Common fixed point and weak commuting mappings

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
43
Common Fixed Point and Weak**
Commuting Mappings
Shweta Gagrani
e-mail: shweta_gagrani@yahoo.com
Abstract
Existence of common fixed points of weak** commuting mappings which satisfies the contractive condition
involving pair of mappings in a complete metric space under certain is shown.
Key words: commuting mappings, weak ** commuting mapping.
1. Introduction
A study of the common fixed points and weak**
commuting mappings is fascinating field of research
lying at the intersection of non-linear analysis. A wide spread interest in the domain and vast amount of
mathematical activity have led to many remarkable new results.
In 1976, Jungck [4] investigated and found interdependence between commuting mappings and
common fixed points and proved the followings:
Let T be a continuous mapping of a complete metric space (X, d) into itself. Then T has a fixed point in
X, if and only if there exists an α ∈ (0, 1) and a mapping S : X → X which commutes with T and satisfies:
(1) S(X) ⊂ T(X) and d(Sx, Sy) ≤ d(Tx, Ty)
For all x, y in X. Indeed, S and T have a unique common fixed point if and only if (1) holds for some α ∈ (0,
1).
Further, in 1977, Singh [10] generalized the above result and proved that two continuous and
commuting mappings from a complete metric space into itself satisfies some conditions, then two commuting
mappings have a unique common fixed point.
Das and Vishwanathana Naik [1] have proved a theorem for two commuting mappings. Fisher [2]
proved a common fixed point of commuting mappings, Rhoades and Seesa [8] established some fixed point
theorems for three pair wise weakly commuting self maps satisfying a very general contractive definitions. Khan
and Imdad [5], considering a pair of self maps {A, T} of metric space (X, d) satisfying a weaker condition the
commutativity: namely weak*
commuting pair of mappings, that is
d(ATx, TAx) ≤ d(A2
x, T2
x)
For each x in X.
B. Fisher [2] has been proved following theorem for two commuting mappings T and S.
If S is a mapping and T is a continuous mapping of the complete metric space into itself and satisfying
the inequality :
(2) d(STx, TSy) ≤ k {d(Tx, TSy) + d(Sy, STx)}
for all x, y in X, where 0 ≤ k ≤ 1/2, then S and T have a unique common fixed point.
In 1986, Pathak [7] has been further generalized a result of Khan and Imdad [5] by considering a pair of
self maps {A, T} of a metric space (X, d) satisfying a weaker condition, then commutativity: namely, weak*
commuting pair of mappings, that is
d(ATx, TAx) ≤ d(A2
x, T2
x)
for each x ∈ X.
In 1995, Lohani and Badshah [6] further generalized the result of B. Fisher[2, 3]
The purpose of this note is to prove some results concerning fixed points of weak**
commuting
mappings defined on complete metric spaces and satisfying some new functional inequality.
Definition 1.1. According to Seesa [9] two self maps S and T defined on metric space (X, d) are said to be
weakly commuting maps iff
d(STx, TSx) ≤ d(Sx, Tx)
for all x in X.
Defintion 1. 2. Two self mappings S and T of metric space (X, d) is called weak**
commuted, if S(X)
⊂ T(X) and for any x ∈ X,
d(S2
T2
x, T2
S2
x) ≤ d(S2
Tx, TS2
x) ≤ d(ST2
x, T2
Sx) ≤ d(STx, TSx) ≤ d(S2
x,
T2
x)
Definition 1.3. A map S : X → X, X being a metric space, is called an idempotent, if S2
= S.
We further generalize the result of Fisher [2, 3], Pathak [7] and Lohani & Badshah [6] by using another
type of rational expression.
Theorem 1.1. If S is a mapping and T is a continuous mapping of complete metric space {S, T} is
weak**
commuting pair and the following condition :

44
),(
),(),(
),(),(),(),d(T
),(
22
222222
22222222222
2222
2
ySxTd
xTSySdySTxTd
xTSySdySTySdySTxTdxTSx
ySTxTSd
β
α
+
+
+
≤
for all x, y in X, where 0 ≤ α + β < 1, then S and T have a unique common fixed point.
Proof. Let x be an arbitrary point in X. Define
(S2
T2
)n
x = x2n or T2
(S2
T2
)n
x = x2n+1
Where n =0, 1, 2, 3…., by contractive condition (A),
( ) ( )
( ) ( )
( )
( )xTSTSxTSTd
xTSTSxTSTSdxTSTSTxTSTd
xTSTSxTSTSdxTSTSTxTSTSd
TSTSTxTSTdxTSTSxTSTd
xTSTSTxTSTSdxxd
nn
nnnn
nnnn
xnnnn
n
nn
122221222
12222122221222221222
122221222212222212222
1222221222122221222
1222222222
122
)(,)(
))(,)(())((,)((
)(,)()((,)(
)((,)()(,)(
)))((,)((),(
−−
−−−−
−−−−
−−−−
−
+
+
+
+
=
≤
β
α
))(,)(())(,)((),(
))(,)((
))((,)((
))((,)(())(,)((
122221222122221222
122
122221222
122222122122221222 2
xTSTSxTSTdxTSTSxTSTdxxd
xTSTSxTSTd
xSTTTSxSTTd
xTSTSTxTSTdxTSTSxTSTd
nnnn
nn
nn
nnnn
−−−−
+
−−
−−−−
+≤
+
≤
βα
β
α
( )
( ) ).,(
))(,(
212
22
12
nn
n
n
xxd
xTSxd
−
−
+≤
+≤
βα
βα
Proceeding in the same manner
d(x2n, x2n+1) < (α + β)2n-1
d(x1, x2).
Also d(xn, xm) ≤ )( 1, +
=
∑ i
m
ni
i xxd for m > n.
Since k < 1, it follows that the sequence {xn} is Cauchy sequence in the complete metric space X and so it
has a limit in X, that is
limn→∞x2n = u = limn→∞x2n+1
and since T is continuous, we have
u = limn→∞x2n+1 = limn→∞T2
(x2n) = T2
u.
Further,
d(x2n+1, S2
u) = d(T2
(S2
T2
)n+1
x, S2
u)
= d(T2
(S2
T2
)n+1
x, S2
(T2
u)) for u = T2
u

45
),(
),(),(
)],(),(),(),([
),)(
))(,(),)((
)))(,(),())(,)())(,)((
22
32
2
22
32
22
223222
2122
1222222122
12222222221221222122
uxd
xuduSxd
xuduSuduSxdxxd
uTxTSd
xTSTuTduTSxTSd
xTSTuTduTSuTduTSxTSdxTSTxTSd
n
nn
nnnn
n
nn
nnnn
+
++
++++
+
++
++++
+
+
+
=
+
+
+
≤
βα
β
α
taking limit as n → ∞, it follows that
d(u, S2
u) = 0.
which implies
d(u, S2
u) = 0 and so u = S2
u = T2
u.
Now consider weak**
commutativity of pair {S, T} implies that S2
T2
u = T2
S2
u, S2
Tu = TS2
u, ST2
u =
T2
Su and so S2
Tu = Tu and T2
Su = Su. Now
d(u, Su) = d(S2
T2
u, T2
S2
(Su))
( )
0),(
0),(1
0
),(
),(),(
),(),(),(),(
),(
),())(,(
),(),())(,(),(
))(,(
)),(())(,(
)]),(())(),(())(,(),([
2
222
22
222222
222222222222
≤⇒
≤−⇒
=
+
+
+
=
+
+
+
=
+
+
+
≤
Suud
Suud
Suud
uSudSuud
uSudSuSudSuuduud
Suud
uSudSuSud
uSudSuSudSuTSuduSud
SuSuTd
uTSSuSdSuSTuTd
uTSSuSduSSTuSSduSSTuTduTSuTd
β
βα
βα
β
α
Hence Su = u, similarly we can show that Tu = u. Hence u is a common fixed point of S and T.
Now suppose that x is an another common fixed point of S and T. Then
),(),( 2222
vSTuTSdvud =
),(
),(),(
),(),(),(),( 22
222222
222222222222
vSuTd
uTSvSdvSTuTd
uTSvSdvSTvSdvSTuTduTSuTd
βα +
+
+
≤
( )
0),(
0),(1
),(
),(),(
),(),(),(),(
≤
≤−
+
+
+
≤
vud
vud
vud
uvdvud
vudvudvudvud
β
βα
Then it follows that u = v. Hence S and T have a unique common fixed point.
Theorem 1.2. If S is mapping and T is a continuous mapping of a complete metric space X into itself
and satisfying {S, T} is weak**
commuting pair and the following condition :
),(
),(),(
)],([)],([
),( 22
222222
222222
2222
ySxTd
ySTySdxTSxTd
ySTySdxTSxTd
ySTxTSd βα +
+
+
≤ (B)
for all x, y in X, where α , β ≥ 0 with 2α + β < 1, then S and T have a unique common fixed point.

46
Proof. Let x be an arbitrary point in X. Define
(S2
T2
)n
x = x2n or T2
(S2
T2
)n
x = x2n+1
Where n =0, 1, 2, 3…., by contractive condition (B),
( )xTSTSxTSTd
xTSTSTxTSTSdxTSTSxTSTd
TSTSTxTSTSdTSTSxTSTd
xTSTSTxTSTSd
xTSTxTSdxxd
nn
nnnn
nnnn
nn
nn
nn
122221222
12222212222122221222
2122222122222122221222
12222212222
22222
122
)(,)(
)))((,)(())(,)((
,)}])((,)({([}])(,)({([
))((,.)((
)))(,)((),(
−−
−−−−
−−−−
−−
+
+
+
+
=
=
β
),()],(),([
),(
),(),(
)],[()],([
212122212
212
122212
2
122
2
212
nnnnnn
nn
nnnn
nnnn
xxdxxdxxd
xxd
xxdxxd
xxdxxd
−+−
−
+−
+−
++≤
+
+
+
≤
βα
βα
( ) ),()(),(1
),(),()(
212122
122212
nnnn
nnnn
xxdxxd
xxdxxd
−+
+−
+≤−
++≤
βαα
αβα
),(
1
),( 212122 nnnn xxdxxd −+
−
+
≤
α
βα
),( 212 nn xxkd −≤
where k =
α
βα
−
+
1
.
Proceeding in the same manner, we have
d(x2n, x2n+1) ≤ k2n-1
d(x1,x2).
Also
d(xn, xm) ≤ ∑=
+
m
ni
ii xxd ),( 1 for m > n.
Since k < 1, it follows that the sequence {xn} is Cauchy sequence in the complete metric space X and so it has a
limit in X, that is
limn→∞x2n = u = limn→∞x2n+1
and since T is continuous, we have
u = limn→∞x2n+1 = limn→∞T2
(x2n)m = T2
u.
Further
d(x2n+3, S2
u) = d(T2
(S2
T2
)n+1
x, S2
u)
= d(T2
(S2
T2
)n+1
x, S2
(T2
u)) (since u = T2
u)
))(,(
})(,){(),(
])(,)[()],([ 1222
1222122222
212221222222
xTSuTd
xTSTxTSduTSuTd
xTSTxTSduTSuTd n
nn
nn
+
++
++
+
+
+
≤ βα
),()],())(,([ 2
223222
222
uTxdxxduTSuTd nnn +++ ++≤ βα
),()),(),(( 223222
2
uxdxxduSud nnn +++ ++≤ βα
Taking limit as n → ∞, it follows that
d(u, S2
u) ≤ 0,
which implies that d(u, S2
u) = 0 and so that u = S2
u = T2
u.
Now consider weak**
commutativity of pair {S, T}, implies that S2
T2
u = T2
S2
u , S2
Tu, TS2
u, ST2
u = T2
Su and
so S2
Tu = Tu and T2
Su= Su.
Now d( u, Su) = d(S2
T2
u, T2
S2
(Su))
))(,(
))(),((),(
)](),([)],([ 22
222222
22222222
SuSuTd
SuSTSuSduTSuTd
SuSTSuSduTSuTd
βα +
+
+
≤

47
),(
),(),(
)],([)],([
Suud
SuSuduud
SuSuduud
βα +
+
+
≤
(1- β) d(u, Su) ≤ 0
this implies that (1-β) ≠ 0. Hence d(u, Su) = 0 or Su = u.
Similarly we can show that Tu = u. Hence u is a common fixed point of S and T. Now suppose that v is another
common fixed point of S and T, then
d(u, v) = d(S2
T2
u, T2
S2
v)
),(
),(),(
)],([)],([ 22
222222
22222222
vSuTd
vSTvSduTSuTd
vSTvSduTSuTd
βα +
+
+
≤
≤ ),()],[()],([ vudvvduud βα ++
(1- β) d(u, v) ≤0.
Since (1-β) ≠ 0, then d(u, v) = 0. Thus it follows that u = v. Hence S and T have a unique common fixed point.
Example 1.1. Let X = [0,1] with Euclidean metric space and define S and T by
2
,
2
x
Tx
x
x
Sx =
+
=
for all x ∈ X, then [0, 1/5] ⊂ [0,1/4], where Sx = [0, 1/5] and Tx = [0, 1/4]
168163
),( 2222
+
−
+
=
x
x
x
x
xSTxTSd
)168()163(
5 2
+−+
=
xx
x
≤
)84)(82(
2
++ xx
x
=
8482 +
−
+ x
x
x
x
= d (S2
Tx,TS2
x)
⇒ d(S2
T2
x, T2
S2
x) ≤ d(S2
Tx, TS2
x)
d(S2
Tx, TS2
x) =
8482 +
−
+ x
x
x
x
=
)84)(82(
2 2
++ xx
x
≤
)84)(8(
3 2
++ xx
x
=
848 +
−
+ x
x
x
x
= d(ST2
x , T2
Sx)
⇒ d(S2
Tx, TS2
x) ≤ d(ST2
x, T2
Sx)
d(ST2
x, T2
Sx) =
848 +
−
+ x
x
x
x
=
)8)((8(
3 2
++ xx
x
≤
)42)(4(
2
++ xx
x
=
424 +
−
+ x
x
x
x
= d(STx, TSx)

48
⇒ d(ST2
x, T2
Sx) ≤ d(STx, TSx)
d(STx, TSx) =
424 +
−
+ x
x
x
x
=
)42)(4(
2
++ xx
x
≤
)43(4
3 2
+x
x
=
434 +
−
x
xx
= d(T2
x, S2
x)
⇒ d(STx, TSx) ≤ d(T2
x, S2
x)
using [0, 1] for x ∈X, we conclude that definition (1.2) as follows :
d(S2
T2
x, T2
S2
x) ≤ d(S2
Tx, TS2
x) ≤ d(ST2
x, T2
Sx) ≤ d(STx, TSx) ≤ d(T2
x, S2
x) for any x∈ X.
References
[1] Dass K.M. and Vishwanathan, Naik, K. Common fixed point theorem for commuting maps on a metric
space, Proc. Amer. Math. Soc. 77(1979), 369.
[2] Fisher, B. Common fixed point theorem, Indian Jour. Pure Appl. Math. 20(1978), 136.
[3] Fisher, B. Common fixed point of commuting mappings, Bull. Inst. Math. Acad. Sincia 9(1981), 399.
[4] Jungck, G. Commuting maps and fixed points, Amer. Math. Monthly 83(1976), 261.
[5] Khan, M.S. and Imdad, M. Fixed point theorems for a class of mappings, Indian Jour. Pure Apll.
Math.14(1983), 1220.
[6] Lohani P.C. and Badshah, V.H. Common fixed point and weak**
commuting mappings, Bull. Cal.
Math. Soc. 87(1995), 289.
[7] Pathak, H.K. Weak*
commuting mappings and fixed points, Indian Jour. Pure Appl. Math. 17(1986),
201.
[8] Rhoades, B.E. and Seesa, S. Common fixed point theorems for three mappings under a weak
commutativity condition, Indian Jour. Pure Appl. Math. 17 (1986), 47.
[9] Seesa, S. On a weak commutativity condition of mappings in a fixed point consideration, Publ. Inst.
Math. 32(1982), 149.
[10] Singh, S.L. On common fixed points of commuting mappings, Math. Sem. Notes, Kobe Univ. 5(1977),
131.

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