Fuzzy contractive maps and fuzzy fixed points (Q698801)

The authors show how one can deduce a number of fuzzy fixed point results from new (as well as old) fixed point theorems for multivalued contractions. Sample of the authors' new results: let \((X, d)\) be a complete metric space, let \(FC(X)= \{A\in [0,1]^X: [A]_1:= A^{-1}\{1\}\) is closed\}, and let \(D_1(A, B)= H([A]_1, [B]_1)\) for all \(A,B\in FC(X)\) where \(H\) is the generalized Hausdorff metric on \(FC(X)\). If \(T: B(x_0,r)\to FC(X)\) is such that \(D_1(Tx, Ty)\leq kd(x,y)\) for all \(x,y\in \overline{B(x_0, r)}\), where \(0< k< 1\), and \(\text{dist}(x_0,[Tx_0]_1)< (1- k)r\), then there exists \(x\in\overline{B(x_0, r)}\) such that \(Tx(x)= 1\) (this is what is called a fuzzy fixed point). As an easy corollary, the authors get Heilpern's fixed point theorem [\textit{S. Heilpern}, J. Math. Anal. Appl. 83, 566-569 (1981; Zbl 0486.54006)] with weaker contractive condition and without certain convexity and compactness type assumptions. In addition to this, the latter corollary is shown to be a consequence of the well-known Nadler's contraction principle. Then the authors go on to produce their own fuzzy analogues of some fixed point results for contractive or nonexpansive mappings, e.g., of two theorems from [\textit{M. Frigon} and \textit{A. Granas}, Topol. Methods Nonlinear Anal. 4, No. 1, 197-208 (1994; Zbl 0829.47047)]. Reviewer's remark: The authors did a good job, although it may be hoped that their paper will not become seminal.