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Coincidenceandcommonfixedpoint...
Applying(6)and(01)wehave
(7)
0≥0(H(Fx2njGp)jd(Tx2njSp)jD(Tx2njFx2n)j
D(SpjGp)jD(Tx2njGp)jD(SpjFx2n))
≥0(D(y2n+1jGp)jd(y2njz)jd(y2njy2n+1)j
D(SpjGp)jD(y2njGp)jd(Spjy2n+1)).
Lettingn→∞weget
0(D(SpjGp)j0j0jD(SpjGp)jD(SpjGp)j0)≤0.
11
By(0a)weobtainSp∈Gp.AsG(X)⊂T(X),thereexistsq∈Xsuch
thatz=Sp=Tq.
Using(6)and(01)wehave
(8)
0≥0(H(FqjGx2n11)jd(TqjSx2n11)jD(TqjFq)j
D(Sx2n11jGx2n11)jD(TqjGx2n11)jD(Sx2n11jFq))
≥0(D(Fqjy2n)jd(Tqjy2n11)jD(TqjFq)j
d(y2n11jy2n)jd(Tqjy2n)jD(y2n11jFq)).
Lettingn→∞weget
0(D(FqjTq)j0jD(FqjTq)j0j0jD(FqjTq))≤0.
By(0b)weobtainTq∈Fq.SinceTandFareR-weaklycommutingof
type(AT)atq∈C(FjT),thereexistsanR>0suchthatD(TTqjFTq)≤
RD(TqjFq)andsoTz∈Fz.Inthesamemanner,Sz∈Gz.IfSz=Tz,
thenSz=Tz∈FznGzandifSz=Tz=z,thenzisacommonfixed
pointofSjTjFandG.
SupposethatT(X)iscomplete.Therefore,{y2n}convergestoz∈T(X)
andsothereexistsq∈Xsuchthatz=Tq.Applying(6)and(01)wehave
theinequality(8).Lettingn→∞weget
0(D(FqjTq)j0jD(FqjTq)j0j0jD(FqjTq))≤0.
By(0b)weobtainTq∈Fq.AsF(X)⊂S(X),thereexistsp∈Xsuch
thatz=Sp=Tq.
Using(6)and(01)wegettheinequality(7).Lettingn→∞weget
0(D(SpjGp)j0j0jD(SpjGp)jD(SpjGp)j0)≤0.
By(0a)weobtainSp∈Gp.Therestoftheprooffollowsasinthecase
S(X)iscomplete.
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