Skip to main content
SpringerLink
Log in

On some generalized non-linear functional integral equations of two variables via measures of noncompactness and numerical method to solve it

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

Existence of solution for functional integral equations of two variables is established in this article under some weaker conditions in a Banach algebra space \(C([0, b]\times [0, b],\mathbb {R}), b>0\) in the form of two operators. We applied the concept of measure of non-compactness (in short, MNC) and Petryshyn fixed point theorem for the operators in the above-mentioned space. For applicability of the obtained results of our theorem, an interesting example is given. To compute the solution of the example, we used an iterative algorithm which was constructed by modified homotopy perturbation method and Adomian polynomials with an acceptable accuracy.

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

References

  1. Adomian, G.: Solving Frontier Problem of Physics: The Decomposition Method. Kluwer Academic Press, Dordrecht the Netherlands (1994)

    Book  Google Scholar 

  2. Agarwal, R.P., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  3. Banas, J., Lecko, M.: Fixed points of the product of operators in Banach algebra. Panam. Math. J. 12, 101–109 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Banaś, J., Jleli, M., Mursaleen, M., Samet, B., Vetro, C.: Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer, Singapore (2017)

    Book  Google Scholar 

  5. Biazar, J., Eslami, M.: Modified HPM for solving systems of volterra integral equation of second kind. J. King Saud Univ-Sci. 23(1), 35–39 (2011)

    Article  Google Scholar 

  6. Chandrasekhar, S.: Radiative Thznsfe. Oxford. Univ. Press, London (1950)

    Google Scholar 

  7. Corduneanu, C.: Integral Equations and Applications. Cambridge Univ. Press, New York (1990)

    MATH  Google Scholar 

  8. Das, A., Rabbani, M., Hazarika, B., Arab, R.: Solvability of infinite systems of non-linear singular integral equations in the \(C(I \times I, c)\) space and using semi-analytic method to find a closed-form of solution. Int. J. Non-linear. Anal. Appl. 10, 63–76 (2019)

    MATH  Google Scholar 

  9. Deepmala, Pathak H.K: A study on some problems on existence of solutions for some non-linear functional-integral equations. Acta. Math. Sci. 33(5), 1305–1313 (2013)

  10. Deepmala, Pathak H.K: Study on existence of solutions for some non-linear functional-integral equations with applications. Math. Commu 18, 97–107 (2013)

  11. Deimling, K.: Non-linear Functional Analysis. Springer-Verlag, Berlin (1985)

    Book  Google Scholar 

  12. Glayeri, A., Rabbani, M.: New technique in semi analytic method for solving non-linear differential equation. Math. Sci. 5, 395–404 (2011)

    MathSciNet  Google Scholar 

  13. Hazarika, B., Arab, R., Mursaleen, M.: Applications of measure of noncompactness and operator type contraction for existence of solution of functional integral equations. Complex Anal. Operator Theory 13, 3837–3851 (2019)

    Article  MathSciNet  Google Scholar 

  14. Hu, S., Khavanin, M., Zhuang, W.: Integral equations arising in the kinetic theory of gases. Appl. Anal. 34, 261–266 (1989)

    Article  MathSciNet  Google Scholar 

  15. He, J.H.: A new approach to non-linear partial differential equations. Commun. Non Linear. Sci. Number. Simulation 2(4), 230–235 (1997)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Kelly, C.T.: Approximation of solutions of some quadratic integral equations in transport theory. J. Integral Eq. 4, 221–237 (1982)

    MathSciNet  Google Scholar 

  18. Maleknejad, K., Mollapourasl, R., Nouri, K.: Study on existence of solutions for some non-linear functional integral equations. Non-linear Anal. 69, 2582–2588 (2008)

    Article  Google Scholar 

  19. Matani, B., Rezaei Roshan, J., Hussain, N.: An extension of Darbo theorem via measure of non-compactness with its application in the solvability of a system of integral equations. Filomat 33(19), 6315–6334 (2019)

    Article  MathSciNet  Google Scholar 

  20. Nashine, H.K., Arab, R., Agarwal, R.P., Shole Haghigh, A.: Darbo type fixed and coupled fixed point results and its application to integral equation. Periodica Math. Hung. 77, 94–107 (2018)

    Article  MathSciNet  Google Scholar 

  21. Mishra, L.N., Sen, M., Mohapatra, R.N.: On existence theorems for generalized nonlinear functional-integral equations with applications. Filomat. 31, 2081–2091 (2017)

    Article  MathSciNet  Google Scholar 

  22. Nussbaum, R.D.: The fixed point index and fixed point theorem for k-set contractions. ProQuest LLC. Ann Arbor MI. Thesis(Ph.D)-The University of Chicago (1969)

  23. Petryshyn, W.V.: Structure of the fixed points sets of k-set-contractions. Arch. Rational Mech. Anal. 40, 312-328 (1970-1971)

  24. Rabbani, M.: Modified homotopy method to solve non-linear integral eqautions. Int. J. Nonlinear Anal. Appl. 6, 133–136 (2015)

    MATH  Google Scholar 

  25. Rabbani, M., Das, A., Hazarika, B., Arab, R.: Existence of solution for two dimensional non-linear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. J. Comput. Appl. Math. 370, 1–13 (2020)

    Article  Google Scholar 

  26. Rabbani, M., Zarali, B.: Solution of Fredholm Integro-Differential Equations System by Modified Decomposition Method. J. Math. Comput. Sci. 5, 258–264 (2012)

    Article  Google Scholar 

  27. Srivastava, H.M., Das, A., Hazarika, B., Mohiuddine, S.A.: Existence of solutions for non-linear functional integral equation of two variables in Banach Algebra. symmetry 11(674), 1–16 (2019)

    MATH  Google Scholar 

Download references

Acknowledgements

This research is supported by Government of India CSIR JRF Fellowship, Program No. 09/1174(0003)/2017-EMR-1, New Delhi.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran

    Mohsen Rabbani

  2. Mathematics Discipline, PDPM IIITDM, Jabalpur-482005, India

    Amar Deep &  Deepmala

Authors
  1. Mohsen Rabbani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amar Deep
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deepmala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohsen Rabbani.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rabbani, M., Deep, A. & Deepmala On some generalized non-linear functional integral equations of two variables via measures of noncompactness and numerical method to solve it. Math Sci 15, 317–324 (2021). https://doi.org/10.1007/s40096-020-00367-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-020-00367-0

Keywords

Mathematics Subject Classification

Search

Navigation