A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

Abstract

By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equation , where is a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equation and prove that the equation has a unique positive definite solution when and and . For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.

1. Introduction

We consider the matrix equation where is an positive definite matrix, is arbitrary matrix, and is a monotone map on .

The study of matrix equation has a long history, involving in particular the study of algebraic Riccati equations for discrete time optimal control and for the stochastic realization problem. Motivated by these equations, somewhat simpler versions, namely, (1) with , were studied in [1–3]. Those three papers were the start of a development. Later on came papers on a number of other specific matrix equations, such as papers [4–10].

Of particular interest is the equation , where is the Kronecker product of with for some . This equation is connected to an interpolation problem proposed by Sakhnovich in [11]. This equation was first studied in [12], and a perturbation analysis was discussed in [13]. Recently, in [14], the author provided a new proof for the uniqueness of the positive definite solution of this equation using a change of variable and a fixed point theorem, which is an easier argument than the one used in [12].

This development leads to consideration of a general class of matrix equations, which started with the paper by El-Sayed and Ran [15], and was developed further by Ran and Reurings [16–19].

In this paper, we are interested in positive definite solutions of (1), where is a monotone map. We obtain a uniqueness theorem by using the fixed point theorem for monotone maps in a normal cone. The uniqueness theorem can be widely used in nonlinear matrix equations involving monotonicity. In addition, we apply the uniqueness theorem to a special equation and discuss the basic fixed point iteration for this special case.

The following notations are used throughout this paper. Let denote Hermitian matrices, let denote matrices, let denote positive definite matrices, and let denote positive semidefinite matrices. For , we write if is positive semidefinite (definite). denotes the conjugate transpose of a matrix . Let denote a solid cone of a real Banach space . denotes the interior points set of . A cone is said to be a solid cone if .

2. Preliminaries

In this section, we introduce some definitions and properties for monotone operators in a normal cone which are the theoretical basis of this paper.

Definition 1 (see [20]). A cone is said to be normal if there exists a constant such that implies . That is, the norm is semimonotone.

Definition 2 (see [20]). The operator , , is said to be an increasing operator if is said to be an decreasing operator if

Definition 3 (see [21]). Let be a solid cone of a real Banach space and . Let . Then is said to be -concave if is said to be -convex if

Lemma 4 (see [21]). Let be a normal cone of a real Banach space and let be -concave and increasing (or -convex and decreasing) for . Then has exactly one fixed point in .

Lemma 5 (see [20]). A cone is normal if and only if , , and imply .

Lemma 6 (see [22]). For all and , the operator given by is an increasing operator. Then if , , and .

In the following, we will apply Lemma 4 to the map .

3. The Application to

We define the spectral norm in ; then is a real Banach space. It is well known that is a cone in and the interior points set is . Since the spectral norm is monotone, we have from Definition 1 that the set is normal cone. So we can apply the results in Section 2 to the maps from into . In the following, we will consider the positive definite solutions of the equation or equivalently the fixed points of the map where is a monotone map on induced by a real valued map on . The following theorem is our main result in this section.

Theorem 7. Let , , . Then has exactly one fixed point in if(1);(2) is increasing and -concave or decreasing and -convex.

Proof. For the application of Lemma 4, we set , , , and . Now we will prove that the map satisfies the conditions of Lemma 4.
For all with , by Lemma 6, we have the following:(1), because maps into itself and .(2)If is increasing and -concave, then is increasing. For all , we have Hence the map is increasing and -concave.
If is decreasing and -convex, then is decreasing. For all , we have Hence the map is decreasing and -concave.
So the map satisfies all the conditions in Lemma 4. According to Lemma 4, has exactly one fixed point in .
The conditions in this theorem which has to satisfy are easy to check if is simple. Now, we will give two simple examples.

Example 8. If , , then has exactly one fixed point in .

Proof. By Lemma 6  is increasing. Also Let ; then is -concave. According to Theorem 7, has exactly one fixed point in .

Example 9. If , , then has exactly one fixed point in .

Proof. By Lemma 6  is decreasing. Also Let ; then is -convex. According to Theorem 7, has exactly one fixed point in .

These two examples have been discussed in several papers by other methods; see, for example, [7, 8]. It seems that the argument presented here is simpler than the arguments of [7, 8].

4. The Case of

In this section, we will discuss a more complex map; namely, . Here is the block diagonal matrix defined by , in which is an matrix. Also, is an positive definite matrix, is an positive semidefinite matrix, and is arbitrary matrix. we always assume that , , and . under these conditions we discuss the positive definite solutions of the equation or equivalently the fixed points of the map . In this case, . The function is increasing, but it seems to be hard to prove that is -concave. Therefore, we will use a change of variable to study an equivalent form of the map .

Let be the set defined by Let . For , we know that is a positive semidefinite matrix. Then (12) turns into Let . Then (12) eventually becomes

Apparently, (12) is equivalent to (15) when . Thus, we can obtain the following conclusion.

Lemma 10. Suppose that , ; then is a positive definite solution of (12) if and only if is a positive definite solution of (15).

Theorem 11. Equation (12) with and always has a unique positive definite solution.

Proof. According to Lemma 10, we first consider (15). Define by and . Now we will prove that the operator satisfies the conditions of Theorem 7.
For all with , by Lemma 6, we have the following:(1);(2);(3)for all , we have Hence, the operator is increasing and -concave. From Theorem 7 we get that the operator has a unique fixed point in , which is the unique positive definite solution of (15). According to Lemma 10, (12) has a unique positive definite solution.

Now we consider the following iterative method for (15) and (12). Let For the matrix sequence defined by (18), we have the following theorem.

Theorem 12. Suppose that , ; then for arbitrary initial matrix , the matrix sequence defined by (18) converges to the unique positive definite solution of (12).

Proof. We first consider the matrix sequence defined by (17). Let , ; then is the unique positive definite solution of (15). For and , there exists a positive number satisfying We will use mathematical induction to prove the following inequality: From (19) it follows that inequality (20) holds for . Assume that (20) is true for . That is, Now we need to prove (20) is true for . From (21), we get that That is, From it follows that , . Then Therefore That is, Hence we get that inequality (20) holds for any positive integer . Let ; we have Therefore, from Lemma 5 it follows that Since , , we have

5. Numerical Examples

We now present some numerical examples to illustrate our results. All computations were performed using MATLAB, version 7.01. In this section, we will use to denote the relative iteration error, errtol to denote the stopping criterion, and to denote the iteration number.

Example 13. Consider (12) with and Then the matrices and satisfy . Consider the iterative method (18) with several values of and several values of the stopping criterion. The experiment data are listed in Table 1.

Table 1 

MATLAB function -file is shown in Algorithm 1.

Gaofixedpoint.m function  [X, counter, err]=Gaofixedpoint(A, Q, C, k, q, errtol) % Solving X = kQ + A⁢∧⁢⁢(Xhat − C)⁢∧⁢q⁢A, with Xhat = kronecker(I,X) % Input: matrices  , , , k > 1, 0 < q < 1, errtol % Output: solution  , iteration counter,  final equation relative error m, n = size (A); [p, w] = size (Q); [r, s] = size (C); % Input size etc checks if floor (m/n) ~= m/n n ~= p n ~= w r ~= m s ~= m k   <= 1 … q <= 0 q>= 1,   error (’incompatible inputs’),   return, end I = eye (m/n); X = kQ; counter = 0; err = 10000;  % Initialize while err >= errtol Iterate  X = kQ + A’(kron(I, X) − C)∧qA; update X  S = X − kQ − A’(kron(I, X) − C)∧qA; form error matrix S  err = norm(S, 1)/norm(X, 1); relative iteration error  counter = counter + 1; Iteration counter end X = (X + X’)/2; make sure X is symmetric S = X − kQ − A’(kron(I, X) − C)∧qA; form final error matrix S err = norm(S, 1)/norm(X, 1);

Algorithm 1 

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author wishes to thank Professor Frank Uhlig and the anonymous referees for providing very useful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China (11071141), the Natural Science Foundation of Shandong Province of China (ZR2015AL017ZR2014AM032), and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02).