Stability And Data Dependence Results For The Mann Iteration Schemes on n-Banach Space

Let be an n-Banach space, M be a nonempty closed convex subset of , and S:M→M be a mapping that belongs to the class mapping. The purpose of this paper is to study the stability and data dependence results of a Mann iteration scheme on n-Banach space .


Introduction
Let be a vector space over a field , which is either a real or complex space with a finite dimension or an infinite dimension, and a real-valued function : ℝ satisfying the following properties for all ,…, and λ ℝ.
= │λ│ . 4. then is called an n-norm on , and is called a linear n-normed space. For simplicity, we will call an n-normed space [1]. In the following, we need the concept of n-Banach space. H. Gunawan [3], roughly speaking of a fixed point iteration procedure which is numerically stable if a small modification in the initial data involved in the computation process will produce a small influence on the computed value of the fixed point.
The outcomes on the data dependence results for the Mann iteration on normed space using contraction mapping were proved by Solutz [4]. Also, data dependence results were discussed by many authors, including Rus and Muresan [5], Espi ́nola and Petrusel [6], and Soltuz [7].
In this paper, we generalize the concept of stability on an n-normed space and we prove the stability of the Mann iteration. We establish data dependence results of the Mann iteration scheme on n-Banach space, under a mapping that belongs to the class : Let be a subset of an n-normed space . A mapping belongs to the class if for all and .
This paper consists of two sections. In section one, we introduce the concept of the stability on an n-normed space and prove that the Mann iteration scheme is stable under different types of mappings. In section two, we prove a data dependence result for a fixed point under a mapping that belongs to the class with the help of the Mann iteration scheme.

Notation:
We will abbreviate as and as

$1 Stability Results
This section focuses on the stability of the Mann iteration scheme [8]. We define the Mann iteration as follows: ; where { } is a sequence1satisfying 0 < 1 for all and ∑ = ∞. But before discussing the stability, we need to generalize the definition of the stability to the n-normed space.

Definition 1.1:
Let be an n-normed space, S: Y Y is a mapping, and { } is the sequence generated by an iteration procedure = f ( S, ) ; (1) where . Suppose that { } converges to a fixed point of S, { } is an arbitrary sequence in , and the set = ; . Then, the iteration (1) is said to be S-Stable or stable with respect to S if 0 if and only if . The following lemma is essential to the main result of the section.

Lemma 1.2 [9]:
If is a real number such that and { } is a sequence of positive numbers such that then for any sequence of positive numbers { } satisfying + ; we have The next result is to prove that the Mann iteration scheme converges to a fixed point of Proposition 1.3: Let M be a convex subset of an n-Banach space and a mapping be a mapping that belongs to the class with . Then, the Mann iteration scheme converges to a fixed point of Proof.
Using the Mann iteration scheme and since S belongs to the class , we obtain that -= = = By combining (1) and (2), we have - Observe that , as From equation (3), we have Because ∑ and taking the limit of both sides of inequality (4) yields -. Now, we discuss the stability for the Mann iteration scheme with respect to belongs to the class mapping.

Theorem 1.4:
Let be an n-normed space and S: Y Y belongs to the class where . Suppose that S has a fixed point . For arbitrary with , if there is ℝ, where , then the Mann iteration scheme is S-stable.

$2 Data Dependence Results
In some cases, to compute a fixed point of we use a certain approximate operator ̃ of to approximate a fixed point ̃ of ̃. So the natural question arises: Does ̃ approximate ? If yes, how can we compute ̃ ? In this section, we try to answer this question. But first we introduce the definition of approximate mapping and lemma.

Definition 2.1:
Let be an n-normed space and and ̃ be two mappings. We say that ̃ is an approximate mapping of if for all and for fixed we have: