zbMATH — the first resource for mathematics

A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. (English) Zbl 0963.54031
Let \(X\) be a nonempty set. If a nonnegative symmetric function \(d\) defined on \(X^2\) disappears only on the diagonal and satisfies the following condition: for all \(x,y,\xi,\eta\in X\), \(\xi,\eta\notin \{x,y\}\), \(\xi\neq \eta\), \[ d(x,y)\leq d(x,\xi)+d(\xi,\eta)+d(\eta,y) \] then \((X,d)\) is called a generalized metric space (of order 4). The author proves that every contractive selfmapping of a complete generalized metric space has a unique fixed point. An extension of this theorem for generalized metric spaces of arbitrary finite order is also true.

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems