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Somefixedpointtheoremsformappings...
(3)
M(x,y)1max{d(x,y),d(x,fx),d(y,fy)}
andGisrealvaluedfunctionsatisfyingthecondition(a)-(d).
ThenfhasauniquefixedpointinX.
7
Proof.Letx∈Xand,forbrevity,definexn1fnx.Foreachinteger
n≥1,from(1)
(4)
Using(3),
/
o
d(xn,xn+1)
o(t)dt≤G(/M(xn11,xn)
o
o(t)dt).
M(xn11,xn)1max{d(xn11,xn),d(xn,xn+1)}.
Substitutinginto(4),oneobtains
(5)
/
o
d(xn,xn+1)
o(t)dt≤G(/max{d(xn11,xn),d(xn,xn+1)}
o
o(t)dt)
1G(max{/d(xn11,xn)
o
o(t)dt,/
o
d(xn,xn+1)
o(t)dt}).
If∫
o
d(xn11,xn)
o(t)dt≤∫
o
d(xn,xn+1)
o(t)dt,thenfrom(5)wehave
/
o
d(xn,xn+1)
o(t)dt≤G(/d(xn,xn+1)
o
o(t)dt)</d(xn,xn+1)
o
o(t)dt,
whichisacontradiction.Thus∫
o
d(xn11,xn)
o(t)dt>∫
o
d(xn,xn+1)
o(t)dtandso
from(5)
(6)
/
o
d(xn,xn+1)
o(t)dt≤G(/d(xn11,xn)
o
o(t)dt)forn≥1.
Nextwedefineasequence{Sn}ofrealnumbersbySn+11G(Sn)with
S11∫
Moreover,by(b)and(c),theseriesΣ
o
d(x,fx)
o(t)dt>0.By(a),wethenhave0<Sn+1<Sn<S1,n≥1.
∞
nl1Snconverges(see[1]).We
shallshowthat∫
o
d(xn,xn+1)
o(t)dt≤Sn+1,n≥1.From(6),wehave
∫
o
d(x1,x2)
o(t)dt≤G(∫d(x,fx)
o
o(t)dt)1G(S1)1S2andthedesiredinequal-
ityisvalidforn11.So,assumethatitistrueforsomen>1.From(6)
again,wehave∫
o
d(xn,xn+1)
o(t)dt≤G(∫d(xn11,xn)
o
o(t)dt)≤G(Sn)1Sn+1.
