Some convergence results for iterative sequences of Prešić type and applications

Khan Mohammad, Berzig Maher, Samet, Samet Bessem

Description

In this article, we study the convergence of iterative sequences of Prešić type involving new general classes of operators in the setting of metric spaces. As application, we derive some convergence results for a class of nonlinear matrix difference equations. Numerical experiments are also presented to illustrate the convergence algorithms. Mathematics Subject Classification 2000 : 54H25; 47H10; 15A24; 65H05.

Sujets

• Convergence
• Recurrence relation
• Fixed point
• Matrix

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 8
• Langue: English

Extrait

Khan et al . Advances in Difference Equations 2012, 2012 :38 http://www.advancesindifferenceequations.com/content/2012/1/38

R E S E A R C H Open Access Some convergence results for iterative sequences of Pre š i ć type and applications Mohammad Saeed Khan 1 , Maher Berzig 2 and Bessem Samet 3*

* Correspondence: bessem. [e-mail] 3 Department of Mathematics, King Saud University, Riyadh, Saudi Arabia Full list of author information is available at the end of the article

Abstract In this article, we study the convergence of iterative sequences of Pre š i ć type involving new general classes of operators in the setting of metric spaces. As application, we derive some convergence results for a class of nonlinear matrix difference equations. Numerical experiments are also presented to illustrate the convergence algorithms. Mathematics Subject Classification 2000 : 54H25; 47H10; 15A24; 65H05. Keywords: iterative sequence, convergence, difference equation, fixed point, matrix

1 Introduction In 1922, Banach proved the following famous fixed point theorem. Theorem 1.1 (Banach [1] ) Let ( X, d ) be a complete metric space and f : X ® X be a contractive mapping, that is, there exists δ Î [0, 1) such that d ( fx , fy ) ≤ δ d ( x , y ), for all x , y ∈ X . Then f has a unique fixed point, that is, there exists a unique x* Î X such that x* = fx*. Moreover, for any x 0 Î X, the iterative sequence x n+ 1 = fx n converges to x* . This theorem called the Banach contraction principle is a simple and powerful theo- rem with a wide range of application, including iterative methods for solving linear, nonlinear, differential, integral, and diff erence equations. Many generalizations and extensions of the Banach contraction principle exist in the literature. For more details, we refer the reader to [2-28]. Consider the k -th order nonlinear difference equation x n +1 = f ( x n − k +1 , . . . , x n ), n = k − 1, k , k + 1, . . . (1) with the initial values x 0 ,..., x k -1 Î X , where k is a positive integer ( k ≥ 1) and f : X k ≤ X . Equation (1) can be studied by means of fixed point theory in view of the fact that x * Î X is a solution to (1)) if and only if x * is a fixed point of f , that is, x * = f ( x *, ..., x *). One of the most important results in this direction has been obtained by Pre š i ć in [22] by generalizing the Banach contraction principle in the following way. Theorem 1.2 (Pre š i ć [22] ) Let ( X , d ) be a complete metric space, k a positive integer and f : X k ® X. Suppose that

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