Advertisement

The behaviour of measures of noncompactness in \(L^\infty ({\mathbb {R}}^n)\) with application to the solvability of functional integral equations

  • Reza Allahyari
Original Paper

Abstract

In this work, we define a new measure of noncompactness on the space \(L^\infty ({\mathbb {R}}^n)\). In addition, we study the existence of solutions for a class of nonlinear functional integral equations of Fredholm type by using an extension of Darbo’s fixed point theorem associated with this new measure of noncompactness. Also, we will include an example which shows the efficiency of our results. The obtained results improve several ones obtained earlier.

Keywords

Measure of noncompactness Darbo’s fixed point theorem Fixed point Modulus of continuity Integral equations 

Mathematics Subject Classification

47H09 47H10 

Notes

Compliance with ethical standards

Conflict of interest

The author declares that they have no competing interests.

References

  1. 1.
    Aghajani, A., Allahyari, R., Mursaleen, M.: A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aghajani, A., Banaś, J., Jalilian, Y.: Existence of solution for a class nonlinear Volterra sigular integral. Comput. Math. Appl. 62, 1215–1227 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetMATHGoogle Scholar
  4. 4.
    Aghajani, A., Jalilian, Y.: Existence and global attractivity of solutions of a nonlinear functional integral equation. Commun. Nonlinear Sci. Numer. Simulat. 15, 3306–3312 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Aghajani, A., Mursaleen, M., Shole Haghighi, A.: Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness. Acta Mathematica Scientia 35B(3), 552–566 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Aghajani, A., Shole Haghighi, A.: Existence of solutions for a class of functional integral equations of volterra type in two variables via measure of noncompactnes. IJST 38(A1), 1–8 (2014)MATHGoogle Scholar
  7. 7.
    Arab, R.: Some generalizations of Darbo fixed point theorem and its application. Miskolc Math. Notes. (Accepted) Google Scholar
  8. 8.
    Arab, R.: The existence of fixed points via the measure of noncompactness and its application to functional integral equations. Mediterr. J. Math. (2015). doi: 10.1007/s00009-014-0506-ycSpringerBasel
  9. 9.
    Arab, R.: Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations. J. Korean Math. Soc. 52(1), 125–139 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Arab, R., Allahyari, R., Shole, A.: Haghighi, Construction of a measure of noncompactness on \(BC(\Omega )\) and its application to Volterra integral equations. Mediterr. J. Math., pp. 1–14 (2015) doi: 10.1007/s00009-015-0547-x
  11. 11.
    Banaś, J., Dhage, B.C.: Global asymptotic stability of solutions of a functional integral equation. Nonlinear. Anal. 69, 1945–1952 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Banaś, J., Goebel, K.: Measures of noncompactness in Banach spaces. In: Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York (1980)Google Scholar
  13. 13.
    Banaś, J., Knap, Z.: Measures of weak noncompactness and nonlinear integral equations of convolution type. J. Math. Anal. Appl. 146, 353–362 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Banaś, J., O’Regan, D., Sadarangani, K.: On solutions of a quadratic Hammerstein integral equation on an unbounded interval. Dynam. Syst. Appl. 18, 251–264 (2009)MathSciNetMATHGoogle Scholar
  15. 15.
    Banaś, J., Paslawska-Poludnik, M.: Monotonic solutions of Urysohn integral equation on unbounded interval. Appl. Math. Comput. 47, 1947–1954 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Banaś, J., Rzepka, R.: An application of a measure of noncompactness in the study of asymptotic stability. Appl. Math. Lett. 16, 1–6 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Banaś, J., Mursaleen, M.: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, Berlin (2014)CrossRefMATHGoogle Scholar
  18. 18.
    Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)MathSciNetMATHGoogle Scholar
  19. 19.
    Darwish, M.A.: Monotonic solutions of a convolution functional integral equation. Appl. Math. Comput. 219, 10777–10782 (2013)MathSciNetMATHGoogle Scholar
  20. 20.
    Darwish, M.A.: On a perturbed functional integral equation of Urysohn type. Appl. Math. Comput. 218, 8800–8805 (2012)MathSciNetMATHGoogle Scholar
  21. 21.
    Darwish, M.A., Henderson, J., O’Regan, D.: Existence and asymptotic stability of solutions of a perturbed fracttional functional integral equation with linear modification of the arrgument. Korean. Math. Soc. 48, 539–553 (2011)CrossRefMATHGoogle Scholar
  22. 22.
    Dhage, B.C., Bellale, S.S.: Local asymptotic stability for nonlinear quadratic functional integral equations. Electron. J. Q. Theory Differ. Equ. 10, 1–13 (2008)MATHGoogle Scholar
  23. 23.
    El-Abd, E.M.: An existence theorem of monotonic solutions for a nonlinear functional integral equation of convolution type. Funct. Anal. Approx. Comput. 4, 77–83 (2012)MathSciNetMATHGoogle Scholar
  24. 24.
    Gomaa, W., Gomaa El-Sayed, W.: Nonlinear functional integral equations of convolution type. Port. Math. 54, 449–456 (1997)MathSciNetGoogle Scholar
  25. 25.
    Ha-Olsen, H., Holden, H.: The Kolmogorov–Riesz compactness theorem. Expos. Math. 28, 385–394 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Jleli, M., Mursaleen, M., Samet, B.: On a class of q-integral equations of fractional orders. Electron. J. Differ. Equ. 2016(17), 1–14 (2016)MATHGoogle Scholar
  27. 27.
    Khosravi, H., Allahyari, R., Shole Haghighi, A.: Existence of solutions of functional integral equations of convolution type using a new construction of a measure of noncompactness on \(L^p ({\mathbb{R}}_{+})\). Appl. Math. Comput. 260, 140–147 (2015)MathSciNetGoogle Scholar
  28. 28.
    Mursaleen, M., Rizvi, S.M.H.: Solvability of infinite system of second order differential equations in \(c_0\) and \(l_1\) by Meir–Keeler condensing operators. Proc. Am. Math. Soc. 144(10), 4279–4289 (2016)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Mashhad BranchIslamic Azad UniversityMashhadIran

Personalised recommendations