Cmjv05i01p0041

J. Comp. & Math. Sci. Vol.5 (1), 41-44 (2014)

Fixed Point Theorems for Asymptotically Nonexpansive Mappings in Banach Space D. K. Singh1, S. K. Pandey2 and Pankaj Kumar3 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 extended the work of Pathak9. We establish fixed point theorems in Banach space by iteration scheme for asymptotically nonexpansive mappings which satisfies Opial's condition. Iteration scheme in {xn} is defined by (2.1) Keywords: Iterates, Opial's condition, Fixed point, Weak convergence, Banach space.

1. INTRODUCTION

sequence {kn} in [0, ) with lim k n  0 such

Let E be a closed convex bounded subset of a Banach space X and T : E  E be a mapping. Then T is called nonexpansive mapping if ||T(x) - T(y) ||  ||x-y|| ,  x, y  E (1.1)

that ||Tn(x) - Tn(y) ||  (1+kn) ||x-y|| ,  x, y  E and n ≥ 1 (1.3) T is called an asymptotically quasinonexpansive mapping if there exists a sequence {kn} in [0, ) with lim k n  0 such

The mapping T is called nonexpansive mapping if

that ||Tn(x) - p ||  (1+kn) ||x-p|| ,  x  E, p F(T), n ≥ 1 (1.4)

n

a

quasi-

||T(x) - p ||  || x - p ||  xE and p F (T).

(1.2) T is called an asymptotically nonexpansive mapping if there exists a

n

Let F (T) = {x  E : T (x) = x}, then F (T) is called the set of fixed points of a mapping T. If E is a closed and convex subset of

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D. K. Singh, et al., J. Comp. & Math. Sci. Vol.5 (1), 41-44 (2014)

a Hilbert space H and T has a fixed point, then for every x  E, {Tn (x)} is weakly almost convergent to a fixed point of T, as n  . This theorem is called the first ergodic theorem, which was proved by Baillon1 for general nonexpansive mappings in Hilbert space H. It was also shown by Pazy [8] that if H is a real Hilbert space and

1 n 1 i  T (x) n i 0

converges weakly to y  E, as n  , then y  F(T). Rhoades and Temir11 introduced the concept of a quasi-nonexpansive mapping was initiated by Tricomi in 1941 for real functions.

Reich13, Dotson3 and Diaz and 2 Metcalf studied fixed points for nonexpansive mappings and quasinonexpansive mappings in Banach spaces.

Recently, this concept was given by Kirk5 in metric spaces and Olusegun7 in Banach spaces. We establish fixed point theorems in Banach space using iterates for asymptotically nonexpansive mappings in Banach space which satisfies Opial's condition and iterates extend of Pathak9 and also the iterative scheme {xn} is defined by (2.1).

Nonexpansive mapping with at least one fixed point is called quasi-nonexpansive mapping.

2. PRELIMINARIES AND DEFINITIONS Let X be a Banach space and let E be a nonempty convex subset of X. Let T, S: E  E be two given asymptotically nonexpansive mappings. The iterative scheme {xn} is defined by xo= x  E and xn1 = nTn (xn) + nTn (yn) + n xn

yn = n' Sn (xn)  n'Tn (xn)  n' xn

(2.1)

where n + n + n = 1, n'  n'  n' = 1. and {n}, {n}, {n}, {n'}, {n'}, {n'} are real sequences in (0, 1).

A Banach space X is said to satisfy Opial's condition6, if for each sequence { x n} in X, xn  x implies that

lim x n  x  lim x n  y

n

n 

(2.2)

for all y  X with y  x.

The strong and weak convergence of the sequence of certain iterates to a fixed point of quasi-nonexpansive mappings was studied by Petryshyn and Williamson10.

Their analyses were related to the convergence of Mann iterates studied by Dotson3. Ghosh and Debnath4 discussed the convergence of Ishikawa iterates of quasinonexpansive mappings in Banach space.

In12 the weakly convergence theorem for I-asymptotically quasi-nonexpansive defined in Hilbert space was proved.

In this paper, we consider that T and S are asymptotically nonexpansive mappings in a Banach space. We establish the weak convergence of the sequence of iterates to a common fixed point of F(T)  F(S).

3. THE MAIN RESULTS 3.1 Theorem Let E be a closed bounded subset of uniformly convex Banach space X, which satisfies Opial's condition, and let T, S be self mappings of E. T and S are asymptotically nonexpansive mappings on

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D. K. Singh, et al., J. Comp. & Math. Sci. Vol.5 (1), 41-44 (2014)

E. Then for x0  E, The sequence {xn} of iterative scheme (2.1) converges weakly to common fixed point of F(T)  F(S).

Proof Let F(T)  F(S) be nonempty and a singleton, then the proof is obvious. So, we assume that F(T)  F(S) is nonempty and F(T)  F(S) is not a singleton. || xn1 - f || = || nTn (xn) + nTn (yn) + n xn - f || = || n Tn (xn)  nTn(yn)  n xn - (n + n+n) f ||  n || T n (xn) - f ||  n || T n (yn) - f ||  n || xn - f ||  n (1+kn) || x n - f ||  n (1+kn) || yn - f ||  n || xn - f || = {n(1+kn) + n)} || xn - f || +n(1+kn)|| yn-f|| = (n + n) || xn - f || +n|| yn-f|| since, lim k n  0 (3.1) n

n

The sequence {xn} contains a subsequence which converges weakly to a point in E. Let {x n k } and {x m k } be two subsequences of {xn} which converge weakly to f and q respectively. We shall show that f = q.

Suppose that X satisfies Opial's condition and that f  q is in weak limit set of the sequence {xn}. Then {x n k }  f and {x m k }  q respectively. Since lim x n  f n 

exists for any f  F(T)  F(S). By Opial's condition, we find that lim x n  f n 

= lim x n k  f  lim x n k  q k

k

 lim x n  q  lim x m j  q  lim x m j  f  lim x n  f n

j

j

n 

This is a contradiction. Thus {xn} converges weakly to an element of F (T)  F(S).

3.2 Theorem

n

||yn -f||=||'n S (xn) + 'nT (xn) + 'n xn - f || = ||'n Sn (xn) + 'nTn (xn) + 'n xn - ('n + 'n+'n) f || ≤ 'n ||Sn (xn) - f || + 'n ||Tn (xn) - f || + 'n ||xn -f|| ≤ 'n (1+kn) ||xn - f || + 'n (1+kn) ||xn - f || + 'n ||xn - f || = 'n ||xn - f || + 'n ||xn - f || + 'n ||xn - f || since, lim k n  0 n

= ('n + 'n + 'n ) ||xn - f || = ||xn - f || From (3.1) || xn1 - f || ≤ (n + n) || xn - f || +n|| xn-f|| = (n + n + n) || xn - f || = || xn - f ||  || xn1 - f || ≤ || xn - f || { || xn - f || } is a non increasing sequence.

Then lim x n  f exists. n 

Now we show that {xn} converges weakly to a common fixed point of T and S.

Let E be a closed convex bounded subset of uniformly convex Banach space X, which satisfies Opial's condition and let T, S be self mappings of E. T and S are asymptotically quasi-nonexpansive mappings on E. Then for xo  E, the sequence {xn} of iteration scheme (2.1) converges weakly to common fixed point of F(T)  F(S).

Proof Since every asymptotically nonexpansive mapping is asymptotically quasi-nonexpansive mapping. The proof of this theorem is similar to theorem (3.1).

ACKNOWLEDGMENTS The author is thankful to Pathak for his valuable suggestions which improved the presentation of the paper.

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