Some convergence results for iterative sequences of Prešić type and applications

  • Mohammad Saeed Khan1,

    Affiliated with

    • Maher Berzig2 and

      Affiliated with

      • Bessem Samet3Email author

        Affiliated with

        Advances in Difference Equations20122012:38

        DOI: 10.1186/1687-1847-2012-38

        Received: 22 November 2011

        Accepted: 29 March 2012

        Published: 29 March 2012

        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 : 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equa_HTML.gif

        Then f has a unique fixed point, that is, there exists a unique x*X such that x* = fx*. Moreover, for any x0X, 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ1_HTML.gif
        (1)

        with the initial values x0,..., xk-1X, 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* ∈ 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 : X k X. 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equb_HTML.gif

        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-1X, 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 : X k X. 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 ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equc_HTML.gif

        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-1X, 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 : X k X. 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equd_HTML.gif

        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-1X, 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 : X k X. 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Eque_HTML.gif
        for all x0, ..., x k X, where δ1, ..., δ k are positive constants such that i = 1 k δ i < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq1_HTML.gif 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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ2_HTML.gif
        (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- 1X, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq2_HTML.gif. 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equf_HTML.gif

        then there exist L ≥ 0 and τ ∈ (0,1) such that Δ n n for 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equg_HTML.gif
           
        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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq3_HTML.gif, 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 : X k X. 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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ3_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq4_HTML.gif and δ = i = 1 k δ i http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq5_HTML.gif. 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 z0X, the iterative sequence {z n } defined by
        z n + 1 = f ( z n , , z n ) , n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equh_HTML.gif

        converges to x*.

        Proof. Define the mapping F : XX by
        F x = f ( x , , x ) , for all x X . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equi_HTML.gif
        Using (3), for all x, yX, 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equj_HTML.gif
        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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ4_HTML.gif
        (4)
        where
        M ( x , y ) = k θ d ( x , F x ) , d ( y , F y ) , d ( x , F y ) , d ( y , F x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equk_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equl_HTML.gif
        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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equm_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equn_HTML.gif
        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 ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equo_HTML.gif
        for all n = 1, 2,.... This implies that
        d ( z n + 1 , z n ) A + δ 1 - A d ( z n , z n - 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equp_HTML.gif
        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 * ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equq_HTML.gif
        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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equr_HTML.gif
        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 * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equs_HTML.gif
        Letting n → ∞ in the above inequality, and using the properties (i) and (ii) of θ, we obtain
        ( 1 - A ) d ( x * , F x * ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equt_HTML.gif

        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 * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equu_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equv_HTML.gif
        Then we get
        ( 1 - δ ) d ( x * , y * ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equw_HTML.gif

        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 : X k X. 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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ5_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq5_HTML.gif, 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ6_HTML.gif
          (6)
           

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

        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-1X 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 * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equx_HTML.gif
        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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equy_HTML.gif
        Then we get
        d ( x k , x * ) E 0 + δ d ( x k - 1 , x * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equz_HTML.gif
        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 * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equaa_HTML.gif
        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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equab_HTML.gif
        we get
        d ( x k + 1 , x * ) E 1 + δ d ( x k , x * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equac_HTML.gif
        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 * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equad_HTML.gif
        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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equae_HTML.gif
        the above inequality becomes
        d ( x n , x * ) δ d ( x n - 1 , x * ) + E n - k , n k . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ7_HTML.gif
        (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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equaf_HTML.gif

        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 ) ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equag_HTML.gif
        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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equah_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equai_HTML.gif

        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 : X k X.

        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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ8_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq6_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ9_HTML.gif
          (9)
           
        converges to x* for any x0, ..., xk-1X, with a rate estimated by
        d x n + 1 , x * a L 1 - A M τ n , n k , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ10_HTML.gif
        (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- 1X 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ11_HTML.gif
        (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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equaj_HTML.gif
        This implies that
        d x i , F x i 1 1 - A d x i , x * , i = 0 , 1 , . . . , k - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ12_HTML.gif
        (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 * . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equak_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ13_HTML.gif
        (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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equal_HTML.gif
        Define the sequence { Δ p } by
        Δ p = d x p , x * , for all p 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equam_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equan_HTML.gif
        Since i = 1 k i a 1 - A = A 1 - A ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_IEq7_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ14_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equao_HTML.gif
        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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ15_HTML.gif
        (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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equap_HTML.gif
        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 * http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ16_HTML.gif
        (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 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ17_HTML.gif
        (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 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ18_HTML.gif
        (18)

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

        Lemma 3.1 [40] For all A, B, C, DP(N), we have
        d A + B , C + D max d A , C , d B , D . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equaq_HTML.gif
        In particular,
        d A + B , A + C d ( B , C ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equar_HTML.gif

        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 UP(N) such that
          U = Q + A * U α A + B * U β B ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equas_HTML.gif
           
        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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equat_HTML.gif
        Using Lemma 3.1 and properties (16)-(18), for all X, Y, ZP(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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equau_HTML.gif
        Thus we proved that
        d f ( X , Y ) , f ( Y , Z ) λ max d X , Y , d Y , Z http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equav_HTML.gif

        for all X, Y, ZP(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equ19_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equaw_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equax_HTML.gif
        It is clear that from our Theorem 3.1, Eq.(19) has a unique equilibrium point UP(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equay_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equaz_HTML.gif
        with residual error
        R 40 = 1 . 624 × 1 0 - 14 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Equba_HTML.gif
        The convergence history of the algorithm (19) is given by Figure 1.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-38/MediaObjects/13662_2011_Article_149_Fig1_HTML.jpg
        Figure 1

        Convergence history for Equation (19).

        Declarations

        Authors’ Affiliations

        (1)
        Department of Mathematics and Statistics, College of Science, Sultan Qaboos University
        (2)
        Ecole Supérieure des Sciences et Techniques de Tunis
        (3)
        Department of Mathematics, King Saud University

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