ilea and D. Otrocol, On the Burton method of progressive contractions for Volterra integral equations, Fixed Point Theory, 21 (2020), 585–593.
  • [19] H. Khan, C. Tunc¸, and A. Khan, Stability results and existence theorems for nonlinear delay-fractional differential equations with Φp-operator, J. Appl. Anal. Comput., 10 (2020), 584–597.
  • [20] H. Khan, C. Tunc¸, W. Chen, and A. Khan, Existence theorems and Hyers-Ulam stability for a class of Hybrid fractional differential equations with p-Laplacian operator, J. Appl. Anal. Comput., 8 (2018), 1211–1226.
  • [21] R. K. Miller, Nonlinear Volterra Integral Equations, Math. Lecture Note Series, Benjamin, Menlo Park, 1971.
  • [22] K. Shah and C. Tunc¸, Existence theory and stability analysis to a system of boundary value problem, J. Taibah Univ. Sci., 11 (2017), 1330–1342.
  • [23] M. Shoaib, M. Sarwar, and C. Tunc¸, Set-valued fixed point results with application to Fredholm-type integral inclusion, J. Taibah Univ. Sci., 14 (2020), 1077–1088.
  • [24] D. R. Smart, Fixed Point Theorems, Cambridge Tracts in Math., No. 66, Cambridge Press, London-New York, 1974.
  • [25] C. Tun¸c and O. Tun¸c, On behaviours of functional Volterra integro-differential equations with multiple time lags. Journal of Taibah University for Science, 12(2) (2018), 173-179.
  • [26] C. Tunc¸, F. S. Alshammari, and F. T. Akyıldız, Existence and uniqueness of solutions of Hammerstein type functional integral equations. Symmetry, 15(12) (2023), 2205.
  • [27] O. Tunc¸ and C. Tun¸c, On Ulam stabilities of delay Hammerstein integral equation. Symmetry, 15 (2023), 1736.
  • [28] O. Tunc¸, C. Tun¸c, and J. C. Yao, Global existence and uniqueness of solutions of integral equations with multiple variable delays and integro differential equations: progressive contractions, Mathematics, 12(2) (2024), 171.
  • [29] O. Tun¸c, C. Tun¸c, and J. C. Yao, New results on Ulam stabilities of nonlinear integral equations, Mathematics, (2024) (in press).
  • [30] M. B. Zada, M. Sarwar, and C. Tunc¸, Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations, J. Fixed Point Theory Appl., 20(25) (2018), 19.
Computational Methods for Differential Equations
Volume 13, Issue 1
January 2025
Pages 183-189

Files

History

  • Receive Date: 13 November 2023
  • Revise Date: 01 February 2024
  • Accept Date: 26 February 2024

Share

How to cite

Statistics