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; matrix1 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
Then f has a unique fixed point, that is, there exists a unique x* ∈ X such that x* = fx*. Moreover, for any x_{0 }∈ X, the iterative sequence x_{n+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 kth order nonlinear difference equation
with the initial values x_{0},..., x_{k1 }∈ X, where k is a positive integer (k ≥ 1) and f : X^{k }≤ X. Equation (1) can be studied by means of fixed point theory in view of the fact that x* ∈ 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
for all x_{0}, ..., x_{k }∈ X, where δ_{1}, ..., δ_{k }are positive constants such that δ_{1 }+ ... + 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 x_{0}, ..., x_{k1 }∈ 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 : X^{k }→ X. Suppose that
for all x_{0}, ..., 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 x_{0}, ..., x_{k1 }∈ 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 : X^{k }→ X. Suppose that
for all x_{0}, ..., 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 x_{0}, ..., x_{k1 }∈ 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 : X^{k }→ X. Suppose that
for all x_{0}, ..., x_{k }∈ X, where δ_{1}, ..., δ_{k }are positive constants such that and
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 x_{0},..., x_{k1 }∈ 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 α_{1}, α_{2}, ..., α_{k }≥ 0 such that . If { Δ_{n}} is a sequence of positive numbers satisfying
then there exist L ≥ 0 and τ ∈ (0,1) such that Δ _{n }≤ Lτ^{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 a_{n+1 }≤ qa_{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:
(i) θ is continuous,
(ii) for all t_{1}, t_{2}, t_{3}, t_{4 }∈ [0, ∞),
Example 2.1 The following functions belong to Θ:
(1) θ(t_{1}, t_{2}, t_{3}, t_{4}) = L min{t_{1}, t_{2}, t_{3}, t_{4}}, L > 0 t_{1}, t_{2}, t_{3}, t_{4 }≥ 0.
(2) θ(t_{1}, t_{2}, t_{3}, t_{4}) = L ln(1 + t_{1}t_{2}t_{3}t_{4}), L > 0 t_{1},t_{2},t_{3},t_{4 }≥ 0.
(3) θ(t_{1}, t_{2}, t_{3}, t_{4}) = L ln(1 + t_{1}) ln(1 + t_{2}) ln(1 + t_{3}) ln(1 + t_{4}), L > 0 t_{1},t_{2},t_{3},t_{4 }≥ 0.
(4) θ(t_{1}, t_{2}, t_{3}, t_{4}) = Lt_{1}t_{2}t_{3}t_{4}, L > 0 t_{1}, t_{2}, t_{3}, t_{4 }≥ 0.
(5) , L > 0 t_{1}, t_{2}, t_{3}, t_{4 }≥ 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
for all x_{0},..., x_{k }∈ X, where δ_{1},..., δ_{k + 1 }are positive constants such that 2A + δ ∈ (0,1) with and . 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 z_{0 }∈ X, the iterative sequence {z_{n}} defined by
converges to x*.
Proof. Define the mapping F : X → X by
Using (3), for all x, y ∈ X, we have
Thus, we have
where
Now, let z_{0 }be an arbitrary element of X. Define the sequence {z_{n}} by
Using (4), we have
On the other hand, from the property (ii) of the function θ, we have
Then we get
for all n = 1, 2,.... This implies that
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
where
Thus we have
Letting n → ∞ in the above inequality, and using the properties (i) and (ii) of θ, we obtain
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
On the other hand, we have
Then we get
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
for all x_{0}, ..., x_{k }∈ X, where δ_{1}, ..., δ_{k }are positive constants such that δ ∈ (0,1) with , and B ≥ 0. Then
(a) there exists a unique x* ∈ X such that x* = f(x*,..., x*);
(b) the sequence {x_{n}} defined by
converges to x* for any x_{0}, ..., x_{k1 }∈ X.
Proof. Applying Theorem 2.1 with δ_{k + 1 }= 0, and remarking that Bθ ∈ Θ, we obtain immediately (a). Now, we shall prove (b). Let x_{0},..., x_{k1 }∈ X and x_{n }= f(x_{nk},..., x_{n1}), n ≥ k. Then by (5), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have
Since k is a fixed positive integer, then we may denote
Then we get
Similarly we get that
Denoting
we get
Continuing this process, for n ≥ k, we obtain
Denoting
the above inequality becomes
Now, we shall prove that the sequence {E_{n}} given by
converges to 0 as n → ∞.
For n ≥ k, from (5), we have
As d(x_{n}, f(x_{nk},..., x_{n1}) = 0, the above inequality leads to
According to Lemma 2.1, this implies the existence of τ ∈ (0,1) and L ≥ 0 such that
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 }= E_{n + 1k}, 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(t_{1}, t_{2}, t_{3}, t_{4}) = min{t_{1}, t_{2}, t_{3}, t_{4}}, 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
for all x_{0}, ..., x_{k }∈ X, where a is a positive constant such that A ∈ (0, 1/2) with . Then
(a) there exists a unique x* ∈ X such that x* = f(x*, ..., x*);
(b) the sequence {x_{n}} defined by
converges to x* for any x_{0}, ..., x_{k1 }∈ X, with a rate estimated by
where L ≥ 0, τ ∈ (0, 1) and M = τ^{1k }+ 2τ^{2k }+ ⋯ + k.
Proof. (a) follows immediately from Theorem 2.1 with δ = 0 and δ_{k+1 }= a. Now, we shall prove (b). Let x_{0}, ..., x_{k1 }∈ X and x_{n }= f(x_{nk}, ..., x_{n1}), n ≥ k. Then by (8), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have
Using (4), for all i = 0,1,..., k  1, we get
This implies that
Now, combining (12) with (11), we obtain
Similarly, one can show that
This implies that
Define the sequence { Δ_{p}} by
We get that
Since , we can apply Lemma 2.1 to deduce that there exist L ≥ 0 and τ ∈ (0,1) such that
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
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
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
where M(A/B) = inf{λ > 0 : A ≤ λB} = λ^{+}(B^{1/2}AB^{1/2}), the maximal eigenvalue of B^{1/2}AB^{1/2}. Here, X ≤ Y means that Y  X is positive semidefinite 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^{1/2}BA^{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,
for any nonsingular matrix M. The other useful result is the nonpositive curvature property of the Thompson metric, that is,
By the invariant properties of the metric, we then have
for any X, Y ∈ P(N) and nonsingular matrix M.
Lemma 3.1 [40]For all A, B, C, D ∈ P(N), we have
In particular,
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
(ii) for any X_{0}, X_{1 }> 0, the iterative sequence {X_{n}} defined by (15) converges to U.
Proof. Define the mapping f : P(N) × P(N) → P(N) by
Using Lemma 3.1 and properties (16)(18), for all X, Y, Z ∈ P(N), we have
Thus we proved that
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
where
and
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,
where ⋅ is the spectral norm.
After 40 iterations, we obtain
with residual error
The convergence history of the algorithm (19) is given by Figure 1.
Figure 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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