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23: Paper Source PDF document

Paper's Title:

Traub-Potra-Type Method for Set-Valued Maps

Author(s):

Ioannis K. Argyros and Sad Hilout

Cameron University,
Department of Mathematics Sciences,
Lawton, OK 73505,
USA

iargyros@cameron.edu

URL: http://www.cameron.edu/~ioannisa/

Poitiers University,
Laboratoire de Mathematiques et Applications,
Bd. Pierre et Marie Curie, Teleport 2, B.P. 30179,
86962 Futuroscope Chasseneuil Cedex,
France

said.hilout@math.univ-poitiers.fr

http://www-math.univ-poitiers.fr/~hilout/

Abstract:

We introduce a new iterative method for approximating a locally unique solution of variational inclusions in Banach spaces by using generalized divided differences of the first order. This method extends a method considered by Traub  (in the scalar case) and by Potra  (in the Banach spaces case) for solving nonlinear equations to variational inclusions. An existence-convergence theorem and a radius of convergence are given under some conditions on divided differences operator and Lipschitz-like continuity property of set-valued mappings. The R-order of the method is equal to the unique positive root of a certain cubic equation, which is $1.839..., and as such it compares favorably to related methods such as the Secant method which is only of order $1.618....



8: Paper Source PDF document

Paper's Title:

An Improved Mesh Independence Principle for Solving Equations and their Discretizations using Newton's Method

Author(s):

Ioannis K. Argyros

Cameron university,
Department of Mathematics Sciences,
Lawton, OK 73505,
USA
iargyros@cameron.edu
 

Abstract:

We improve the mesh independence principle [1] which states that when Newton's method is applied to an equation on a Banach space as well as to their finite--dimensional discretization there is a difference of at most one between the number of steps required by the two processes to converge to within a given error tolerance. Here using a combination of Lipschitz and center Lipschitz continuity assumptions instead of just Lipschitz conditions we show that the minimum number of steps required can be at least as small as in earlier works. Some numerical examples are provided whereas our results compare favorably with earlier ones.



2: Paper Source PDF document

Paper's Title:

Algorithms for Nonlinear Problems Involving Strictly Pseudocontractive Mappings

Author(s):

Mathew Olajiire Aibinu1, Surendra Colin Thakur2, Sibusiso Moyo3

1Institute for Systems Science & KZN E-Skill CoLab,
Durban University of Technology,
Durban 4000,
South Africa.

1DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS),
Johannesburg,
South Africa.
E-mail: 
moaibinu@yahoo.com mathewa@dut.ac.za

2 KZN E-Skill CoLab,
Durban University of Technology,
Durban 4000,
South Africa.
E-mail: 
thakur@dut.ac.za

3Institute for Systems Science & Office of the DVC Research, Innovation & Engagement Milena Court,
Durban University of Technology,
 Durban 4000,
South Africa.
E-mail:
dvcrie@dut.ac.za

 

Abstract:

The puzzles in approximating a fixed point of nonlinear problems involving the class of strictly pseudocontractive mappings are conquered in this paper through viscosity implicit rules. Using generalized contraction mappings, a new viscosity iterative algorithm which is implicit in nature is proposed and analysed in Banach spaces for the class of strictly pseudocontractive mappings. The computations and analysis which are used in the proposed scheme are easy to follow and this gives rooms for a broad application of the scheme. It is obtained that the proposed iterative algorithm converges strongly to a fixed point of a μ-strictly pseudocontractive mapping which also solves a variational inequality problem. The result is also shown to hold for finite family of strictly pseudocontractive mappings. A numerical example is given to show the skillfulness of the proposed scheme and its implementation.


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