Abstract
In this article, firstly, the necessary and sufficient condition of upper (lower) completeness is attained. Secondly, the notion of upper (lower) contraction of a mapping between quasi-metric spaces is put forward. Considering the asymmetric of quasi-metrics, the concept of left (right) fixed point of a mapping from a quasi-metric space to itself is proposed. Finally, two fixed point theorems in quasi-metric spaces are obtained.
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