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An extension of Sadovskii’s fixed-point theorem with applications to integral equations

Abstract

In this paper, we give a generalization of Sadovski\(\breve{\text {i}}\)’s fixed-point theorem for condensing operators, which is slightly more flexible than this result in applying to some different problems. We apply our extension to prove some results in integral equations. At the end, we illustrate our results by concrete examples to confirm that our method can be used effectively to solve some integral equations.

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References

  1. Agarwal, R., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications, vol. 141. Cambridge university press (2001)

  2. Aghajani, A., Banaś, J., Sabzali, N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin 20(2), 345–358 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Akhmerov, R., Kamenskii, M., Potapov, A., Rodkina, A., Sadovskiĭ, B., IACOB, A.: Measures of noncompactness and condensing operators: Oper. Theory 55, 1–244 (1992)

    MathSciNet  Google Scholar 

  4. Banaś, J.: On measures of noncompactness in Banach spaces. Commentationes Mathematicae Universitatis Carolinae 21(1), 131–143 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Banaś, J., Rzepka, B.: An application of a measure of noncompactness in the study of asymptotic stability. Appl. Math. Lett. 16(1), 1–6 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. Banaś, J., Jleli, M., Mursaleen, M, Samet, B., Vetro, C. (eds.): Advances in nonlinear analysis via the concept of measure of noncompactness. Springer, Singapore (2017)

  7. Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)

    MathSciNet  MATH  Google Scholar 

  8. Folland, G.: Real Analysis: Modern Techniques and Their Applications. John Wiley & Sons, New York (2013)

    MATH  Google Scholar 

  9. Kuratowski, C.: Sur les espaces complets. Fundamenta Mathematicae 15(1), 301–309 (1930)

    MathSciNet  Article  MATH  Google Scholar 

  10. Munkres, J.R.: Topology. Prentice Hall Inc., Upper Saddle River (2000)

    MATH  Google Scholar 

  11. Reich, S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41(2), 460–467 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  12. Rudin, W.: Principles of mathematical analysis, Vol. 3. McGraw-hill, New York (1964)

  13. Sadovskiĭ, B.: A fixed-point principle. Funct. Anal. Appl. 1(2), 151–153 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  14. Samadi, A., Ghaemi, M.: An extension of Darbo fixed point theorem and its applications to coupled fixed point and integral equations. Filomat 28(4), 879–886 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  15. Toledano, J., Benavides, T., Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory, vol. 99. Springer Science & Business Media (1997)

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Acknowledgements

The author would like to thank Mahabad Branch, Islamic Azad University for the financial support of this study, which is based on a research with project code: 51663931110002.

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Correspondence to Hassan Khandani.

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This work was completed with the support of Islamic Azad university of Mahabad.

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Khandani, H. An extension of Sadovskii’s fixed-point theorem with applications to integral equations. J. Fixed Point Theory Appl. 20, 15 (2018). https://doi.org/10.1007/s11784-017-0481-6

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  • DOI: https://doi.org/10.1007/s11784-017-0481-6

Keywords

  • Integral equations
  • Measure of noncompactness
  • Kuratowski measure of noncompactness
  • Condensing operators
  • Sadovski\(\breve{\text {i}}\) fixed-point theorem

Mathematics Subject Classification

  • Primary 47H08
  • Secondary 47H10