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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.

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