Abstract
Our purpose in this paper is to consider a sequence {x n } defined as (2) and a asymptotically k n − strict pseudocontractive mapping in the intermediate sense Ishikawa iterative process with errors. The results presented in this paper mainly improved and extend the corresponding results announced in [D
Introduction
The class of asymptotically nonexpansive mappings was introduced as an important generalization of the class of nonexpansive mappings£and the existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [1972] as below:
Theorem 1. If C is a nonempty closed convex bounded subset of a uniformly convex Banach Space, then every asymptotically nonexpansive mapping T : C −→ C has a fixed point in C.

Mann iteration process:
x n+1 = (1 − α n )x n + α n T n x n , n ∈ N.,
Mann iteration process:
for (1) the approximation of fixed points of asymptotically nonexpansive mappings was developed and it's weak convergence was obtained by Schu [1991] .
Mann iteration process:
Iterative methods for approximation of fixed points of asymptotically nonexpansive mappings have been widely studied by authors [C.E. Chidume, 2003; S.S.Chang, 2001 ; S.H. Khan, 2005; Z.Q.Liu, 2004; K.Nammanee, 2000; B.E.Rhoades, 1994; M.O.Osilike, 2000; L.Wang, 2006] .
Mann iteration process:
The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck, Kuczumow and Reich [1993] .
Mann iteration process:
In 2008, Kim and Xu [2008] introduced the concept of asymptotically k-strict pseudocontractive mappings in Hilbert space.
Mann iteration process:
In 2009, Sahu and Xu [2009] study some properties and convergence of (1) for the class of asymptotically k-strict pseudocontractive mappings in the intermedidate sense which are not necessarily Lipschitzian.
Mann iteration process:
Our purpose in this paper is to consider a sequence {x n } defined as (2) and a asymptotically k n − strict pseudocontractive mapping in the intermediate sense Ishikawa iterative process with errors. The results presented in this paper mainly improved and extend the corresponding results announced in [D.R. Sahu, 2009; T.H.Kim, 2008; K.Nammanee, 2000; T.H.Kim, 2008] .
Preliminaries
Let C be a nonempty subset of a normed space X and T : C → C a mapping. We need the following concepts,
Preliminaries
(ii) T is asymptotically nonexpansive(cf.[K. Goebel, 1972] )if there exists a sequence k n of positive numbers satisfying the property
Preliminaries
for all integers n ≥ 1 and x, y ∈ C.
Preliminaries
(iii) T is uniformly Lipschitzian if there exists a constant L > 0 such that,
Preliminaries
for all integers n ≥ 1 and x, y ∈ C.
Preliminaries
(iv) T is asymptotically nonexpansive in the intermediate sense [R.E.Bruck, 1993] provided T is uniformly continuous and,
Preliminaries
It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian.
Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · , respectively and let C be a closed convex subset of H, ω ω ({x n }) denotes the weak ω − limit set of {x n } .
Preliminaries
Definition 1 Let C be a nonempty subset of a Hilbert space H. A mapping T : C −→ C is said to be an asymptotically k n -strict pseudocontractive mapping in the intermediate sense with sequence {γ n } if there exist sequences {k n } in [0, 1) and
Preliminaries
for all x, y ∈ C and n ∈ N.
Preliminaries
we need some facts and tools which are listed as follows, Osilike, 2000; K.K.Tan, 1992] ) Let {γ n }, {β n }, {δ n } be three sequences of nonnegative numbers satisfying the recursive inequality, (Agarwal, O'Regon. [2007] ) Let {x n } be a bounded sequence in a reflexive Banach space X. if ω ω ({x n }) = {x}, then x n converges weakly to x. Lemma 3. Let H be a real Hilbert space. Then the following hold:
Preliminaries
(c) let {x n } ∈ H such that x n weak converges to x, then lim sup n→∞ x n − y 2 = lim sup n→∞ x n − x 2 + x − y , ∀y ∈ H. 
Preliminaries
for all x, y ∈ C and n ∈ N. 
Proof. It is easy to see that F(T ) is closed and convex.
We prove that I − T is cassandra97@example.net. Assume that {x n } be a sequence in C. By Lemma 4. we obtain
Proof. It is easy to see that F(T ) is closed and convex.
for all n, m ∈ N and some constant M > 0. Let f (x) = lim sup n→∞ x n − x 2 , for all x ∈ H. Since {x n } weakly converges to x,
Proof. It is easy to see that F(T ) is closed and convex.
Hence we have
Proof. It is easy to see that F(T ) is closed and convex.
By lim sup n→∞ x n − T m x n = 0, we have lim sup
Proof. It is easy to see that F(T ) is closed and convex.
It follows that T m x → x(m → ∞) and (I − T )x = 0 . 2 (2) satisfy the following conditions:
Theorem 3. Let C be a nonempty subset of a Hilbert space H and T : C → C be uniformly continuous asymptotically k n − pseudocontracitive in the intermediate sense with sequence
(ii) {x n } be bounded.
Then {x n } converges weakly to an element of F(T ).
Proof. First we show that { y n − T n y n } be bounded.
Then {x n } converges weakly to an element of F(T ).
Journal of Mathematics Research Vol. 3, No. 4; November 2011 where σ := 2 (1 − β n )(x n − p) + β n (T n x n − p), τ n , we simply σ as follows.
Then {x n } converges weakly to an element of F(T ).
hence
Then {x n } converges weakly to an element of F(T ).
Which implies that
Then {x n } converges weakly to an element of F(T ).
By condition (i), (ii) and (7), we have { y n − p } be bounded, and so {y n }.
Then {x n } converges weakly to an element of F(T ).
From lemma 4, we obtained
Then {x n } converges weakly to an element of F(T ).
hence { T n y n − p } be bounded. By y n − T n y n ≤ T n y n − p + y n − p , we have { y n − T n y n } be bounded.
Then {x n } converges weakly to an element of F(T ).
Next we prove that lim n→∞ x n − p , for each p ∈ F(T ) exists. Indeed,
Then {x n } converges weakly to an element of F(T ).
In the other hand,
Then {x n } converges weakly to an element of F(T ).
By substituting (10) into (9), we have
Then {x n } converges weakly to an element of F(T ).
−2α n (1 − α n ) T n y n − x n 2 + 3 σ n 2 .
Then {x n } converges weakly to an element of F(T ).
Substituting (6) into (11), we have,
Then {x n } converges weakly to an element of F(T ).
≤ 2[(1 − α n ) + α n (1 + γ n )(1 + β n γ n )] x n − p 2 + 2α n (1 + γ n )(β n c n + 2 τ n 2 )
Then {x n } converges weakly to an element of F(T ).
+2α n c n + 3 σ n 2 + 2α n k n y n − T n y n − 2δ 2 T n y n − x n 2 − 2δ 2 x n − T n x n 2 ≤ 2ε n x n − p 2 + θ n − 2δ 2 T n y n − x n 2 − 2δ 2 x n − T n x n 2 .
Then {x n } converges weakly to an element of F(T ).
where ε n = [(1 − α n ) + α n (1 + γ n )(1 + β n γ n )], θ n = 2α n (1 + γ n )(2β n c n + 3 τ n 2 ) + 2α n c n + 3 σ n 2 + 2α n k n y n − T n y n . Hence,

{'paper_id': '34322919', 'title': 'Weak Convergence of Ishikawa Iteration with Error for Pseudo Contractive Mappings in Hilbert Spaces', 's2_url': 'https://api.semanticscholar.org/CorpusID:34322919', 'source': 'S2ORC'}