A common approach in the theory of generalized metric spaces, particularly in spaces which lack symmetry, is to pass to a symmetrized function. In the literature on quasi-partial b-metric spaces, it has been repeatedly assumed without proof that the standard symmetrization <Equation ID="Equa"><EquationSource Format="MATHML"><math><msub><mi>d</mi><mi>q</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>q</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$ d_{q}(x,y)=q(x,y)+q(y,x)-q(x,x)-q(y,y) $</EquationSource></Equation> is always a b-metric. We first provide a definitive counterexample to disprove this general claim. Next, from an elementary estimate derived from the axiom (QPb4), we obtain a convenient sufficient criterion: if a self-distance modulus <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi mathvariant="fraktur">M</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$\mathfrak{M}(q)$</EquationSource></InlineEquation> is finite, then <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><msub><mi>d</mi><mi>q</mi></msub></math></EquationSource><EquationSource Format="TEX">$d_{q}$</EquationSource></InlineEquation> is a b-metric with explicit constant <InlineEquation ID="IEq3"><EquationSource Format="MATHML"><math><msup><mi>K</mi><mo>∗</mo></msup><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo>+</mo><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="fraktur">M</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$K^{*}(q)=s+(s-1)\mathfrak{M}(q)$</EquationSource></InlineEquation>. We then show by example that this condition is not necessary. Finally, once <InlineEquation ID="IEq4"><EquationSource Format="MATHML"><math><msub><mi>d</mi><mi>q</mi></msub></math></EquationSource><EquationSource Format="TEX">$d_{q}$</EquationSource></InlineEquation> is known to be a b-metric, we explain how fixed point results from complete b-metric spaces may be transferred back to the quasi-partial b-metric setting.

A counterexample for the symmetrization of quasi-partial b-metrics

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2026年8月3日

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Ledia Subashi,
Florion Cela,
Tedis Ramaj

摘要

A common approach in the theory of generalized metric spaces, particularly in spaces which lack symmetry, is to pass to a symmetrized function. In the literature on quasi-partial b-metric spaces, it has been repeatedly assumed without proof that the standard symmetrization $d_{q} (x, y) = q (x, y) + q (y, x) - q (x, x) - q (y, y)$ $ d_{q}(x,y)=q(x,y)+q(y,x)-q(x,x)-q(y,y) $ is always a b-metric. We first provide a definitive counterexample to disprove this general claim. Next, from an elementary estimate derived from the axiom (QPb4), we obtain a convenient sufficient criterion: if a self-distance modulus $M (q)$ $\mathfrak{M}(q)$ is finite, then $d_{q}$ $d_{q}$ is a b-metric with explicit constant $K^{*} (q) = s + (s - 1) M (q)$ $K^{*}(q)=s+(s-1)\mathfrak{M}(q)$. We then show by example that this condition is not necessary. Finally, once $d_{q}$ $d_{q}$ is known to be a b-metric, we explain how fixed point results from complete b-metric spaces may be transferred back to the quasi-partial b-metric setting.