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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 $x,y,\xi,\eta\in X$, $\xi,\eta\notin \{x,y\}$, $\xi\ne \eta$, $$d(x,y)\le d(x,\xi)+d(\xi,\eta)+d(\eta,y)$$ 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.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces