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Classification of completeness of quasi metric space and some fixed point results. (English) Zbl 1378.54041
Let $$X$$ be a quasi metric space. In this paper, nine concepts of completeness are introduced, under the general scheme:
$$(X,d)$$ is (weakly/left/right) $$\Gamma$$-complete, if each (weakly/left/right) $$A$$-Cauchy sequence is $$B$$-convergent (where $$\Gamma\in \{\zeta, \eta, \theta, K, {\mathcal M}, Smyth\}$$, $$A\in \{d,d^s,K\}$$, $$B\in \{d,d^{-1},d^s\}$$).
Further, the relationships between these completeness concepts are discussed via two diagrams and several examples. Then, as a by-product of these, the following fixed point theorem is stated.
Theorem. Let $$(X,d)$$ be a left $${\mathcal M}$$-complete quasi-metric space, $$q$$ be a $$Q$$-function on $$X$$, $$T:X\to X$$ be a mapping, and $$F\in {\mathcal F}$$. If $$T$$ is an $$F_q$$-contraction, then $$T$$ has a unique fixed point $$z\in X$$; with, in addition, $$q(z,z)=0$$.
Here, $${\mathcal F}$$ is the class of all Wardowski functions, cf. [D. Wardowski, Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)]; and the argument is essentially the one described in the quoted paper.

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