Abstract
We discuss several properties of -functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any weighted quasipseudometric space is a -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 -function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.
Highlights
Introduction and PreliminariesKada et al introduced in 1 the concept of w-distance on a metric space and extended the Caristi-Kirk fixed point theorem 2, the Ekeland variation principle 3 and the nonconvex minimization theorem 4, for w-distances
Al-Homidan et al introduced in 5 the notion of Q-function on a quasimetric space and successfully obtained a CaristiKirk-type fixed point theorem, a Takahashi minimization theorem, an equilibrium version of Ekeland-type variational principle, and a version of Nadler’s fixed point theorem for a Qfunction on a complete quasimetric space, generalizing in this way, among others, the main results of 1 because every w-distance is, a Q-function
The following consequence of Theorem 3.3, which is illustrated by Example 3.4, improves and generalizes in several directions the Banach Contraction Principle for partial metric spaces obtained in Theorem 5.3 of 14
Summary
Introduction and PreliminariesKada et al introduced in 1 the concept of w-distance on a metric space and extended the Caristi-Kirk fixed point theorem 2 , the Ekeland variation principle 3 and the nonconvex minimization theorem 4 , for w-distances. A Q-function on a T0 qpm space X, d is a function q : X × X → 0, ∞ satisfying the following conditions: Q1 q x, z ≤ q x, y q y, z , for all x, y, z ∈ X, Q2 if x ∈ X, M > 0, and yn n∈Æ is a sequence in X that τd−1 -converges to a point y ∈ X
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