Fixed point theorems for fuzzy mappings in quasi-metric spaces
Let \((X,d)\) be a quasi-metric space and let \(d^s=d\vee d^{-1}\). Let \({\mathcal A}(X)\) be the set of all \(A\in[0,1]^X\) such that \(A_r=A^{-1} [\) \(r,1]\) \((r \in[0,1])\) and \(A_0=\text{Cl}_{d^s} A^{-1}(0,1]\) are compact in \((X,d^s)\) and such that \(\sup A(X)=1\). Further, given \(A,B\in {\mathcal A}(X)\), let \(p(A,B)= \sup \{\inf \{d(x,y): x\in A_r,y\in B_r\}: r\in[0,1]\}\) and let \(D(A,B)= \sup\{H_d (A_r,B_r): r\in[0,1]\}\) where \(H_d\) is the Hausdorff quasi-metric on the collection of all nonempty compact subsets of \((X,d^s)\). Theorem. Let \((X,d)\) be Smyth-complete [\textit{M. B. Smyth}, J. Lond. Math. Soc., II. Ser. 49, No. 2, 385-400 (1994; Zbl 0798.54036)]. Let \(F:X\to{\mathcal A}(X)\) satisfy the following properties: (1) if \(F(z)(a)=1\) and \(p(a,F(a))=0\), then \((F(a))(a) =1\), and (2) \(D(Fx,Fy)\leq h\max \{d(x,y),p(x,Fx), p(y,Fy),{1\over 2}(p(x,Fy) +p (y,Fx)\}\) for all \(x,y\in X\) \((0\leq h<1)\). Then there exists an \(a\in X\) such that \((F(a))(a)=1\).
scientific article; zbMATH DE number 1545768