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8
IshakAltunandDuranTurkoglu
SinceΣ
vergenttoo.From(2)theseriesΣ
∞
nl1Snisconvergent,itfollowsthatΣ
∞
nl1d(xn,xn+1)converges.
∞
nl1∫
o
d(xn,xn+1)
o(t)dtiscon-
Again,sinceXisd-complete{xn}convergestosomez∈X.
From(1),
/
o
d(fz,xn+1)
o(t)dt≤G(/M(z,xn)
o
o(t)dt)
1G(max{/d(z,xn)
o
o(t)dt,/
o
d(z,fz)
o(t)dt,
/
o
d(xn,xn+1)
o(t)dt}).
Takingthelimitasn→∞,oneobtains
/
o
d(fz,z)
o(t)dt≤G(/d(z,fz)
o
o(t)dt),
whichimpliesthat∫
o
d(fz,z)
o(t)dt10whichfrom(2)impliesthatd(z,fz)10
orz1fz.
Supposethatzandwarefixedpointsoff.Thenfrom(1),
/
o
d(z,w)
o(t)dt1/
o
d(fz,fw)
o(t)dt≤G(/d(z,w)
o
o(t)dt)
whichimpliesthat∫
o
d(z,w)
o(t)dt10,whichfrom(2),impliesd(z,w)10or
z1wandthefixedpointisunique.
.
Theorem3.Let(X,T)beHausdorffd-completetopologicalspace,f,h
w-continuousself-mappingsofXsatisfying
(7)
/
o
d(hx,hy)
o(t)dt≤G(/M
o
∗(x,y)
o(t)dt)
forallx,y∈X,whereoandGareasinTheorem2and
M
∗(x,y)1max{d(fx,fy),d(fx,hx),d(fy,hy)}.
Supposealsothat
(ź)fandhcommute,
(źź)h(X)⊆f(X).
ThenfandhhaveauniquecommonfixedpointinX.
