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Theory and Modern Applications

Some convergence results for iterative sequences of Prešić type and applications

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.

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 : XX be a contractive mapping, that is, there exists δ [0, 1) such that

d ( f x , f y ) δ d ( x , y ) , f o r a l l x , y X .

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= fx n 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 [228].

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 x0,..., xk-1 X, where k is a positive integer (k ≥ 1) and f : XkX. 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 : XkX. Suppose that

d f x 0 , , x k - 1 , f x 1 , , x k i = 1 k δ i d ( x i - 1 , x i ) ,

for all x0, ..., x k X, where δ1, ..., δ k are positive constants such that δ1 + ... + δ k (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 {x n } 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 : XkX. Suppose that

d ( f ( x 0 , , x k - 1 ) , f ( x 1 , , x k ) ) λ max { d ( x 0 , x 1 ) , , d ( x k - 1 , x k ) } ,

for all x0, ..., x k 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 {x n } 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 : XkX. Suppose that

d f x 0 , , x k - 1 , f x 1 , , x k a i = 0 k d x i , f ( x i , , x i ) ,

for all x0, ..., x k 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 {x n } 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 : XkX. Suppose that

d f x 0 , , x k - 1 , f x 1 , , x k i = 1 k δ i d ( x i - 1 , x i ) + M ( x 0 , x k ) ,

for all x0, ..., x k X, where δ1, ..., δ k are positive constants such that i = 1 k δ i < 1 and

M ( x 0 , x k ) = L min d ( x 0 , f ( x 0 , , x 0 ) ) , d ( x k , f ( x k , , x k ) ) , d ( x 0 , f ( x k , , x k ) ) , d ( x k , f ( x 0 , , x 0 ) ) , d ( x k , f ( x 0 , , x k - 1 ) )
(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 {x n } defined by (1) converges to x*.

In the particular case k = 1, the contractive condition (2) reduces to strict almost contraction (see [47]).

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 [2931]. We refer the reader to [3234] 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, ..., α k ≥ 0 such that i = 1 k α i =α<1. If { Δ n } is a sequence of positive numbers satisfying

Δ n + k α 1 Δ n + α 2 Δ n + 1 + + α k Δ n + k - 1 , n 1 ,

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

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

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

  1. (i)

    θ is continuous,

  2. (ii)

    for all t1, t2, t3, t4 [0, ∞),

    θ ( t 1 , t 2 , t 3 , t 4 ) = 0 t 1 t 2 t 3 t 4 = 0 .

Example 2.1 The following functions belong to Θ:

  1. (1)

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

  2. (2)

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

  3. (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. (4)

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

  5. (5)

    θ ( t 1 , t 2 , t 3 , t 4 ) = L ( e t 1 t 2 t 3 t 4 - 1 ) , 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 : XkX. Suppose that

d ( f ( x 0 , , x k - 1 ) , f ( x 1 , , x k ) ) i = 1 k δ i d ( x i - 1 , x i ) + δ k + 1 i = 0 k d ( x i , f ( x i , , x i ) ) + θ d ( x 0 , f ( x 0 , , x 0 ) ) , d ( x k , f ( x k , , x k ) ) , d ( x 0 , f ( x k , , x k ) ) , d ( x k , f ( x 0 , , x 0 ) )
(3)

for all x0,..., x k X, where δ1,..., δk + 1are positive constants such that 2A + δ (0,1) with A= k ( k + 1 ) 2 δ k + 1 and δ= i = 1 k δ i . 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 {z n } defined by

z n + 1 = f ( z n , , z n ) , n 0

converges to x*.

Proof. Define the mapping F : XX by

F x = f ( x , , x ) , for all x X .

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

d ( F x , F y ) = d ( f ( x , , x ) , f ( y , , y ) ) d ( f ( x , , x ) , f ( x , , x , y ) ) + d ( f ( x , , x , y ) , f ( x , , x , y , y ) ) + + d ( f ( x , y , , y ) , f ( y , , y ) ) [ δ k d ( x , y ) + δ k - 1 d ( x , y ) + + δ 1 d ( x , y ) ] + ( 1 + + k ) δ k + 1 [ d ( x , F x ) + d ( y , F y ) ] + k θ d ( x , F x ) , d ( y , F y ) , d ( x , F y ) , d ( y , F x ) .

Thus, we have

d ( F x , F y ) i = 1 k δ i d ( x , y ) + k ( k + 1 ) 2 δ k + 1 [ d ( x , F x ) + d ( y , F y ) ] + M ( x , y ) ,
(4)

where

M ( x , y ) = k θ d ( x , F x ) , d ( y , F y ) , d ( x , F y ) , d ( y , F x ) .

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

z n = F z n - 1 = f ( z n - 1 , , z n - 1 ) , n 1 .

Using (4), we have

d z n + 1 , z n = d ( F z n , F z n - 1 ) i = 1 k δ i d ( z n , z n - 1 ) + k ( k + 1 ) 2 δ k + 1 [ d ( z n , z n + 1 ) + d ( z n - 1 , z n ) ] + M ( z n , z n - 1 ) .

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

M ( z n , z n - 1 ) = k θ d ( z n , z n + 1 ) , d ( z n - 1 , z n ) , 0 , d ( z n - 1 , z n + 1 ) = 0 .

Then we get

d ( z n + 1 , z n ) δ d ( z n , z n - 1 ) + k ( k + 1 ) 2 δ k + 1 [ d ( z n , z n + 1 ) + d ( z n - 1 , z n ) ]

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

d ( z n + 1 , z n ) A + δ 1 - A d ( z n , z n - 1 )

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

d ( x * , F x * ) d ( x * , z n + 1 ) + d ( F z n , F x * ) d ( x * , z n + 1 ) + i = 1 k δ i d ( z n , x * ) + k ( k + 1 ) 2 δ k + 1 [ d ( z n , z n + 1 ) + d ( x * , F x * ) ] + M ( z n , x * ) ,

where

M ( z n , x * ) = k θ d ( z n , z n + 1 ) , d ( x * , F x * ) , d ( z n , F x * ) , d ( x * , z n + 1 ) .

Thus we have

( 1 - A ) d ( x * , F x * ) d ( x * , z n + 1 ) + δ d ( z n , x * ) + A d ( z n , z n + 1 ) + M ( z n , x * ) .

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

( 1 - A ) d ( x * , F x * ) 0 ,

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

d ( x * , y * ) = d ( F x * , F y * ) δ d ( x * , y * ) + k ( k + 1 ) 2 δ k + 1 [ d ( x * , F x * ) + d ( y * , F y * ) ] + M ( x * , y * ) = δ d ( x * , y * ) + M ( x * , y * ) .

On the other hand, we have

M ( x * , y * ) = k θ d ( x * , F x * ) , d ( y * , F y * ) , d ( x * , F y * ) , d ( y * , F x * ) = k θ 0 , 0 , d ( x * , y * ) , d ( y * , x * ) = 0 .

Then we get

( 1 - δ ) d ( x * , y * ) 0 ,

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 : XkX. Suppose that

d ( f ( x 0 , , x k - 1 ) , f ( x 1 , , x k ) ) i = 1 k δ i d ( x i - 1 , x i ) + B min d ( x k , f ( x 0 , , x k - 1 ) ) , θ d ( x 0 , f ( x 0 , , x 0 ) ) , d ( x k , f ( x k , , x k ) ) , d ( x 0 , f ( x k , , x k ) ) , d ( x k , f ( x 0 , , x 0 ) )
(5)

for all x0, ..., x k X, where δ1, ..., δ k are positive constants such that δ (0,1) with δ= i = 1 k δ i , and B ≥ 0. Then

  1. (a)

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

  2. (b)

    the sequence {x n } defined by

    x n + 1 = f ( x n - k + 1 , , x n ) , n = k - 1 , k , k + 1 ,
    (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 x n = f(x n-k ,..., xn- 1), nk. Then by (5), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have

d ( x k , x * ) = d ( f ( x 0 , , x k - 1 ) , f ( x * , , x * ) ) d ( f ( x 0 , , x k - 1 ) , f ( x 1 , , x k - 1 , x * ) ) + d ( f ( x 1 , , x k - 1 , x * ) , f ( x 2 , , x k - 1 , x * , x * ) ) + + d ( f ( x k - 1 , x * , , x * ) , f ( x * , , x * ) ) δ 1 d ( x 0 , x 1 ) + ( δ 1 + δ 2 ) d ( x 1 , x 2 ) + + ( δ 1 + + δ k - 1 ) d ( x k - 2 , x k - 1 ) + δ d ( x k - 1 , x * ) .

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

E 0 = δ 1 d ( x 0 , x 1 ) + ( δ 1 + δ 2 ) d ( x 1 , x 2 ) + + ( δ 1 + + δ k - 1 ) d ( x k - 2 , x k - 1 ) .

Then we get

d ( x k , x * ) E 0 + δ d ( x k - 1 , x * ) .

Similarly we get that

d ( x k + 1 , x * ) δ 1 d ( x 1 , x 2 ) + ( δ 1 + δ 2 ) d ( x 2 , x 3 ) + + ( δ 1 + + δ k - 1 ) d ( x k - 1 , x k ) + δ d ( x k , x * ) .

Denoting

E 1 = δ 1 d ( x 1 , x 2 ) + ( δ 1 + δ 2 ) d ( x 2 , x 3 ) + + ( δ 1 + + δ k - 1 ) d ( x k - 1 , x k ) ,

we get

d ( x k + 1 , x * ) E 1 + δ d ( x k , x * ) .

Continuing this process, for nk, we obtain

d ( x n , x * ) δ 1 d ( x n - k , x n - k + 1 ) + ( δ 1 + δ 2 ) d ( x n - k + 1 , x n - k + 2 ) + + ( δ 1 + + δ k - 1 ) d ( x n - 2 , x n - 1 ) + δ d ( x n - 1 , x * ) .

Denoting

E n - k = δ 1 d ( x n - k , x n - k + 1 ) + ( δ 1 + δ 2 ) d ( x n - k + 1 , x n - k + 2 ) + + ( δ 1 + + δ k - 1 ) d ( x n - 2 , x n - 1 ) ,

the above inequality becomes

d ( x n , x * ) δ d ( x n - 1 , x * ) + E n - k , n k .
(7)

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

E n = δ 1 d ( x n , x n + 1 ) + ( δ 1 + δ 2 ) d ( x n + 1 , x n + 2 ) + + ( δ 1 + + δ k - 1 ) d ( x n + k - 2 , x n + k - 1 ) ,

converges to 0 as n → ∞.

For nk, from (5), we have

d ( x n , x n + 1 ) = d ( f ( x n k , , x n 1 ) , f ( x n k + 1 , , x n ) ) δ 1 d ( x n k , x n k + 1 ) + δ 2 d ( x n k + 1 , x n k + 2 ) + + δ k d ( x n 1 , x n ) + B min { d ( x n , f ( x n k , , x n 1 ) , θ ( d ( x n k , F ( x n k ) ) , d ( x n , F x n ) , d ( x n k , F x n ) , d ( x n , F x n k ) ) } .

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

d ( x n , x n + 1 ) δ 1 d ( x n - k , x n - k + 1 ) + δ 2 d ( x n - k + 1 , x n - k + 2 ) + + δ k d ( x n - 1 , x n ) .

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

d ( x n , x n + 1 ) L τ n , for all n 1 .

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

Finally, using (7) and applying Lemma 2.2 with a n = d(x n , x*) and b n = En + 1-k, we get that d(x n , x*) → 0 as n → ∞, that is, the iterative sequence {x n } 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 : XkX.

Suppose that

d ( f ( x 0 , , x k - 1 ) , f ( x 1 , , x k ) ) a i = 0 k d ( x i , f ( x i , , x i ) ) + θ d ( x 0 , f ( x 0 , , x 0 ) ) , d ( x k , f ( x k , , x k ) ) , d ( x 0 , f ( x k , , x k ) ) , d ( x k , f ( x 0 , , x 0 ) )
(8)

for all x0, ..., x k X, where a is a positive constant such that A (0, 1/ 2) with A= k ( k + 1 ) 2 a. Then

  1. (a)

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

  2. (b)

    the sequence {x n } defined by

    x n + 1 = f ( x n - k + 1 , , x n ) , n = k - 1 , k , k + 1
    (9)

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

d x n + 1 , x * a L 1 - A M τ n , n k ,
(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 x n = f(x n-k , ..., xn- 1), nk. Then by (8), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have

d ( x k , x * ) = d f x 0 , . . . , x k - 1 , f x * , . . . , x * d f x 0 , . . . , x k - 1 , f x 1 , . . . , x k - 1 , x * + d f x 1 , . . . , x k - 1 , x * , f x 2 , . . . , x k - 1 , x * , x * + + d f x k - 1 , x * , . . . , x * , f x * , . . . , x * a d x 0 , F x 0 + 2 a d x 1 , F x 1 + + k a d x k - 1 , F x k - 1 .
(11)

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

d x i , F x i d x i , x * + d F x i d x i , x * + A d x i , F x i + k θ 0 , d x i , F x i , d x * , F x i , d x i , F x * = d x * , x i + A d x i , F x i .

This implies that

d x i , F x i 1 1 - A d x i , x * , i = 0 , 1 , . . . , k - 1 .
(12)

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

d x k , x * a 1 - A d x 0 , x * + 2 a 1 - A d x 1 , x * + + k a 1 - A d x k - 1 , x * .

Similarly, one can show that

d x n , x * a 1 - A d x n - k , x * + 2 a 1 - A d x n - k + 1 , x * + + k a 1 - A d x n - 1 , x * , n k .
(13)

This implies that

d x p + k , x * a 1 - A d x p , x * + 2 a 1 - A d x p + 1 , x * + + k a 1 - A d x p + k - 1 , x * , p 0 .

Define the sequence { Δ p } by

Δ p = d x p , x * , for all p 0 .

We get that

Δ p + k a 1 - A Δ p + 2 a 1 - A Δ p + 1 + + k a 1 - A Δ p + k - 1 , p 0 .

Since i = 1 k i a 1 - A = A 1 - A ( 0 , 1 ) , we can apply Lemma 2.1 to deduce that there exist L ≥ 0 and τ (0,1) such that

Δ p L τ p , p 1 .
(14)

This implies that Δ p → 0 as p → ∞, that is, x p 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

X n + 1 = Q + A * X n - 1 α A + B * X n β B

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 [3741]). In this section, basing on Theorem 1.3 of Ćirić and Prešić, we shall study the nonlinear matrix difference equation

X n + 1 = Q + A * X n - 1 α A + B * X n β B ,
(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

d A , B = max log M A / B , log M B / 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, XY 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,

d A , B = d A - 1 , B - 1 = d M A M * , M B M *
(16)

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

d X r , Y r r d X , Y , r [ 0 , 1 ] .
(17)

By the invariant properties of the metric, we then have

d M X r M * , M Y r M * r d ( X , Y ) , r [ - 1 , 1 ]
(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

d A + B , C + D max d A , C , d B , D .

In particular,

d A + B , A + C d ( B , C ) .

3.1 A convergence result

We shall prove the following convergence result.

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

  1. (i)

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

    U = Q + A * U α A + B * U β B ;
  2. (ii)

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

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

f ( X , Y ) = Q + A * X α A + B * Y β B , X , Y P ( N ) .

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

d f ( X , Y ) , f ( Y , Z ) = d Q + A * X α A + B * Y β B , Q + A * Y α A + B * Z β B d A * X α A + B * Y β B , A * Y α A + B * Z β B max d A * X α A , A * Y α A , d B * Y β B , B * Z β B max α d ( X , Y ) , β d ( Y , Z ) max α , β max d ( x , Y ) , d ( Y , Z ) = λ max d ( x , Y ) , d ( Y , Z ) .

Thus we proved that

d f ( X , Y ) , f ( Y , Z ) λ max d X , Y , d Y , Z

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 {X n } defined by

X n + 1 = Q + A * X n - 1 1 / 2 A + B * X n 1 / 3 B , X 0 , X 1 > 0 ,
(19)

where

A = 0 . 306 0 . 6894 0 . 6093 0 . 2514 0 . 4285 0 . 7642 0 . 0222 0 . 0987 0 . 8519 , B = 0 . 9529 0 . 645 0 . 4801 0 . 441 0 . 1993 0 . 9823 0 . 9712 0 . 0052 0 . 92

and

X 0 = 1 0 0 0 1 0 0 0 1 , X 1 = Q = 10 3 . 85 - 3 . 85 3 . 85 10 3 . 92 - 3 . 85 3 . 92 10 .

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

R m = X m + 1 - Q + A * X m + 1 1 / 2 A + B * X m + 1 1 / 3 B ,

where || is the spectral norm.

After 40 iterations, we obtain

U X 40 = 17 . 22 7 . 559 4 . 429 7 . 559 14 . 55 10 . 38 4 . 429 10 . 38 26 . 56

with residual error

R 40 = 1 . 624 × 1 0 - 14 .

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

Figure 1
figure 1

Convergence history for Equation (19).

References

  1. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math 1922, 3: 133–181.

    MATH  Google Scholar 

  2. Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications. Cambridge University Press, Cambridge; 2001.

    Chapter  Google Scholar 

  3. Azam A, Arshad M, Beg I: Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos Solitons Fract 2009, 42(5):2836–2841. 10.1016/j.chaos.2009.04.026

    Article  MathSciNet  MATH  Google Scholar 

  4. Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpathian J Math 2008, 24: 8–12.

    MathSciNet  MATH  Google Scholar 

  5. Berinde V: On the approximation of fixed points of weak contractive mappings. Carpathian J Math 2003, 19(1):7–22.

    MathSciNet  MATH  Google Scholar 

  6. Berinde V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal Forum 2004, 9: 43–53.

    MathSciNet  MATH  Google Scholar 

  7. Berinde V: Iterative Approximation of Fixed Points. In Lecture Notes in Mathematics. Springer, Berlin; 2007. 1912

    Google Scholar 

  8. Chen YZ: A Prešić type contractive condition and its applications. Nonlinear Anal 2009, 71: 2012–2017. 10.1016/j.na.2009.01.041

    Article  MathSciNet  Google Scholar 

  9. Ćirić LB: Generalized contractions and fixed point theorems. Publ L'Inst Math 1971, 12(26):19–26.

    MathSciNet  MATH  Google Scholar 

  10. Ćirić LB: On contraction type mappings. Math Balkanica 1971, 1: 52–57.

    MathSciNet  MATH  Google Scholar 

  11. Ćirić LB: On a family of contractive maps and fixed points. Publ L'Inst Math 1974, 17: 45–51.

    MathSciNet  MATH  Google Scholar 

  12. Ćirić LB: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos Solitons Fract 2009, 42(1):146–154. 10.1016/j.chaos.2008.11.010

    Article  MathSciNet  MATH  Google Scholar 

  13. Ćirić LB, Prešić SB: On Presic type generalisation of Banach contraction mapping principle. Acta Math Univ Com 2007, LXXVI(2):143–147.

    MATH  Google Scholar 

  14. Imdad M, Ali J, Tanveer M: Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces. Chaos Solitons Fract 2009, 42(5):3121–3129. 10.1016/j.chaos.2009.04.017

    Article  MathSciNet  MATH  Google Scholar 

  15. Kannan R: Some results on fixed points. Bull Calcutta Math Soc 1968, 60: 71–76.

    MathSciNet  MATH  Google Scholar 

  16. Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull Aust Math Soc 1984, 30: 1–9. 10.1017/S0004972700001659

    Article  MathSciNet  MATH  Google Scholar 

  17. Meir A, Keeler E: A theorem on contraction mappings. J Math Anal Appl 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6

    Article  MathSciNet  MATH  Google Scholar 

  18. Miheţ D: Fixed point theorems in probabilistic metric spaces. Chaos Solitons Fract 2009, 41(2):1014–1019. 10.1016/j.chaos.2008.04.030

    Article  MathSciNet  MATH  Google Scholar 

  19. Nadler SB Jr: Multi-valued contraction mappings. Pac J Math 1969, 30: 475–488.

    Article  MathSciNet  MATH  Google Scholar 

  20. Păcurar M: Approximating common fixed points of Presić-Kannan type operators by a multi-step iterative method. An Şt Univ Ovidius Constanţa 2009, 17(1):153–168.

    MATH  Google Scholar 

  21. Păcurar M: Fixed points of almost Persić operators by a k -step iterative method. An Ştiinţ Univ Al I. Cuza Iaşi. Mat (N.S.) Tomul 2011., LVII:

    Google Scholar 

  22. Prešić SB: Sur une classe d'inéquations aux differences finies et sur la convergence de certaines suites. Pub de l'Inst Math Belgrade 1965, 5(19):75–78.

    MATH  Google Scholar 

  23. Rao KPR, Kishore GNV, Mustaq Ali Md: A generalization of the Banach contraction principle of Presic type for three maps. Math Sci 2009, 3(3):273–280.

    MATH  Google Scholar 

  24. Rhoades BE: A comparison of various definitions of contractive mappings. Trans Am Math Soc 1977, 226: 257–290.

    Article  MathSciNet  MATH  Google Scholar 

  25. Rus IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca; 2001.

    Google Scholar 

  26. Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1974.

    Google Scholar 

  27. Suzuki T: A generalized Banach contraction principle which characterizes metric completeness. Proc Am Math Soc 2008, 136: 1861–1869.

    Article  MATH  Google Scholar 

  28. Vetro F: On approximating curves associated with nonexpansive mappings. Carpathian J Math 2011, 27(1):142–147.

    MathSciNet  MATH  Google Scholar 

  29. Godunov SK, Ryabenku VS: Difference Schemes. North-Holland, Amsterdam; 1987.

    Google Scholar 

  30. Goldberg S: Introduction to Difference Equations. Wiley, New York; 1958.

    Google Scholar 

  31. Johnson RM: Theory and Applications of Linear Differential and Difference Equations. Wiley, New York; 1984.

    Google Scholar 

  32. Agarwal RP: Difference Equations and Inequalities. Marcel Dekker, New York; 1992.

    Google Scholar 

  33. Kelley WG, Peterson A: Diference Equations. Academic press, New York; 1990.

    Google Scholar 

  34. Lakshmikautham V, Frigiante D: Theory of Difference Equations. Academic press, New York; 1990.

    Google Scholar 

  35. Rus IA: An iterative method for the solution of the equation x = f ( x, ..., x ). Rev Anal Numer Theory Approx 1981, 10(1):95–100.

    MathSciNet  MATH  Google Scholar 

  36. Rus IA: An Abstract Point of View in the Nonlinar Difference Equations. Editura Carpatica, Cluj-Napoca; 1999:272–276.

    Google Scholar 

  37. Dehgham M, Hajarian M: An efficient algorithm for solving general coupled matrix equations and its application. Math Comput Model 2010, 51: 1118–1134. 10.1016/j.mcm.2009.12.022

    Article  MathSciNet  MATH  Google Scholar 

  38. Duan X, Liao A:On Hermitian positive definite solution of the matrix equation i = 1 m A i * X r A i =Q. J Comput Appl Math 2009, 229: 27–36. 10.1016/j.cam.2008.10.018

    Article  MathSciNet  MATH  Google Scholar 

  39. Liao A, Yao G, Duan X: Thompson metric method for solving a class of nonlinear matrix equation. Appl Math Comput 2010, 216: 1831–1836. 10.1016/j.amc.2009.12.022

    Article  MathSciNet  MATH  Google Scholar 

  40. Lim Y:Solving the nonlinear matrix equation X=Q+ i = 1 m M i X δ i M i * via a contraction principle. Linear Algebra Appl 2009, 430: 1380–1383. 10.1016/j.laa.2008.10.034

    Article  MathSciNet  Google Scholar 

  41. Zhoua B, Duana G, Li Z: Gradient based iterative algorithm for solving coupled matrix equations. Syst Control Lett 2009, 58: 327–333. 10.1016/j.sysconle.2008.12.004

    Article  MathSciNet  Google Scholar 

  42. Thompson AC: On certain contraction mappings in a partially ordered vector space. Proc Am Math soc 1963, 14: 438–443.

    MathSciNet  MATH  Google Scholar 

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Khan, M.S., Berzig, M. & Samet, B. Some convergence results for iterative sequences of Prešić type and applications. Adv Differ Equ 2012, 38 (2012). https://doi.org/10.1186/1687-1847-2012-38

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