Almost Stability of Modified Iteration Method with Errors for a Fixed Point of Uniformaly L- Lipschitzian

In this paper, we prove strong convergence theorem of modified Mann iteration sequence with errors for uniformaly LLipschitzian mapping in arbitrary Banach space. Our results improve and gernalize the recent results Osilike , Xu and Xie and many others. 1.INTRODUTION AND PRELIMINARY DEFINITIONS Let X be arbitrary real Banach space and C is a nonempty subset of X and T is a selfmapping of C. F(T) is set of fixed point of T. The various mapping appearing in the following Definition (1.1) have been studied widely and deeply be many authors; see e.g., [1-4] for more details. Definition (1.1): Let X be an arbitrary real Banach space and C be a nonempty subset of X. A mapping T:C C is called: (i) Strongly pseudocontractive if there exists (0,1) k such that [( ) ( ) .....(1.1) x y x y r I T kI x I T kI y where I is the identity mapping on C and for all , x y C and r>0. (ii) Lipschitz if there exists a constant L>0 such that ( ) ( ) .....(1.2) T x T y L x y


1.INTRODUTION AND PRELIMINARY DEFINITIONS
Let X be arbitrary real Banach space and C is a nonempty subset of X and T is a selfmapping of C. F(T) is set of fixed point of T.
The various mapping appearing in the following Definition (1.1) have been studied widely and deeply be many authors; see e.g., [1][2][3][4] for more details.

Definition (1.1):
Let X be an arbitrary real Banach space and C be a nonempty subset of X.A mapping T:C C is called: We consider the iteration [1] (  [5,6,7]).
In 1996, Osilike [8], proved that if X puniformly smooth Banach space, C nonempty closed of X and T:C C is Lipschitz strongly pseudocontractive mapping with fixed point q in C, then both the Mann and Ishikawa iteration schemes are stable.Then he extended the results to arbitrary real Banach space in [6].
In 2001, Zeqing, Lili and Shin [9], show that if X is an arbitrary real Banach space and T:C C is a Lipschitz strongly pseudocontractive mapping, then under certain conditions the Ishikawa iterative with errors converges strongly to the unique fixed point of T. we also proved that this iteration procedure is stable with respect to T.
In 2004, Xu and Xie [10], proved necessary and sufficient condition for strongly convergence of Mann iteration process with errors to a fixed point of Lipschitz strongly pseudocontractve mapping in real Banach space.
We consider the iteration

u n
Where n is sequences in (0,1) and n u is sequence in C satisfying 1 .
n n u This iteration is known Modified Mann iteration sequence with random errors.
We consider the iteration [1] Next we recall the definition stability.Let  x f T x is said to be T nstable or stable with respect of T n .
It is our purpose in this paper to show that if X is an arbitrary real Banach space and T:X X is uniformly L-Lipschitzian mapping , then under certain condition the Modified Mann iterative method with errors converges strongly to the unique fixed point of T. we also prove that this iteration procedure is stable with respect of T n .Our results generalize most of the results that have appared recently.
For our result we need the following lemma: Lemma 1. x converges strongly to unique fixed point q of T, (2) n y any sequence in X.Then n y converges strongly to fixed point q of T if and only if n converges to 0.
Proof(1): using (1.4), we have (2 ) ( ) ( (1 It follows for and thecondition and so bylemma we have x q i e x converges stronglyto fixed po q of T If p alsois a fixed po T , 1 (1 ) .
. putting r in we obtain q p k q p It implies that q p Proof(2): Suppose that lim 0, and so by lemma we have y q i e y converges strongly to fixed po q of T Then theiterative process defined by x f T x isT stable ( On the coontrary let y converges strongly to fixed po q of T Then y y T y u y q y q L y q M This implies that procedure which yields a sequence of a points n x in C. Suppose that F(T)and n x converges to a fixed point q of T.Let n y be an arbitrary sequence in C and 1 ( , ) .
This iteration is known Mann iteration sequence with random errors.