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StabilityofthePexiderfunctionalequation
9
LetVbeasymmetric,bounded,andideallyconvexsubsetofaBanachspace
E(thesymmetrymeans−V=V).Forfjgjh:S→Ewesuppose
(2)
f(xg)−g(x)−h(g)∈Vj
xjg∈S.
ThenthereareFjGjH:S→EsatisfyingthePexiderequation
(3)
F(xg)=G(x)+H(g)j
xjg∈Sj
aswellastheconditions
(4)
(5)
F(x)−f(x)∈3Vj
G(x)−g(x)∈4Vj
H(x)−h(x)∈4Vj
x∈S.
Proof.Withg=nandwithx=nin(2)weget
f(x)−g(x)−h(n)∈Vj
f(g)−g(n)−h(g)∈Vj
hencef(x)∈g(x)+h(n)+V,f(g)∈h(g)+g(n)+V,thus
f(xg)−f(x)−f(g)+g(n)+h(n)∈f(xg)−g(x)−h(g)+V+V
⊆V+V+V=3Vj
thelastequalitybeingtrue,sinceVisconvex.For
(6)
thismeans
f(x):=f(x)−g(n)−h(n)j
˜
x∈Sj
f(xg)−˜
˜
f(x)−˜
f(g)∈3Vj
xjg∈Sj
andbyTheorem1thereisafunctionΦ:S→Esuchthat
Φ(xg)=Φ(x)+Φ(g)j
Φ(x)−˜
f(x)∈3Vj
xjg∈S.
NowitiseasilyseenthatforF(x):=Φ(x)+g(n)+h(n),G(x):=Φ(x)+
g(n),H(x):=Φ(x)+h(n),x∈S,weget(3)and(4):
(3)isobvious;F(x)−f(x)∈3Vfollowsfrom(6)andΦ(x)−˜
f(x)∈3V;
theremainingformulaein(4)areconsequencesof(5),(6),andΦ(x)−˜
f(x)∈
3V.
I
