On \(\phi\)-contractive mappings and on the functional equation \(y=f(x,y)\)
Let (S,d) be a complete metric space, and \(f: S\to S\) a map such that \(d(f(x),f(y))\leq \Phi (d(x,y)),\) where \(\Phi (0)=0\), and \(\Phi (t)<t\) for positive t. The main result is: If for any sequence \((t_ n)\) of non- negative real numbers the inequality \(t_{n+1}\leq \Phi (t_ n)\) implies that \(t_ n\to 0\), then there is a point y in S such that \((x_ n\to y\) for any sequence \(x_ n)\) in S such that \(d(x_{n+1},y)\leq d(f(x_ n),y).\)
scientific article; zbMATH DE number 3988244
scientific article