Cmjv05i01p0045

J. Comp. & Math. Sci. Vol.5 (1), 45-48 (2014)

Fixed Point Theorems in Banach Space D. K. Singh, S. K. Pandey and Pankaj Kumar 1,2

Govt. Vivekanand P.G. College, Maihar Distt-Satna, M.P., INDIA. 3 Kendriya Vidyalaya, Bailey Road, Patna, INDIA. (Received on: January 25, 2014) ABSTRACT

In this paper,we establish fixed points theorems in Banach space using Mann Iteration for nonexpansive and asymtotically nonexpansive mappings . Iteration scheme is defined by (2.1) and (2.2). We extended the work of Mann7.

Keywords: Banach space, nonexpansive mapping, asymtotically nonexpansive mappings, fixed point, convex set.

1. INTRODUCTION Let C be a closed convex subset of a Banach space (X, | . | ) and let T : C  C be a nonexpansive mapping.

That is, |Tx - Ty |  |x - y| , for all x, y  C. Let  be the set of nonnegative integers, and suppose A = {ank : n , k  } is an infinite matrix satisfying ank  0, for all n , k  , ank = 0, if k > n, n

a

nk

= 1, for all n  ,

k 0

lim a nk = 0 for all k  .

n

If x0  C, then a sequence S = {xn : n  }  C is defined as

n

xn = an0 x0 +

a

nk

T (xk-1), n  N

(1.1)

k 1

This iteration scheme is due to Mann7. Many eminent Mathematicians like Browder1, Day, James & Swaminathan2, Edelstein4, Kirk5, Lim6, Plubtieng & Wangkeeree8 and Reich9 studied fixed point theorems.

In 1975, S. Reich10 proved the following theorem.

1.1 Theorem Let C be a boundedly weakly compact convex subset of a Banach space X. Suppose that each weakly compact convex subset of C possesses the fixed point property for nonexpansive mappings, and that T : C C is nonexpansive. If the

Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)

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D. K. Singh, et al., J. Comp. & Math. Sci. Vol.5 (1), 45-48 (2014)

sequence S defined by (1.1) is bounded for some x0  C, then T has a fixed point. See3.

Dotson and Mann3 have proved that if X is uniformly convex and if S is bounded for some x0 in C, then T has a fixed point.

A special case of the Dotson-Mann result was independently established by Reinermann11.

2. PRELIMINARIES AND DEFINITIONS Let C be a closed convex subset of a Banach space (X, | . |) and let T : C  C be asymptotically nonexpansive mapping if there exists a sequence {  } in [0, ) with lim    0 such that n  

|T (x) - T(y) |  (1 +  ) |x - y| , for all x, y  C and   1.

T is called an asymptotically quasinonexpansive mapping if there exists a sequence {  } in [0, ) with lim    0  

such that |T(x) - p|  (1 +  ) |x - p| , for all x  C and p  F(T),  1. where F(T) = {x  C : T(x) = x}.

Let N denotes the set of non negative integers and suppose A = {ank: n, k  N} is an infinite matrix satisfying ank  0, for all n, k  N. ank = 0, if k > n,

n

xn = an0 x0 +

a

nk

T (xk-1) + un

(2.1)

k 1

where {un} is bounded sequence in C such that | un|  , 0 <  < 1.

Let T : C  C be asymptotically nonexpansive mapping.

If x0  C, the sequence S = {xn : n  N}  C is defined by n

xn = an0 x0 +

a

nk

T(xk-1)+ un, n  N,

k 1

1 (2.2) where {un} is bounded sequence in C such that | un|  , 0 <  < 1.

In this paper, we have proved the main results by using iteration (2.1) and (2.2).

3. MAIN RESULTS 3.1 Theorem Let C be a boundedly weakly compact convex subset of a Banach space X. Suppose that each weakly compact convex subset of C possesses the fixed point property for nonexpansive mappings, and that T: C C is nonexpansive. If the sequence S = {xn : nN}  C in a Banach space X defined by (2.1) is bounded for some x 0 in C, then T has a fixed point.

Proof

n

a

nk

= 1 for all n  N,

k 0

Consider a point y in C and set R = lim sup y - x n . R is finite because S n 

n

lim a nk = 0 for all k  N.

is bounded. Let K = {z  C: lim sup z - x n  R }. K is a nonempty

If x0 belongs to C, then a sequence S = {xn : n  N}  C can be defined by

bounded closed convex subset of C. i.e. K is a weakly compact subset of C. Let z  K.

n 

Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)

47

D. K. Singh, et al., J. Comp. & Math. Sci. Vol.5 (1), 45-48 (2014)

We have

Proof n

|T(z) - xn|  | T(z) - an0 x0 -

a

Consider a point y in C and set R = lim sup y - x n . R is finite because S is

nk

k 1

T(xk-1) - un|

n 

n

 an0 |T(z) - x0| +

a

nk

|T(z) - T(xk-1) |

bounded. Let K = {z  C : lim sup z - x n  R }. K is a nonempty

|z - xk-1 | + 

bounded closed convex subset of C. i.e. K is weakly compact subset of C. Let z  K. We have

k 1

n 

+ |un| n

 an0 |T(z) - x0| +

a

nk

k 1

For  > 0, m ()  N such that |z - xk| < R +  for all k > m.

Therefore, we obtain for n > m + 1, m 1

|T(z) - xn|  an0 |T(z) - x0| +

a

nk

|z - xk-1|

k 1

n

|T(z) - xn| = |T(z) - an0 x0 -

nk

k 1

T(xk-1)- un| n

 an0 |T(z) - x0| +

a

nk

|T(z) - T (xk-1) |

nk

|T(z) - T(z)

nk

|T(z) - T(z) |

k 1

n

+

a

 a nk (R + ) + .

k m  2

+ | un | n

= h (n) + R +  +  where lim h(n)  0

 an0 |T(z) - x0| +

Thus T (z) belongs to K.  T(z) = z.

 an0 |T(z) - x0| +

n 

a k 1

+ T(z) - T (xk-1) |+  n

Hence, z is a fixed point of T. This completes the proof.

3.2 Theorem

a k 1

n

+

a

nk

|T(z) - T(xk-1) |+ 

k 1 n

 an0 |T(z) - x0| +

a

nk

(1 + l) | z - xk-1 |

k 1

+   lim T(z) - T  (z)  0 n 

Let C be a boundedly weakly compact convex subset of a Banach space X. Suppose that each weakly compact convex subset of C possesses the fixed point property for asymptotically nonexpansive mappings, and that T : C C is asymptotically nonexpansive mapping. The sequence S = {xn:n  N}  C in a Banach space X defined by (2.2) is bounded for some x0  C, then T has a fixed point.

n

 |T(z) - xn|  an0 |Tz - x0| +

a

nk

|

k 1

z - xk-1 |+ 

 lim    0 n 

For  > 0,  m ()  N such that |z - xk| < R + ,  k > m.

Hence, for n > m + 1

Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)

48

D. K. Singh, et al., J. Comp. & Math. Sci. Vol.5 (1), 45-48 (2014) m 1

|T(z) - xn|  an0 |T(z) - x0| +

a

nk

|z - xk-1|

k 1

n

+

a

nk

(R + ) +

k m  2

m 1

 an0 |T(z) - x0| +

a

nk

| z - xk-1 | + R +

k 1

+ = h (n) + R +  + where lim h(n)  0 . n 

Thus T (z) belongs to K.  T(z) = z Hence, z is a fixed point of T. ACKNOWLEDGMENT The author is thankful to Mann for his valuable research papers which improved the presentation of paper.

REFERENCES 1. Browder, F. E. "Nonlinear equations of evolution and nonlinear accretive Operators in Banach spaces," Bull.

Amer. Math. Soc.,73,867-874 (1967).

2. Day, M. M. James R. C. and Swaminathan, S. "Normed linear spaces that are uniformly convex in every direction", Canad. J. Math., 23 10511059 (1971).

3. Dotson,W.G. Jr. and Mann,W. R. "A generalized corollary of the BrowderKirk fixed point theorem", Pacific J. Math., 26, 455-459 (1968).

4. Edelstein, M. "The construction of an asymptotic center with a fixed point property", Bull. Amer. Math. Soc., 78, 206-208 (1972).

5. Kirk, W.A. "A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72, 1004-1006 (1965).

6. Lim, T.C. "A fixed point theorem for families of nonexpansive mappings", Pacific J. Math., 53, 487-493 (1974).

7. Mann, W. R. "Mean value methods in iteration", Proc. Amer. Math. Soc., 4, 506-510 (1953).

8. Plubtieng, S. and Wangkeeree, R. "Fixed point iteration for asymptotically quasi-nonexpansive mapping in Banach space,” Hindawi Publishing Corporation IJMMS,11, 1685-1992 (2005).

9. Reich, S. “Remarks on fixed points ||”, Atti Accad. Nat. Lincei Rend. Cl. Sci. Fis. Mat. Natur.,53,250-254 (1972).

10. Reich, S. “Fixed points iterations of nonexpensive mapping”, Pacific Journal of Mathematics Vol.60,No.2,195-198 (1975).

11. Reinermann, J. “Approximation von Fixpunkten”, Studia Math., 39, 1-15 (1971).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)