YOGITA R SHARMA
ISSN: 2319 - 1163
Volume: 2 Issue: 1
1-4
FIXED POINT AND WEAK COMMUTING MAPPING Yogita R Sharma Abstract In this paper we prove some new fixed point theorems for weak commuting mapping on complete metric space. Our results generalize several corresponding relations in metric space of weak commuting mapping.
Keywords: Fixed point, weak commuting mapping, and metric space. AMS Classification No (2000): 47H10, 54H25 _____________________________________________***_______________________________________________ 1. INTRODUCTION In 1922, Banach [1] introduced very important concept of contraction mapping, which is known as a Banach contraction principle which states that “every contraction mapping of a complete metric space into itself has a unique fixed point”. Ciric [2] gave a generalization of Banach contraction principle as: Let S be a mapping of complete metric space X , d into itself, there exists an (1).
0,1 such that
d Sx, Sy
(3).
S X T X d Sx, Sy ad Tx, Sx d Ty, Sy bd Tx, Sy d Ty, Sx d Tx, Ty
For all x, y X , where
a, b, c are non-negative real numbers satisfying 0 2a 2b c 1, then S and T have a unique common fixed point.
d x, y , d x, Sx , d y, Sy , max d x, Sy , d y, Sx
For all
Let two continuous and commuting mappings S and T from a complete metric space X , d into itself satisfying the following conditions:
x, y X , then S has a unique fixed point.
In 1976, Jungck [4] investigated an interdependence between commuting mappings and fixed points, 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
Ranganathan [7] has further generalized the result of Jungck [4] which gives criteria for the existence of a fixed point as follows: 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 a real number
0,1
and a mapping
S : X X which commutes with satisfying:
S X T X
there exists an 0,1 and a mapping S : X X which commutes with T and satisfies
(4). and
S X T X and d Sx, Sy d Tx, Ty For all x, y X . Indeed S and T have a unique common fixed point if and only if (.2) holds for some 0,1 .
d Sx, Sy maxd Tx, Sx, d Ty, Sy , d Tx, Sy , d Ty, Sx , d Tx, Ty
(2).
Singh [9] further generalized the above result and proved the followings:
For all x, y X . Indeed commuting mapping S and T have a unique common fixed point. Fisher [3] proved following theorem for two commuting mappings S and T.
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YOGITA R SHARMA
ISSN: 2319 - 1163
Volume: 2 Issue: 1
1-4
Theorem 1 If S is a mapping and T is a continuous mapping of the complete metric space X into itself and satisfying the inequality.
d ST , TSy k d Tx, TSy d Sy, STx , For all x, y X , where 0 k 1 2 , then S and T have a
d S 2T 2 x, T 2 S 2 x
32 x 2 4 x 81 36 x 81
8x2 4 x 27 12 x 27
x x 4 x 27 12 x 27
(5).
unique common fixed point.
Fisher further extended his theorem and proved a common fixed point of commuting mappings S and T.
d S 2Tx, TS 2 x .
Theorem 2 If S is a mapping and T is a continuous mapping of the complete metric space X into itself and satisfying the inequality.
Implies that
d ST , TSy k d Tx, STx d Sy, TSy , for all x, y X ,where 0 k 1 2 , then S and T have a
2
(6).
d S 2T 2 x, T 2 S 2 x d S 2Tx, TS 2 x 2
d S Tx, TS x
8x2 4 x 27 12 x 27
8x 2 x 27 9 x 27
x x x 27 9 x 27
2. WEAK** COMMUTING MAPPINGS AND COMMON FIXED POINT
Definition 1 According to Sessa [8] two self maps S and T defined on metric space X , d are said to be weakly commuting maps if and only if d STx, TSx d Sx, Tx ,
d ST 2 x, T 2 Sx
d S 2Tx, TS 2 x d ST 2 x, T 2 Sx
2
2
d ST x, T Sx
For all x X .
Definition 2 wo self mappings S and T of metric space X , d are said to be weak** commuted, if S X T X and for any x X , d S 2T 2 x, T 2 S 2 x d S 2Tx, TS 2 x
d ST x, T Sx d STx, TSx d S x, T x 2
2
2
2
Definition3 A map S : X X , X being a metric space, 2 is called an idempotent, if S S . Example1. Let X [0,1] with Euclidean metric space x x and define S and T by Sx , Tx for all x X , x3 3 then 0,1 0, 9 10 where Sx 0,1 and Tx 0, 9 10
x x 4 x 27 12 x 27
unique common fixed point.
In this section Weak** commuting mappings and unique common fixed point theorem in metric space is established. First we give the following definitions:
x x 4 x 81 36 x 81
x x x 27 9 x 27
8x2 x 27 9 x 27
2 x2 x 9 3x 9
x x x9 3x 9
d STx, TSx
d ST 2 x, T 2 Sx d STx, TSx
d STx, TSx
x x x 9 3x 9
2 x2 x 9 3x 9
4 x2 9 4x 9
x x 9 4x 9
d T 2 x, S 2 x
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YOGITA R SHARMA
ISSN: 2319 - 1163
Volume: 2 Issue: 1
1-4
d x2n1 , x2n d x2n , x2n1 d x2n1 , x2n
d STx, TSx d T 2 x, S 2 x .
d x2n1 , x2n d x2n , x2n1
Using 0, 1 for x X , we conclude that definition (2) as follows:
d ST
d S 2 T 2 x, T 2 S 2 x d S 2 Tx, TS 2 x 2
x, T 2 Sx d STx, TSx d S 2 x, T 2 x
1 d x2n , x2n1 d x2n1 , x2n
d x , x 1 2n1 2n kdx2 n1 , x2n ,
d x2 n , x2 n1
for any x X . We further generalized the result of Fisher [3], Pathak [6], Lohani and Badshah [5] by using another type of rational expression.
where
k
. 1
Proceeding in the same manner
Theorem1. 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: (7).
d S 2T 2 x, T 2 S 2 y
d T d T
d S x d S
2
2
2
x, S T x
2
2
2
x, S T
y
3
2
2
2
y, T S y
2
2
2
2
y, T S
3
2
for all x, y in X , where , 0 with 2 and T have a unique common fixed point.
m
Also
d xn , xm d xi , xi 1 for m n . i n
2
d T x, S y , 2
d x2n , x2n1 k 2n1d x1 , x2 .
1, then S
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
lim x2n u lim x2 n1 n
n
Proof Let x be an arbitrary point X. Define
S T x x 2 n
and since T is continuous, we have
n
T 2 S 2T 2 x x2 n1 , where n 0,1,2,3,..., by contractive condition (7), 2
2n
or
d x2n , x2n1 d S 2T 2 x, T 2 S 2T 2 x
d S 2T 2 S 2T
n
2 n1
n
x, T 2 S 2 T 2 S 2T
x x
d T S T 2
2
2 n1
2
2
2
x, S T S T
2 n1
x
n
n
Further,
2 n1
d T 2 S 2T 2 n 1 x, S 2T 2 S 2T 2 n 1 3 n 1 d S 2T 2 S 2T 2 x, T 2 S 2 T 2 S 2 T 2 d T 2 S 2T 2 n 1 x, S 2T 2 S 2T 2 n 1 x 2 2 2 2 2 n 1 x, T 2 S 2 T 2 S 2 T 2 d S T S T
u lim x2 n1 lim T 2 x2 n T 2u .
d T S T
d x2n1 , S 2 u d T 2 S 2T 2
n 1
n 1
3 x 2 x
d x2n1 , x2n 3 d x2n , x2n1 3 d x , x 2 n 1 2n d x2n1 , x2n 2 d x2n , x2n1 2
2
for
2
n 1
2 n1
x, S 2 u
x, S 2 T 2u
u T 2u
d T u, S T u d S T x, T S T x d T u, S T u d S T x, T S T x d T u, S T x d T u, S T u d x , x d x , T u 2
2
2
3
2
2 n 1
2
2
2 n 1
3
2
2
2
2
2
2 n 1
2
2
2 n 1
2
2
2
2
2
2
2 n1
2 n 2
2 n3
2 n2
2
d u, S 2u d x2n2 , x2n3 d x2n2 , u
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YOGITA R SHARMA
ISSN: 2319 - 1163
Volume: 2 Issue: 1
1-4
n , it follows that d u, S u 0,
Thus, it follows that common fixed point.
which implies
REFERENCES
Taking limit as 2
d u, S u 0 and so u S u T u . 2
2
2
Now consider weak** commutativity of pair
S, T implies
S T u T S u, S Tu TS u, ST u T 2 Su and 2 2 so S Tu Tu and T Su Su . 2
that
2
2
2
2
2
2
Now,
d u, Su d S 2T 2u, T 2 S 2 Su
d T u, S T u d S d T u, S T u d S d T u, S Su
2
2
2
3
2
2
2
2
2
Su , T 2 S 2 Su 3 2 Su , T 2 S 2 Su 2
2
2
d u, u d Su, Su d u, Su d u, u d Su, Su
1 d u, Su 0 .
1 0.
This implies that
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.
u v .Hence S and T have a unique
[1]. Banach, S. Sur les operations dans les ensembles abstraits et leur application aux equations integrals, Fund. Math. 3(1922), 133-181. [2]. Ciric, Lj. B. A generalization of Banach contraction principle, Proc. Amer. Math. Soc. 45(1974), 267-273. [3]. Fisher, B. Common fixed point and constant mappings on metric spaces, Math. Sem. Notes, Kobe University 5(1977), 319-326. [4]. Jungck, G. Commuting mappings and fixed points, Amer. Math. Monthly 83(1976), 261-263. [5]. Lohani, P. C. And Badshah, V. H. Common fixed point and weak**commuting mappings. Bull. Cal. Math. Soc. 87(1995), 289-294. [6]. Pathak, H. K. Weak* commuting mappings and fixed points. Indian J. Pure Appl. Math. 17(1986), 201-211. [7]. Ranganathan, S. A fixed point theorem for commuting mapping. Math. Sem. Notes. Kobe. Univ. 6(1978), 351-357. [8]. Sessa, S. On a weak commutativity condition in fixed point considerations, Publ. Inst. Math. 32(1982), 149-153. [9]. Singh, S. L. On common fixed point of commuting mappings. Math. Sem. Notes. Kobe. Univ. 5(1977), 131-134.
Now suppose that v is another common fixed point of S and T. Then
d u, v d S 2T 2u, T 2 S 2 v
d T u, S T u d S d T u, S T u d S d T u, S v 2
v
2
2
2
3
2
v, T 2 S 2 v
3
2
2
2
2
2
v, T 2 S 2
2
2
d u, u d v, v d u, v
1 d u, v 0 . Since,
1 0 , then d u, v 0 .
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