Reach Us
+44-1474-556909

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

^{2}Department 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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

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

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

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

Where E* denotes the dual space of E and denotes 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]

(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,

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

For all x, y∈K

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

**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 with and it follows that p = Tp

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

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

for

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

exists for all p∈ F;

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

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

such that

(1)

For the estimate of in (1) we get

(2)

Substituting (2) into (1) gives

(3)

For in (3) we have,

(4)

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

Observe that

(5)

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

That is

By theorem 3.2

Taking infimum over all p∈F,we have,

Thus exist we finally prove (iii) suppose that Ffrom (ii) and

We have for and , it follows

From (1.3) that

Consequently,

Therefore is a Cauchy sequence. Suppose for some . Then

Since F is closed set, u∈F

So, Noor iteration process is T –stable.

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

- Browder FE, Petryshyn WV(1967) Construction of fixed points of nonlinear mappings in Hilbert space. JMath Anal Appl 20: 197-228.
- Noor MA (2000)Newapproximations schemes for general variational inequalities. J MathAnal Appl 251: 217-299.
- Chen R, Song Y, Zhou H (2006)Convergence theorems for implicit iteration process for a finite family ofcontinuous pseudo contractive mappings. J Math Anal Appl 314: 701-709.
- TakahashiW (2000) Nonlinear Functional Analysis FixedPoint Theory and its Applications. Yokohama Publishers Inc.
- Zhou H (2008) Convergencetheorems of common fixed points for a finite family of Lipschitz
- Pseudo contractions in Banachspaces. Nonlinear Anal 68:2977-2983.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Hamilton Mechanics
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Physical Mathematics
- Quantum Mechanics
- Quantum electrodynamics
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topology
- mirror symmetry
- vector bundle

- Total views:
**12071** - [From(publication date):

July-2015 - Jul 23, 2019] - Breakdown by view type
- HTML page views :
**8285** - PDF downloads :
**3786**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20