International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 3, Issue 1, Jan 2014, 63-70 © IASET

COMMON FIXED POINT THEOREMS IN HILBERT SPACE ARVIND KUMAR SHARMA1, V. H. BADSHAH2 & V. K. GUPTA3 1

2

3

ABSTRACT The aim of this present paper is to obtain a common fixed point for continuous self mappings on Hilbert space. Our purpose here is to generalize the our previous result [3]

KEYWORDS: Common Fixed Point, Hilbert Space, Continuous Self Mappings 2010 Mathematics Subject Classification: 47H10.

INTRODUCTION Kannan [3] proved that a self mapping T on complete metric space (X,d) satisfying the condition

d ( x, y )    d ( x, Tx)  d ( y, Ty )  for all x, y in X where 0 <  <

1 2

has a unique fixed in (X,d).

Koparde and Waghmode [5] have proved fixed point theorem for a self mappings T on a closed subset S of Hilbert space H, satisfying the Kannan type condition Tx  T y

2

2

  x  T x  y  Ty

2

for all x, y in S and x  y ;0  a 

1

2

Koparde and Waghmode [5] also extended this result to the pair of mappings T 1 and T2 and to their power p,q are some positive integers. In this paper we have obtained a unique fixed point for self mapping satisfying rational inequality  x  Tx 2  y  Ty Tx  Ty    x  Tx  y  Ty 

2

 1    x  y for all x, y in S and x  y ;    0,   0 and 2    1. 2 

In a Hilbert space

MAIN RESULTS We proved followingg fixed point theorems. Theorem 1: Let S be a closed subset of a Hilbert space H. Let T be a self mappings on S satisfying the following condition  x  Tx 2  y  Ty Tx  Ty     x  Tx  y  Ty

2

 1    x  y for all x, y in S and x  y ;    0,   0 and 2    1. 2 

Then T has a unique common fixed point.

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Arvind Kumar Sharma, V. H. Badshah & V. K. Gupta

Proof: Let S be a closed subset of a Hilbert space H . Let T be a self mappings on S. Let x0  S be any arbitrary point in S. Define a sequence xn n1 in S by 

xn1  Txn  Tn1x0 , for n  0,1, 2, .... Suppose that xn1  xn for n  0,1, 2, ....

For any integer n ≥ 1.

xn 1  xn

Txn  Txn 1

2   xn  Txn  xn 1  Txn 1   x  Txn  xn 1  Txn 1   n





xn  xn 1

2

 xn1  xn

xn  xn 1  xn 1  xn

     xn  xn 1  

2

 xn 1  xn

  xn  xn 1

xn  xn 1  xn 1  xn

 xn  xn1

2

2

  xn  xn1

  xn  xn 1   xn 1  xn   xn  xn 1   xn 1  xn   xn  xn 1   xn  xn 1

i.e.

xn 1  xn   xn 1  xn   xn  xn 1   xn  xn 1

 (1   ) xn 1  xn  (   ) xn  xn 1 

If k 

xn1  xn 

  xn  xn1 1

  then k  1. 1- 

xn 1  xn  k xn  xn 1  k xn  xn 1  k 2 xn 1  xn  2  k 3 xn  2  xn 3  ...  k n x1  x0 i.e. xn 1  xn  k n x1  x0

for all n  1 is integer .

Now for any positive integer m  n  1

xn  xm  xn  xn 1  xn 1  xn  2  ...  xm1  xm  k n x1  x0  k n 1 x1  x0  ...  k m 1 x1  x0

65

Common Fixed Point Theorems in Hilbert Space

 k n x1  x0

1 k  ...  k

mn 1

 kn  i.e. xn  xm   x x  1  k  1 0  

 0 as n   (k  1)

Therefore xn n1 is a Cauchy sequence. 

Since S is a closed subset of a Hilbert space H, so xn n1 converges to a point u in S. Now we will show that u is common fixed point of self mappings T from S into S. Suppose that Tu  u . Consider

u  T u  u  xn  xn  T u

 xn  T u  T xn 1  T u 2   xn 1  Txn 1  u  Tu   x  Txn 1  u  Tu   n 1

2

2   xn1  xn  u  Tmu     xn1  xn  u  Tmu

     xn 1  u   2

     xn1  u  

u  T u    xn1  xn  u  T u    xn1  u

i.e.

(1   ) u  T u   xn 1  xn   xn 1  u So

 0 as

u  T u  0.

Hence u  T u . Hence u is a common fixed point of self mappings T. Uniqueness: Suppose that there is u  w such that T w  w . Consider u  w  Tu  Tw      

2  w  Tw     u w u  Tu  w  Tw  

u  Tu

2

  u  Tu   w  Tw   u  w i.e.

uw   uw

n

66

Arvind Kumar Sharma, V. H. Badshah & V. K. Gupta

uw  0

Thus u  w . Hence fixed point is unique. Theorem 2: Let S be a closed subset of a Hilbert space H. Let T1 and T2 are self mappings on S satisfying the following condition

 x  T x 2  y  T y 1 2 T1 x  T2 y     x  T1 x  y  T2 y

2

 1    x  y for all x, y in S and x  y ;    0,   0 and 2    1. 2 

Then T1 and T2 have a unique common fixed point. Proof: Let S be a closed subset of a Hilbert space H . Let T1 and T2 are self mappings on S. Let x0  S be any arbitrary point in S. Define a sequence xn n1 in S by 

x2n1  T1 x2n

   for n  0,1, 2, ... . x2n 2  T2 x2n1   Suppose that xn2  xn1  xn for n  0,1, 2, ....

Consider

x1  x2

T1 x0  T2 x1

2   x0  T1 x0  x1  T2 x1   x0  T1 x0  x1  T2 x1  



x0  x1

2

 x1  x2

x0  x1  x1  x2

2

     x0  x1  

2

  x0  x1

  x0  x1   x1  x2   x0  x1 i.e.

x1  x2   x0  x1   x1  x2   x0  x1

 (1   ) x1  x2  (   ) x0  x1 (   ) x0  x1 (1   )

x1  x2 

If k 

  then k  1. 1- 

x1  x2  k x0  x1 Now

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Common Fixed Point Theorems in Hilbert Space

x2  x3

T2 x1  T1 x2

T1 x2  T2 x1

2   x2  T1 x2  x1  T2 x1   x2  T1 x2  x1  T2 x1  



x2  x3

i.e.

2

 x1  x2

x2  x3  x1  x2

2

     x2  x1  

2

  x2  x1

x2  x3   x2  x3   x1  x2   x2  x1

 (1   ) x2  x3  (   ) x1  x2 (   ) x1  x2 (1   )

x2  x3 

If k 

  then k  1. 1- 

x2  x3  k x1  x2  k 2 x0  x1 i.e.

x2  x3  k 2 x0  x1

Hence by induction we get for integer n ≥ 1

xn  xn1  k n x0  x1 Or

xn1  xn  k n x1  x0 Now for any positive integer

m  n 1

xn  xm  xn  xn 1  xn 1  xn  2  ...  xm1  xm  k n x1  x0  k n 1 x1  x0  ...  k m 1 x1  x0

 k n x1  x0

1 k  ...  k

mn 1

 kn  i.e. xn  xm   x x  1  k  1 0  

 0 as n   (k  1)

Therefore xn n1 is a Cauchy sequence. 

Since S is a closed subset of a Hilbert space H, so xn n1 converges to a point u in S. Now we will show that u is common fixed point of the pair of mappings T1 and T2 from S into S.

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Arvind Kumar Sharma, V. H. Badshah & V. K. Gupta

First we will show that u is fixed point of T1. . Suppose that T1u  u Consider

u  T1u  u  x2n2  x2n2  T1u

 x2n2  T1u  T2 x2 n 1  T1u  T1u  T2 x2 n 1 2 2   u  T1u  x2 n1  T2 x2 n1        u  x2 n1  u  T1u  x2 n1  T2 x2 n1    2   u  T1u  x2 n1  x2 n 2    u  T1u  x2 n1  x2 n 2 

2

     u  x2 n1  

u  T1u    u  T1u  x2 n1  x2 n2

i.e.

(1   ) u  T1u   x2n 1  x2n  2   u  x2n 1 So

u  T1u  0.

Hence u  T1u . Similarly we can prove that u is fixed point of T2. Hence u is a common fixed point of mappings T1 and T2 Uniqueness: Suppose that there is w  u such that T2 w  w . Consider u  w  T1u  T2 w 2 2   u  T1u  w  T2 w       u w u  T u  w  T w 1 2    

  u  T1u   w  T2 w   u  w

i.e.

uw   uw

 uw 0 Thus u  w. Similarly v  S if is such that T1v  v then u  v

 

u  x2 n1

 0 as n  

69

Common Fixed Point Theorems in Hilbert Space

Hence u is unique common fixed point of T1 and T2. Corollary Let p and q be two positive integers such that operators T 1p and T2q :S→S where S is closed subset of Hilbert space H,satisfy the inequality 2 2  p q x  T x  y  T y 1 2 1   T1p x  T2q y       x  y for all x, y in S and x  y ;    0,   0 and 2    1. p q 2  x  T1 x  y  T2 y   

Then T1 and T2 have a unique common fixed point. Proof: By the above theorem, the T1p and T2q have a unique common fixed point, say u  S . u  T1p u

Since

T1 u  T1 T1p u

therefore

 T1p  T1u 

i.e. T1u is also a fixed point of T1p .But by above theorem T1p has a unique fixed point. Therefore, T1 u  u. Similarly T2 u  u. Hence u is common fixed point of T1 and T2. For uniqueness ,if w is a another common fixed point of T 1 and T2,then clearly w is a also a common fixed point p

of T1 and T2q This gives w = u. This completes the proof. Example: Let X = [0, 1], with Euclidean metric d .Then {X, d} is a Hilbert space with the norm defined by

d ( x, y)  x  y .

Let  xn n 1

1   ( n 1)/2 x0 when n is odd  2.6   1 x0 when n is even  6n /2

be the sequence in X and let Let T1 and T2 are self mappings on X defined by

T1  x  

x 3

and T2  x  

x 2

70

Arvind Kumar Sharma, V. H. Badshah & V. K. Gupta

Clearly T1 and T2 are are satisfy the rational inequality. Also sequence

 xn n1 is a Cauchy sequence in X, which 

is converges in X also it has a common point in X.

CONCLUSIONS In this paper we replace Kannan inequality by rational inequality The theorems proved in this paper by using rational inequality is improved and stronger form of some earlier inequality given by Kannan [3], Koparde and Waghmode [5]. We have obtained a unique fixed point for self mapping satisfying rational inequality extended this result to the pair of mappings T1 and T2 and to their power p,q are some positive integers.

REFERENCES 1.

Badshah, V.H. and Meena, G., (2005): Common fixed point theorems of an infinite sequence of mappings, Chh. J. Sci. Tech. Vol. 2, 87-90.

2.

Browder, Felix E. (1965): Fixed point theorem for non compact mappings in Hilbert space, Proc Natl Acad Sci U S A. 1965 June; 53(6): 1272–1276.

3.

Kannan, R., (1968) Some results on fixed points, Bull. Calcutta Math. Soc. 60,71-76.

4.

Koparde, P.V. and Waghmode, B.B. (1991): On sequence of mappings in Hilbert space, The Mathematics Education, XXV, 197.

5.

Koparde, P.V. and Waghmode, B.B. (1991): Kannan type mappings in Hilbert spaces, Scientist Phyl. Sciences Vol.3, No.1, 45-50.

6.

Pandhare, D.M. and Waghmode, B.B. (1998): On sequence of mappings in Hilbert space, The Mathematics Education, XXXII, 61.

7.

Sangar, V.M. and Waghmode, B.B. (1991): Fixed point theorem for commuting mappings in Hilbert space-I, Scientist Phyl. Sciences Vol.3, No.1, 64-66.

8.

Sharma, A.K., Badshah, V.H and Gupta, V.K. (2012): Common fixed point theorems of a sequence of mappings in Hilbert space, Ultra Scientist Phyl. Sciences.

9.

Veerapandhi, T. and Kumar, Anil S. (1999): Common fixed point theorems of a sequence of mappings in Hilbert space, Bull. Cal. Math. Soc.91 (4), 299-308.

7 maths ijamss common fixed point theorems in hilbert space arvind kumar sharma
7 maths ijamss common fixed point theorems in hilbert space arvind kumar sharma

The aim of this present paper is to obtain a common fixed point for continuous self mappings on Hilbert space. Our purpose here...