## 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.

### MSC:

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