$\psi -$weak Contractions in Fuzzy Metric Spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Lahore University of Management Sci- ences, 54792- Lahore, Pakistan

2 Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India

3 Department of Mathematics and Humanities, S. V. National Institute of Technology, Surat, 395007, Gujarat, India

Abstract

In this paper, the notion of $\psi -$weak contraction \cite{Rhoades} is
extended to fuzzy metric spaces. The existence of common fixed points for
two mappings is established where one mapping is $\psi -$weak contraction
with respect to another mapping on a fuzzy metric space. Our result
generalizes a result of Gregori and Sapena \cite{Gregori}.

Keywords

References

\bibitem{Alber} Y. I. Alber and S. Guerre-Delabriere, \textit{Principles of
weakly contractive maps in Hilbert spaces}, In: I. Gohberg, Y. Lyubich
, eds., New Results in Operator Theory, In: Advances and Appl.,
Birkhauser, Basel, {\bf98} (1997), 7-22.
\bibitem{Dutta} P. N. Dutta, B. S. Chaudhury and K. Das, \textit{Some fixed
point results in Menger spaces using a control function}, Surveys in
Mathematics and Its Application 4, (2009), 41-52.
\bibitem{Deng} Z. K. Deng, \textit{Fuzzy pseudo-metric spaces}, J. Math.
Anal. Appl. 86, (1982), 74-95.
\bibitem{Naschie} M. S. El Naschie, \textit{On a fuzzy khaler-like
manifold which is consistent with two slit experiment}, Int. J. Non-linear
Sci. and Numerical Simulation, {\bf6} (2005), 95-98.
\bibitem{Grabiec} M. Grabiec, \textit{Fixed points in fuzzy metric spaces},
Fuzzy Sets and Systems, {\bf27} (1988), 385-389.
\bibitem{Grabiec2} M. Grabiec, Y. J. Cho and V. Radu, \textit{On
nonsymmetric topological and probabilistic structures}, Nova Science
Publisher, New York, 2006.
\bibitem{George} A. George and P. Veeramani, \textit{On some results in
fuzzy metric spaces}, Fuzzy Sets and Systems, {\bf64} (1994), 395-399.
\bibitem{George2} A. George and P. Veeramani, \textit{On some results of
analysis for fuzzy metric spaces}, Fuzzy Sets and Systems, {\bf90} (1997), 365-368.
\bibitem{Gregori} V. Gregori and A. Sapena, \textit{On fixed-point
theorems in fuzzy metric spaces}, Fuzzy Sets and Systems, {\bf125} (2000), 245-252.
\bibitem{Hadzic} O. Hadzic and E. Pap, \textit{Fixed point theory in
probabilistic metric spaces}, Kluwer Academic Publisher, Dordrecht, 2001.
\bibitem{Kaleva} O. Kaleva and S. Seikkala, \textit{On fuzzy metric spaces},
Fuzzy Sets and Systems, {\bf27} (1984), 215-229.
\bibitem{Kramosil} O. Kramosil and J. Michalek, \textit{Fuzzy metric and
statistical metric spaces}, Kybernetika, {\bf11} (1975), 326-334.
\bibitem{Kirk} W. A. Kirk and B. Sims, \textit{Hand book of metric fixed
point theory}, Kluwer Academic Publisher, 2001.
\bibitem{Mihet} D. Mihet, \textit{On fuzzy contractive mappings in fuzzy
metric spaces}, Fuzzy Sets and Systems, {\bf158} (2007), 915-921.
\bibitem{Saadati} R. Saadati, S. Sedghi and H. Zaou, {\it A common fixed point
theorem for $\psi -$ weakly commuting maps in $L-$fuzzy metric spaces},
Iranian Journal of Fuzzy Systems, {\bf5(1)} (2008), 47-53
\bibitem{Schweizer} I. Schweizer and A. Sklar, \textit{Statistical metric
spaces}, Pacific J. Math., {\bf10} (1960), 314-334.
\bibitem{Sedghi} S. Sedghi, K. P. R. Rao and N. Shobe, {\it A common fixed point
theorem for six weakly compatible mappings in M-fuzzy metric spaces},
Iranian Journal of Fuzzy Systems, {\bf5(2)} (2008), 49-62.
\bibitem{Rhoades} B. E. Rhoades, \textit{Some theorems on weakly contractive
maps}, Non Linear Analysis, {\bf47} (2001), 2683-2693.
\bibitem{Vasuki} R. Vasuki and P. Veeramani, \textit{Fixed point theorems
and Cauchy sequences in fuzzy metric spaces}, Fuzzy Sets and Systems
{\bf135} (2003), 415-417.
\bibitem{Zadeh} L. A. Zadeh, \textit{Fuzzy Sets}, Information and Control,
{\bf8} (1965), 338-353.
Iranian Journal of Fuzzy Systems
Volume 8, Issue 5 - Serial Number 5
September and October 2011
Pages 141-148

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