Let G be an abelian group, let ???? be the real or complex field, let X be a normed space over ????, and let u∈X such that ‖u‖=1 be given. We assume that there exists a subspace X1 of X such that
X = Lin(u) ⊕ X1
and
‖αu+x1‖ ≥ max(|α|, ‖x1‖) for α∈????, x1∈X1.
Then we prove that the general solution f: G→???? of the equation
f(x+y) + f(x-y) + ‖f(x+y)-f(x-y)‖u = 2f(x) + 2f(y) for x,y∈G
is given by the formula
f(x) = |a(x)|u for x∈G,
where a: G→ℝ is an additive function.
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Vol. 8 (1994)
Published: 1994-09-30
10.1515/amsil
English