×

zbMATH — the first resource for mathematics

Fixed point theorems for cyclic contraction mappings in fuzzy metric spaces. (English) Zbl 1429.54058
Summary: In the present paper, an extension of the Edelstein contraction theorem for cyclic contractions in a fuzzy metric space is established, which also can be considered as a generalization of the fuzzy Edelstein contraction theorem introduced by M. Grabiec [Fuzzy Sets Syst. 27, No. 3, 385–389 (1988; Zbl 0664.54032)]. Additionally, we extend a fixed point theorem in \(G\)-complete fuzzy metric spaces given by Y.-H. Shen et al. [Iran. J. Fuzzy Syst. 10, No. 4, 125–133 (2013; Zbl 1333.54050)] to \(M\)-complete fuzzy metric spaces. Two examples are constructed to illustrate the corresponding results, respectively.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54A40 Fuzzy topology
54E40 Special maps on metric spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Grabiec, M, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst, 27, 385-389, (1988) · Zbl 0664.54032
[2] Mishra, SN; Sharma, N; Singh, SL, Common fixed points of maps on fuzzy metric spaces, Int. J. Math. Math. Sci, 17, 253-258, (1994) · Zbl 0798.54014
[3] Vasuki, R, A common fixed point theorem in a fuzzy metric space, Fuzzy Sets Syst, 97, 395-397, (1998) · Zbl 0926.54005
[4] Cho, YJ, Fixed points in fuzzy metric spaces, J. Fuzzy Math, 5, 949-962, (1997) · Zbl 0887.54003
[5] Singh, B; Chauhan, MS, Common fixed points of compatible maps in fuzzy metric spaces, Fuzzy Sets Syst, 115, 471-475, (2000) · Zbl 0985.54009
[6] Sharma, S, Common fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst, 127, 345-352, (2002) · Zbl 0990.54029
[7] Gregori, V; Sapena, A, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst, 125, 245-253, (2002) · Zbl 0995.54046
[8] Mihet, D, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets Syst, 144, 431-439, (2004) · Zbl 1052.54010
[9] Mihet, D, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst, 158, 915-921, (2007) · Zbl 1117.54008
[10] Mihet, D, A class of contractions in fuzzy metrics paces, Fuzzy Sets Syst, 161, 1131-1137, (2010) · Zbl 1189.54035
[11] Abbas, M; Imdad, M; Gopal, D, \(φ\)-weak contractions in fuzzy metric spaces, Iran. J. Fuzzy Syst, 8, 141-148, (2011) · Zbl 1260.54056
[12] Shen, YH; Qiu, D; Chen, W, Fixed point theorems in fuzzy metric spaces, Appl. Math. Lett, 25, 138-141, (2012) · Zbl 1232.54042
[13] Kirk, WA; Srinivasan, PS; Veeramani, P, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 79-89, (2003) · Zbl 1052.54032
[14] Păcurar, M; Rus, IA, Fixed point theory for \(φ\)-contractions, Nonlinear Anal, 72, 1181-1187, (2010) · Zbl 1191.54042
[15] Karapinar, E, Fixed point theory for cyclic weak \(ϕ\)-contraction, Appl. Math. Lett, 24, 822-825, (2011) · Zbl 1256.54073
[16] Shen, YH, Qiu, D, Chen, W: Fixed point theory for cyclic \(φ\)-contractions in fuzzy metric spaces. Iran. J. Fuzzy Syst. (2013, to appear) · Zbl 0995.54046
[17] Gopal, D; Imdad, M; Vetro, C; Hasan, M, Fixed point theory for cyclic weak \(ϕ\)-contraction in fuzzy metric spaces, (2012)
[18] Murthy, PP; Mishra, U; Rashmi, R; Vetro, C, Generalized [inlineequation not available: see fulltext.]-weak contractions involving [inlineequation not available: see fulltext.]-reciprocally continuous maps in fuzzy metric spaces, Ann. Fuzzy Math. Inf, 5, 45-57, (2013)
[19] Schweizer, B; Sklar, A, Statistical metric spaces, Pac. J. Math, 10, 385-389, (1960) · Zbl 0091.29801
[20] George, A; Veeramani, P, On some results in fuzzy metric spaces, Fuzzy Sets Syst, 64, 395-399, (1994) · Zbl 0843.54014
[21] Vetro, C, Fixed points in weak non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst, 162, 84-90, (2011) · Zbl 1206.54066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.