On Jensen’s and the quadratic functional equations with involutions
Keywords:Functional equation, Jensen, quadratic, additive function, semigroup, ecuación funcional, cuadrática, función aditiva, semigrupo.
We determine the Solutions f : S → H of the generalized Jensen’s functional equation
f( x + σ(y)) + f( x + τ(y)) = 2f(x), x , y∈ S
and the solutions f : S → H of the generalized quadratic functional equation
f ( x + σ(y)) + f (x + τ(y)) = 2f (x) + 2f (y), x, y ∈ S,
where S is a commutative semigroup, H is an abelian group (2-torsion free in the first equation and uniquely 2-divisible in the second) and σ, τ are two involutions of S.
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