Operator Theory and Fixed Points in Fuzzy Normed Algebras and Applications

  • Yeol Je Cho
  • Themistocles M. Rassias
  • Reza Saadati


In this chapter, first, we consider the concept of fuzzy Banach algebras and fuzzy compact operators in fuzzy normed spaces. Then we apply some fixed point theorems to solve the operator equation AxBx = x in fuzzy Banach algebras under some nonlinear contraction.


  1. 5.
    R.P. Agarwal, Y.J. Cho, R. Saadati, On Random topological structures. Abstr. Appl. Anal. 2011, 41 pp., Article ID 762361 (2011)Google Scholar
  2. 8.
    C. Alsina, B. Schweizer, A. Sklar, Continuity properties of probabilistic norms. J. Math. Anal. Appl. 208, 446–452 (1997)MathSciNetCrossRefGoogle Scholar
  3. 11.
    A.B. Amar, S. Chouayekh, A. Jeribi, New fixed point theorems in Banach algebras under weak topology features and applications to nonlinear integral equations. J. Funct. Anal. 259, 2215–2237 (2010)MathSciNetCrossRefGoogle Scholar
  4. 40.
    B.C. Dhage, Remarks on two fixed point theorems involving the sum and the product of two operators. Comput. Math. Appl. 46, 1779–1785 (2003)MathSciNetCrossRefGoogle Scholar
  5. 60.
    V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485–489 (2000)MathSciNetCrossRefGoogle Scholar
  6. 66.
    O. Hadžić, E. Pap, Fixed Point Theory in PM-Spaces (Kluwer Academic Publishers, Dordrecht, 2001)CrossRefGoogle Scholar
  7. 67.
    O. Hadžić, E. Pap, New classes of probabilistic contractions and applications to random operators, in Fixed Point Theory and Applications, ed. by Y.J. Cho, J.K. Kim, S.M. Kong, vol. 4 (Nova Science Publishers, New York, 2003), pp. 97–119Google Scholar
  8. 117.
    D. O’Regan, R. Saadati, Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195, 86–93 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Yeol Je Cho
    • 1
    • 2
  • Themistocles M. Rassias
    • 3
  • Reza Saadati
    • 4
  1. 1.Department of Mathematical EducationGyeongsang National UniversityJinjuKorea (Republic of)
  2. 2.School of Mathematical ScienceUniversity of Electronic Science and Technology of ChinaChengduChina
  3. 3.Department of MathematicsNational Technical University of AthensAthensGreece
  4. 4.Department of MathematicsIran University of Science and TechnologyTehranIran

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