On the alienation of multiplicative and additive functions
Abstract
Given S a semigroup. We study two Pexider-type functional equations
f(xy) + g(xy) = f(x) + f(y) + g(x)g(y), x,y ∈ S,
and
∫Sf(xyt)dμ(t) + ∫Sg(xyt)dμ(t) = f(x) + f(y) + g(x)g(y), x,y ∈ S,
for unknown functions f and g mapping S into ℂ, where μ is a linear combination of Dirac measures (δz_i})i∈I for some fixed elements (zi)i∈I contained in S such that ∫Sdμ(t) = 1.
The main goal of this paper is to solve the above two functional equations and examine whether or not they are equivalent to the systems of equations
f(xy) = f(x) + f(y), g(xy) = g(x)g(y), x,y ∈ S,
and
∫Sf(xyt)dμ(t) = f(x) + f(y), ∫Sg(xyt)dμ(t) = g(x)g(y), x,y ∈ S,
respectively.
Keywords
alienation; semigroup; multiplicative function; additive function; quadratic equivalence
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Department of Mathematics, Faculty of Sciences, Ibn Zohr University Morocco
https://orcid.org/0009-0002-0297-9390
Department of Mathematics, Faculty of Sciences, Ibn Zohr University Morocco
Department of Mathematics, Faculty of Sciences, Ibn Zohr University Morocco
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