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Stability of Convergence Theorems of the Noor Iteration Method for an Enumerable Class of Continuous Hemi Contractive Mapping in Banach Spaces | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Stability of Convergence Theorems of the Noor Iteration Method for an Enumerable Class of Continuous Hemi Contractive Mapping in Banach Spaces

Akanksha Sharma1*, Kalpana Saxena2 and Namrata Tripathi1

1Department of Mathematics, Technocrats Institute of Technology, Bhopal, Madhya Pradesh, India

2Department of Mathematics, Govt. Motilal Vigyan Mahavidyalya, Bhopal, Madhya Pradesh, India

*Corresponding Author:
Akanksha Sharma
Department of Mathematics, Technocrats Institute of Technology
Bhopal, Madhya Pradesh, India
Tel: 07552751679
E-mail: [email protected]

Received June 25, 2015; Accepted July 27, 2015; Published July 30, 2015

Citation: Sharma A, Saxena K, Tripathi N (2015) Stability of Convergence Theorems of the Noor Iteration Method for an Enumerable Class of Continuous Hemi Contractive Mapping in Banach Spaces. J Appl Computat Math 4: 239. doi:10.4172/2168-9679.1000239

Copyright: © 2015 Sharma A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

The purpose of this is to study the Noor iteration process for the sequence {xn} converges to a common fix point for enumerable class of continuous hemi contractive mapping in Banach spaces.

Keywords

Stability; Noor iterations; Hemicontractive mapping; Convergence theorem; Continuous pseudocontractive mapping

2000 Mathematics Subject Classification: 47J25, 47H10, 54H25

Introduction

Let E be a real Banach space and let J denote the normalized duality mapping from E to E* defined by

equation

Where E* denotes the dual space of E and equationdenotes the generalization duality pair.

It is well known that if E* is strictly convex then J is single–valued. In the sequel, we shall denote the single–valued duality mapping by j. Let k be a nonempty closed convex subset of Banach space E and T : K → K be a self-mapping of K.

Definition 3.1: (i) A mapping T with domain D(T) and range R(T) in a Banach space is called pseudocontrative mapping, if for all x, y∈ D(T), there exists j(x − y)∈ J (x − y) such that [1]

equation     (1)

(ii) A mapping T with domain D(T) and range R(T) in E is called a hemicontractive mapping if F (T) ≠∅ and for all x∈D(T)x* ∈F(T) such that,

equation

(iii) A mapping T : K → K is called L-Lipschitizan there exists L>0 such that

equation For all x, y∈K

Definition 3.2: If equation and equation are sequences of real numbers in [0, 1] [2]. For arbitrary equation be a Noor iteration defined by,

equation

equation

equation

Lemma 3.3: Let E be a real uniformly convex Banach space [3], K is nonempty closed convex subset of E and T a continuous pseudocontractive mapping of K, then I-T is demiclosed at zero, that is, for all sequences equation with equation and equation it follows that p = Tp

Lemma 3.4: Let δ be a number satisfying 0 ≤δ <1 andequation a positive sequence satisfying equation[4,5]. Then, for any positive sequence equation satisfying: equation It follows that equation.

Results

Theorem 4.1: Let equation be defined as above and equation i n equation and let equation be a Banach space, T : E→E a self-map of E with a fixed point p, satisfying the contractive condition

equationfor equation

Let equation is converge to p and defined by the iteration (3.2) where equation is a real sequence in (0, 1) and define as equationThen

equation exists for all p∈ F;

equation

equation converges strongly to a common fixed point of equation if and only if equation

Proof: Let p∈ F and n ≥1 by 3.1 we choose equation

such that

equation

equation

equation

equation

equation

equation 

equation

equation        (1)

For the estimate of in (1) we get

equation

equation   

equation

equation

equation

equation

equation    (2)

Substituting (2) into (1) gives

equation    (3)

For equation in (3) we have,

equation

equation

equation

equation

equation

equation

equation    (4)

Substituting (4) into (3) and using lemma 3.3

equation

equation

equation

Observe that

equation    (5)

Therefore, taking the limit as n→∞ of both sides of the inequality (5) and using lemma 1.6 we get

equation That is equation

By theorem 3.2 equation

Taking infimum over all p∈F,we have,

equation

Thusequation exist we finally prove (iii) suppose thatequation Ffrom (ii) and

equationWe have equation for equation and equation, it follows

From (1.3) that

equation

Consequently,

equation

Therefore equation is a Cauchy sequence. Suppose equation for some equation . Then

equation

Since F is closed set, u∈F

So, Noor iteration process is T –stable.

Conclusion

Thus, the stability of Noor iteration considerable for finding fixed point for enumerable class of continuous hemi contractive mapping in Banach spaces.

References

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