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Zbl 1102.54001
Di Bari, Cristina; Vetro, Calogero
Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space. (English)
[J] J. Fuzzy Math. 13, No. 4, 973-982 (2005). ISSN 1066-8950

The authors use the notion of fuzzy metric spaces introduced by Kramosil and Michalek and further modified by George and Veeramani. In such a fuzzy metric space $(X,M,*)$, if $f:X\to X$ is a map, then $f$ is a fuzzy contraction if there exists $k\in(0,1)$ such that $$\left(\frac{1}{M(f(x),f(y), t)}-1\right)\le k\left(\frac{1}{M(x,y,t)}-1 \right)$$ for every $x,y\in X$ and every $t>0$. The authors prove the following result: Let $(X,\tau)$ be a topological space that admits a compatible fuzzy metric, $a\in X$ and $f:X\to X$ a continuous mapping. Then the following conditions are equivalent: (1) there exists a compatible fuzzy metric for $(X,\tau)$ such that $f$ is fuzzy contractive and the sequence $f^n(x)\to a$ for every $x\in X$, (2) $a$ is a fuzzy attractor under $f$ for compact subsets of $X$. (Let $(X,\tau)$ be a topological space, $a\in X$ and $f:X\to X$. The point $a$ is an attractor under $f$ for compact subsets of $X$, if for every open $nbd$ $U$ of $a$ and for every compact $C\subset X$, there exists $n_0\in N$ such that $f^n(C)\subset U$ for every $n\ge n_0$; if $(X,\tau)$ admits a compatible fuzzy metric then $a$ is a fuzzy attractor under $f$ for compact subsets of $X$.) The authors further define weak fuzzy contractions and prove the following: Let $(X,M,*)$ be a complete fuzzy metric space and suppose that $M$ is triangular. If $f:X\to X$ is a weak fuzzy contraction, then (1) $f$ has a unique fixed point $a$, (2) $f^n(x)\to a$ for all $x\in X$, $$\left(\frac{1}{M\bigl (a,f^n(x),t\bigr)}-1\right)\le\frac {q^n}{1-q}\left(\frac{1}{M\bigl(x,t(x),t\bigr)}-1\right)\text{ for every }x\in X.\tag 3$$
[M. N. Mukherjee (Calcutta)]

MSC 2000:: *54A40 Fuzzy topology
54E35 Metric spaces, metrizability
54H25 Fixed-point theorems in topological spaces

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