Common fixed points of fuzzy mappings
Let \((X,d)\) be a complete metric linear space and \(T_ 1\), \(T_ 2\) be fuzzy mappings from \(X\) into the set \(W(X)\) of the approximate quantities of \(X\) such that \(D(T_ 1x_ n,T_ 2x) \to 0\) for any sequence \(\{x_ n\}\) of \(X\) converging to \(x\). Main theorem: if there exist non-negative real numbers \(\alpha_ i,\alpha_ 1 + \alpha_ 2 + \alpha_ 4 + \alpha_ 5 \leq 1\), such that for all \(x\), \(y\) in \(X\), \[ \begin{aligned} [D(T_ 1 x,T_ 2y)]^ 2 & \leq \alpha_ 1 \cdot [d(x,y)]^ 2 + \alpha_ 2 \cdot p(x,T_ 1 x) \cdot p(y,T_ 2 y)\;+\\ & + \alpha_ 3 \cdot p(x,T_ 2 y)\cdot p(y,T_ 1 x) + \alpha_ 4 \cdot d(x,y) \cdot p(x,T_ 1 x) + \alpha_ 5 \cdot d(x,y) \cdot p(y,T_ 2 y)\end{aligned} \] then \(\{z\} \subseteq T_ 1(z) \cap T_ 2(z)\) for some point \(z\) in \(X\). Here \(D\) and \(p\) are suitable distances defined on the fuzzy sets of \(X\). Two other related results are given for \(S\)-convex structures on \(X\).
scientific article; zbMATH DE number 516638