2015
DOI: 10.15672/hjms.2015449441
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Functional equations related to weightable quasi-metrics

Abstract: Starting from the definition of a weightable quasi-metric we observe that several functional equations are induced in a natural way. Studying these equations we characterize weightable quasi-metrics and show that they define representable total preorders. We also analyze how to retrieve weightable quasi-metrics from real-valued functions satisfying suitable functional equations.

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Cited by 4 publications
(15 citation statements)
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“…In our main results we will assume that X is a topological space and G : X × X → R is continuous, or it satisfies a weaker regularity condition. We will prove that, either G is in a sense trivial solution, or there is a map which lies below G on X × X, is equal to G at a given point (x 0 , y 0 ) ∈ X × X and solves Sincov's equation (1). From this we will derive a representation of solutions of (3) as supremum of functions of the form (2).…”
Section: Multiplicative Sincov's Inequalitymentioning
confidence: 94%
See 2 more Smart Citations
“…In our main results we will assume that X is a topological space and G : X × X → R is continuous, or it satisfies a weaker regularity condition. We will prove that, either G is in a sense trivial solution, or there is a map which lies below G on X × X, is equal to G at a given point (x 0 , y 0 ) ∈ X × X and solves Sincov's equation (1). From this we will derive a representation of solutions of (3) as supremum of functions of the form (2).…”
Section: Multiplicative Sincov's Inequalitymentioning
confidence: 94%
“…Künzi [5] it is termed quasimetric. Let us note that M.J. Campión, E. Induráin, G. Ochoa and O. Valero [1] studied weightable quasi-metric in connection with several functional equations, in particular with additive Sincov's equation.…”
Section: Additive Sincov's Inequalitymentioning
confidence: 99%
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“…Thus, in particular 0 ∈ F(A, A) for all A ⊆ X. For some more recent investigations, see [4,33,38,27,34,35,7,8,3,14,13]. The most relevant ones are the set-valued considerations of Smajdor [38] and Augustová and Klapka [3].…”
Section: Theoremmentioning
confidence: 99%
“…In the following we shall consider a metric approach to asymmetry that is more in the spirit of paper [5] where the function F (x, y) = d(x, y) − d(y, x) (whenever x, y ∈ X) of disymmetry is considered. The following sets might be of special interest for a more detailed study on asymmetry, which will be conducted elsewhere.…”
Section: The Dierence Approach To the Skewness Of A Quasi-pseudometricmentioning
confidence: 99%