International Journal of Scientific & Engineering Research Volume 2, Issue 10, Oct-2011 1

ISSN 2229-5518

A Fixed Point Theorem in Hausdorff Space

Rajesh Shrivastava, Manavi Kohli

### Keywords: Fixed point, Hausdorff space, mapping, continuous maping, contraction principle, minimax principle, commuting mapping

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# 1. INTRODUCTION:

In 1972, Brouwer  proved his well known fixed point theorem and next Schauder extended the validity of Brouwer’s fixed point theorem to normed

# PROOF:

For any x 0

 X we choose x  X , we define a sequence { x n }

linear spaces. Jungck ,, established some fixed
of elements of X, such that
and common fixed point theorems for continuous commuting mappings and gave criterion of the existence of fixed points for Cgf in compact metric

x n+1  Tx n

Now,
, for n= 0,1,2,….
spaces. Kakutani  generalized Brouwer’s fixed point theorem to multimaps and applied the result to prove a version of the Von Neumann minimax principle in R n . In this paper the results of Park and, Singh and Rao  have also been extended.

# 2. THEOREM:

Let T be a continuous mapping of a Hausdorff space

d(x n+1 , xn+2 )  d(Tx n , Tx n+1 )

From (1) we have d(Tx , Tx )  [{d(xn , Txn ).d(xn+1 , Txn+1 )  d(xn , Txn+1 ).d(xn+1, Txn )}

n n+1 d(x , Tx ) +{d(xn+1, Tx n ).d(xn+1, Txn )  d(xn , Txn+1 ).d(xn , Txn )}
d(xn , Txn+1 ) +{d(xn , xn+1 ).d(xn+1, Txn )  d(xn , xn+1 ).d(xn , Txn+1 )}]
d(xn+1, Txn+1 )  d(xn , Txn ) d(Tx,Ty)  [{ d(x,Tx).d(y,Ty)+d(x,Ty).d(y,Tx)

d(y,Ty)

 {d(x n , Tx n )  d(x n , Tx n )  d(x n , Tx n )}

d(x , Tx ){}

= n n

+{ d(y.Tx).d(y,Tx)+d(x,Ty).d(x,Tx)}

d(x,Ty)

Proceeding in the same manner, we get

d(x , x )  {}n+1 d(x , x ) +{d(x,y).d(y,Tx)+d(x,y).d(x,Ty) }] ..............(1)

d(y,Ty)+d(x,Tx)

as n  , lim d(x n , x n+1 )  0

n 

X into itself and let d:X X  R +
be a continuous
mapping such that for x, y X
and x  y , satisfying
Hence,{ x n } converges to limit x (say).By completeness of X, the
for all n=0,1,2,………..; x,yX and

0  ()  1.

Then T has a unique fixed point.

, ,  0. Also

sequence { x n } is Cauchy.

# CLAIM:

x is a fixed point of T.
On the contrary if we assume that x  Tx , then

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d(x,Tx) 

d(x,x n+1 )  d(x n+1 , Tx)

Dr. Rajesh Shrivastava is Professor and Head of Department Mathematics in Govt. Science & Commerce College Benazir Bhopal (MP)

E-mail: rajeshraju0101@rediffmail.com

Mrs. Manavi Kohli is a research scholar in Department of Mathematics in

Govt. Science & Commerce college Benazir, Bhopal (MP)

E-mail: manvi.kohli@rediffmail.com

= d(x,xn+1 )  d(Tx n , Tx)

By using (1), as n  

d(x,Tx)  0

Hence proved , x is the fixed point.

International Journal of Scientific & Engineering Research Volume 2, Issue 10, Oct-2011 2

ISSN 2229-5518

# 3. CONCLUSION:

In this paper we have proved a fixed point theorem for contractive mapping. This work can further be used for establishing results for generalized contractive and contractive type set valued mapping in other metric spaces.

# 4. ACKNOWLEDGEMENT:

I express my sincere gratitude to my respected Guide Prof. Dr. Rajesh Shrivastava for correcting my work and giving me valuable suggestions. I am thankful to Dr. Umakant Mishra, the Principal of our college. I also extend my thanks to all my respected teachers and dear colleagues.

# 5. REFERENCES:

 F.E. Browder, A generalization of the

Schauder fixed point theorem, Math.Ann.

174(1967) , 285-390.

 G.Jungck, Commuting mappings and fixed

points, Amer.Math.Monthly 83(1976),no.4,

261-263.

 G.Jungck, Periodic and fixed points, and commuting mappings, Proc. Amer.Math.Soc.

76(1979), no.2, 333-338.

 G.Jungck, Common fixed points for

commuting and compatible maps on compacta, Proc. Amer.Math.Soc.103(1988), no. 3, 977-983.

 S.Kakutani, A generalization of Brouwer’s

fixed point theorem, Duke Math.J.vol.7(1941)

pp. 457-459.

 S.Park, Fixed points of f-contractive maps, Rocky Mountain J.Math.8(1978),no.4, 743-

750.

 S.L.Singh and K.P.R. Rao, coincidence and

fixed points for four mappings, Indian

J. Math.31(1989). No.3, 215-223.