Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
10
IshakAltunandDuranTurkoglu
acontradiction.ThusM∗(xo,x1)1d(fxo,hxo),andthedesiredinequality
isvalidforn11,infact
/
o
d(hxo,hx1)
o(t)dt≤G(/d(fxo,hxo)
o
o(t)dt)1G(T1)<T1.
Assumethatitistrueforsomen>1.Then
/
o
d(hxn,hxn+1)
o(t)dt≤G(/M
o
∗(xn,xn+1)
o(t)dt),
where
M
∗(xn,xn+1)1max{d(hxn11,hxn),d(hxn,hxn+1)}.
Byassumption,M∗(xn,xn+1)/10foreachn.IfM∗(xn,xn+1)1d(hxn,hxn+1),
thenweget
/
o
d(hxn,hxn+1)
o(t)dt≤G(/M
o
∗(xn,xn+1)
o(t)dt)
</
o
d(hxn,hxn+1)
o(t)dt,
acontradiction.Therefore,M∗(xn,xn+1)1d(hxn11,hxn)and
/
o
d(hxn,hxn+1)
o(t)dt≤G(/M
o
∗(xn,xn+1)
o(t)dt)
1G(/d(hxn11,hxn)
o
o(t)dt)
≤G(Tn)1Tn+1.
SinceΣ
convergenttoo.ThereforetheseriesΣ
∞
nl1Tnisconvergent,itfollowsthatΣ
∞
nl1d(hxn,hxn+1)converges.
∞
nl1∫
o
d(hxn,hxn+1)
o(t)dtis
NowXisd-completeso{hxn}convergestosomez∈X.Thenw-continuity
offimpliesthatfhxn→fz.Sincefandhcommute,andhisw-continuous,
fhxn1hfxn1hhxn11→hz.SinceXisHausdorff,hz1fz.Againusing
(7),
/
o
d(hhz,hz)
o(t)dt≤G(/M
o
∗(hz,z)
o(t)dt)
and
M
∗(hz,z)1d(fhz,hz)1d(hfz,hz)1d(hhz,hz),
