In this paper we obtain some results on the inhomogeneous Sincov’s equation <Equation ID="Equ17"> <EquationSource Format="TEX">$\begin{aligned} f(x,z)-f(x,y)-f(y,z)=I(x,y,z) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f:S^2 \rightarrow G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mi>S</mi> <mn>2</mn> </msup> <mo stretchy="false">→</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> is the unknown function and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$I:S^3 \rightarrow G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>:</mo> <msup> <mi>S</mi> <mn>3</mn> </msup> <mo stretchy="false">→</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> is a given function (S is a nonempty set and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(G, +)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an Abelian group). We give necessary and sufficient conditions on I for the existence of a solution for the inhomogeneous Sincov’s equation. Moreover, we give a result on Ulam stability for the Sincov’s equation and obtain the best Ulam constant of it. In this way we give an answer to a problem formulated by L. Reich at the 61st International Symposium on Functional Equations.

On the Inhomogeneous Sincov’s Equation

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2026年4月8日

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Alexandra Măduţa,
Dorian Popa,
Alexandra-Maria Sîngeorzan

摘要

In this paper we obtain some results on the inhomogeneous Sincov’s equation $\begin{aligned} f(x,z)-f(x,y)-f(y,z)=I(x,y,z) \end{aligned}$ $\begin{matrix} f (x, z) - f (x, y) - f (y, z) = I (x, y, z) \end{matrix}$ where $f:S^2 \rightarrow G$ $f : S^{2} \to G$ is the unknown function and $I:S^3 \rightarrow G$ $I : S^{3} \to G$ is a given function (S is a nonempty set and $(G, +)$ $(G, +)$ is an Abelian group). We give necessary and sufficient conditions on I for the existence of a solution for the inhomogeneous Sincov’s equation. Moreover, we give a result on Ulam stability for the Sincov’s equation and obtain the best Ulam constant of it. In this way we give an answer to a problem formulated by L. Reich at the 61st International Symposium on Functional Equations.