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Coincidenceandcommonfixedpoint...
(4)
Thepairs(TjF)and(SjG)areR-weaklycommutingoftype(AT)
attheircoincidencepoints.
7
(5)
H(FxjGy)≤a
D2(FxjSy)+D2(GyjTx)
D(FxjSy)+D(GyjTx)
+bd(TxjSy)j
forallxjy∈X,x/=y,Fx/=FyandGx/=Gy,whereajb>0and
a+2b<1,wheneverD(FxjSy)+D(GyjTx)/=0andH(FxjGy)=0
wheneverD(FxjSy)+D(GyjTx)=0.Then,thereexistsz∈Xsuchthat
z=Tz=Sz∈FznGz.
ThistheoremgeneralizesTheorems3.1and3.2of[1].
In[13]and[14],thestudyoffixedpointsformappingssatisfyingim-
plicitrelationswasintroducedandthestudyofapairofhybridmappings
satisfyingimplicitrelationswasinitiatedin[15].
Itisourpurposeinthispapertoprovecoincidenceandcommonfixed
pointtheoremsfortwopairsofhybridmappingssatisfyingimplicitrelations
usingtheconceptofR−weakcommutativityoftypeATwhichgeneralize
theresultsof[1-3],[8],[12-16]and[21].
2.Implicitrelations
Let06thefamilyofallrealcontinuousmappings0(t1jt2jt3jt4jt5jt6):
R6
+→Rsatisfyingthefollowingconditions:
(01):0isincreasinginvariablet1anddecreasinginvariablest3jt4jt5
andt6.
(02):thereexists0≤h<1andk>1suchthat
(0a):u≤ktand0(tjujujuju+uj0)≤0or
(0b):u≤ktand0(tjujujuj0ju+u)≤0
impliesu≤hu.
Example1.0(t1jt2jt3jt4jt5jt6)=t1−at2−b(t3+t4)−C(t5+t6),
ajC>0,b≥0anda+2b+2C<1.
(01):Obviously.
(02):Let1<k<
a+2b+2C
1
,u≤ktand0(tjujujuju+uj0)=
t−au−b(u+u)−C(u+u)≤0.Then,u≤kt≤kau+kb(u+u)+kC(u+u)]
andsou≤hu,whereh=
1−(kb+kC)
k(a+b+C)
<1.Similarly,u≤ktand
0(tjujujuj0ju+u)≤0impliesu≤hu.
a<1.
Example2.0(t1jt2jt3jt4jt5jt6)=t1−amax{t2jt3jt4j
t5+t6
2
},0<
