Minimization theorems for fixed point theorems in fuzzy metric spaces and applications
The authors prove a minimization theorem in fuzzy metric spaces. The theorem goes as follows: If \(X\) and \(Y\) are complete fuzzy metric spaces of a given type, \(f: X\to Y\) is closed and continuous and \(\varphi: f(X)\to (-\infty, \infty]\)\ is lower semicontinuous and bounded from below then under certain assumptions \(\varphi \circ f\) attains its minimum. Using the theorem above the authors obtain a variational principle as well as a fixed point theorem for fuzzy metric spaces. Finally the equivalence of these theorems is proved. The variational principle, the fixed point theorem and their equivalence also appear in \textit{M. Stojaković} and \textit{Z. Ovcin} [ibid. 66, 353-356 (1994; Zbl 0842.54020)].
scientific article; zbMATH DE number 788341