A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. (English) Zbl 0963.54031

Branciari, A.

Publ. Math. Debr. 57, No. 1-2, 31-37 (2000).

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.

Reviewer: J.Matkowski (Bielsko-Biala)

MSC:

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

Keywords:

contraction mappings; generalized metric spaces; triangular inequality