Advertisement

On generalization of Darbo–Sadovskii type fixed point theorems for iterated mappings in Fréchet spaces

  • Leszek Olszowy
  • Szymon Dudek
Article
  • 64 Downloads

Abstract

In this paper, we give new Darbo–Sadovskii type fixed point theorems for iterated mappings in Fréchet spaces. Moreover, we solve two open questions proposed in Ariza-Ruiz and Garcia-Falset (Fixed Point Theory, 2018).

Keywords

Darbo–Sadovskii type fixed point theorems Fréchet spaces measures of noncompactness reflexivity 

Mathematics Subject Classification

Primary 47H10 Secondary 47H08 

References

  1. 1.
    Ariza-Ruiz, D., Garcia-Falset, J.: Abstract measures of noncompactness and fixed points for nonlinear mappings points. Fixed Point Theory (2018) (in press) Google Scholar
  2. 2.
    Ayerbe-Toledano, J.M., Dominguez-Benavides, T., López-Aceda, G.: Measure of Noncompactness in Metric Fixed Point Theory. Birkhauser, Basel (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Banaś, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)Google Scholar
  4. 4.
    Dudek, S.: Fixed point theorems in Fréchet algebras and Fréchet spaces and applications to nonlinear integral equations. Appl. Anal. Discrete Math. 11(2), 340–357 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Geraghty, M.A.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    James, R.C.: Reflexivity and the supremum of linear functionals. Ann. Math. 66(1), 159–169 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, L., Guo, F., Wu, C., Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl. 309, 638–649 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Olszowy, L.: Fixed point theorems in the Fréchet space C(R+) and functional integral equations on an unbounded interval. Appl. Math. Comput. 218, 9066–9074 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Olszowy, L.: A family of measures of noncompactness in the space \(L^1_{loc}(\mathbb{R}+)\) and its application to some nonlinear Volterra integral equation. Mediterr. J. Math. 11(2), 687–701 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Reich, S.: A fixed point theorem for Fréchet spaces. J. Math. Anal. Appl. 78, 33–35 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Applied PhysicsRzeszów University of TechnologyRzeszowPoland

Personalised recommendations