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ON COMPACTNESS AND G-COMPLETENESS IN FUZZY METRIC SPACES | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 12، دوره 9، شماره 4، دی 2012، صفحه 151-158 اصل مقاله (321.12 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2012.139 | ||
| نویسنده | ||
| Pedro Tirado* | ||
| Instituto Universitario de Matematica Pura y Aplicada, Universidad Politecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain | ||
| چکیده | ||
| In [Fuzzy Sets and Systems 27 (1988) 385-389], M. Grabiec in- troduced a notion of completeness for fuzzy metric spaces (in the sense of Kramosil and Michalek) that successfully used to obtain a fuzzy version of Ba- nachs contraction principle. According to the classical case, one can expect that a compact fuzzy metric space be complete in Grabiecs sense. We show here that this is not the case, for which we present an example of a compact fuzzy metric space that is not complete in Grabiecs sense. On the other hand, Grabiec used a notion of compactness to obtain a fuzzy version of Edelstein s contraction principle. We present here a generalized version of Grabiecs version of the Edelstein xed point theorem and di erent interesting facts on the topology of fuzzy metric spaces. | ||
| کلیدواژهها | ||
| Fuzzy metric space؛ Cauchy sequence؛ G-completeness؛ Compactness؛ Fixed point theorem | ||
| مراجع | ||
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[1] I. Altun, Some xed point theorems for single and multi valued mappings on ordered non- archimedean fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 7(1) (2010), 91-96. [2] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399. [3] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389. [4] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Sys- tems, 115 (2000), 485-489. [5] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11 (1975), 336-344. [6] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems, 144 (2004), 431-439. [7] D. Mihet, Fuzzy quasi-metric versions of a theorem of Gregori and Sapena, Iranian Journal of Fuzzy Systems, 7(1) (2010), 59-64. [8] S. Romaguera, A. Sapena and P. Tirado, The Banach xed point theorem in fuzzy quasi- metric spaces with application to the domain of words, Topology Appl., 154 (2007), 2196- 2203. [9] R. Saadati, S. Sedghi and H. Zhou, A common xed point theorem for -weakly commuting maps in L-fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 5(1) (2008), 47-54. [10] A. Sapena, A contribution to the study of fuzzy metric spaces, Appl. Gen. Topology, 2 (2001), 63-76. [11] B. Schweizer and A. Sklar, Statistical metric spaces, Paci c J. Math., 10 (1960), 314-334. [12] R. Vasuki and P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems, 135 (2003), 415-417. | ||
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