Bounded-Coefficient Stability and Approximation of Partial b-Metrics

Abstract

Partial \(b\)-metrics combine the relaxed triangle inequality of \(b\)-metrics with the nonzero self-distance of partial metrics.  We study a stability question for this class.  If a sequence of partial \(b\)-metrics has one common coefficient and converges pointwise to a partial distance, then the limit is again a partial \(b\)-metric with the same coefficient.  We also prove a converse approximation phenomenon: every bounded partial distance satisfying the usual symmetry, separation and small self-distance axioms is uniformly approximable by partial \(b\)-metrics whose coefficients may depend on the approximation parameter.  An explicit bounded partial distance with nonconstant self-distance shows that the divergence of these coefficients is sometimes unavoidable.  We add a local version for unbounded partial distances, discuss what the approximation does not preserve topologically, and derive fixed point consequences from Shukla's contraction theorem for complete partial \(b\)-metric spaces, including a stability statement for fixed points of approximating contractions.