Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations
Journal Title: Annales Mathematicae Silesianae - Year 2018, Vol 32, Issue
Abstract
We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations $$∫_{S}f(xyt)dμ(t) +∫_{S}f(xσ(y)t)dμ(t) = 2f(x)f(y), x,y∈S;$$ $$∫_{S}f(xσ(y)t)dμ(t)−∫_{S}f(xyt)dμ(t) = 2f(x)f(y), x,y∈S,$$ where $S$ is a semigroup, $σ$ is an involutive automorphism of $S$ and $μ$ is a linear combination of Dirac measures $(δ_{z_i})_{i∈I}$, such that for all $i∈I$, $z_i$ is in the center of $S$. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where $σ$ is an involutive morphism.
Authors and Affiliations
Elhoucien Elqorachi, Ahmed Redouani
Keywords
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