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

Department of Engineering Mathematics, M.I.T.M, Ujjain, Madhya Pradesh, India

2

School of Studies in Mathematics, Vikram University, Ujjain, Madhya Pradesh, India

3

Department of Mathematics, Government Madhav Science College, Ujjain, Madhya Pradesh, India

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.

64

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 n1 in S by

xn1 Txn Tn1x0 , for n 0,1, 2, .... Suppose that xn1 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

xn1 xn

xn xn 1 xn 1 xn

xn xn 1

2

xn 1 xn

xn xn 1

xn xn 1 xn 1 xn

xn xn1

2

2

xn xn1

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

xn1 xn

xn xn1 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 ... xm1 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

mn 1

kn i.e. xn xm x x 1 k 1 0

0 as n (k 1)

Therefore xn n1 is a Cauchy sequence.

Since S is a closed subset of a Hilbert space H, so xn n1 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 xn1 xn u Tmu xn1 xn u Tmu

xn 1 u 2

xn1 u

u T u xn1 xn u T u xn1 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.

uw uw

n

66

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

uw 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 n1 in S by

x2n1 T1 x2n

for n 0,1, 2, ... . x2n 2 T2 x2n1 Suppose that xn2 xn1 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

67

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 xn1 k n x0 x1 Or

xn1 xn k n x1 x0 Now for any positive integer

m n 1

xn xm xn xn 1 xn 1 xn 2 ... xm1 xm k n x1 x0 k n 1 x1 x0 ... k m 1 x1 x0

k n x1 x0

1 k ... k

mn 1

kn i.e. xn xm x x 1 k 1 0

0 as n (k 1)

Therefore xn n1 is a Cauchy sequence.

Since S is a closed subset of a Hilbert space H, so xn n1 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.

68

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 x2n2 x2n2 T1u

x2n2 T1u T2 x2 n 1 T1u T1u T2 x2 n 1 2 2 u T1u x2 n1 T2 x2 n1 u x2 n1 u T1u x2 n1 T2 x2 n1 2 u T1u x2 n1 x2 n 2 u T1u x2 n1 x2 n 2

2

u x2 n1

u T1u u T1u x2 n1 x2 n2

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.

uw uw

uw 0 Thus u w. Similarly v S if is such that T1v v then u v

u x2 n1

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 n1 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

Published on Jan 23, 2014

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

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