Skip to main content
SpringerLink
Log in

New iterative methods for finding solutions of Hammerstein equations

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Let \(G:H\rightarrow H\) and \(K: H\rightarrow H\) be monotone mappings that are either sequentially weakly continuous or continuous, where H is a real Hilbert space. In this work, we introduce two new iterative methods for approximating solutions of the Hammerstein equation \(u+GKu=0\), if they exist. The first iterative method is shown to always converge weakly to an element in the solution set of the Hammerstein equation if this solution set is nonempty. The second iterative method is a modification of the first method to upgrade weak convergence to strong convergence. Convergence results are obtained without requiring the maps to be bounded. Numerical examples are provided to demonstrate the convergence of one of these methods. Comparisons with some existing methods show that the method is cost effective in terms of the number of iterations required to obtain a solution and the computational time.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data availability

Not applicable.

Code availability

Not applicable.

References

  1. Allouch, C., Sbibih, D., Tahrichi, M.: Legendre superconvergent Galerkin-collocation type methods for Hammerstein equations. J. Comput. Appl. Math. 353, 253–264 (2019). https://doi.org/10.1016/j.cam.2018.12.040

    Article  MathSciNet  MATH  Google Scholar 

  2. Appell, J., Benavides, T.D.: Nonlinear Hammerstein equations and functions of bounded Riesz–Medvedev variation. Topol. Methods Nonlinear Anal. 47, 319–332 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Bello, A.U., Omojola, M.T., Yahaya, J.: An inertial-type algorithm for approximation of solutions of Hammerstein integral inclusions in Hilbert spaces. Fixed Point Theory Algorithms Sci. Eng. 2021(8), 22 (2021)

    MathSciNet  MATH  Google Scholar 

  4. Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MATH  Google Scholar 

  5. Boikanyo, O.A.: The generalized contraction proximal point algorithm with square-summable errors. Afr. Mat. 28, 321–332 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Brézis, H., Browder, F.E.: Some new results about Hammerstein equations. Bull. Am. Math. Soc. 80, 568–572 (1974)

    MathSciNet  MATH  Google Scholar 

  7. Brézis, H., Browder, F.E.: Existence theorems for nonlinear integral equations of Hammerstein type. Bull. Am. Math. Soc. 81, 73–78 (1975)

    MathSciNet  MATH  Google Scholar 

  8. Brézis, H., Browder, F.E.: Nonlinear integral equations and systems of Hammerstein type. Adv. Math. 18, 115–147 (1975)

    MathSciNet  MATH  Google Scholar 

  9. Browder, F.E.: The solvability of nonlinear functional equations. Duke Math. J. 30, 557–566 (1963)

    MathSciNet  MATH  Google Scholar 

  10. Browder, F.E.: Nonlinear mappings of non-expansive and accretive operators in Banach spaces. Bull. Am. Math. Soc. 73, 875–882 (1967)

    MATH  Google Scholar 

  11. Browder, F.E.: Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis, pp. 425–500. Academic Press, New York (1971)

    Google Scholar 

  12. Browder, F.E., Gupta, C.P.: Nonlinear monotone operators and integral equations of Hammerstein type. Bull. Am. Math. Soc. 75, 1347–1353 (1969)

    MATH  Google Scholar 

  13. Browder, F.E., De Figueiredo, D.G., Gupta, C.P.: Maximal monotone operators and nonlinear integral equations of Hammerstein type. Bull. Am. Math. Soc. 76, 700–705 (1970)

    MathSciNet  MATH  Google Scholar 

  14. Chepanovich, R.S.: Nonlinear Hammerstein equations and fixed points. Publ. Inst. Math. (Belgr.) 35, 119–123 (1984)

    MathSciNet  Google Scholar 

  15. Chidume, C.E., Bello, A.U.: An iterative algorithm for approximating solutions of Hammerstein equations with monotone maps in Banach spaces. Appl. Math. Comput. 313, 408–417 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Chidume, C.E., Djitte, N.: Iterative approximation of solutions of nonlinear equations of Hammerstein-type. Nonlinear Anal. 70, 4086–4092 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Chidume, C.E., Djitte, N.: Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operator. Nonlinear Anal. 70, 4071–4078 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Chidume, C.E., Djitte, N.: An iterative method for solving nonlinear integral equations of Hammerstein type. Appl. Math. Comput. 219, 5613–5621 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Chidume, C.E., Idu, K.O.: Approximation of zeros of bounded maximal monotone maps, solutions of Hammerstein integral equations and convex minimization problems. Fixed Point Theory Appl. (2016). https://doi.org/10.1186/s13663-016-0582-8

    Article  MATH  Google Scholar 

  20. Chidume, C.E., Ofoedu, E.U.: Solution of nonlinear integral equations of Hammerstein type. Nonlinear Anal. 74, 4293–4299 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Chidume, C.. E., Shehu, Y.: Strong convergence theorem for approximation of solutions of equations of Hammerstein type. Nonlinear Anal. Theory Methods Appl. 75(14), 5664–5671 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Chidume, C.E., Shehu, Y.: Approximation of solutions of generalized equations of Hammerstein type. Comput. Math. Appl. 63, 966–974 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Chidume, C.E., Shehu, Y.: Iterative approximation of solutions of equations of Hammerstein type in certain Banach spaces. Appl. Math. Comput. 219, 5657–5667 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Chidume, C.E., Shehu, Y.: Approximation of solutions of equations Hammerstein type in Hilbert spaces. Fixed Point Theory 16(1), 91–102 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Chidume, C.E., Zegeye, H.: Approximation of solutions of nonlinear equations of monotone and Hammerstein-type. Appl. Anal. 82(8), 747–758 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Chidume, C.E., Zegeye, H.: Iterative approximation of solutions of nonlinear equation of Hammerstein-type. Abstr. Appl. Anal. 6, 353–367 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Chidume, C.E., Zegeye, H.: Approximation of solutions nonlinear equations of Hammerstein type in Hilbert space. Proc. Am. Math. Soc. 133, 851–858 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Chidume, C.E., Adamu, A., Chinwendu, L.O.: Approximation of solutions of Hammerstein equations with monotone mappings in real Banach spaces. Carpath. J. Math. 35(3), 305–316 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Chidume, C.E., Nnakwe, M.O., Adamu, A.: A strong convergence theorem for generalized-\(\Phi \)-strongly monotone maps, with applications. Fixed Point Theory Appl. (2019). https://doi.org/10.1186/s13663-019-0660-9

    Article  MathSciNet  MATH  Google Scholar 

  30. Chidume, C.E., Adamu, A., Okereke, L.C.: Iterative algorithms for solutions of Hammerstein equations in real Banach spaces. Fixed Point Theory Appl. 2020, 4 (2020). https://doi.org/10.1186/s13663-020-0670-7

    Article  MathSciNet  MATH  Google Scholar 

  31. Daman, O., Tufa, A.R., Zegeye, H.: Approximating solutions of Hammerstein type equations in Banach spaces. Quaest. Math. 42(5), 561–577 (2019)

    MathSciNet  MATH  Google Scholar 

  32. De Figueiredo, D.G., Gupta, C.P.: On the variational methods for the existence of solutions to nonlinear equations of Hammerstein type. Bull. Am. Math. Soc. 40, 470–476 (1973)

    MathSciNet  Google Scholar 

  33. Djitte, N., Sene, M.: An iterative algorithm for approximating solutions of Hammerstein integral equations. Numer. Funct. Anal. Optim. 34(12), 1299–1316 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Dolezal, V.: Monotone operators and applications in control and network theory. In: Studies in Automation and Control, Vol. 2. Elsevier Scientific, New York (1978)

  35. Goebel, K., Kirk, W.A.: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  36. Kazemi, M.: Triangular functions for numerical solution of the nonlinear Volterra integral equations. J. Appl. Math. Comput. 1–24 (2021)

  37. Kazemi, M.: On existence of solutions for some functional integral equations in Banach algebra by fixed point theorem. Int. J. Nonlinear Anal. Appl. 13(1), 451–466 (2022)

    Google Scholar 

  38. Kazemi, M., Ezzati, R.: Existence of solution for some nonlinear two-dimensional Volterra integral equations via measures of noncompactness. Appl. Math. Comput. 275, 165–171 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Kürkçü, O.K.: An evolutionary numerical method for solving nonlinear fractional Fredholm–Volterra–Hammerstein integro-differential-delay equations with a functional bound. Int. J. Comput. Math. https://doi.org/10.1080/00207160.2022.2095510

  40. Maingé, P.E.: Strong convergence of projected subgradiant method for nonsmooth and nonstrictily convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Micula, S., Cattani, C.: On a numerical method based on wavelets for Fredholm–Hammerstein integral equations of the second kind. Math. Methods Appl. Sci. 41, 9103–9115 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Minjibir, M.S., Mohammed, I.: Iterative algorithms for solutions of Hammerstein integral inclusions. Appl. Math. Comput. 320, 389–399 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Neamprem, K., Klangrak, A., Kaneko, H.: Taylor–Series expansion methods for multivariate Hammerstein integral equations. Int. J. Appl. Math. 47(4), 1–5 (2017)

    MathSciNet  Google Scholar 

  44. Ofoedu, E.U., Onyi, C.E.: New implicit and explicit approximation methods for solutions of integral equations of Hammerstein type. Appl. Math. Comput. 246, 628–637 (2014)

    MathSciNet  MATH  Google Scholar 

  45. Opial, Z.: Weak convergence of the sequence of successive approximations for non-expansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    MATH  Google Scholar 

  46. Pascali, D., Sburlan, S.: Nonlinear mappings of monotone type. Editura Academia, Bucharest (1978)

    MATH  Google Scholar 

  47. Shehu, Y.: Strong convergence theorem for integral equations of Hammerstein type in Hilbert spaces. Appl. Math. Comput. 231, 140–147 (2014)

    MathSciNet  MATH  Google Scholar 

  48. Tufa, A.R., Zegeye, H., Thuto, M.: Iterative solutions of nonlinear integral equations of Hammerstein type. Int. J. Anal. Appl. 9(2), 129–141 (2015)

    MATH  Google Scholar 

  49. Uba, M.O., Uzochukwu, M.I., Onyido, M.A.: Algorithm for approximating solutions of Hammerstein integral equations with maximal monotone operators. Indian J. Pure Appl. Math. 48(3), 391–410 (2017)

    MathSciNet  MATH  Google Scholar 

  50. Wang, J.: Numerical algorithm for two-dimensional nonlinear Volterra–Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Eng. Comput. 38(9), 3548–3563 (2021). https://doi.org/10.1108/EC-06-2020-0353

    Article  Google Scholar 

  51. Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109–113 (2002)

    MathSciNet  MATH  Google Scholar 

  52. Zegeye, H., Malonza, D.M.: Hybrid approximation of solutions of integral equations of the Hammerstein type. Arab. J. Math. 2, 221–232 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous referees for their comments and suggestions which have helped to improve this manuscript.

Funding

No funding was used to undertake this work.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana

    Oganeditse A. Boikanyo & Habtu Zegeye

Authors
  1. Oganeditse A. Boikanyo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Habtu Zegeye
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally. Authors read and approved the final manuscript.

Corresponding author

Correspondence to Oganeditse A. Boikanyo.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Ethics approval

Not applicable.

Consent to participate

Authors agree to participate in the revision of the manuscript to the format required by the journal.

Consent for publication

Authors give their consent for the manuscript to be published.

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 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

Boikanyo, O.A., Zegeye, H. New iterative methods for finding solutions of Hammerstein equations. J. Appl. Math. Comput. 69, 1465–1490 (2023). https://doi.org/10.1007/s12190-022-01795-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-022-01795-y

Keywords

Mathematics Subject Classification

Search

Navigation