# Fixed point theory for cyclic $\varphi$-contractions in fuzzy metric spaces

Document Type : Research Paper

Authors

1 School of Mathematics and Statistics, Tianshui Normal Univer- sity, Tianshui 741001, People's Republic of China

2 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, People's Republic of China

3 School of Information, Capital University of Economics and Business, Beijing, 100070, People's Republic of China

Abstract

In this paper, the notion of cyclic $\varphi$-contraction in fuzzy
metric spaces is introduced and a fixed point theorem for this type
of mapping is established. Meantime, an example is provided to
illustrate this theorem. The main result shows that a self-mapping
on a G-complete fuzzy metric space has a unique fixed point if it
satisfies the cyclic $\varphi$-contraction. Afterwards, some results in
connection with the fixed point are given.

Keywords

#### References

bibitem{AbImGo:wcfms} M. Abbas, M. Imdad and D. Gopal {it $psi$-weak contractions in fuzzy metric spaces}, Iranian Journal of Fuzzy Systems, {bf 8(5)} (2011), 141-148.
bibitem{Al:sfptsmmonfms} I. Altun, {it Some fixed point theorems for single and multivalued mappings on ordered non-Archimedean fuzzy metric spaces}, Iranian Journal of Fuzzy Systems, {bf 7(1)} (2010), 91-96.
bibitem{Ch:Fpfms} Y. J. Cho, {it Fixed points in fuzzy metric spaces}, Journal of Fuzzy Mathematics, {bf 5(4)} (1997), 949-962.
bibitem{GeVe:Osrfms} A. George, P. Veeramani, {it On some results in fuzzy metric spaces}, {bf 64} (1994), 395-399.
bibitem{Gr:Fpfms} M. Grabiec, {it Fixed points in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 27} (1988), 385-389.
bibitem{GrSa:Ofptfms} V. Gregori and A. Sapena, {it On fixed point theorems in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 125} (2002), 245-253.
bibitem{HaPaBu:Ofptfms} O. Had$check{z}$i$acute{c}$, E. Pap and M. Budin$check{c}$evi$acute{c}$, {it Countable extension of trianular norms and their applications to the fixed point theory in probabilistic metric spaces}, Kybernecika, {bf 38} (2002), 363-382.
bibitem{Mi:Abctfms} D. Mihet, {it A Banach contraction theorem in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 144} (2004), 431-439.
bibitem{Mi:Ofcmfms} D. Mihet, {it On fuzzy contractive mappings in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 158} (2007), 915-921.
bibitem{Mi:Accfms} D. Mihet, {it A class of contractions in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 161} (2010) 1131-1137.
bibitem{MiShSi:Cfpmfms} S. N. Mishra, N. Sharma and S. L. Singh, {it Common fixed points of maps on fuzzy metric spaces}, International Journal of Mathematics and Mathematical Sciences, {bf 17} (1994), 253-258.
bibitem{PaRu:Fptc} M. P$breve{a}$curar and I. A. Rus, {it Fixed point theory for $varphi$-contractions}, Nonlinear Analysis, {bf 72} (2010), 1181-1187.
bibitem{QiSh:Smtsfscfptfm} D. Qiu and L. Shu, {it Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings}, Information Sciences, {bf 178} (2008), 3595-3604.
bibitem{QiShGu:Cfptfmcc} D. Qiu, L. Shu and J. Guan, {it Common fixed point theorems for fuzzy mappings under $phi$-contraction condition}, Chaos, Solitons and Fractals, {bf 41} (2009), 360-367.
bibitem{SaSeZh:Acfptwclfms} R. Saadsti, S. Sedghi and H. Zhou {it A common fixed point theorem for $psi$-weakly commuting maps in L-fuzzy metric spaces}, Iranian Journal of Fuzzy Systems, {bf 5(1)} (2008), 47-53.
bibitem{ScSk:Smp} B. Schweizer and A. Sklar, {it Statistical metric spaces}, Pacific Journal of Mathematics, {bf 10} (1960) 385-389.
bibitem{SeRaSh:Acfptswcmmfms} 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, {bf 5(2)} (2008), 49-62.
bibitem{Sh:Cfptfms} S. Sharma, {it Common fixed point theorems in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 127} (2002), 345-352.
bibitem{ShQiCh:Fptfms} Y. H. Shen, D. Qiu and W. Chen, {it Fixed point theorems in fuzzy metric spaces}, Applied Mathematics Letters, {bf 25} (2012), 138-141.
bibitem{SiCh:Cfpcmfms} B. Singh and M. S. Chauhan, {it Common fixed points of compatible maps in fuzzy metric spaces}, Fuzzy Sets and Systems, {bf 115} (2000), 471-475.
bibitem{Va:Acfptafms} R. Vasuki, {it A common fixed point theorem in a fuzzy metric space}, Fuzzy Sets and Systems, {bf 97} (1998), 395-397.