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Uniformlycontinuouscomposition...
9
Sinceforanyx∈Ctheconstantfunctiontl→x(t∈I)belongsto
RVł(IjC)andHmapsRVł(IjC)intoRVψ(IjY),thefunctionh(·jx),be-
longstoRVψ(IjY)foranyx∈C.Sincew∈F,wehave
P→∞
lim
w(ρ)
ρ
1lim
r→o
Tw
11(
1
T)10.
Hence
lim
(β−O)w11(1/(β−O))10.
;10→o
Nowtaket∈IandO≤t≤β,O<β,Ojβ∈I.Lettingβ−Otendto
zeroin(3),andmakinguseofthecontinuityofthefunctionh(·jx)forany
x∈C(cf.Remark1),weget
h(tjx1
+x2
2
)1h(tjx1)
+h(tjx2)
2
j
forallt∈Iandx1jx2∈C.
Thus,foreacht∈Ithefunctionh(tj·)satisfiestheJensenfunctional
equationinC.Hence,bythestandardargument(cf.Kuczma[5]),we
concludethatthereexistanadditivefunctionA(t):X−→YandB(t)∈Y
suchthat
h(tjx)1A(t)x+B(t)j
t∈Ijx∈C
whichfinishestheproofofthefirstpartofourresult.
Since0∈C,theconstantzerofunctionbelongstoRVł(IjC).Set-
tingthisfunctioninthejustprovedformulaandtakingintoaccountthat
HmapsRVł(IjC)intoRVψ(IjY),weinferthatH(0)1h(·j0)1Bbe-
longstoRVψ(IjY).TheuniformcontinuityofoperatorH:RVł(IjC)−→
RVψ(IjC)impliesthecontinuityoftheadditivefunctionA(t)fort∈I.
Consequently,A(t)∈L(XjY)foreacht∈I.Thiscompletesofproof.
.
Remark2.Intheproofofthetheoremweapplytheuniformcontinuity
oftheoperatorHonlyonthesetZ⊂RVł(IjC)suchthatf∈Zifthere
areOjβ∈I,O<βsuchthat
f(t)1
1
2
[η0,;(t)(x1−x2)+x+x2]jt∈Ij
whereη0,;isdefinedby(2),x1jx2∈Candx1x1orx1x2.
ThustheassumptionoftheuniformcontinuityofHonRVł(IjC)inthe
theoremcanbereplacedbyaweakerconditionoftheuniformcontinuityof
HonZ.
