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

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 X = Q + Af(X)A, where f is a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equation X = kQ + A∗(X − C)qA and prove that the equation has a unique positive definite solution when Q ≥ C and k > 1and 0 < q < 1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.


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 () = ± −1 , were studied in [1][2][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][5][6][7][8][9][10].
Of particular interest is the equation  =  +  * ( X − ) −1 , where X 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][17][18][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 M(, ) denote  ×  matrices, let P() denote  ×  positive definite matrices, and let P() denote  ×  positive semidefinite matrices.For ,  ∈ P(), we write  ≥  ( > ) if  −  is positive semidefinite (definite). * denotes the conjugate transpose of a matrix .Let  denote a solid cone of a real Banach space . 0 denotes the interior points set of .A cone is said to be a solid cone if  0 ̸ = .

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. 2
Definition 2 (see [20]).The operator Γ :  → ,  ⊂ , is said to be an increasing operator if Γ is said to be an decreasing operator if Definition 3 (see [21]).Let  be a solid cone of a real Banach space  and Γ :  0 →  0 .Let 0 ≤  < 1.Then Γ is said to be -concave if Γ is said to be (−)-convex if Lemma 4 (see [21]).Let  be a normal cone of a real Banach space  and let Γ :  0 →  0 be -concave and increasing (or (−)-convex and decreasing) for 0 ≤  < 1.Then Γ has exactly one fixed point  in  0 .
In the following, we will apply Lemma 4 to the map () =  +  * ().

The Application to 𝐺(𝑋) = 𝑄 + 𝐴 * 𝑓(𝑋)𝐴
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 (0, ∞).The following theorem is our main result in this section.(2)  is increasing and -concave or decreasing and (−)convex.
(2) If  is increasing and -concave, then  is increasing.For all  ∈ (0, 1), we have Hence the map  is increasing 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.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].

The Case of 𝐺(𝑋) = 𝑘𝑄 + 𝐴 * ( X − 𝐶) 𝑞 𝐴
In this section, we will discuss a more complex map; namely, () =  +  * ( X − )  .Here X is the block diagonal matrix defined by X = diag(, , . . ., ), in which  is an  ×  matrix.Also,  is an  ×  positive definite matrix,  is an  ×  positive semidefinite matrix, and  is arbitrary  ×  matrix.we always assume that Q ≥ ,  > 1, and 0 <  < 1. under these conditions we discuss the positive definite solutions of the equation or equivalently the fixed points of the map ().In this case, () = ( X − )  .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 .
Lemma 10.Suppose that Q ≥ ,  > 1; then  is a positive definite solution of (12) if and only if  =  −  is a positive definite solution of (15).Theorem 11.Equation ( 12) with Q ≥  and  > 1 always has a unique positive definite solution.

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 err() = ‖ −  −  * ( X − )  ‖ 1 /‖‖ 1 to denote the relative iteration error, errtol to denote the stopping criterion, and  to denote the iteration number.) . (30) Then the matrices  and  satisfy Q > .Consider the iterative method (18) with several values of  and several values of the stopping criterion.The experiment data are listed in Table 1.
MATLAB function -file is shown in Algorithm 1.