Fixed points and completeness in metric and generalized metric spaces. (English. Russian original) Zbl 1451.54011
J. Math. Sci., New York 250, No. 3, 475-535 (2020); translation from Fundam. Prikl. Mat. 22, No. 1, 127-215 (2018).
As is well-known, completeness of the underlying space is not a necessary condition for the Banach contraction principle to hold.
The present paper gives a survey of situations when fixed point results imply completeness. The cases of normed, metric, quasi-metric, ultrametric, partial metric and dislocated metric (i.e. metric-like) spaces are covered, as well as spaces endowed with an additional order structure. Some results in the case of multivalued mappings are also presented. Some other kinds of spaces are briefly mentioned. The connection with Ekeland’s variational principle and the Caristi-Kirk fixed point theorem is treated with more detail, including some refinements of known results. In a separate section the connection between
the existence of fixed points and completeness is treated for mappings in partially ordered sets and lattices endowed with the respective topologies. Most of the known results are presented without proof, but several of them, including original ones, are given with full proofs.
In conclusion, one can say that this is a well-written article which can be very useful, both as a survey of known results in metric fixed point theory and as a collection of new results in this field.
In conclusion, one can say that this is a well-written article which can be very useful, both as a survey of known results in metric fixed point theory and as a collection of new results in this field.
Reviewer: Zoran Kadelburg (Beograd)
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H10 | Fixed-point theorems |
| 54E50 | Complete metric spaces |
| 54H12 | Topological lattices, etc. (topological aspects) |
| 06F30 | Ordered topological structures |
Keywords:
fixed point; completeness; Ekeland’s variational principle; Caristi-Kirk fixed point theorem; fixed points in partially ordered setsReferences:
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