<'lfiiPTl!tiX
CO/NC/OIN(l RNO CQMMON FIXID PQINTS_j_f! SRHS SPHCI
Y./.Uoebcls [64 J results were extended to L-space, metrk space, 2-mctric space and
multi valued contraction maps on metric spaces hy Okndo [I JJ], Singh nnd Vircndra
[198], Kulshreshta [109] and Naimpally et al. [12!J] respectively. In [88] Jungck
contraction principle appeared for a pair of continuous and commuting self maps .
Afier Jungck a spate of research papers appeared using these concepts in various ways
with several contractive type mappings by many authors. (Sec [25]-[27], [35] ,[41]
. [44-45), [53-54] ,[56-57]. [69], [91], [97], [99], [102[,[104-105), [123], [142]
. [141]. [136], [143]. [158], [H,7J, [183J,[I!Jij,[I!JJ],[J!J7j ,[198]).
Cho and Singh [32] and Murthy and Sharma [ 126] introduced the concepts of
commuting and wekly uniformly contraction maps respectively in saks space and
proved several fixed point theorems using these concepts.
In this chapter we introduce the concept of compatible mappings and compatible
mappings of type (A-I) and type (A-2) in saks space and give some relationship
bet ween these mappings. Wc have also established a scriesofcoincidcncc and common
fixed point theorems for compatible mappings of type (A-I) and type (A-2) in saks
space which extends and generalises many known results in saks space as well as metric
spaces. Our theorem extends the results ofDiviccaro et al[45], Fisher and Sessa (58]
, Gregus (64], Jungck (66] Mukherjee et al.(l23] and many others.
9. 2. In this section we introduce the concept of compatible mappings as well as
compatible mappings of type (A· I) and type (A-2) in saks space and derive some
relationship between them.
/Jejinition 9./fl26/: Let S and T be self maps of a saks space (X , N, ,N 2). The
mappings S and T arc said to be weakly uniformly contraction pair if
N,(STx - TTx) < N2(Sx • Tx) and N,(TSx - SSx) < N,(Sx - Tx)
Definition 9.2: LetS and T be self maps of a saks space (X, N, ,N2). The mappings
SandT are said to be compatible iflim, __ ~N,(STx, • TSx,) = 0, whenever {x") is a
sequence in X such that lim" ... Sx, =lim, ... Tx, = t for some t in X.
89
/Jefinitum 'J.3: LetS and T be sclfnu1ps of a saks spare (X, N, ,N 1). The pair c
mappings (S, T) is said to be compatible of type (A-I) if lim. ,. N1(STx. -lTx.) •'
• IVIII'IO<'VI'I
I~
It
I iN II NC''(III'ill'l' ioo X'"''" thnt
Iiiii
"
s~
. lion
II
TK II -I
fnr
Nlllllll
in X
/Jefinition 9. .J: Let SandT be self maps of a saks space (X, N1 ,N 1). The pair c
mappings (S ,T) is said to be compatible of type (A-2) iflim....n NI(TSx.- ssx.) =
, whenever (xn ) is a se<(Uencc in X such that lim
Sx = limn -. w
Txn = t for some
in X.
n<~>n
Clearly if a pair of mappings (S,T) is compatible of type (A-I) then the pair (T,S)
compatible of type (A-2). Further from the definitions its clear that ifS and Tweak!
uniformly contraction pair of mappings then the pair (S,T) are compatible of type (f.
I) as well as type (A-2). The following example illustrates that the implication is n<
reversible.
Example 9.1. Let X= [0, oo) and N, =d be the euclidean metlic. ConsiderthemappinJ
S and T defined by ,
Sx = 2.x 2 and Tx = 3.x 2•
Uy
routine check up one can easily verify that the pair (S,T) a1c compatible ol'lypc (1
I) and type (A-2) but S and Tare not weakly uniformly contraction maps.
/;xample 9.2. Let X= (O,oo) and N2 = d be the euclidean metric.
Consider the mappings S and T defined by ,
I
Sx
=
ifx E (0,1)
[ l+xifx>l.
I +x ifx E (0,1)
Tx
=
[I
ifxE[l,oo).
Consider the {x,) defined by x. = 1/n for all n. We see that lim •-·oo Sxn =lim nTx n =I.
90
Lim" .... N1(STx. - TTx.) ~ J/2 nnd Lim • .N,(TSx. - SSx.) • 0. lienee the pair
td 1111oppi11H~ (S,'I') i• l'ttlll)'lllddt• uf'IYJI" (i\·2) 1>111 11111 ttiiYJII' (I\· I)
We now cite the following p1upositions which gives the condition under which
definitions 9.2,9.3, and 9.4 becomes equivalent
l'roposition 'J. I: LetS and T be self maps of a saks space (X, N, ,N,).
a)
b)
c)
1fT is continuous then the pair ofmuppings (S, T) is compatible of type (AI) iff S and T are compatible.
lfS is continuous then the pair of mappings (S, T) is compatible of type (A2) iff S and Tare compatible
IfS and Tare continuous then the pair (S, T) is compatible of type (A-I) itT
the pair (S, T) is compatible of type (A-2).
l'roof: a) Let lim•. >~ Sx, =lim,_.," Tx. = t for some tin X, and let the pair(S, T)
be compatible of type (A-I). Since Tis continuous we have limn--,• (()TSx n = Tt and
limn - ..., TTx n = Tt.
lienee N2(STx.- TSx.) :5: N2(STx_- TTx,) + N1(TTx •.. TSxJ
Hence lim._ ·~N/STx.- TSx_) = 0.
Now let S and T be compatible. Then we have
N2(STxn - TTxn) :5: N2(STxn - TSx n) + N2(TSx n - TTx n)
b)
Let lim. ->wsx. = lim,_ Jx. = t for some t in X , and let the pair (S , T) be
compatible of type (A-2). Since Sis continuous we have limn->w STx. =Stand
lim,->«> SSx, =St. Hence N2(STx,- TSx,) :5: N2(STx.- SSx,) + N2(SSx.- TSx.).
Hence lim, _,wN2(STx,- TSx.) = 0.
Now letS and T be compatible. Then we have
:5: N,(TSx n - STx)
+ N,(STxn - SSx n).
N2(TSxn - SSx)
n
n
Hence lim._,~N,(TSx,- SSx.) = 0.
91
c) Let lim, .~Sx, -lim, .. _ Tx, at lor some 1in X ,and lc:t the pair (S, T) be compatible
of type (A-I) We have:
N,(TSx.- SSx n) !> N2(TSx n - STx n) + Nl (STx n - TTx n) + NJ(TTx n - SS n).
Since S and T arc continuous we see thntlim n- SSx n • lim n---·w STxn and
limn ...TSx, =lim, ..•,TTx,.llcncc lim, __ N,CfSx,- SSx 11 ) = 0. Therefore the pair
(S,T) iN compntil>lc of type (i\-2).
Next suppose the pair (S,T) is compatible of type (A-2). We have
N,(STx n - TTxn ) !> N,(STx n - TSx n) + N2(TSx n - SSxn ) + N2(SSx n - TTn).
Again using continuity ofS and T we get lim. _...N,(STx,.- TTx,) = 0. Therefore the
pair (S,T) is compatible of type (A-I).
As a direct consequence of proposition 9.1 we have the following
Proposition9.2. LetS and T be self maps of a metric space (X, N 1 ,N2). IfS and T
are continuous then the following statements are equivalent.
a. The pair (S,T) is compatible of type (A-I).
b. The pair (S,T) is compatible of type (A-2).
c. The mappings S and T are compatible.
Next we give some properties of compatible mappings of type (A-I} and type (A-2)
which will be used in our main theorem.
Proposition 9.3: LetS and T be self maps of a saks space (X, N1 ,N,). If the pair (S
, T) are compatible of type (A-I) and Sz = Tz for some z in X then STz = TTz.
l'rrH!f: Let {x,} be a sequence in X defined by x.
= z for n = 1,2, ... and let Tz = Sz.
Then we have lim,_ ..• Sx. = Sz & lim, .• Tx, = Tz. Since the pair (S, T) is compatible
of type (A-I) we have
N2(STz- TTz) =lim, __ ~ N2(STx,- TTx.) = 0.
Hence STz = TTz.
!'roposition9. .f: LetS and T be self maps of a metric space(X, N1 ,N2). Ifthe pair(S, T)
92
is compatible of type (A-2) ;md Sz ~ Tz for som~
1.
in X then TSz • SSz.
Y. J. In this section we prove some wincidencc point theorems nnd common fixed point
theo1 en1s in saks space which impwvcs many well known 1csults
We require the following well known lemma for our main theorem.
!.emma Y.l.fl3./f Let (X,, d)= (X, ,N 1 ,N,) be a saks space. Then the following
statements arc equivalent.
(I) N 1 is equivalent to N, on X.
(2) (X, N1) is n Banach space and N1 s N, on X.
(3) (X, N2) is a Frechet space and N2 s N1 on X.
Let A,B,S,T be mappings from a saks space (X, N1 ,N 2) into itself such that
(9.3. I) A(X) U B(X) S S(X) n T(X)
(9.3 .2) N2(Ax- By)=> a N,(Sx- Ty)}] + b max(N 2(Ax- Sx) , N2(By- Ty),
l/2. {N2(Ax- Ty) + N,(By- Sx)} ]
for all x,y in X , a,b > 0 and 0 < a+b < I .
For some arbitrary x0 in X , by (9.3. I) we choose x1 in X such that Ax0 = Tx 1 , and for
this x 1 there exists x2 such that Sx, = Bx 1• Continuing this process we define the
sequence {yJ in X such that
(9.3.3) Y2n =Ax'"= Tx2n+l and Y2n>1 = Bx,o+l
= Sx 2_. 2
Lemma 9.2. Let A.B,S and T be mappings from saks space (X ,N 1 ,N2) into itself
satisfying (4.1) and (4.2). Then the sequence ( y"} defined by (4.3) is a cauchy sequence.
Proof By (9.3.2} we have,
N2(Ax 2"- 8"'"+ 1) => a.N,(Sx'"- Tx'", 1)] + max[N2(Axa.- Sx'"), N2(Bxm+l - Txm+ 1),
1/2. {N 2(Ax1"- Tx,.,) + N2(Bx2n+l - Sx,.)}
or equivalently
93
lfN,(y,. - Y,.;.) > N,(Y,. - Y,._,) in the above inequality then we get
N,(Y,. - Y,•• ,) :5 (a+ b).N 2(Y,. - Y,•. ,), a contradiction.
lienee N2(y 2n
-
y2ntl ) :5 N2(y 2n - y~n-1 ) ·
Hence we get N2(yln - y2n•l ) <- (a+b) · Nl (yln - yln-1 )
It follows that N,(y, - Y•• ,) :5 (a+b)".N 2(y0
-
y1).
If m <. n then the repeated use of above inequality gives
:5
{(a+b)m + (a+b)"'" + (a+b)'"'' + ... + (a+b)"''}.N 2(y 1 - y0)
= (a+b)"-'/(1 - (a+b)). N,(y,- yJ Hence since 0 < a+b <I we see that sequence {y,}
is a cauchy sequence.
l">)"eP!I-\?'
Theorem 9.1. Let A.B,S and T be,From saks space (X, N1 ,N2) into itself satisfYing
(9.3 .I) , (9.3.2) and
(9.3.4) S(X) n T(X) is a complete subspace of X.
Then a) A and S have a coincidence point in X.
b) B and T have a coincidence point in X.
Proof By Lemma 9.2 sequence {y,} defined by (9.3.3) is a Cauchy sequence in
S(X) n T(X). Since S(X) n T(X) is a complete subspace of X, the sequence {y.}
must converge to some point say w in S(X) n T(X). On the other hand since
subsequences {y,.} and {y'"''} are also Cauchy sequences in S(X)nT(X), they should
also converge to he same point win X. Hence there should exist two points u and v
in X such that Su =wand Tv= w. Using (9.3.2) we get,
N,(Au- Bx,•• ,) :5 a.N,(Su - Tx,.. ,) + b.max[N 2(Au- Su) , N,(Bx'"'' - Tx'"',),
94
I/2.(N 1(Au- Tx,•• ,) + N1(13x,•. ,.Su))
J
as n -->oo we gel
N,(Au- w) :s n.O +I> max[N,(i\u- w) ,0, I/2.(N 1(i\u- w)
t
OJJ
i.e. N,(Au- w) s b.N 2(Au- w) a contradiction. Thcrcfhre Au= w.Hence Su • Au •
w. Similarly it canl>c shown that Uv =Tv= w .
......... {'i>'"')
•
1hew·em 9.2. Let A.B,S and T be)rom saks space (X, N, ,N,) into itself satisfYing
(9.31), (9.3.2), (9.3.4) and (9.35).
(9.3.5) The pairs (A,S) and (U,T) are compatible of type (A-I) or type (A-2).
Then A,B,S, and T have a unique common fixed point.
P1;ooj By Theorem 9.1 , Au= Su = w and Bv =Tv= w. If the pairs (A,S) and (B,T)
are compatible of type (A-I) then by proposition (9.3) we get ASu = SSu and BTv =
TTv, i.e. Aw = Sw and Bw = Tw. If the pairs (A,S) and (B,T) are compatible of type
(A-2) then by proposition (9.4) we get SAu = AAu and TBv= BBv, i.e. Sw= Aw and
Tw = Bw. i.e. in both the cases we get, Aw = Sw and Bw= Tw. Using (9.3.2) we get
N2(Ax 2, - Bw)
s a.N,(Sx'"- Tw) + b.max(N 2(Ax,,- Sx,.), N2(Bw- Tw),
l/2.{N 2(Ax,- Tw) + N,(Bw- Sx'"))]
as n ->oo we get ,
N2(w-13w)
we have
s (a+b).N 2(w -l3w) a contradiction. lienee l3w = w. Again by(9.3.2),
N,(Aw- Bw)
s a.N2(Sw- Tw) + b.,max[N 2(Aw- Sw), N2(Bw- Tw),
l/2.{N2(Axw- Tw) + N,(Bw- Sw))],
1.c. N,(Aw- w)
s (a+b).N 2(Aw- w), a contradiction.
Hence Aw = w. Therefore w is common fixed point of A,B,S and T.
The following results follows immediately upon noting that under
95
the given hypothesis the scllucnce given by (9.3 J) is a Cauchy sequence in
S(X) n T(X)
.........
,.,, ...
,~
'171<'m'<'lll IU. Let A. U.S and T be,.from snks space (X. N, .N,) into itself satisfying
(9.3.1). (9.3,4). (9.3 5) llllll
('I I
r,) N,(Ax -By) •.,j.(N 1 (S~- ly). N,(A~- Sx). N,(lly- Ty), N,(Ax • Ty),
N,(IJy- Sx))
for all x,y in X where .p : [O,w)' --> [O,oo) is
I. nondecrensing and upper scmicontinuous in each coordinate variable.
2. for each t > 0, y(t) =max{ ~·(O,O,t,t,t), tj>(t,t,t,2t,O), tjJ(t,t,t,0,2t) }< t
Then A,B,S, and T have a unique common fixed point.
Remark 1. For S =T Theorem 9.3 includes the results ofCho and Singh [32].
Remark 2. IfX is a metric space and N2(x-y) is replaced by d(x,y) in (9.3.6) we get the
result ofKang et al.[93].
.-..,e.p~
... ')
Theorem 9.4. Lct.A.B,S and T bcjrom saks space (X. N1 ,N 2) into itself satisfYing
(9.3.1), (9.3.4), (9.3.5) and
(9.3. 7) N2(Ax- By)
s ktj>(N 2(Sx- Ty), N2(Ax- Sx), N2(By- Ty) ,
(l/2) (N2(Ax- Ty) + N2(By- Sx)))
for all x,y in X where .p: [O,CXJ)'--> [O,oo) is
I. nondecreasing and upper semicontinuous in each coordinate variable .
2. for each t > 0, y(t) = max{tjJ(O,O,t,t,t), tj>(t,t,t,2t,O), tjJ(t,t,t,0,2t)) < ~
•
Then A,B,S, and T have a unique common fixed point.
Remark 3. Theorem 9.4 extends and generalises theorem 3 of Murthy and Sharma
[126] by replacing weakly uniformly contraction pair of maps with compatible maps
of type (A-I) or type (A-2).
96
·~··~rr··,~
1h<·m·em 'J. 5. Let A.ll,S and l bc~rom Silks spa<:~: (X . N, . N,J intuit self satisfying
('I 3 I) , ('U 4), ('I 3 5) and
(9 3 !!) N,'(Ax- By)$ <j>(N,'(Sx- Ty), N1(Ax- Sx), N,(lly- Ty), N,(Ax- Ty),
N2(lly- Sx))
for all x,y in X where .p:
[Oy~)' ---> [O,oo)
is
I. nondecreasing and upper semicontinuous in each coordinate variable .
2. for each t > 0, y(t) = max{<j>(O,O,t,t,t), lj>(t,t,t,2t,O), lj>(t,t,t,0,2t)}< l
Then A,B,S, and T have a unique common fixed point.
Remark .f. Theorem 9.5 extends and generalises Theorem 4 of Murthy and Sharma
[ 126] by replacing weakly uniformly contraction pair of maps with compatible maps
of type (A-I) or type (A-2).
As an immediate consequence of Theorem 9.2 we have the following :
rw')"\FP~o·
Corollary 9.6. Let A.B,S and T ber{rom saks space (X, N, ,N 1) into itself satisfying
(9.3.1), (9.3.4), (9.3.5) and
(9.3. 9) (N2 2(Ax- By))• :$ a.(N2 2(Sx- Ty))• for all x,y in X, 0 <a< 1 and p 1.
Then A,B,S, and T have a unique common fixed point in X.
97