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10
AbdelkrimAlioucheandValeriuPopa
Now,assumethatD(TxjGy)+D(SyjFx)/=0forallxjy∈X.Let
xo∈Xbeanarbitrarypoint.By(3)andLemma1,wedefineasequence
{yn}inXby
y2n=Tx2n∈Gx2n11j
and
y2n+1=Sx2n+1∈Fx2n
d(y2njy2n+1)≤kH(Fx2njGx2n11)j
d(y2n+1jy2n+2)≤kH(Fx2njGx2n+1)jforn=1j2j...
.
Using(6)and(01),wehave
0≥0(H(Fx2njGx2n11)jd(Tx2njSx2n11)jD(Tx2njFx2n)j
D(Sx2n11jGx2n11)jD(Tx2njGx2n11)jD(Sx2n11jFx2n))
≥0(H(Fx2njGx2n11)jd(y2n11jy2n)jd(y2njy2n+1)j
d(y2n11jy2n)j0jd(y2n11jy2n+1))
≥0(H(Fx2njGx2n11)jd(y2n11jy2n)jd(y2njy2n+1)j
d(y2n11jy2n)j0jd(y2n11jy2n)+d(y2njy2n+1)).
By(0b),weobtain
d(y2njy2n+1)≤hd(y2n11jy2n).
Inthesamemanner,applying(6)weget
0≥0(H(Fx2njGx2n+1)jd(Tx2njSx2n+1)jD(Tx2njFx2n)j
D(Sx2n+1jGx2n+1)jD(Tx2njGx2n+1)jD(Sx2n+1jFx2n))
≥0(H(Fx2njGx2n+1)jd(y2njy2n+1)jd(y2njy2n+1)j
d(y2n+1jy2n+2)jd(y2njy2n+2)j0)
≥0(H(Fx2njGx2n+1)jd(y2njy2n+1)jd(y2njy2n+1)j
d(y2n+1jy2n+2)jd(y2njy2n+1)+d(y2n+1jy2n+2)j0).
Therefore
d(y2n+1jy2n+2)≤hd(y2njy2n+1).
andso
d(ynjyn+1)≤hd(yn11jyn).
Then,{yn}isaCauchysequenceinX.AssumethatS(X)iscomplete.
Then,{y2n+1}convergestoz∈S(X)andsothereexistsp∈Xsuchthat
z=Sp.Also,{y2n}convergestozsince
d(y2njz)≤d(y2njy2n+1)+d(y2n+1jz)
