Fuzzy estimates for Cairic's fixed point theorems

Document Type : Original Paper

Authors

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Abstract

In this paper, we investigate the estimation of fixed point errors for two important fixed point

theorems of quasi-contractive mappings in fuzzy norm spaces. For this purpose, we define 􀀀generalized

contraction and ("; )􀀀uniformly generalized contraction functions on fuzzy metric spaces and show

that these definition are generalization of contractive functions defined by Ćirić on classical metric space.

Also, when Picard’s iteration is used to approximate fixed points in fuzzy norm spaces, we obtain complete

expressions for Ćirić’s fixed point theorems, including estimates of the fuzzy error.

Highlights

  1. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (2) (1976) 620–709.
    T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (3) (2003) 687–705.
     T. Bag, S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (3) (2005) 513–547.
  2. Bag, S.K. Samanta, Some fixed point theorems in fuzzy normed linear spaces, Information Sciences 177 (2007) 3271–3289.
    V. Berinde, Iterative approximation of fixed points, Springer-Verlag, Berlin Heidelberg, 2007.
    Lj.B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math. 12 (1971) 19–26.
     A. Chitra, P.V. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc. 74 (1969) 660–665.
     J. Franklin, Methods of Mathematical Economics, Springer Verlag, New York, 1980.
    V. Gregori, J.-J. Mi˜nana, D. Miravet, Extended fuzzy metrics and fixed point theorems, Mathematics 7 (3) (2019), Article 303.
    R. Kannan, On certain sets and fixed-point theorems, Roum. Math. pure appl. 14 (1969) 51–54.
    S.A.M. Mohsenalhosseini, H. Mazaheri, Approximate fixed point theorems in fuzzy norm spaces for an operator, Advances in Fuzzy Systems Volume 2013, Article ID 613604, 8 pages.
     S.A.M. Mohsenalhosseini, H. Mazaheri, M.A. Dehghan, Approximate fixed point in fuzzy normed spaces for nonlinear maps, Iranin Journal of Fuzzy Systems 10 (1) (2013) 135–142.
     H.K. Pathak, N. Hussain, Common fixed points for Banach pairs with applications, Non-linear Anal. 69 (2008) 2788–2802.
     Sh. Rezapour, M.E. Samei, Some fixed point results for α-ψ-contractive type mappings on intuitionistic fuzzy metric spaces, Journal of Advanced Mathematical Studies 7 (1) (2014) 176-–178.
     B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–290.
     M.E. Samei, Convergence of an iterative scheme for multifunctions on fuzzy metric spaces, Sahand Communications in Mathematical Analysis 15 (1) (2019) 91-–106.
    M.E. Samei, Some fixed point results on intuitionistic fuzzy metric spaces with a graph, Sahand Communications in Mathematical Analysis 13 (1) (2019) 141–-152.

 

Keywords

Main Subjects

References

 H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (2) (1976) 620–709.
 T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (3) (2003) 687–705.
 T. Bag, S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (3) (2005) 513–547.
 T. Bag, S.K. Samanta, Some fixed point theorems in fuzzy normed linear spaces, Information Sciences 177 (2007) 3271–3289.
V. Berinde, Iterative approximation of fixed points, Springer-Verlag, Berlin Heidelberg, 2007.
 Lj.B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math. 12 (1971) 19–26.
 A. Chitra, P.V. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc. 74 (1969) 660–665.
 J. Franklin, Methods of Mathematical Economics, Springer Verlag, New York, 1980.
V. Gregori, J.-J. Mi˜nana, D. Miravet, Extended fuzzy metrics and fixed point theorems, Mathematics 7 (3) (2019), Article 303.
R. Kannan, On certain sets and fixed-point theorems, Roum. Math. pure appl. 14 (1969) 51–54.
S.A.M. Mohsenalhosseini, H. Mazaheri, Approximate fixed point theorems in fuzzy norm spaces for an operator, Advances in Fuzzy Systems Volume 2013, Article ID 613604, 8 pages.
 S.A.M. Mohsenalhosseini, H. Mazaheri, M.A. Dehghan, Approximate fixed point in fuzzy normed spaces for nonlinear maps, Iranin Journal of Fuzzy Systems 10 (1) (2013) 135–142.
 H.K. Pathak, N. Hussain, Common fixed points for Banach pairs with applications, Non-linear Anal. 69 (2008) 2788–2802.
 Sh. Rezapour, M.E. Samei, Some fixed point results for α-ψ-contractive type mappings on intuitionistic fuzzy metric spaces, Journal of Advanced Mathematical Studies 7 (1) (2014) 176-–178.
 B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–290.
 M.E. Samei, Convergence of an iterative scheme for multifunctions on fuzzy metric spaces, Sahand Communications in Mathematical Analysis 15 (1) (2019) 91-–106.
M.E. Samei, Some fixed point results on intuitionistic fuzzy metric spaces with a graph, Sahand Communications in Mathematical Analysis 13 (1) (2019) 141–-152.
Journal of Advanced Mathematical Modeling
Volume 13, Issue 2 - Serial Number 30
September 2023
Pages 188-202

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History

  • Receive Date: 05 December 2022
  • Revise Date: 19 June 2023
  • Accept Date: 23 June 2023

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