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10
MohamedAkkouchi
fromwhich,withthehelpoftheassumption(A3),wededucethatthat
d(u,v)=0,orequivalently,thatu=v,whichisacontradiction.We
concludethatzistheuniquefixedpointofT.Thuswehaveprovedthe
assertion(i).
(ii)Weshowthewell-posedness.Let{yn}beanyarbitrarysequenceof
pointsinXsuchthat
(14)
n→∞
lim
d(Tyn,yn)=0.
Wehavetoprovethatthesequence{yn}convergestotheuniquefixedpoint
zofT.
Byusing(6),foreverynonnegativeintegern,wehave
(15)
d(yn,z)≤d(yn,Tyn)+d(Tyn,Tz)
≤d(yn,Tyn)+φ(max{d(yn,z),d(Tyn,yn),
d(Tz,z),d(Tyn,z),d(Tz,yn)})
≤d(yn,Tyn)+φ(max{d(yn,z),d(Tyn,yn),0,
d(Tyn,yn)+d(yn,z),d(z,yn)})
=d(yn,Tyn)+φ((d(Tyn,yn)+d(yn,z))).
From(15),weget
(16)
d(yn,z)+d(yn,Tyn)≤2d(yn,Tyn)+φ((d(Tyn,yn)+d(yn,z))).
Byusingtheconditions(a)and(b)oftheassumption(A4),wededucethat
d(yn,z)+d(yn,Tyn)≤ψ(2(d(Tyn,yn))),
whichimpliesthatlimn→∞d(yn,z)=0.Thisprovesthatthefixedpoint
problemofTiswell-posed.Thuswehaveestablishedtheassertion(ii).
(iii)ItremainstoshowthatTiscontinuousatz.Tothisend,let{wn}
beanyarbitrarysequenceinXsuchthatwn→z=Tz(i.e.,{wn}converges
toz).Thenfrom(6),wehave
(17)
d(Twn,z)=d(Twn,Tz)
≤φ(max{d(wn,z),d(Twn,wn),d(Tz,z),
d(Twn,z),d(Tz,wn)})
≤φ(max{d(wn,z),d(Twn,z)+d(z,wn),0,
(Twn,z),d(z,wn)})
=φ(d(Twn,z)+d(z,wn))
From(17),weobtainthat
(18)
d(Twn,z)+d(z,wn)≤d(z,wn)+φ(d(Twn,z)+d(z,wn)).
