International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 1- 5 © TJPRC Pvt. Ltd.,
COMMON FIXED POINT THEOREM IN G-METRIC SPACE VISHAL GUPTA, A. K. TRIPATHI, RAMAN DEEP & RICHA SHARMA Department of Mathematics, Maharishi Markandeshwar University,Mullana, Ambala, Haryana, India
ABSTRACT In this paper, we give some common fixed point theorems for weakly compatible self mappings satisfying certain contractive condition in G-metric space
KEYWORDS: Weakly compatible mapping, G-metric space, common fixed point INTRODUCTION Metric fixed point theory is an important mathematical discipline because of its applications in areas such as variational and linear inequalities, optimization, and approximation theory. The concept of 2-metric space was initially given by Gahler [3] whose abstract properties were suggested by the area of function in Euclidean space. In [4] the author has introduced a new structure of a D-metric space which is a generalized idea of the ordinary metric spaces. Dhage [1] introduced the definition of D-metric space and proved many new fixed point theorems in D-metric spaces. Recently, Mustafa and Sims [2] presented a new definition of G-metric space and made great contribution to the development of Dhage theory. Moreover, many theorems were proved in this new setting with most of them recognizable as a counterpart of well-known metric space theorems.
PRELIMINARIES Definition 2.1: Let X be a non empty set and let G: X × X × X → R+ be a function satisfying the following 1) G(x, y, z) = 0 if x = y = z 2) 0 < G(x, x, y) for all x, y ∈ X with x ≠ y 3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with z ≠ y 4) G(x, y, z) = G(x, z, y) = G(y, z, x) = …. 5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X Then the function G is called a generalized metric or a G-metric on X and the pair (X, G) is called G-metric space. Definition 2.2: A sequence
{ xn } in a G-metric space X is:
(i) A G-Cauchy sequence if, for any ε >0, there is an numbers)
such that
n0 in N (the set of natural
G ( xn , xm , xl ) < ε for all n, m, l ≥ n0
(ii) a G-convergent sequence if, for any ε >0, there is an x in X and an
n0 in N, such
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Vishal Gupta, A. K. Tripathi, Raman Deep & Richa Sharm
that for all
n, m ≥ n0 , G ( x, xn , xm ) < ε .
A G-metric space on X is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X. Definition 2.3: Let A and S be mappings from a G-metric space
( X ,G)
into itself. Then the mappings are
said to be weakly compatible if they commute at their coincidence point, that is, Ax=Sx implies that ASx=SAx.
( X , G ) be a G-metric space. Then the following are equivalent: (1) { xn } is G-convergent to x, (2) G ( xn , xn , x ) → 0 as n → ∞ , (3) G ( xn , x, x ) → 0 as n → ∞ , (4) G ( xm , xn , x ) → 0 as m, n → ∞ .
Proposition 2.4[6]: Let
( X , G ) be a G-metric space. Then, for any x, y, z, a in X it follows that: If G ( x, y, z ) = 0 , then x = y = z , G ( x , y , z ) ≤ G ( x , x, y ) + G ( x, x, z ) , G ( x, y, y ) ≤ 2G ( y, x, x ) , G ( x , y , z ) ≤ G ( x , a , z ) + G ( a, y , z ) , G ( x, y , z ) ≤ 2 3 ( G ( x , y , a ) , G ( x, a , z ) , G ( a , y , z ) ) G ( x, y , z ) ≤ ( G ( x, a , a ) , G ( y , a , a ) , G ( z , a , a ) )
Proposition 2.5[6]: Let (1) (2) (3) (4) (5) (6) Definition 2.6: Let
( X ,G)
be a G-metric space, for
A, B, C ⊂ X define
δ G ( A, B, C ) = sup {G ( a, b, c ) ; a ∈ A, b ∈ B, c ∈ C}
If A consists of a single point a, we write
δ G ( A, B, C ) = δ G ( a, B, C )
If B and C also consist of a single point b and c respectively, we write
δ G ( A, B, C ) = G ( a, b, c )
In particular for A =
B = C ∈ X , an = δ G ( A) = sup {G ( a, b, c ) , a, b, c ∈ A}
It follows that from the definition that (i)
If
A ⊆ B then δ G ( A) ≤ δ G ( B )
For a sequence
An = { xn , xn+1 , xn + 2 .........} in G-metric space ( X , G ) , let
an = δ G ( An ) ,for all n ∈ N ,then (a) Since (b) (c)
A n ⊇ An +1 then an +1 ≤ an
G ( xl , xm , xk ) ≤ δ G ( An ) = an for every l , m, k ≥ n 0 ≤ δ G ( An ) = an and an +1 ≤ an for every n ≥ 1
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Common Fixed Point Theorem in G-Metric Space
{an }
Therefore
is decreasing and bounded for all
n ∈ N and so there exist an 0 ≤ a such
that lim an = a . n →∞
1. Main Results Lemma 3.1: Let
( X ,G)
be a G-metric space .If lim an = 0 then sequence n →∞
sequence. Proof: Since lim an = 0 , we have that for every ε n →∞
n > m, an − 0 < ε that is an = δ G ( An ) < ε Then for l , m, k ≥ n > m , we have
{
{ xn }
is a Cauchy
> 0 , there exist a m ∈ N such that for every
}
G ( xl , xm , xk ) ≤ sup G ( xi , x j , x p ) : xi , x j , x p ∈ An = an < ε
Therefore
{ xn }
is a Cauchy sequence in X.
Theorem 3.1: Let S, R, T, U be self mapping of a complete G metric space (i)
SR ⊆ TU and TU ( X ) is closed subset of X.
(ii)
The pair
(iii)
( X ,G)
satisfying
( SR, TU ) is weakly compatible, G ( SRx, SRy, SRz ) ≤ Φ ( G (TUx, TUy, TUz ) ) for every x, y , z ∈ X , where Φ :[0, ∞) → [0, ∞ ) is a non-decreasing continuous function with Φ ( t ) < t
for
t >0 ( S , R ) , (T ,U ) are commutative , then S,R,T,U have a common fixed point in X. every
(iv) Proof: Let
x0 be an arbitrary point in X. by (i) we can choose a point x1 in X such that
y0 = SRx0 = TUx1 and y1 = SRx1 = TUx2 . In general, there exist a sequence yn = SRxn = TUxn +1 ,
{ yn }
such that
for n= 0,1,2,3…
{ yn } is a Cauchy sequence An = { yn , yn+1 , yn+ 2 .......} and an = δ G ( An ) , n ∈ N .Then we know that
We prove that the sequence Let
lim an = a for some n →∞
a ≥ 0 . Taking x = xn + k , y = ym + k and z = zl + k in (iii) for k ≥ 1 and m, n, l ≥ 0 , we have
G ( yn + k , ym+ k , yl + k ) = G ( SRxn + k , SRxm+ k , SRxl + k ) ≤ Φ ( G (TUxn+ k , TUxm+ k TUxl + k ) ) = Φ ( G ( yn+ k −1 , ym+ k −1 , yl + k −1 ) )
Now we claim Since Ak −1
G ( yn + k −1 , ym + k −1 , yl + k −1 ) ≤ ak −1 for every n, m, l ≥ 0 .
= { yk −1 , yk , yk +1.........} , ak −1 = sup {G ( a, b, c ) , a, b, c ∈ Ak −1}
Also yk + n −1 , yk + m −1 , yk +l −1 Also
(1.1)
Φ is increasing in t,
⊆ Ak −1 , implies G ( yn + k −1 , ym + k −1 , yl + k −1 ) ≤ ak −1
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Vishal Gupta, A. K. Tripathi, Raman Deep & Richa Sharm
sup G ( yn + k −1 , ym+ k −1 , yl + k −1 ) ≤ Φ ( ak −1 )
From (1.1) we get
m , n ,l ≥ 0
Therefore we have ak
≤ Φ ( ak −1 ) , letting k → ∞ , we get a ≤ Φ ( a ) . If a ≠ 0 , then a ≤ Φ ( a ) < a ,
which is a contradiction. Thus Thus by lemma,
{ yn }
a = 0 . Hence lim an = 0 , n →∞
is a Cauchy sequence in X. by completeness of X, there exist
lim yn = lim SRxn = lim TUxn +1 = y1 . Also TU ( X ) is closed, there exist n →∞
n →∞
n →∞
TUz = y1 , Now we show that SRz = y1 . For this set xn , xn , z replacing
y1 ∈ X such that z ∈ X such that
x, y, z respectively in
( SRxn , SRxn , SRz ) ≤ Φ ( G (TUxn , TUxn , TUz ) ) Taking n → ∞ , we get G ( y1 , y1 , SRz ) ≤ Φ ( G ( y1 , y1 , y1 ) ) = 0 Implies SRz = y1 . Since the pair ( SR, TU ) is weakly compatible, ( SR )(TU ) = (TU )( SR ) z . Thus SRy1 = TUy1
equation (iii) , we get G
Now we prove that
hence
we
get
SRy1 = y1 , If we substitute x, y, z in (iii) by xn , xn , y1 respectively
G ( SRxn , SRxn , SRy1 ) ≤ Φ ( G (TUxn , TUxn , TUy1 ) ) , Taking n → ∞ , we get G ( y1 , y1SRy1 ) ≤ Φ ( G ( y1 , y1TUy1 ) ) = Φ ( G ( y1 , y1 , SRy1 ) ) If SRy1
≠ y1 , then G ( y1 , y1SRy1 ) < G ( y1 , y1SRy1 ) , is a contradiction.
Therefore
SRy1 = TUy1 = y1 . y1 and y2 be fixed points of SR, TU . x = y = y1 and z = y2 in (iii) we have
For uniqueness let Taking
G ( y1 , y1 , y2 ) = G ( SRy1 , SRy1 , SRy2 ) ≤ Φ ( G (TUy1 , TUy1 , TUy2 ) )
= Φ ( G ( y1 , y1 , y2 ) ) < G ( y1 , y1 , y2 ) , a contradiction. Thus we have y1 = y2
( S , R ) , (T ,U ) are mutually commutative pair of mapping. Consider Sy1 = S ( SRy1 ) = S ( RSy1 ) = SR ( Sy1 ) , implies Sy1 is the unique
Now by (iv)
y1 is the unique fixed point of SR , hence Sy1 = y1 . Also Ry1
SR , but
= R ( SRy1 ) = ( RS )( Ry1 ) = SR ( Ry1 ) , implies Ry1 is the fixed point of SR, but y1 is the
unique fixed point of Thus
fixed point of
SR . Hence Ry1 = y1 .
Sy1 = Ry1 = y1 , In the same way we have Ty1 = Uy1 = y1 .
Hence the result.
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Common Fixed Point Theorem in G-Metric Space
REFERENCES 1.
Dhage, B.C., 1992. Generalized
metric spaces and mappings with fixed point. Bull. Calcutta Math.
Soc., 84(4): 329-336. 2.
Mustafa, Z. and B. Sims, 2006. A new approach to generalized metric spaces. J. Nonlinear Convex Anal., 7: 289-297.
3.
S. Gahler, 2-metricsche raume ihre topologische structur, Math. Nachr. 26 (1963/64), 115-148.
4.
Dhage, B.C., A Study of Some Common Fixed Point Theorems, PH.D. Thesis, Marathwada Univ. Aurangabad, India (1984).
5.
B. C. Dhage, “Generalised metric spaces and mappings with fixed point,” Bulletin of the Calcutta Mathematical Society, vol. 84, no. 4, pp. 329–336, 1992.
6.
Z.Mustafa and B.Sims, A new approach to a generalized metric spaces, J. Nonlinear Convex Anal., 7(2006), 289-297.