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

Authors: Dongjie Gao
Journal: Journal of Mathematics
 (2015)

---

## 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 equationX
=
Q
+
A
⁎
f
(
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
=
k
Q
+
A
⁎
(
X
^
-
C
)
q
A and prove that the equation has a unique positive definite solution when Q
^
≥
C and k
>
1 and 0
<
q
<
1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.


---

## Body

## 1. Introduction

We consider the matrix equation(1)
X
=
Q
+
A
∗
f
X
A
,where Q is an n
×
n positive definite matrix, A is arbitrary m
×
n matrix, and f is a monotone map on P
(
m
).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 f
(
X
)
=
±
X
-
1, 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 equationX
=
Q
+
A
∗
(
X
^
-
C
)
-
1
A, where X
^ is the Kronecker product of I
p with X for some p. 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 f 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. LetH
(
n
) denote n
×
n Hermitian matrices, let M
(
m
,
n
) denote m
×
n matrices, let P
(
n
) denote n
×
n positive definite matrices, and let P
¯
(
n
) denote n
×
n positive semidefinite matrices. For X
,
Y
∈
P
(
n
), we write X
≥
Y
(
X
>
Y
) if X
-
Y is positive semidefinite (definite). A
∗ denotes the conjugate transpose of a matrix A. Let P denote a solid cone of a real Banach space E. P
0 denotes the interior points set of P. A cone is said to be a solid cone if P
0
≠
ϕ.


## 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 coneP
⊂
E is said to be normal if there exists a constant N
>
0 such that 0
≤
x
≤
y implies x
≤
N
y. That is, the norm · is semimonotone.Definition 2 (see [20]).
The operatorΓ
:
D
→
E, D
⊂
E, is said to be an increasing operator if (2)
x
1
≥
x
2
,
x
1
,
x
2
∈
D
⟹
Γ
x
1
≥
Γ
x
2
.
Γ is said to be an decreasing operator if (3)
x
1
≥
x
2
,
x
1
,
x
2
∈
D
⟹
Γ
x
1
≤
Γ
x
2
.Definition 3 (see [21]).
LetP be a solid cone of a real Banach space E and Γ
:
P
0
→
P
0. Let 0
≤
α
<
1. Then Γ is said to be α-concave if (4)
Γ
t
x
≥
t
α
Γ
x
∀
x
∈
P
0
,
0
<
t
<
1
.
Γ is said to be (
-
α
)-convex if (5)
Γ
t
x
≤
t
-
α
Γ
x
∀
x
∈
P
0
,
0
<
t
<
1
.Lemma 4 (see [21]).
LetP be a normal cone of a real Banach space E and let Γ
:
P
0
→
P
0 be α-concave and increasing (or (
-
α
)-convex and decreasing) for 0
≤
α
<
1. Then Γ has exactly one fixed point x in P
0.Lemma 5 (see [20]).
A coneP is normal if and only if x
n
≤
z
n
≤
y
n, x
n
→
x, and y
n
→
x imply z
n
→
x.Lemma 6 (see [22]).
For allA
∈
H
(
n
) and r
∈
[
0,1
], the operator Φ given by Φ
(
A
)
=
A
r is an increasing operator. Then A
r
≥
B
r if A
,
B
∈
H
(
n
), A
≥
B, and 0
≤
r
≤
1.In the following, we will apply Lemma4 to the map G
(
X
)
=
Q
+
A
∗
f
(
X
)
A.


## 3. The Application toG
(
X
)
=
Q
+
A
∗
f
(
X
)
A

We define the spectral norm· in H
(
n
); then H
(
n
) is a real Banach space. It is well known that P
¯
(
n
) is a cone in H
(
n
) and the interior points set is P
(
n
). Since the spectral norm is monotone, we have from Definition 1 that the set P
¯
(
n
) is normal cone. So we can apply the results in Section 2 to the maps from P
(
n
) into P
(
n
). In the following, we will consider the positive definite solutions of the equation (6)
X
=
Q
+
A
∗
f
X
Aor equivalently the fixed points of the map (7)
G
X
=
Q
+
A
∗
f
X
A
,where f is a monotone map on P
(
m
) induced by a real valued map on (
0
,
∞
). The following theorem is our main result in this section.Theorem 7.
LetG
(
X
)
=
Q
+
A
∗
f
(
X
)
A, Q
∈
P
(
n
), A
∈
M
(
m
,
n
). Then G has exactly one fixed point X in P
(
n
) if(1)
f
:
P
¯
(
m
)
→
P
¯
(
m
);
(2)
f is increasing and α-concave or decreasing and (
-
α
)-convex.Proof.
For the application of Lemma4, we set E
=
H
(
n
), P
=
P
¯
(
n
), P
0
=
P
(
n
), and Γ
=
G. Now we will prove that the map G satisfies the conditions of Lemma 4.
For allX
,
Y
∈
P
(
n
) with X
≥
Y, by Lemma 6, we have the following:(1)
G
:
P
(
n
)
→
P
(
n
), because f maps P
¯
(
m
) into itself and Q
∈
P
(
n
).
(2)
Iff is increasing and α-concave, then G is increasing. For all t
∈
(
0,1
), we have (8)
G
t
X
=
Q
+
A
∗
f
t
X
A
≥
t
α
Q
+
t
α
A
∗
f
X
A
=
t
α
Q
+
A
∗
f
X
A
=
t
α
G
X
.Hence the map G is increasing and α-concave.
Iff is decreasing and (
-
α
)-convex, then G is decreasing. For all t
∈
(
0,1
), we have (9)
G
t
X
=
Q
+
A
∗
f
t
X
A
≤
t
-
α
Q
+
t
-
α
A
∗
f
X
A
=
t
-
α
Q
+
A
∗
f
X
A
=
t
-
α
G
X
.Hence the map G is decreasing and (
-
α
)-concave.
So the mapG satisfies all the conditions in Lemma 4. According to Lemma 4, G has exactly one fixed point X in P
(
n
).
The conditions in this theorem whichf has to satisfy are easy to check if f is simple. Now, we will give two simple examples.Example 8.
Iff
(
X
)
=
X
q, 0
<
q
<
1, then G
(
X
)
=
Q
+
A
∗
X
q
A has exactly one fixed point X in P
(
n
).Proof.
By Lemma6  f
:
P
¯
(
n
)
→
P
¯
(
n
) is increasing. Also (10)
f
t
X
=
t
X
q
=
t
q
X
q
=
t
q
f
X
.Let α
=
q; then f is α-concave. According to Theorem 7, G
(
X
)
=
Q
+
A
∗
X
q
A has exactly one fixed point X in P
(
n
).Example 9.
Iff
(
X
)
=
X
-
q, 0
<
q
<
1, then G
(
X
)
=
Q
+
A
∗
X
-
q
A has exactly one fixed point X in P
(
n
).Proof.
By Lemma6  f
:
P
¯
(
n
)
→
P
¯
(
n
) is decreasing. Also (11)
f
t
X
=
t
X
-
q
=
t
-
q
X
-
q
=
t
-
q
f
X
.Let α
=
q; then f is (
-
α
)-convex. According to Theorem 7, G
(
X
)
=
Q
+
A
∗
X
-
q
A has exactly one fixed point X in P
(
n
).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 ofG
(
X
)
=
k
Q
+
A
∗
(
X
^
-
C
)
q
A

In this section, we will discuss a more complex map; namely,G
(
X
)
=
k
Q
+
A
∗
(
X
^
-
C
)
q
A. Here X
^ is the block diagonal matrix defined by X
^
=
diag
⁡
(
X
,
X
,
…
,
X
), in which X is an n
×
n matrix. Also, Q is an n
×
n positive definite matrix, C is an m
n
×
m
n positive semidefinite matrix, and A is arbitrary m
n
×
n matrix. we always assume that Q
^
≥
C, k
>
1, and 0
<
q
<
1. under these conditions we discuss the positive definite solutions of the equation (12)
X
=
k
Q
+
A
∗
X
^
-
C
q
Aor equivalently the fixed points of the map G
(
X
). In this case, f
(
X
)
=
(
X
^
-
C
)
q. The function f is increasing, but it seems to be hard to prove that f is α-concave. Therefore, we will use a change of variable to study an equivalent form of the map G.LetS
1
(
n
)
⊂
P
(
n
) be the set defined by (13)
S
1
n
=
X
∈
P
n
∣
X
>
Q
.Let C
1
=
Q
^
-
C. For Q
^
≥
C, we know that C
1 is a positive semidefinite matrix. Then (12) turns into (14)
X
-
Q
=
k
-
1
Q
+
A
∗
X
^
-
Q
^
+
C
1
q
A
.Let Y
=
X
-
Q. Then (12) eventually becomes (15)
Y
=
k
-
1
Q
+
A
∗
Y
^
+
C
1
q
A
.Apparently, (12) is equivalent to (15) when X
=
Y
+
Q. Thus, we can obtain the following conclusion.Lemma 10.
Suppose thatQ
^
≥
C, k
>
1; then X is a positive definite solution of (12) if and only if Y
=
X
-
Q is a positive definite solution of (15).Theorem 11.
Equation (12) with Q
^
≥
C and k
>
1 always has a unique positive definite solution.Proof.
According to Lemma10, we first consider (15). Define G
1 by G
1
(
Y
)
=
(
k
-
1
)
Q
+
A
∗
(
Y
^
+
C
1
)
q
A and f
1
(
Y
)
=
(
Y
^
+
C
1
)
q. Now we will prove that the operator G
1 satisfies the conditions of Theorem 7.
For allX
,
Y
∈
P
(
n
) with X
≥
Y, by Lemma 6, we have the following:(1)
G
1
:
P
(
n
)
→
P
(
n
);
(2)
f
1
(
X
)
=
(
X
^
+
C
1
)
q
≥
(
Y
^
+
C
1
)
q
=
f
1
(
Y
);
(3)
for allt
∈
(
0,1
), we have (16)
f
1
t
Y
=
t
Y
^
+
C
1
q
=
t
q
Y
^
+
t
-
1
C
1
q
≥
t
q
Y
^
+
C
1
q
=
t
q
f
1
Y
.Hence, the operator f
1 is increasing and α-concave. From Theorem 7 we get that the operator G
1 has a unique fixed point Y in P
(
n
), 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 (17)
Y
n
+
1
=
k
-
1
Q
+
A
∗
Y
^
n
+
C
1
q
A
,
n
=
0,1
,
2
,
…
,
(18)
X
n
+
1
=
k
Q
+
A
∗
X
^
n
-
C
q
A
,
n
=
0,1
,
2
,
…
.For the matrix sequence {
X
n
} defined by (18), we have the following theorem.Theorem 12.
Suppose thatQ
^
≥
C, k
>
1; then for arbitrary initial matrix X
0
∈
S
1
(
n
), the matrix sequence {
X
n
} defined by (18) converges to the unique positive definite solution X of (12).Proof.
We first consider the matrix sequence{
Y
n
} defined by (17). Let Y
=
X
-
Q, Y
0
=
X
0
-
Q; then Y is the unique positive definite solution of (15). For Y and Y
0, there exists a positive number 0
<
α
≤
1 satisfying (19)
α
Y
≤
Y
0
≤
α
-
1
Y
.We will use mathematical induction to prove the following inequality: (20)
α
q
n
Y
≤
Y
n
≤
α
-
1
q
n
Y
,
n
=
0,1
,
2
,
…
.From (19) it follows that inequality (20) holds for n
=
0. Assume that (20) is true for n
=
k. That is, (21)
α
q
k
Y
≤
Y
k
≤
α
-
1
q
k
Y
.Now we need to prove (20) is true for n
=
k
+
1. From (21), we get that (22)
α
q
k
Y
^
+
C
1
≤
Y
k
^
+
C
1
≤
α
-
1
q
k
Y
k
^
+
C
1
,
α
q
k
Y
^
+
C
1
q
≤
Y
k
^
+
C
1
q
≤
α
-
1
q
k
Y
^
+
C
1
q
,
k
-
1
Q
+
A
∗
α
q
k
Y
^
+
C
1
q
A
≤
k
-
1
Q
+
A
∗
Y
k
^
+
C
1
q
A
≤
k
-
1
Q
+
A
∗
α
-
1
q
k
Y
^
+
C
1
q
A
.That is, (23)
k
-
1
Q
+
A
∗
α
q
k
Y
^
+
C
1
q
A
≤
Y
k
+
1
≤
k
-
1
Q
+
A
∗
α
-
1
q
k
Y
^
+
C
1
q
A
.From 0
<
α
≤
1 it follows that 0
<
α
q
k
+
1
≤
1, (
α
-
1
)
q
k
+
1
≥
1. Then (24)
k
-
1
Q
+
A
∗
α
q
k
Y
^
+
C
1
q
A
≥
α
q
k
+
1
k
-
1
Q
+
A
∗
α
q
k
Y
^
+
α
q
k
C
1
q
A
=
α
q
k
+
1
k
-
1
Q
+
A
∗
Y
^
+
C
1
q
A
,
k
-
1
Q
+
A
∗
α
-
1
q
k
Y
^
+
C
1
q
A
≤
α
-
1
q
k
+
1
k
-
1
Q
+
A
∗
α
-
1
q
k
Y
^
+
α
-
1
q
k
C
1
q
A
=
α
-
1
q
k
+
1
k
-
1
Q
+
A
∗
Y
^
+
C
1
q
A
.Therefore (25)
α
q
k
+
1
k
-
1
Q
+
A
∗
Y
^
+
C
1
q
A
≤
Y
k
+
1
≤
α
-
1
q
k
+
1
k
-
1
Q
+
A
∗
Y
^
+
C
1
q
A
.That is, (26)
α
q
k
+
1
Y
≤
Y
k
+
1
≤
α
-
1
q
k
+
1
Y
.Hence we get that inequality (20) holds for any positive integer n. Let n
→
∞; we have (27)
α
q
n
⟶
1
,
α
-
1
q
n
⟶
1
.Therefore, from Lemma 5 it follows that (28)
Y
n
⟶
Y
.Since X
n
=
Y
n
+
Q, X
=
Y
+
Q, we have (29)
X
n
⟶
X
.


## 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 usee
r
r
(
X
)
=
X
-
k
Q
-
A
∗
(
X
^
-
C
)
q
A
1
/
X
1 to denote the relative iteration error, errtol to denote the stopping criterion, and n to denote the iteration number.Example 13.
Consider (12) with k
=
2 and (30)
A
=
3
-
2
2
0
-
8
3
7
2
-
5
1
7
6
,
Q
=
23
-
2
-
2
15
,
C
=
9
-
1
2
0
4
1
-
1
8
-
1
-
1
3
-
2
2
-
1
6
1
-
1
2
0
-
1
1
7
0
-
2
4
3
-
1
0
8
0
1
-
2
2
-
2
0
5
.Then the matrices Q and C satisfy Q
^
>
C. Consider the iterative method (18) with several values of q and several values of the stopping criterion. The experiment data are listed in Table 1.Table 1
errtol
q
err(X)
n
Iteration solutionX
1
×
1
0
-
5
1
/
3
9.4202
×
1
0
-
6
11
2.13388783159709
-
0.00086471161567
-
0.00086471161567
0.53349253880560
×
1
0
3
1
/
2
5.2724
×
1
0
-
6
19
2.28733182761235
-
0.13190912090591
-
0.13190912090591
0.47447253811381
×
1
0
4
1
×
1
0
-
10
1
/
3
3.7231
×
1
0
-
11
22
2.13391707124142
-
0.00086517759943
-
0.00086517759943
0.53349925037736
×
1
0
3
1
/
2
7.6719
×
1
0
-
11
35
2.28735583246287
-
0.13191055029808
-
0.13191055029808
0.47447745556329
×
1
0
4MATLAB functionm-file is shown in Algorithm 1.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  A
_
m
,
n, Q
_
n
,
n, C
_
m
,
m, k > 1, 0 < q < 1, errtol
% Output: solution  X
_
n
,
n, 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 = k∗Q; counter = 0; err = 10000;  % Initialize
while err >= errtolh
h
h
h
h
h
h
i
h
h
h
h
h
h
h
h
h
h
h
h
%
Iterate
X = k∗Q + A’∗(kron(I, X) − C)∧q∗A;i
i
x
i
i
i
%
update X
S = X − k∗Q − A’∗(kron(I, X) − C)∧q∗A;l
%
form error matrix S
err =norm(S, 1)/norm(X, 1);h
h
i
i
h
h
h
h
h
h
h
%
relative iteration error
counter = counter + 1;h
h
h
h
h
h
h
i
i
i
h
h
h
h
h
h
h
%
Iteration counter
end
X = (X + X’)/2;h
h
h
h
h
h
h
h
h
i
i
i
i
h
h
h
h
h
h
h
h
h
h
h
%
make sure X is symmetric
S = X − k∗Q − A’∗(kron(I, X) − C)∧q∗A;i
i
i
i
i
%
form final error matrix S
err =norm(S, 1)/norm(X, 1);



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*Source: 167049-2015-12-24.xml*