1. Shailesh T. Patel, Sunil Garg, Ramakant Bhardwaj / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1459-1461
Some Results Concerning Fixed Point In Hilbert Spaces
Shailesh T. Patel, Sunil Garg, Dr.Ramakant Bhardwaj
The student of Singhania University
Sinior Scientist,MPCST Bhopal
Truba Institute of Engineering & Information Technology,Bhopal(M.P.)
ABSTRACT
In the present paper we will find some  min{
results concerning fixed point and common
y  Tx
2
fixed point theorems in Hilbert spaces for
x  Ty , y  Tx }  
2 2
.........3.1.1
rational expression, which will satisfy the well 1  x  Tx x  Ty
known results.
For all x,y in X where α,β,γ and δ
Key words: Common Fixed Point, Hilbert spaces are non-negative real with
α+β+γ+4δ<1.Then T has a unique fixed
point in X.
INTRODUCTION
PaSndhare and Waghmode [3] proved the Proof : Let x0єC and arbitrary we define a
following result in Hilbert space The study of sequence {xn} as follows:
properties and application of fixed point of various
type of contractive mapping in Hilbert spaces were x1=Tx0, x2=Tx1= 𝑇 2 x0….xn=Txn-
obtained among others by Browder, F.E.,and 𝑛 𝑛+1
1=𝑇 x0, xn+1=Txn=𝑇 x0
Petryshyn [1], Hicks,T.L. and Huffman,Ed.W’[2],
Huffman[3],Koparde and Waghmode[4], Smita If for some n,xn+1=xn,then it
Nair and Shalu Shrivastava[5,6].In this paper we immediately follows xn is a fixed
present a common fixed point theorem involingself point of T, i.e.Txn=xn
mapping.
The study of properties and applications of We suppose that xn+1≠xn for
fixed point of various type of contractive mapping every n=0,1,2…..,Then appling (3.1.1),
in also Banach spaces were obtained among others We have for all n≥1
by Browder, F.E.[7], Browder, F.E.,and Petryshyn,
W.V.[8].
xn1  xn  Txn  Txn1
2 2
Theorem A Let C be a close subset of
 a xn  Txn 
Hilbert spaceX and let T be a self 2
mapping on C satisfying.
 xn1  Txn1   xn  xn1
2 2
+
Tx  Ty  a[ x  Tx  y  Ty ]  b x  y
2 2 2 2
xn1  Txn
2
For all x,yєC, x  y,0 < b, 0 <1  min{ xn  Txn1 , xn1  Txn }  
2 2
.........3.1.1
and 2a+b<1. 1  xn  Txn xn  Txn1
Then T has a unique fixed point.
In the present paper , we first extend
Theorem A as follows:
 a xn  xn1 
2
Theorem3.1Let C be a closed subset of Hilbert
space X and T be a mapping on
 xn1  xn   xn  xn1
2 2
X into it self satisfying +
Tx  Ty  a x  Tx 
2 2
 y  Ty   x  y
2 2
1459 | P a g e

2. Shailesh T. Patel, Sunil Garg, Ramakant Bhardwaj / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1459-1461
 a xn1  xn 2 
2
xn1  xn1
2
 min{ xn  xn , xn1  xn1 }  
2 2
1  xn  xn1 xn  xn  xn  xn1   xn1  xn
2 2
{ xn  xn1  xn1  xn 2 }
2 2
 a xn  xn1 
2
 xn1  xn   xn  xn1
2 2
  xn1  xn  xn  xn1 
2 2
 (     ) xn  xn1 
2
 
(   ) xn1  xn2
2
 (     ) xn1  xn 
2
xn 2  xn1 
2
(   ) xn  xn1
2
(     )
xn  xn 1
2
(1    ) xn1  xn (1     )
2
 (     ) xn1  xn
2
xn 2  xn1  k 2 xn1  xn
2 2
(     )
xn1  xn 
2
where k= (1     )
(     )
xn1  xn
2
(1     ) Proceeding in this way, we obtain
xn1  xn
2
 k xn1  xn
2  k n x0  x1
n=1,2,……..
(     )
where k= (1     ) Where α+β+γ+2η<1,
because k < 1
→0 as n→∞
xn  xn1  k n x0  x1
2 2
xn 2  xn1  Txn1  Txn
2 2
Hence for any positive integer p,
 a xn1  Txn1 
2
xn  xn1  xn  xn1  xn1  xn2  ......  xn p1  xn p
 xn  Txn   xn1  xn
2 2
xn  Txn1
2  (k n  k n1  ........  k n p1 ) xn1  xn
 min{ xn1  Txn , xn  Txn1 }  
2 2
1  xn1  Txn1 xn1  Txn
kn
x  xn
= 1  k n 1
in 0<k<1
 a xn1  xn 2 
2
 xn  xn1   xn1  xn
2 2
xn  xn p  0
Thus as
n
.
xn  xn2
2
Therefore {xn} is a Cauchy sequence in
 min{ xn1  xn1 , xn  xn2 }  
2 2
C. Since C is a closed subset of a Hilbert space X
1  xn1  xn2 xn1  xn1 ,there exists an element u Є C which is the limit of
{xn}.
1460 | P a g e

3. Shailesh T. Patel, Sunil Garg, Ramakant Bhardwaj / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1459-1461
lim x n References:
i.e. u = n Now, further we have, 1. Browder, F.E.,and Petryshyn,
W.V.,Construction of Fixed point of non-
linear mapping in Hilbert
u  Tu  (u  xn )  ( xn  Tu )
2 2
paces.J.Math.Nal.Appl.,20,197-228(1967).
2. Hicks,T.L. and Huffman,Ed.W., Fixed
point theorems of generalized Hilbert
 u  xn  xn  Tu  2 Reu  xn, xn  Tu
2 2
spaces. J.Math.Nal.Appl.,64,381-
385(1978).
3. Huffman,Ed.W.,Striet convexity in locally
convex spaces and Fixed point theorems
 u  xn   xn1  Txn1 
2 2
of generalized Hilbert
spaces,Ph.D.,Thesis,Univ.of Missouri-
 u  Tu   xn1  u
2
2
Rolla,Missouri(1977).
4. Koparde,P.V.and Waghmode,D.B.,Kanan
type mappings in Hilbert spaces,Scientist
Txn1  u
2
of Physical
 min{ xn1  Tu , u  Txn1 }  
2 2
1  Txn1  xn1 xn1  Tu sciences,3(1),45-50(1991).
5. Smita Nair and Shalu Shrivastava,
Common Fixed point theorems in Hilbert
 2 Reu  xn, xn  Tu space of Acta Ciencia Indica,Vol.XXXII
M,No.3,997(2006).
Letting n→∞, so that xn-1,xn→u, we get 6. Smita Nair and Shalu Shrivastava,
Common Fixed point theorems in Hilbert
u  Tu   u  Tu spaces of Acta Ciencia Indica,Vol. XXXII
2 2
M,No 3,995(2006).
7. Browder, F.E., Fixed point theorems for
This implies u= Tu, since β<1 non-linear Semi-contractive mapping in
If follows that u is a fixed point of T. We now show that u is unique , Banach aces.Arch.Rat.Nech.Anal.,21,259-
269 (1965-66)
For that let v є C another common fixed point T such that v  u , then 8. Browder, F.E.,and Petryshyn, W.V.,The
Folutio by Heration of non-linear
functional equations in
v u  Tv  Tu
2 2
Banach
spaces.Bull.Amel.Math.Soc.,72,571-
576(1966).
  v  Tv   u  Tu
2 2
 v u
2
u  Tv
2
 min{ v  Tu , u  Tv }  
2 2
1  v  Tv v  Tu
vu
2
≤ (γ +δ+ η)
This implies
v  u  (     ) v  u
2 2
Since     < 1
Remark: On taking α=β=a, γ=b, and δ=0, η=0 in theorem 1, we get
theorem A.
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