We obtain a quasi-metric generalization of Caristi's fixed point theorem for a kind of complete quasi-metric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the recurrence equation of certain algorithms.MSC: 54H25; 47H10; 54E50; 68Q25
We discuss several properties of Q-functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any T 0 weighted quasipseudometric space is a Q-function and show that both the Sorgenfrey line and the Kofner plane provide significant examples of quasimetric spaces for which the associated supremum metric is a Q-function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.
We introduce and study a probabilistic quasi-metric on the set of complexity functions, which provides an efficient framework to measure the distance from a complexity function f to another one g in the case that f is asymptotically more efficient than g. In this context we also obtain a version of the Banach fixed point theorem which allows us to show that the functionals associated both to Divide and Conquer algorithms and Quicksort algorithms have a unique fixed point.
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