Mediterranean Journal of Mathematics

, Volume 11, Issue 2, pp 687–701 | Cite as

A Family of Measures of Noncompactness in the Space \({L^1_{\text{loc}}(\mathbb{R}_+)}\) and its Application to Some Nonlinear Volterra Integral Equation

Open Access


The aim of this paper is to study a new family of measures of noncompactness in the space \({L^1_{\text{loc}}(\mathbb{R}_+)}\) consisting of all real functions locally integrable on \({\mathbb{R}_+}\) , equipped with a suitable topology. As an example of applications of the technique associated with that family of measures of noncompactness, we study the existence of solutions of a nonlinear Volterra integral equation in the space \({L^1_{\text{loc}}(\mathbb{R}_+)}\) . The obtained result generalizes several ones obtained earlier with help of other methods.

Mathematics Subject Classification (2010)

Primary 47H10 Secondary 47H30 


Carathéodory condition Lebesgue locally integrable function limit of an inverse system Fréchet space measure of noncompactness weak topology 


  1. 1.
    Aghajani A., Banaś J., Jalilian J.: Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62(3), 1215–1227 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aghajani A., Jalilian Y., Sadarangani K.: Existence of solutions for mixed volterra-fredholm integral equations. Electron. J. Differ. Equ. 2012(137), 1–12 (2012)MathSciNetGoogle Scholar
  3. 3.
    Appell J., De Pascale E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili. Boll. Un. Mat. Ital. 6(3-B), 497–515 (1984)MathSciNetGoogle Scholar
  4. 4.
    Appell, J., Zabrejko, P.P.: Nonlinear superposition operators. In: Cambridge Tracts in Mathematics, vol. 95. Cambridge University Press, Cambridge (1990)Google Scholar
  5. 5.
    Banaś J., Chlebowicz A.: On existence of integrable solutions of a functional integral equation under Carathéodory conditions. Nonlinear Anal. 70, 3172–3179 (2009)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Banaś J., Chlebowicz A.: On integrable solutions of a nonlinear Volterra integral equation under Carathéodory conditions. Bull. London Math. Soc. 41, 1073–1084 (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Banaś, J., Goebel, K.: Measure of noncompactness in banach spaces. In: Lecture Notes in Pure and Applied Math, vol. 60, Marcle Dekker, New York (1980)Google Scholar
  8. 8.
    Banaś J., Knap Z.: Measure of weak noncompactness and nonlinear integral equations of convolution type. J. Math. Anal. Appl. 146, 353–362 (1990)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Banaś J., Paslawska-Poludniak M.: Monotonic solutions of Urysohn integral equation on unbounded interval. Comput. Math. Appl. 47, 1947–1954 (2004)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Darwish M.A.: On a perturbed functional integral equation of Urysohn type. Appl. Math. Comput. 218, 8800–8805 (2012)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Djebali S., Sahnoun Z.: Nonlinear alternatives of schauder and Krasnosel’skij types with applications to Hammerstein integral equations in L 1 spaces. J. Differ. Equ. 249, 2061–2075 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    El-Sayed A.M.A., Sherif N., El-Farag I.A.: A nonlinear operator functional equation of Volterra type. Appl. Math. Comput. 148, 665–979 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Engelking R.: General Topology. Heldermann, Berlin (1989)MATHGoogle Scholar
  14. 14.
    Krasnosel’skii M.A., Zabrejko P.P., Pustyl’nik J.I., Sobolevskii P.J.: Integral operators in space of summable functions. Noordhoff, Leyden (1976)CrossRefGoogle Scholar
  15. 15.
    Salhi N., Taoudi M.A.: Existence of integrable solutions of an integral equation of Hammerstein type on an unbounded interval. Mediterr. J. Math. 9(4), 729–739 (2012)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Taoudi M.A.: Integrable solutions of a nonlinear functional integral equation on an unbounded interval. Nonlinear Anal. 71, 4131–4136 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Zabrejko P.P., Koshelev A.I., Krasnosel’skii M.A., Mikhlin S.G., Rakovshchik L.S., Stecenko V.J.: Integral Equations. Noordhoff, Leyden (1975)MATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsRzeszów University of TechnologyRzeszowPoland

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