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Some convergence results for iterative sequences of Prešić type and applications

Mohammad Saeed Khan1, Maher Berzig2 and Bessem Samet3*

Author Affiliations

1 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, Al-Khod, Post Box 36, PCode 123, Muscat, Sultanate of Oman

2 Ecole Supérieure des Sciences et Techniques de Tunis, 5, Avenue Taha Hussein-Tunis, B.P. 56, Bab Menara-1008, Tunisie

3 Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

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Advances in Difference Equations 2012, 2012:38  doi:10.1186/1687-1847-2012-38

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2012/1/38


Received:22 November 2011
Accepted:29 March 2012
Published:29 March 2012

© 2012 Khan et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

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 ∈ [0, 1) such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M1">View MathML</a>

Then f has a unique fixed point, that is, there exists a unique x* X such that x* = fx*. Moreover, for any x0 X, the iterative sequence xn+1 = fxn converges to x*.

This theorem called the Banach contraction principle is a simple and powerful theorem with a wide range of application, including iterative methods for solving linear, nonlinear, differential, integral, and difference 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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M2">View MathML</a>

(1)

with the initial values x0,..., xk-1 X, where k is a positive integer (k ≥ 1) and f : Xk X. Equation (1) can be studied by means of fixed point theory in view of the fact that x* ∈ 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 Prešić in [22] by generalizing the Banach contraction principle in the following way.

Theorem 1.2 (Prešić [22]) Let (X,d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M3">View MathML</a>

for all x0, ..., xk X, where δ1, ..., δk are positive constants such that δ1 + ... + δk ∈ (0,1). Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*, ..., x*). Moreover, for any initial values x0, ..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

It is easy to show that for k = 1, Theorem 1.2 reduces to the Banach contraction principle. So, Theorem 1.2 is a generalization of the Banach fixed point theorem.

In [13], Ćirić and Prešić generalized Theorem 1.2 as follows.

Theorem 1.3 (Ćirić and Prešić [13]) Let (X,d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M4">View MathML</a>

for all x0, ..., xk X, where λ ∈ (0,1) is a constant. Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*,..., x*). Moreover, for any initial values x0, ..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

The applicability of the result due to Ćirić and Prešić to the study of global asymptotic stability of the equilibrium for the nonlinear difference Equation (1) is revealed, for example, in the recent article [8].

Other generalizations were obtained by Păcurar in [20,21].

Theorem 1.4 (Păcurar [20]) Let (X, d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M5">View MathML</a>

for all x0, ..., xk X, where a is a constant such that 0 < ak(k + 1) < 1. Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*, ..., x*). Moreover, for any initial values x0, ..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

In the particular case k = 1, from Theorem 1.4, we obtain Kannan's fixed point theorem for discontinuous mappings in [15].

Theorem 1.5 (Păcurar [21]) Let (X, d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M6">View MathML</a>

for all x0, ..., xk X, where δ1, ..., δk are positive constants such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M7">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M8">View MathML</a>

(2)

with L ≥ 0. Then f has a unique fixed point x* X, that is, there exists a unique x* X such that x* = f(x*, ..., x*). Moreover, for any initial values x0,..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

In the particular case k = 1, the contractive condition (2) reduces to strict almost contraction (see [4-7]).

Note that these approaches are motivated by the currently increasing interest in the study of nonlinear difference equations which appear in many interesting examples from system theory, economics, inventory analysis, probability models for learning, approximate solutions of ordinary and partial differential equations just to mention a few [29-31]. We refer the reader to [32-34] for a detailed study of the theory of difference equations.

For other studies in this direction, we refer the reader to [23,25,35,36].

In this article, we study the convergence of the iterative sequence (1) for more general classes of operators. Presented theorems extend and generalize many existing results in the literature including Theorems 1.1, 1.2, 1.4, and 1.5. We present also an application to a class of nonlinear difference matrix equations and we validate our results with numerical experiments.

2 Main results

In order to prove our main results we shall need the following lemmas.

Lemma 2.1 Let k be a positive integer and α1, α2, ..., αk ≥ 0 such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M9">View MathML</a>. If { Δn} is a sequence of positive numbers satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M10">View MathML</a>

then there exist L ≥ 0 and τ ∈ (0,1) such that Δ n n for all n ≥ 1.

Lemma 2.2 Let {an}, {bn} be two sequences of positive real numbers and q ∈ (0,1) such that an+1 qan + bn, n ≥ 0 and bn → 0 as n → ∞. Then an → 0 as n → ∞.

Let Θ be the set of functions θ : [0, ∞)4 → [0, ∞) satisfying the following conditions:

(i) θ is continuous,

(ii) for all t1, t2, t3, t4 ∈ [0, ∞),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M11">View MathML</a>

Example 2.1 The following functions belong to Θ:

(1) θ(t1, t2, t3, t4) = L min{t1, t2, t3, t4}, L > 0 t1, t2, t3, t4 ≥ 0.

(2) θ(t1, t2, t3, t4) = L ln(1 + t1t2t3t4), L > 0 t1,t2,t3,t4 ≥ 0.

(3) θ(t1, t2, t3, t4) = L ln(1 + t1) ln(1 + t2) ln(1 + t3) ln(1 + t4), L > 0 t1,t2,t3,t4 ≥ 0.

(4) θ(t1, t2, t3, t4) = Lt1t2t3t4, L > 0 t1, t2, t3, t4 ≥ 0.

(5) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M12">View MathML</a>, L > 0 t1, t2, t3, t4 ≥ 0.

Our first result is the following.

Theorem 2.1 Let (X,d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M13">View MathML</a>

(3)

for all x0,..., xk X, where δ1,..., δk + 1 are positive constants such that 2A + δ ∈ (0,1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M14">View MathML</a>and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M15">View MathML</a>. Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*,..., x*). Moreover, for any z0 X, the iterative sequence {zn} defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M16">View MathML</a>

converges to x*.

Proof. Define the mapping F : X X by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M17">View MathML</a>

Using (3), for all x, y X, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M18">View MathML</a>

Thus, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M19">View MathML</a>

(4)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M20">View MathML</a>

Now, let z0 be an arbitrary element of X. Define the sequence {zn} by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M21">View MathML</a>

Using (4), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M22">View MathML</a>

On the other hand, from the property (ii) of the function θ, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M23">View MathML</a>

Then we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M24">View MathML</a>

for all n = 1, 2,.... This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M25">View MathML</a>

for all n = 1, 2,.... Since we have 2A + δ ∈ (0,1), then {zn} is a Cauchy sequence in (X, d). Now, since (X, d) is complete, there exists x* X such that zn x* as n → ∞. We shall prove that x* is a fixed point of F, that is, x* = Fx*. Using (4), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M26">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M27">View MathML</a>

Thus we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M28">View MathML</a>

Letting n → ∞ in the above inequality, and using the properties (i) and (ii) of θ, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M29">View MathML</a>

which implies (since 1 - A > 0) that x* = Fx* = f(x*, ..., x*).

Now, we shall prove that x* is the unique fixed point of F. Suppose that y* ∈ X is another fixed point of F, that is, y* = Fy* = f(y*,..., y*). Using (4), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M30">View MathML</a>

On the other hand, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M31">View MathML</a>

Then we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M32">View MathML</a>

which implies (since δ < 1) that x* = y*.

Theorem 2.2 Let (X, d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M33">View MathML</a>

(5)

for all x0, ..., xk X, where δ1, ..., δk are positive constants such that δ ∈ (0,1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M15">View MathML</a>, and B ≥ 0. Then

(a) there exists a unique x* ∈ X such that x* = f(x*,..., x*);

(b) the sequence {xn} defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M34">View MathML</a>

(6)

converges to x* for any x0, ..., xk-1 X.

Proof. Applying Theorem 2.1 with δk + 1 = 0, and remarking that ∈ Θ, we obtain immediately (a). Now, we shall prove (b). Let x0,..., xk-1 X and xn = f(xn-k,..., xn-1), n k. Then by (5), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M35">View MathML</a>

Since k is a fixed positive integer, then we may denote

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M36">View MathML</a>

Then we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M37">View MathML</a>

Similarly we get that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M38">View MathML</a>

Denoting

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M39">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M40">View MathML</a>

Continuing this process, for n k, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M41">View MathML</a>

Denoting

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M42">View MathML</a>

the above inequality becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M43">View MathML</a>

(7)

Now, we shall prove that the sequence {En} given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M44">View MathML</a>

converges to 0 as n → ∞.

For n k, from (5), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M45">View MathML</a>

As d(xn, f(xn-k,..., xn-1) = 0, the above inequality leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M46">View MathML</a>

According to Lemma 2.1, this implies the existence of τ ∈ (0,1) and L ≥ 0 such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M47">View MathML</a>

Now, En is a finite sum of sequences converging to 0, so it is convergent to 0.

Finally, using (7) and applying Lemma 2.2 with an = d(xn, x*) and bn = En + 1-k, we get that d(xn, x*) → 0 as n → ∞, that is, the iterative sequence {xn} converges to the unique fixed point of f

Remark 2.1 In the particular case θ(t1, t2, t3, t4) = min{t1, t2, t3, t4}, from Theorem 2.2 we obtain Păcurar's result (see Theorem 1.5).

Now, we shall prove the following result.

Theorem 2.3 Let (X, d) be a complete metric space, k a positive integer and f : Xk X.

Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M48">View MathML</a>

(8)

for all x0, ..., xk X, where a is a positive constant such that A ∈ (0, 1/2) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M49">View MathML</a>. Then

(a) there exists a unique x* ∈ X such that x* = f(x*, ..., x*);

(b) the sequence {xn} defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M50">View MathML</a>

(9)

converges to x* for any x0, ..., xk-1 X, with a rate estimated by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M51">View MathML</a>

(10)

where L ≥ 0, τ ∈ (0, 1) and M = τ1-k + 2τ2-k + ⋯ + k.

Proof. (a) follows immediately from Theorem 2.1 with δ = 0 and δk+1 = a. Now, we shall prove (b). Let x0, ..., xk-1 X and xn = f(xn-k, ..., xn-1), n k. Then by (8), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M52">View MathML</a>

(11)

Using (4), for all i = 0,1,..., k - 1, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M53">View MathML</a>

This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M54">View MathML</a>

(12)

Now, combining (12) with (11), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M55">View MathML</a>

Similarly, one can show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M56">View MathML</a>

(13)

This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M57">View MathML</a>

Define the sequence { Δp} by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M58">View MathML</a>

We get that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M59">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M60">View MathML</a>, we can apply Lemma 2.1 to deduce that there exist L ≥ 0 and τ ∈ (0,1) such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M61">View MathML</a>

(14)

This implies that Δp → 0 as p → ∞, that is, xp x* as p → ∞. Finally, (10) follows from (14) and (13).

Remark 2.2 Many results can be derived from our Theorems 2.1, 2.2 and 2.3 with respect to particular choices of θ (see Example 2.1).

Remark 2.3 Clearly, Theorem 1.4 of Păcurar is a particular case of our Theorem 2.3.

3 Application: convergence of the recursive matrix sequence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M62">View MathML</a>

In the last few years there has been a constantly increasing interest in developing the theory and numerical approaches for Hermitian positive definite (HPD) solutions to different classes of nonlinear matrix equations (see [37-41]). In this section, basing on Theorem 1.3 of Ćirić and Prešić, we shall study the nonlinear matrix difference equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M63">View MathML</a>

(15)

where Q is an N × N positive definite matrix, A and B are arbitrary N × N matrices, α and β are real numbers. Here, A* denotes the conjugate transpose of the matrix A.

We first review the Thompson metric on the open convex cone P(N) (N ≥ 2), the set of all N × N Hermitian positive definite matrices. We endow P(N) with the Thompson metric defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M64">View MathML</a>

where M(A/B) = inf{λ > 0 : A ≤ λB} = λ+(B-1/2AB-1/2), the maximal eigenvalue of B-1/2AB-1/2. Here, X Y means that Y - X is positive semi-definite and X < Y means that Y - X is positive definite. Thompson [42] has proved that P(n) is a complete metric space with respect to the Thompson metric d and d(A, B) = |log( A-1/2BA-1/2)|, where |⋅| stands for the spectral norm. The Thompson metric exists on any open normal convex cones of real Banach spaces; in particular, the open convex cone of positive definite operators of a Hilbert space. It is invariant under the matrix inversion and congruence transformations, that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M65">View MathML</a>

(16)

for any nonsingular matrix M. The other useful result is the nonpositive curvature property of the Thompson metric, that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M66">View MathML</a>

(17)

By the invariant properties of the metric, we then have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M67">View MathML</a>

(18)

for any X, Y P(N) and nonsingular matrix M.

Lemma 3.1 [40]For all A, B, C, D P(N), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M68">View MathML</a>

In particular,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M69">View MathML</a>

3.1 A convergence result

We shall prove the following convergence result.

Theorem 3.1 Suppose that λ = max{|α|, |β|} ∈ (0,1). Then

(i) Equation (15) has a unique equilibrium point in P(N), that is, there exists a unique U P(N) such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M70">View MathML</a>

(ii) for any X0, X1 > 0, the iterative sequence {Xn} defined by (15) converges to U.

Proof. Define the mapping f : P(N) × P(N) → P(N) by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M71">View MathML</a>

Using Lemma 3.1 and properties (16)-(18), for all X, Y, Z P(N), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M72">View MathML</a>

Thus we proved that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M73">View MathML</a>

for all X, Y, Z P(N). Since λ ∈ (0, 1), (i) and (ii) follow immediately from Theorem 1.3.

3.2 Numerical experiments

All programs are written in MATLAB version 7.1.

We consider the iterative sequence {Xn} defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M74">View MathML</a>

(19)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M75">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M76">View MathML</a>

It is clear that from our Theorem 3.1, Eq.(19) has a unique equilibrium point U P(3). We denote by Rm (m ≥ 1) the residual error at the iteration m, that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M77">View MathML</a>

where |⋅| is the spectral norm.

After 40 iterations, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M78">View MathML</a>

with residual error

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/38/mathml/M79">View MathML</a>

The convergence history of the algorithm (19) is given by Figure 1.

thumbnailFigure 1. Convergence history for Equation (19).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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