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Well-posednessofafixedpoint...
9
Byvirtueoftheproperties(a)and(b)oftheassumption(A4),theprevious
inequalityimpliesthat
(9)
diam(OT(xo,n))≤ψ(d(xo,Txo)),
∀n≥1.
From(9),wededucethatdiam(OT(xo))isfiniteandthat
diam(OT(xo))≤ψ(d(xo,Txo)).
Byusing(ii)ofLemma2,weobtainthat
(10)
diam(OT(T
mxo))≤φm(ψ(d(xo,Txo)))
holdstrueforallpositiveintegerm.Inparticular,(10)implies
(11)
d(Tpxo,Tpxo)≤φm(ψ(d(xo,Txo))),forallintegersp,q≥m.
ByLemma1,wehave
m→∞
lim
φm(s)=0,∀s∈[0,fl).
Weconcludefrom(11),thatthePicardsequence{Tnxo}isaCauchyse-
quence.Since(X,d)isaT−orbitallycompletemetricspace,thereissome
zinXsuchthat
(12)
n→∞
lim
xn=z.
NowweshowthatzisafixedpointofT.Byusing(6),wehave
(13)
d(Tz,xn+1)=d(Tz,Txn)
≤φ(max{d(z,xn),d(Tz,z),d(xn+1,xn),
d(Tz,xn),d(xn+1,z)}).
Bymakingn→flandusingrightupper-semi-continuityofthefunctionφ,
weobtainfrom(13)that
d(Tz,z)≤φ(max{0,d(Tz,z),0,d(Tz,z),0})
=φ(d(Tz,z)),
fromwhich,withthehelpoftheassumption(A3),wededucethatd(Tz,z)=
0,orequivalently,thatzisafixedpointofT.
Tocompletetheproofoftheassertion(i),weneedtoprovetheuniqueness
ofz.LetussupposethatuandvaretwodifferentfixedpointsofT.From
(6),wehave
d(u,v)=d(Tu,Tv)
≤φ(max{d(u,v),d(Tu,u),d(Tv,v),d(Tu,v),d(Tv,u)})
=φ(d(u,v)),
