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Somefixedpointtheoremsformappings...
13
≤G(/M(xm(k)11,xn(k)11)
o
o(t)dt)
≤G(/max{d(xm(k)11,xn(k)11),d(xm(k)11,xm(k)),d(xn(k)11,xn(k))}
o
o(t)dt)
≤G(/max{d(xm(k)11,xm(k))+ε,d(xm(k)11,xm(k)),d(xn(k)11,xn(k))}
o
o(t)dt).
Using(11),(12),(13)and(14),wehave
/
o
ε
o(t)dt≤/
o
d(xm(k),xn(k))
o(t)dt≤G(/
o
ε
o(t)dt),
whichisacontradiction.Therefore{xn}isCauchy.SinceXiscomplete
{xn}convergestosomez∈X.Thereforewecancompletetheproofasin
theproofofTheorem2.
.
WecanprovethefollowingtheoremusingtheproofsofTheorem3and
Theorem4.
Theorem5.Let(X,d)becompletemetricspace,f,hcontinuousself-
mappingsofXsatisfying
(15)
/
o
d(hx,hy)
o(t)dt≤G(/M
o
∗(x,y)
o(t)dt)
forallx,y∈X,whereoandGareasinTheorem4and
M
∗(x,y)1max{d(fx,fy),d(fx,hx),d(fy,hy)}.
Supposealsothat
(ź)fandhcommute,
(źź)h(X)⊆f(X).
ThenfandhhaveauniquefixedpointinX.
Remark3.Ifo(t)11inTheorem5,wehaveageneralizationofmain
theoremof[8].
Example.LetX1{1
n:n12,3,...}∪{0}withthemetricinducedby
d(x,y)1|x−y|,thussinceXisaclosedsubsetofitisacompletemetric
space.Weconsidernowtwomappingsh,f:X→Xdefinedby
hx1{
0,
n+1,x11
1
x10
n
andfx1x.
