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Somefixedpointtheoremsformappings...
Proof.Letx∈XanddefineT11∫
o
d(fxo,hxo)
o(t)dt.IfT110,then
/
o
d(hhxo,hxo)
o(t)dt≤G(/M
o
∗(hxo,xo)
o(t)dt),
where
M
∗(hxo,xo)1max{d(fhxo,fxo),d(fhxo,hhxo),d(fxo,hxo)}.
9
Sincefandhcommuteandfxo1hxo,d(fhxo,fxo)10.Therefore
M∗(hxo,xo)1d(hhxo,hxo)andM∗(hxo,xo)mustbezero.For,otherwise
wehave
/
o
d(hhxo,hxo)
o(t)dt≤G(/M
o
∗(hxo,xo)
o(t)dt)
1G(/d(hhxo,hxo)
o
o(t)dt)</d(hhxo,hxo)
o
o(t)dt
acontradiction.ThusM∗(hxo,xo)10andhxoisafixedpointofh.But
thenfhxo1hfxo1hhxo1hxoandhxoisalsoafixedpointoff.
SupposethatT1>0.By(źź)thereexistsanx1∈Xsuchthatfx11hxo.
Ingeneraldefine{xn}⊂Xsothatfxn1hxn11forn≥1.
Withoutlossofgeneralitywemayassumethatfxn/1hxnforeachn.
For,iffxn1hxnforsomen,theaboveargument,withxoreplacedwith
xn,yieldsfxnasacommonfixedpointoffandh.
Define{Tn}byTn+11G(Tn),withT11∫
o
d(fxo,hxo)
o(t)dt>0.By(a),
0<Tn+1<Tn<T1,n≥1.
Moreover,by(b)and(c)theseriesΣ
∞
nl1Tnconverges.Weshallshow
that∫
o
d(hxn11,hxn)
o(t)dt≤Tn,n≥1.
Forn11,wehave
/
o
d(hxo,hx1)
o(t)dt≤G(/M
o
∗(xo,x1)
o(t)dt),
where
M
∗(xo,x1)1max{d(fxo,fx1),d(fxo,hxo),d(fx1,hx1)}
1max{d(fxo,hxo),d(hxo,hx1)}.
IfM∗(xo,x1)1d(hxo,hx1),then
/
o
d(hxo,hx1)
o(t)dt≤G(/M
o
∗(xo,x1)
o(t)dt)
</
o
d(hxo,hx1)
o(t)dt,
