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<p>We solve the functional equation <Equation ID="Equa"> <EquationSource Format="TEX">\( f(x\sigma (y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\ x,y\in S, \)</EquationSource> </Equation>for complex-valued functions <i>f</i>,&#xa0;<i>g</i> on semigroups, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma :S\rightarrow S\)</EquationSource> </InlineEquation> is an automorphism. Furthermore, we show that its variant <Equation ID="Equb"> <EquationSource Format="TEX">\( f(\sigma (y)x) = f(x)g(y)+f(y)g(x)-g(x)g(y),\ x,y\in S, \)</EquationSource> </Equation>have the same solutions. As an application, we solve the functional equation <Equation ID="Equc"> <EquationSource Format="TEX">\( f(x\sigma (y))=f(x)g(y)+f(y)g(x)+\alpha g(x\sigma (y)),\ x,y\in S, \)</EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \in \mathbb {C}\backslash \lbrace 0\rbrace \)</EquationSource> </InlineEquation> is a fixed constant.</p>

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A Levi-Civita functional equation with an automorphism

  • 期刊论文

  • 发表日期：

    2026年1月8日

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  • Youssef Aserrar,
  • Elhoucien Elqorachi

摘要

We solve the functional equation \( f(x\sigma (y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\ x,y\in S, \) for complex-valued functions f, g on semigroups, where \(\sigma :S\rightarrow S\) is an automorphism. Furthermore, we show that its variant \( f(\sigma (y)x) = f(x)g(y)+f(y)g(x)-g(x)g(y),\ x,y\in S, \) have the same solutions. As an application, we solve the functional equation \( f(x\sigma (y))=f(x)g(y)+f(y)g(x)+\alpha g(x\sigma (y)),\ x,y\in S, \) where \(\alpha \in \mathbb {C}\backslash \lbrace 0\rbrace \) is a fixed constant.