## A generalized Banach contraction principle that characterizes metric completeness.(English)Zbl 1145.54026

An example by Connell of a metric space $$X$$ such that $$X$$ is not complete and every contraction on $$X$$ has a fixed point shows that the Banach contraction principle can not be used to characterize completeness of metric spaces. Hence, one can look for some kind of a fixed point theorem that can serve this purpose. This was done earlier by Kannan but his fixed point theorem is not a generalization of the Banach contraction principle. In this elegant paper the author proves a nice fixed point theorem of this kind. The Banach contractions are included in his contractions while Kannan contractions are completely unrelated to the new class. In the last section a generalization of the Meir-Keeler fixed point theorem is also established.

### MSC:

 54E50 Complete metric spaces 54H25 Fixed-point and coincidence theorems (topological aspects)
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### References:

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