Kannappan–Wilson and Van Vleck–Wilson functional equations on semigroups

期刊论文
发表日期：

2024年5月6日

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Y. Aserrar,
E. Elqorachi

摘要

Let $S$ $S$ be a semigroup, $Z(S)$ $Z (S)$ the center of $S$ $S$ and $\sigma \colon S \rightarrow S$ $σ : S \to S$ is aninvolutive automorphism. Our main results is that we describe the solutions ofthe Kannappan-Wilson functional equation

$\int_{S} f(xyt)\, d\mu(t) + \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,$ $\int_{S} f (x y t) d μ (t) + \int_{S} f (σ (y) x t) d μ (t) = 2 f (x) g (y), x, y \in S,$

and the Van Vleck-Wilson functional equation

$\int_{S} f(xyt)\, d\mu(t) - \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,$ $\int_{S} f (x y t) d μ (t) - \int_{S} f (σ (y) x t) d μ (t) = 2 f (x) g (y), x, y \in S,$

where $\mu$ $μ$ is a measure that is a linear combination of Dirac measures $(\delta_{z_i})_{i\in I}$ ${(δ_{z_{i}})}_{i \in I}$ ,such that $z_i\in Z(S)$ $z_{i} \in Z (S)$ for all $i\in I$ $i \in I$ . Interesting consequences of these results arepresented.