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A general common fixed point theorem for multi-maps satisfying an implicit relation on fuzzy metric spaces. (English) Zbl 1199.54246

Let \((X,M,*)\) denote a complete fuzzy metric space, where \(X\) is a nonempty set, \(*:[0,1]\times[0,1]\to[0,1]\) is a continuous \(t\)-norm, and \(M\) is a fuzzy set on \(X^2\times(0,+\infty)\). Let \(CB(X)\) denote the set of all nonempty closed and bounded subsets of \(X\). Consider the mappings \(F,G:X\to CB(X)\) and \(f,g:X\to X\). Using appropriate iterations, and under some specific conditions, the authors prove that there exists the unique \(p\in X\) such that \(\{p\}=\{fp\}=\{gp\}=Fp=Gp\). Thus, the general common fixed point for multimaps is found.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54A40 Fuzzy topology
54C60 Set-valued maps in general topology
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