Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
DOI 10.1186/s13660-016-1083-3
RESEARCH
Open Access
Investigation of the existence and
uniqueness of extremal and positive definite
solutions of nonlinear matrix equations
Abdel-Shakoor M Sarhan and Naglaa M El-Shazly*
*
Correspondence:
[email protected]
Department of Mathematics,
Faculty of Science, Menoufia
University, Shebin El-Koom, Egypt
Abstract
∗ δi
We consider two nonlinear matrix equations X r ± m
i=1 Ai X Ai = I, where –1 < δi < 0,
and r, m are positive integers. For the first equation (plus case), we prove the existence
of positive definite solutions and extremal solutions. Two algorithms and proofs of
their convergence to the extremal positive definite solutions are constructed. For the
second equation (negative case), we prove the existence and the uniqueness of a
positive definite solution. Moreover, the algorithm given in (Duan et al. in Linear
Algebra Appl. 429:110-121, 2008) (actually, in (Shi et al. in Linear Multilinear Algebra
52:1-15, 2004)) for r = 1 is proved to be valid for any r. Numerical examples are given
to illustrate the performance and effectiveness of all the constructed algorithms. In
Appendix, we analyze the ordering on the positive cone P(n).
Keywords: positive definite; existence; uniqueness; extremal solutions
1 Introduction
Consider the two nonlinear matrix equations
Xr +
m
A∗i X δi Ai = I,
– < δi < ,
(.)
A∗i X δi Ai = I,
– < δi < ,
(.)
i=
and
Xr –
m
i=
where Ai are n × n nonsingular matrices, I is the n × n identity matrix, and r, m are positive
integers, whereas X is an n × n unknown matrix to be determined (A∗i stands for the conjugate transpose of the matrix Ai ). The existence and uniqueness, the rate of convergence,
and necessary and sufficient conditions for the existence of positive definite solutions of
similar kinds of nonlinear matrix equations have been studied by several authors [–].
El-Sayed [] considered the matrix equation X + A∗ F(X)A = Q with positive definite matrix Q and has shown that under some conditions an iteration method converges to the
positive definite solution. Dehghan and Hajarian [] constructed an iterative algorithm
to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y , Z.
© 2016 Sarhan and El-Shazly. This article is distributed under the terms of the Creative Commons Attribution 4.0 International
License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any
medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons
license, and indicate if changes were made.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 2 of 21
Also, they obtained an optimal approximation reflexive solution pair to a given matrix
pair [Y , Z] in the reflexive solution pair set of the generalized coupled Sylvester matrix
equations (AY – ZB, CY – ZD) = (E, F).
Dehghan and Hajarian [] constructed an iterative method to solve the general coupled
p
matrix equations j= Aij Xj Bij = Mi , i = , , . . . , p (including the generalized (coupled) Lyapunov and Sylvester matrix equations as particular cases) over generalized bisymmetric
matrix group (X , X , . . . , Xp ) by extending the idea of conjugate gradient (CG) method.
They determined the solvability of the general coupled matrix equations over generalized
bisymmetric matrix group in the absence of roundoff errors. In addition, they obtained the
optimal approximation generalized bisymmetric solution group to a given matrix group
(X , X , . . . , Xp ) in Frobenius norm by finding the least Frobenius norm of the generalized
bisymmetric solution group of new general coupled matrix equations.
Hajarian [] derived a simple and efficient matrix algorithm to solve the general coupled
p
matrix equations j= Aij Xj Bij = Ci , i = , , . . . , p (including several linear matrix equations
as particular cases) based on the conjugate gradients squared (CGS) method.
Hajarian [] developed the conjugate gradient squared (CGS) and biconjugate gradient
stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the
Sylvester-transpose matrix equation ki= (Ai XBi + Ci X T Di ) = E and the periodic Sylvester
matrix equation (Aj Xj Bj + Cj Xj+ Dj ) = Ej for j = , , . . . , λ.
Shi [] considered the matrix equation
X=
m
A∗i X δi Ai ,
|δi | < ,
(.)
i=
and Duan [] considered the matrix equation
X–
m
A∗i X δi Ai = Q,
 < |δi | < , with Q positive definite,
(.)
i=
and Duan [] considered the matrix equation
X–
m
A∗i X r Ai = Q,
r ∈ [–, ) ∪ (, ).
(.)
i=
The main results of [] are the following:
(a) The proof of the uniqueness of a positive definite solution of Equation (.).
(a) An algorithm for obtaining that solution with an original proof of convergence.
The main results of [] are the following:
(b) The uniqueness of a positive definite solution of Equation (.), depending on
Lemma ..
(b) The same algorithm given by [] is used with the same slightly changed proof.
In this paper, we show that the proof given in [] can be deduced directly from []. In
[], the authors proved that Equation (.) always has a unique Hermitian positive definite solution for every fixed r. We would like to assure that Equations (.) and (.) are
a nontrivial generalization of the corresponding problems with r = . In fact, Equations
(.) and (.) correspond to infinitely many (r = )-problems with nested intervals of δi
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 3 of 21
as – r < δi < , as shown by the following - transformation: Putting X r = Y , Equations

∗ δi
(.) and (.) become Y ± m
i= Ai Y Ai = I with – r < δi <  for r = , , . . . . Our results
show that some properties are invariant for all the corresponding problems such as the
existence of the positive definite and extremal solutions of Equation (.) and the property
of uniqueness of a solution of Equation (.). Moreover, it is clear that the set of solutions
of Equation (.) differs with respect to the corresponding problems. Furthermore, in our
paper [], we obtained a sufficient condition for Equation (.) to have a unique solution. Since this condition depends on r, the uniqueness may hold for some corresponding
problems and fails for others.
This paper is organized as follows. First, in Section , we introduce some notation, definitions, lemmas, and theorems that will be needed for this work. In Section , the existence
of positive definite solutions of Equation (.) beside the extremal (maximal and minimal)
solutions, which is of a more general form than other existing ones, is proved. In Section ,
two algorithms for obtaining the extremal positive definite solutions of Equation (.) are
proposed. The merit of the proposed method is of iterative nature, which makes it more
efficient. In Section , some numerical examples are considered to illustrate the performance and effectiveness of the algorithms. In Section , the existence and uniqueness of a
positive definite solution of Equation (.) is proved. Finally, the algorithm in [] is adapted
for solving this equation. At the end of this paper, in Appendix, we analyze the defined ordering on the positive cone P(n) showing that the ordering is total. This important result
shows that problem (.) has one maximal and one minimal solution. Also, we explain
the effect of the value of α on the number of iterations of the given algorithms, shown in
Tables  and .
2 Preliminaries
The following notation, definitions, lemmas, and theorems will be used herein:
. For A, B ∈ C n×n , we write A >  (≥ ) if the matrix A is Hermitian positive definite
(HPD) (semidefinite). If A – B >  (A – B ≥ ), then we write A > B (A ≥ B).
. If a Hermitian positive definite matrix X satisfies A ≤ X ≤ B, then we write
X ∈ [A, B].
. By a solution we mean a Hermitian positive definite solution.
. If Equation (.) has the maximal solution XL (minimal solution XS ), then for any
solution X, XS ≤ X ≤ XL .
. P(n) denotes the set of all n × n positive definite matrices.
. Let E be a real Banach space. A nonempty convex closed set P ⊂ E is called a cone
if:
(i) x ∈ P, λ ≥  implies λx ∈ P.
(ii) x ∈ P, –x ∈ P implies x = θ , where θ denotes the zero element.
We denote the set of interior points of P by Po . A cone is said to be solid if Po = φ. Each
cone P in E defines a partial ordering in E given by x ≤ y if and only if y – x ∈ P.
In this paper, we consider P to be the cone of n × n positive semidefinite matrices, denoted P(n); its interior is the set of n × n positive definite matrices P(n).
Definition . ([]) A cone P ⊂ E is said to be normal if there exists a constant M >  such
that θ ≤ x ≤ y implies x ≤ My.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 4 of 21
Definition . ([]) Let P be a solid cone of a real Banach space E, and : Po → Po . Let
 ≤ a < . Then is said to be a-concave if (tx) ≥ t a (x) ∀x ∈ Po ,  < t < .
Similarly, is said to be (–a)-convex if (tx) ≤ t –a (x) ∀x ∈ Po ,  < t < .
Lemma . ([]) Let P be a normal cone in a real Banach space E, and let : Po → Po
be a-concave and increasing (or (–a)-convex and decreasing) for an a ∈ [, ). Then has
exactly one fixed point x in Po .
Lemma . ([]) If A ≥ B >  (or A > B > ), then Aγ ≥ Bγ >  (or Aγ > Bγ > ) for all
γ ∈ (, ], and Bγ ≥ Aγ >  (or Bγ > Aγ > ) for all γ ∈ [–, ).
Definition . ([]) A function f is said to be matrix monotone of order n if it is monotone with respect to this order on n × n Hermitian matrices, that is, if A ≤ B implies
f (A) ≤ f (B). If f is matrix monotone of order n for all n, then we say that f is matrix monotone or operator monotone.
Theorem . ([]) Every operator monotone function f on an interval I is continuously
differentiable.
Definition . ([]) Let D ⊂ E. An operator f : D → E is said to be an increasing operator if y ≥ y implies f (y ) ≥ f (y ), where y , y ∈ D. Similarly, f is said to be a decreasing
operator if y ≥ y implies f (y ) ≤ f (y ), where y , y ∈ D.
Theorem . (Brouwer’s Fixed Point, []) Every continuous map of a closed bounded
convex set in Rn into itself has a fixed point.
3 On the existence of positive definite solutions of X r +
The map F associated with Equation (.) is defined by
F(X) = I –
m
m
∗ δi
i=1 Ai X Ai
=I
r
A∗i X δi Ai
.
(.)
i=
Theorem . The mapping F defined by (.) is operator monotone.
m ∗

∗ δi
r
Proof Suppose X ≥ X > . Then, F(X ) = (I – m
i= Ai X Ai ) and F(X ) = (I –
i= Ai ×

δ
Xi Ai ) r .
δ
δ
Since X ≥ X , we have X i ≤ Xi for all i = , , . . . , m.
m ∗ δi
δ
δ
∗ δi
Then A∗i X i Ai ≤ A∗i Xi Ai and m
i= Ai X Ai ≤
i= Ai X Ai .
m ∗ δi
m ∗ δi
Therefore, I – i= Ai X Ai ≥ I – i= Ai X Ai , and since r is a positive integer,  < r ≤ .
m ∗ δi 

∗ δi
r
r
Hence, (I – m
i= Ai X Ai ) ≥ (I –
i= Ai X Ai ) , that is, F(X ) ≥ F(X ). Thus, F(X) is
operator monotone.
The following theorem proves the existence of positive definite solutions for Equation (.), based on the Brouwer fixed point theorem.
Theorem . If a real number β <  satisfies ( – β r )I ≥
has positive definite solutions.
m
i= β
δi
A∗i Ai , then Equation (.)
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 5 of 21
Proof It can easily be proved that the condition of the theorem leads to βI ≤ (I –
m ∗ 
m ∗ 
r
r
i= Ai Ai ) . Let D = [βI, (I –
i= Ai Ai ) ]. It is clear that D is closed, bounded, and convex. To show F : D → D , let X ∈ D . Then βI ≤ X, and thus β δi I ≥ X δi , – < δi < . Therem ∗ δ
m δ ∗ 


δi ∗
i
i
r
r
r
fore, (I – m
i= β Ai Ai ) ≤ (I –
i= Ai X Ai ) . By the definition of β, (I –
i= β Ai A) ≥
m ∗ δ

βI, and thus βI ≤ (I – i= Ai X i Ai ) r = F(X), that is,
βI ≤ F(X).
(.)
It is clear that X ≤ I. Then X δi ≥ I and (I –
F(X) ≤ I –
m
m

∗ δi
r
i= Ai X Ai )
≤ (I –
m

∗
r
i= Ai Ai ) ,
that is,
r
A∗i Ai
.
(.)
i=
From (.) and (.) we get that F(X) ∈ D ; therefore, F : D → D . F is continuous since
it is operator monotone. Therefore, F has a fixed point in D , which is a solution of Equation (.).
The following remark, in addition to Examples . and . in Section , assures the validity of this theorem.
Remark Let us consider the simple case r = m = n = , δ = –  . It is clear that β does not
exist for a >
a ≤

√
 
,

√
 

(where a is A∗ A in this simple case) and also that no solution exists. For
there exist β and solutions (i.e., the theorem holds). For instance, for a =
there exist β =


√
 
,

and the solution, namely x =  .
Theorem . The mapping F has the maximal and the minimal elements in D = [βI, (I –
m ∗ 
r
i= Ai Ai ) ], where β is given in Theorem ..
Proof By Theorems . and . the mapping F is continuous and bounded above since
F(X) < I. Let supX∈D F(X) = Y . So, there exists an X̂ in D satisfying Y – εI < F(X̂) ≤ Y . We
can choose a sequence {Xn } in D satisfying Y – ( n )I < F(Xn ) ≤ Y . Since D is compact, the
sequence {Xn } has a subsequence {Xnk } convergent to X ∈ D . So, Y – ( n )I < F(Xnk ) ≤ Y .
Taking the limit as n → ∞, by the continuity of F we get limn→∞ F(Xnk ) = F(X) = Y =
max{F(X) : X ∈ D }. Hence, F has a maximal element X ∈ D .
Similarly, we can prove that F has a minimal element in D , noting that F is bounded
below by the zero matrix.
4 Two algorithms for obtaining extremal positive definite solution of
∗ δi
Xr + m
i=1 Ai X Ai = I
In this section, we present two algorithms for obtaining the extremal positive definite solutions of Equation (.). The main idea of the algorithms is to avoid computing the inverses
of matrices.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 6 of 21
Algorithm . (INVERSE-FREE Algorithm) Consider the iterative algorithm
Xk+ = I –
m
r
δi
A∗i Ykδ
,
Ai
i=
(.)
– Yk+ = Yk I – Xk Yk δ ,
k = , , , . . . , i = , , . . . , m,
Y = α δ I,
X = αI,
α > ,
where δ is a negative integer such that
|δ|
r
< .
Theorem . Suppose that Equation (.) has a positive definite solution. Then the iterative Algorithm . generates subsequences {Xk } and {Xk+ } that are decreasing and
converge to the maximal solution XL .
Proof Suppose that Equation (.) has a solution. We first prove that the subsequences
{Xk } and {Xk+ } are decreasing and the subsequences {Yk } and {Yk+ } are increasing.
Consider the sequence of matrices generated by (.).
For k = , we have
X = I –
m
r
δi
A∗i Yδ
= I–
Ai
i=
Since (α r – )I +
I–
m
r
m
α
δi
A∗i Ai
.
i=
m
i= α
δi
A∗i Ai > , we have I –
m
i= α
δi
A∗i Ai < α r I. Then
r
α
δi
A∗i Ai
< αI = X .
(.)
i=
–
So we get X < X . Also, Y = Y [I – X Y δ ] = α δ [I – αα – I] = α δ I = Y .
For k = , we have
X = I –
m
r
δi
A∗i Y δ
= I–
Ai
m
i=
r
α
δi
A∗i Ai
= X .
i=
So we have X < X . Also,
Y = Y I
– – X Y δ = α δ
I – I –
m
r
α δi A∗i Ai
α –
i=
= α δ I – α δ– I –
m
r
α δi A∗i Ai
.
i=
From (.) we get α δ I – α δ– (I –
For k = , we have
X = I –
m
i=
r
δi
A∗i Yδ
Ai
.
m
i= α
δi

A∗i Ai ) r > α δ I. Thus, Y > Y .
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
δi
δi
Since Y > Y , we have Yδ > Yδ , so we get (I –
Page 7 of 21
m
δi
∗ δ
i= Ai Y
–

Ai ) r < (I –
m
δi
∗ δ
i= Ai Y

Ai ) r .
– δ
Thus, X < X . Also, Y = Y [I – X Y δ ]. Since Y > Y and X < X , we have Y
–
Y δ
and –X > –X , so
–
– X Y δ ] > Y [I
–
–
that –X Y δ > –X Y δ and
–
– X Y δ ]. Thus, Y > Y .
I
–
– X Y δ
> I
–
– X Y δ ,
>
and we get
Y [I
Similarly, we can prove:
X > X > X > · · ·
and
X  > X > X > · · · ,
Y < Y < Y < · · ·
and
Y < Y < Y < · · · .
Hence, {Xk } and {Xk+ }, k = , , , . . . , are decreasing, whereas {Yk } and {Yk+ }, k =
, , , . . . , are increasing.
Now, we show that {Xk } and {Xk+ } are bounded from below by XL (Xk > XL ) and that
{Yk } and {Yk+ } are bounded from above by XLδ .
By induction on k we get
X – XL = αI – I –
m
r
δ
A∗i XLi Ai
.
i=

∗ δi
r
Since XL is a solution of (.), we have that (I – m
i= Ai XL Ai ) < I and αI – (I –
m ∗ δi 
Ai XL Ai ) r > αI – I = (α – )I > , α > , that is, X > XL . Also, X – XL = (I –
i=
m ∗ δi 

m
∗ δi
r
r
i= Ai X Ai ) – (I –
i= Ai XL Ai ) .
m ∗ δi 

δi
δi
∗ δi
r
r
Since X > XL , we have X < XL , and therefore (I – m
i= Ai X Ai ) > (I –
i= Ai XL Ai ) .
m ∗ δi δ
δ
δ
r
So we get X > XL . Also, XL – Y = (I – i– Ai XL Ai ) – α I.
δ
Since X > XL , we have αI > XL and α δi I < XLi ∀i = , , . . . , m, so we have
I–
m
α δi A∗i Ai > I –
m
i=
I–
A∗i XLi Ai
δ
and
i=
m
δr
α δi A∗i Ai
< I–
i=
m
(.)
δr
δ
A∗i XLi Ai
.
i=
From (.) we get
I–
m
δr
α
δi
A∗i Ai
> α δ I.
(.)
i=
δ
∗ δi
δ
δ
δ
r
From (.) and (.) we get (I – m
i= Ai XL Ai ) > α I, so we have XL > Y and XL > Y .
Assume that Xk > XL , Xk+ > XL at k = t that is, Xt > XL , Xt+ > XL . Also, Yt < XLδ and
Yt+ < XLδ .
Now, for k = t + ,
Xt+ – XL = I –
m
i=
δi
δ
A∗i Yt+
Ai
r
– I–
m
i=
r
δ
A∗i XLi Ai
.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
δi
δ
δ
Since Yt+ < XLδ , we have Yt+
< XLi and thus
m ∗ δδi


∗ δi
r
(I – i= Ai Yt+ Ai ) r > (I – m
i= Ai XL Ai ) .
Page 8 of 21
m
δi
∗ δ
i= Ai Yt+ Ai
<
m
∗ δi
i= Ai XL Ai . Therefore,
–
δ
Hence, Xt+ – XL > , that is, Xt+ > XL . Also, XLδ – Yt+ = XLδ – Yt+ [I – Xt+ Yt+
].
–
δ
< XL– and –Xt+ < –XL ,
Since Yt+ < XLδ and Xt+ > XL , we have Yt+
– δ
–Xt+ Yt+i < –XL– XL = –I,
– δ
I – Xt+ Yt+i < I – I = I.
and
–
δ
] < XLδ . Hence, Yt+ < XLδ , and thus
Then Yt+ [I – Xt+ Yt+
Xt+ – XL = I –
m
r
δi
δ
A∗i Yt+ Ai
– I–
m
i=
r
δ
A∗i XLi Ai
.
i=
δi
δ
δ
< XLi and
Since, Yt+ < XLδ , we have Yt+
δ
i
m ∗ δi 
m ∗ δ

r
r
i= Ai Yt+ Ai ) > (I –
i= Ai XL Ai ) .
m
δi
∗ δ
i= Ai Yt+ Ai <
m
∗ δi
i= Ai XL Ai . Therefore, (I –
–
δ
Hence, Xt+ – XL > , that is, Xt+ > XL , and thus XLδ – Yt+ = XLδ – Yt+ [I – Xt+ Yt+
].
–
δ
< XL– and –Xt+ < –XL , and thus
Since Yt+ < XLδ and Xt+ > XL , we have Yt+
–
δ
< –XL– XL = –I
–Xt+ Yt+
–
δ
I – Xt+ Yt+
< I – I = I.
and
–
δ
Then Yt+ [I – Xt+ Yt+
] < XLδ . Hence, Yt+ < XLδ .
Since {Xk } and {Xk+ } are decreasing and bounded from below by XL and {Yk } and
{Yk+ } are increasing and bounded from above by XLδ , it follows that limk→∞ Xk = X and
limk→∞ Yk = Y exist.

∗ δi
r
Taking limits in (.) gives Y = X δ and X = (I – m
i= Ai X Ai ) , that is, X is a solution.
Hence, X = XL .
Remark We have proved that the maximal solution is unique; see the Appendix.
Now, we consider the case  < α < .
Algorithm . (INVERSE-FREE Algorithm) Consider the iterative (simultaneous) algorithm
Xk+ = I –
m
r
δi
A∗i Ykδ
Ai
,
i=
– Yk+ = Yk I – Xk Yk δ ,
X = αI,
Y = α δ I,
(.)
k = , , , . . . , i = , , . . . , m,
 < α < ,
where δ is a negative integer such that
|δ|
r
< .
Theorem . Suppose that Equation (.) has a positive definite solution such that
m δ ∗
r
i
i= α Ai Ai < ( – α )I. Then the iterative Algorithm . generates the subsequences {Xk }
and {Xk+ } that are increasing and converge to the minimal solution XS .
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 9 of 21
Proof Suppose that Equation (.) has a solution. We first prove that the subsequences
{Xk } and {Xk+ } are increasing and the subsequences {Yk } and {Yk+ } are decreasing.
Consider the sequence of matrices generated by (.).
For k = , we have
X = I –
m
r
δi
δ
A∗i Y Ai
= I–
r
m
i=
α δi A∗i Ai
.
i=
From the condition of the theorem we have
m
I–
α δi A∗i Ai > α r I.
(.)
i=
Then,
I–
m
r
α δi A∗i Ai
> αI = X .
(.)
i=
–
So we get X > X . Also, Y = Y [I – X Y δ ] = α δ [I – αα – I] = α δ I = Y .
For k = , we have
X = I –
m
r
δi
δ
A∗i Y Ai
= I–
m
i=
r
α δi A∗i Ai
= X .
i=
So we have X > X . Also,
Y = Y I
– – X Y δ = α δ
= α δ I – α δ– I –
m
I – I –
m
α
A∗i Ai
α
–
i=
r
α δi A∗i Ai
r
δi
.
i=
From (.) we get α δ I – α δ– (I –
For k = , we have
X = I –
m
i= α
δi

A∗i Ai ) r < α δ I. Thus, Y < Y .
r
δi
A∗i Yδ
m
.
Ai
i=
δi
δi
Since Y < Y , we have Yδ < Yδ , so we get (I –
– δ
Thus, X > X . Also, Y = Y [I – X Y ].
–
m
δi
∗ δ
i= Ai Y

Ai ) r > (I –
m
–
δi
∗ δ
i= Ai Y
–

Ai ) r .
–
Since Y < Y and X > X , we have Y δ < Y δ and –X < –X , so that –X Y δ < –X Y δ
–
–
–
X < X  < X < · · ·
and
X  < X < X < · · · ,
Y > Y > Y > · · ·
and
Y > Y > Y > · · · .
–
and I – X Y δ < I – X Y δ , and we get Y [I – X Y δ ] < Y [I – X Y δ ]. Thus, Y < Y .
Similarly, we can prove that
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 10 of 21
Hence {Xk } and {Xk+ }, k = , , , . . . , are increasing, whereas {Yk } and {Yk+ }, k =
, , , . . . , are decreasing.
Now, we show that {Xk } and {Xk+ } are bounded from above by XS (XS > Xk ), and {Yk }
and {Yk+ } are bounded from below by XSδ .
By induction on k we obtain
XS – X  = I –
m
r
A∗i XSi Ai
δ
– αI > ,
i=
that is, XS > X . Also,
XS – X = I –
m
r
A∗i XSi Ai
δ
– I–
i=
m
r
A∗i Xi Ai
δ
.
i=
m ∗ δi 

δ
δ
∗ δi
r
r
Since XS > X , we have XSi < Xi , and therefore (I – m
i= Ai XS Ai ) > (I –
i= Ai X Ai ) .
δ
∗ δi
r
So we get XS > X . Also, Y – XSδ = α δ I – (I – m
A
X
A
)
.
i– i S i
δ
Since XS > X , we have XS > αI and XSi < α δi I ∀i = , , . . . , m, and we get
I–
m
α δi A∗i Ai < I –
m
i=
I–
A∗i XSi Ai
δ
and
i=
m
δr
α δi A∗i Ai
> I–
i=
m
(.)
δr
A∗i XSi Ai
δ
.
i=
From (.) we get
I–
m
δr
α
δi
A∗i Ai
< α δ I.
(.)
i=
δ
∗ δi
δ
δ
r
From (.) and (.) we get α δ I > (I – m
i= Ai XS Ai ) , and thus Y > XS and Y > XS .
Assume that Xk < XS and Xk+ < XS at k = t, that is, Xt < XS and Xt+ < XS . Also,
Yt > XSδ and Yt+ > XSδ .
Now, for k = t + , we have
XS – Xt+ = I –
m
r
δ
A∗i XSi Ai
– I–
i=
m
δi
δ
A∗i Yt+
Ai
r
.
i=
δi
δi
m ∗ δi
δ
δ
∗ δ
> XSi and m
Since Yt+ > XSδ , we have Yt+
i= Ai Yt+ Ai >
i= Ai XS Ai . Therefore, (I –
δ
m ∗ δi 
m ∗ δi

r
r
i= Ai Yt+ Ai ) < (I –
i= Ai XS Ai ) .
Hence, XS – Xt+ > , that is, XS > Xt+ . Also,
– δ – XSδ .
Yt+ – XSδ = Yt+ I – Xt+ Yt+
–
δ
Since Yt+ > XSδ and Xt+ < XS , we have Yt+
> XS– and –Xt+ > –XS , and then
– δ
–Xt+ Yt+i > –XS XS– = –I
– δ
and I – Xt+ Yt+i > I – I = I.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 11 of 21
–
δ
Then Yt+ [I – Xt+ Yt+
] > XSδ . Hence, Yt+ > XSδ . Consider
XS – Xt+ = I –
m
r
δ
A∗i XSi Ai
m
– I–
i=
δi
δ
A∗i Yt+ Ai
r
.
i=
δi
δi
m ∗ δi
δ
δ
∗ δ
> XSi and m
Since Yt+ > XSδ , we have Yt+
i= Ai Yt+ Ai >
i= Ai XS Ai . Therefore, (I –
m ∗ δi 
m ∗ δδi

r
r
i= Ai Yt+ Ai ) < (I –
i= Ai XS Ai ) .
Hence, XS – Xt+ > , that is, XS > Xt+ . Now consider
– δ – XSδ .
Yt+ – XSδ = Yt+ I – Xt+ Yt+
–
δ
> XS– and –Xt+ > –XS , and thus
Since Yt+ > XSδ and Xt+ < XS , we have Yt+
–
δ
> –XS XS– = –I
–Xt+ Yt+
and
–
δ
I – Xt+ Yt+
> I – I = I.
–
δ
Then Yt+ [I – Xt+ Yt+
] > XSδ , and hence Yt+ > XSδ .
Since {Xk } and {Xk+ } are increasing and bounded from above by XS and {Yk } and
{Yk+ } are decreasing and bounded from below by XSδ , it follows that limk→∞ Xk = X and

∗ δi
r
limk→∞ Yk = Y exist. Taking the limits in (.) gives Y = X δ and X = (I – m
i= Ai X Ai ) ,
that is, X is a solution of Equation (.). Hence, X = XS .
Remark We have proved that the minimal solution is unique; see the Appendix.
5 Numerical examples
In this section, we report a variety of numerical examples to illustrate the accuracy and
efficiency of the two proposed Algorithms . and . to obtain the extremal positive definite solutions of Equation (.). The solutions are computed for different matrices Ai ,
i = , , . . . , m, and different values of α, r, δ, and δi , i = , , . . . , m. All programs are written
∗ δi
in MATLAB version .. We denote ε(Xk ) = Xkr + m
i= Ai Xk Ai – I∞ = Xk+ – Xk ∞ for
the stopping criterion, and we use ε(Xk ) < tol for different chosen tolerances.
Example . Consider the matrix equation (.) with the following two matrices A and
A :
⎛
⎞
–. . –. –.
⎜ . –. .
. ⎟
⎜
⎟
A = . × ⎜
⎟,
⎝ –.

.
. ⎠

.

.
⎛
⎞
. . . .
⎜ . . . . ⎟
⎜
⎟
A = ⎜
⎟,
⎝ 
. . . ⎠
. . . .
where r = , δ = –  , δ = –  , δ = –.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 12 of 21
Table 1 INVERSE-FREE iterative Algorithm 4.1
α
1.2
1.4
1.6
1.8
2
k
tol = 10–4
tol = 10–6
tol = 10–8
7
8
9
10
10
14
15
16
17
17
21
22
23
24
24
k : The number of iterations.
We applied Algorithm . for different values of the parameter α > . The number of
iterations needed to satisfy the stopping condition (required accuracy) in order the sequence of positive definite matrices to converge to the maximal solution of Equation (.)
are listed in Table .
⎛
⎞
.
. –. –.
⎜ .
. –. . ⎟
⎜
⎟
XL ∼
⎟.
=⎜
⎝ –. –. . –. ⎠
–. . –. .
The eigenvalues of XL are (., ., ., .).
Note: Theorem . holds for β =  ,  , . . . , etc.
Example . Consider the matrix equation (.) with the following two matrices A and
A :
⎛
.
⎜ –.
⎜
A = ⎜
⎝ –.
.
⎛
.
⎜ .
⎜
A = ⎜
⎝ .
–.
⎞
–. .
. –. ⎟
⎟
⎟,
.
. ⎠
.
.
⎞
–. –. .
. –. . ⎟
⎟
⎟,
.
. . ⎠
.
. .
–.
.
.
–.
which satisfy the conditions of Theorem ., where r = , δ = –  , δ = –  , δ = –. We
applied Algorithm . for different values of the parameter α ( < α < ). The obtained
results are summarized in Table , where k is the number of iterations needed to satisfy the
stopping condition (required accuracy) in order the sequence of positive definite matrices
converge to the minimal solution of Equation (.).
⎛
⎞
. –. –. –.
⎜ –. . –. –. ⎟
⎜
⎟
XS ∼
=⎜
⎟.
⎝ –. –. . –. ⎠
–. –. –. .
The eigenvalues of XS are (., ., ., .).
Also, Theorem . holds for β =  ,  , . . . , etc.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 13 of 21
Table 2 INVERSE-FREE iterative Algorithm 4.3
α
k
0.5
0.7
0.9
tol = 10–4
tol = 10–6
tol = 10–8
12
6
3
19
13
10
26
20
17
k : The number of iterations.
Table 3 INVERSE-FREE iterative Algorithm 4.1
α
k
1.2
1.4
1.6
1.8
2
tol = 10–6
tol = 10–8
tol = 10–10
8
10
11
12
13
20
22
23
24
25
31
33
34
35
36
k : The number of iterations.
Remark From Table  we see that the number of iterations k increases as the value of α
(α > ) increases, and from Table  we see that the number of iterations k decreases as the
value of α ( < α < ) increases. For details, see the Appendix in the end of this paper.
Example . Consider the matrix equation (.) with the following two matrices A and
A :
⎛
–.
⎜
⎜ .
⎜
⎜ –.
A = ⎜
⎜ –.
⎜
⎜
⎝ –.
.
⎛
–.
⎜
⎜ .
⎜
⎜ .
A = ⎜
⎜ .
⎜
⎜
⎝ –.
.
⎞
.
–. –. . –.
⎟
–.
.
–. .
–. ⎟
⎟
.
.
–.
.
. ⎟
⎟,
. –.
.
.
. ⎟
⎟
⎟
.
.
.
. –. ⎠
.
–.
.
.
–.
⎞
. . . .
.
⎟
.
. –. –. –. ⎟
⎟
. . .
.
. ⎟
⎟,
. .
.
.
. ⎟
⎟
⎟
–. . .
.
. ⎠
–. . . . –.
where r = , δ = –  , δ = –  , δ = –. We applied Algorithm . for different values of the
parameter α (α > ). The number of iterations needed to satisfy the stopping condition
(required accuracy) in order the sequence of positive definite matrices to converge to the
maximal solution of Equation (.) are listed in Table .
⎛
.
⎜ .e–
⎜ –.e–
XL ∼
=⎜
⎜ –.e–
⎝
–.e–
.e–
.e–
.
.e–
–.e–
–.e–
.e–
–.e– –.e–
.e– –.e–
.
.e–
.e–
.
.e– –.e–
–.e– .
–.e–
–.e–
.e–
–.e–
.
–.e–
⎞
.e–
.e– ⎟
–.e– ⎟
⎟
. ⎟
⎠
–.e–
.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 14 of 21
Table 4 INVERSE-FREE iterative Algorithm 4.3
α
0.5
0.7
0.9
k
tol = 10–4
tol = 10–6
tol = 10–8
tol = 10–10
13
3
3
20
10
9
27
17
16
33
23
23
k : The number of iterations.
The eigenvalues of XL are: (., ., ., ., ., .).
Example . Consider the matrix equation (.) with the two matrices A and A as follows:
⎛
–.
.
⎜
–.
⎜ .
⎜
⎜ .
–.
A = ⎜
⎜ .
.
⎜
⎜
⎝ –. .
–. –.
⎛
.
–.
⎜
.
⎜ .
⎜
⎜ .
–.
A = ⎜
⎜ –.
.
⎜
⎜
⎝ –.
.
.
–.
⎞
–.
.
. –.
⎟
–.
.
–. –. ⎟
⎟
–. –. . –. ⎟
⎟,
–. –. . –. ⎟
⎟
⎟
. –. –. –. ⎠
.
–. . –.
⎞
. –. –.
.
⎟
–. –. –. . ⎟
⎟
.
–.
.
–. ⎟
⎟
.
.
.
–. ⎟
⎟
⎟
.
–.
.
. ⎠
.
.
–.
.
which satisfy the conditions of Theorem ., where r = , δ = –  , δ = –  , δ = –. Applying Algorithm ., for different values of the parameter α ( < α < ). The obtained
results are summarized in Table , where k is the number of iterations needed to satisfy
the stopping condition (required accuracy) in order the sequence of positive definite matrices converge to the minimal solution of Equation (.).
⎛
.
⎜ .
⎜ .
XS ∼
=⎜
⎜ .
⎝
.
.e–
.
.
.
.
–.
–.
.
.
.
.
.
.
.
.
.
.
.
.e–
.
–.
.
.
.
.
⎞
.e–
–. ⎟
. ⎟
⎟.
.e– ⎟
⎠
.
.
The eigenvalues of XS are (., ., ., ., ., .).
Remarks
. The obtained results for Examples . and . shown in Tables  and , respectively,
indicate that increasing the dimension of the problem does not affect the efficiency
of the proposed algorithms.
. From Tables , , , and , it is clear that we obtained a high accuracy for different
values of α after a few numbers of steps; see the number of iterations, which
indicate that our algorithms have high efficiency.
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 15 of 21
6 On the existence and the uniqueness of a positive definite solution of
∗ δi
Xr – m
i=1 Ai X Ai = I
In this section, we prove that Equation (.) has a unique positive definite solution and
construct an interval that includes that solution. Associated with Equation (.) is the operator G defined by
G(X) = I +
m
r
A∗i X δi Ai
.
(.)
i=
Theorem . If Equation (.) has a positive definite solution X, then X ∈ [G (I), G(I)].
Proof Let X be the positive definite solution of Equation (.). Then X = (I +

Ai ) r , and we have
X ≥ I,
m
∗ δi
i= Ai X
×
(.)
which implies that X δi ≤ I, – < δi < , ∀i = , , . . . , m.
Then
X= I+
m
r
A∗i X δi Ai
≤ I+
i=
m
r
A∗i Ai
= G(I).
(.)
i=
Hence, from (.) and (.) we get
I ≤ X ≤ G(I).
(.)
δi
m ∗
m ∗ δi
m ∗ δ
∗
δi
i
r
r
Then (I + m
i= Ai Ai ) ≤ X ≤ I, and thus
i= Ai (I +
i= Ai Ai ) Ai ≤
i= Ai X Ai ≤
m ∗
i= Ai Ai .
m ∗
m ∗ δi
m ∗
∗ δi
r
r
r
We have m
i= Ai X Ai = X – I. Then
i= Ai (I +
i= Ai Ai ) Ai ≤ X – I ≤
i= Ai Ai .
m ∗
m ∗ δi
m ∗ 


Therefore, (I + i= Ai (I + i= Ai Ai ) r Ai ) r ≤ X ≤ (I + i= Ai Ai ) r , that is, G (I) ≤ X ≤
G(I). Hence, X ∈ [G (I), G(I)].
We use the Brouwer fixed point theorem to prove the existence of positive definite
solutions of Equation (.). Since P(n) is not complete, we consider the subset D =
[G (I), G(I)] ⊂ P(n), which is compact.
Theorem . Equation (.) has a Hermitian positive definite solution.
Proof It is obvious that D is closed, bounded, and convex. To show that G : D → D ,

∗
δi
r
let X ∈ D . Then G (I) ≤ X ≤ G(I), that is, X ≤ (I + m
i= Ai Ai ) , and thus X ≥ (I +
m ∗ δi
m ∗ δ
m ∗
m ∗ δi

i
r
r
r
i= Ai Ai ) , – < δi < . Therefore, (I +
i= Ai X Ai ) ≥ (I +
i= Ai (I +
i= Ai Ai ) ×

r
Ai ) .
Hence,
G(X) ≥ G (I).
(.)
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Similarly, since X ≥ I, we have X δi ≤ I and (I +
is,
Page 16 of 21
m

∗ δi
r
i= Ai X Ai )
≤ (I +
G(X) ≤ G(I).
m

∗
r
i= Ai Ai ) ,
that
(.)
From (.) and (.) we get G(X) ∈ D , that is, G : D → D .
∗ δi
According to Lemma .. in [], m
i= Ai X Ai is continuous on D . Hence, G(X) is
continuous on D . By Brouwer’s fixed point theorem, G has a fixed point in D , which is a
solution of Equation (.).
Remark Applying Lemma ., in [], it is proved that Equation (.) has a unique positive
definite solution for r = . The following theorem shows that the uniqueness holds for any
r, that is, for Equation (.).
Theorem . Equation (.) has a unique positive definite solution.
Proof We show that G has a unique fixed point. For all X, Y ∈ P(n) such that X ≥ Y , by
Definition . we have G : P(n) → P(n) and
G(X) = I +
m
r
A∗i X δi Ai
≤ I+
i=
m
r
A∗i Y δi Ai
= G(Y ),
i=
that is, the operator G is decreasing. We define θ = max{|δi |, i = , , . . . , m}. Then  < θ < .
For all t ∈ (, ), we have
G(tX) =
I+
m
r
A∗i (tX)δi Ai
i=
=
I+
m
r
t δi A∗i X δi Ai
i=
≤ t –θ I + t –θ
m
r
t δi A∗i X δi Ai
i=
=t
– θr
I+
m
r
t
δi
A∗i X δi Ai
,
i=
where  < θr < . Let γ = θr . Then G(tX) ≤ t –γ G(X), which means that the operator G is
(–γ )-convex. We obtain that G has a unique fixed point X in P(n), which is the unique
positive definite solution of Equation (.).
Remarks
. In [], the authors considered the nonlinear matrix equation
X=
m
i=
ATi X δi Ai ,
|δi | < , i = , , . . . , m.
(.)
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 17 of 21
They proved that the recursively defined matrix sequence
Xn+m+ =
m
δ
i
ATi Xn+i
Ai ,
n ≥ ,
(.)
i=
where X , X , . . . , Xm are arbitrary initial positive definite matrices, converge to the
unique positive definite solution of (.).
In [], the authors considered the nonlinear matrix equation
X =Q+
m
A∗i X δi Ai ,
 < |δi | < , i = , , . . . , m,
(.)
i=
where Q is a known positive definite matrix. They considered the same formula
(.) as
Xn+m+ = Q +
m
i
A∗i Xn+i
Ai ,
δ
n ≥ .
(.)
i=
.
They used the algorithm given in [] and used the main steps for the proof of
convergence with slight changes to suit their problem (.).
Since Q has no effect on the convergence of the algorithm and since in [] it is
proved that both Xn+m+ and Xn+i converge to the unique solution X, we see that the
proof given in [] is redundant.
For our Equation (.), we use the recursive formula
Xn+m+ = I +
m
r
δi
A∗i Xn+i
Ai
(.)
i=
according to the following proposition.
Proposition . The matrix sequence defined by (.) converges to the unique positive
definite matrix solution X of Equation (.) for arbitrary initial positive definite matrices
X , X , . . . , Xm , provided that it is valid for r = .
m ∗ δi
∗ δi
Proof We have that m
i= Ai Xn+i Ai is continuous and thus Gn (X) = I +
i= Ai Xn+i Ai is

r
continuous. Define F(Gn (X)) = (Gn (X)) , where r is a positive integer. Then F is also continuous. So, limn→∞ F(Gn (X)) = F limn→∞ Gn (X). Taking the limit of (.) as n → ∞ and

∗ δi
r
using (.) and (.) with Q = I, we obtain X = (I + m
i= Ai X Ai ) .
7 Numerical examples
Example . Consider the matrix equation (.), with the following two matrices A and
A :
A =
. .
,
. .
A =
. .
,
. .
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 18 of 21
where r = , δ = –  , δ = –  . Let the initial positive definite matrices be
X =
.
.
.
,
.
X =
. .
.
. .
Applying the recursive formula (.), after nine iterations of (.), we get the unique
positive definite solution of (.)
X = X =
. .
.
. .
We see that X ∈ [G (I), G(I)], where G is defined by (.).
Example . Consider the matrix equation (.) with three matrices
. .
,
A =
. .
. .
A =
,
. .
A =
. .
,
. .
and
where r = , δ = –  , δ = –  , δ = –  . Let the initial positive definite matrices be
. .
,
X =
. .
. .
X =
.
. .
X =
. .
,
. .
and
Applying the recursive formula (.), after  iterations of (.), we get the unique
positive definite solution of (.)
X = X =
. .
.
. .
We see that X ∈ [G (I), G(I)], where G is defined by (.).
Remarks
. If we use X and X from Example ., then we get the same solution X = X , which
proves the uniqueness of the solution of Equation (.).
. If we put
. .
,
X =
. .
. .
X =
. .
X =
.
.
.
,
.
and
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 19 of 21
in Example ., then we get the same solution
X = X =
. .
,
. .
which proves the uniqueness of the solution of Equation (.).
The above examples show that the recursive formula defined by (.) is feasible and
effective to compute the unique positive definite solution of Equation (.).
8 Conclusion
∗ δi
In this paper we considered two nonlinear matrix equations X r ± m
i= Ai X Ai = I. For the
first equation (plus case), the proofs of the existence of positive definite solutions beside
the extremal solutions are given. Also two algorithms are suggested for computing the
extremal solutions. For the second equation (negative case), the existence and uniqueness
of a positive definite solution are proved. The algorithm in [] is adapted for solving this
equation. Numerical examples are introduced to illustrate the obtained theoretical results.
Appendix
The paper is concerned with the positive definite solutions of our two problems (.) and
(.), that is, the solutions that lie in the set P(n) of positive definite matrices, which is the
interior of the cone P(n) of positive (semidefinite) matrices.
Definition A. (Positive operator) A bounded self-adjoint linear operator T from the
Hilbert space H into it self, T : H → H, is said to be positive, written T ≥ , if
Tx, x ≥ 
∀x ∈ H.
(A.)
• If T is self-adjoint, then Tx, x is real.
• The matrices are linear bounded operators defined on the Hilbert space Rn .
• One of our goals is to prove the existence of the extremal (maximal-minimal) positive
definite solutions of the matrix equation (.). So we need the definition of an ordered
Hilbert space: If T and T are linear bounded self-adjoint operators, then
 ≤ T ≤ T
if T – T ∈ P(n),
(A.)
where P(n) is the cone of positive semidefinite matrices.
The classical definition an ordered Hilbert space in functional analysis:  ≤ T ≤ T if
T x, x ≥ T x, x ≥  ∀x ∈ H, which leads to
(T – T )x, x ≥ 
⇒
T – T ≥ ,
(A.)
that is, (A.) is equivalent to (A.).
Definition (A.) gives us an important result: The ordering of linear bounded selfadjoint operators on P(n) is - corresponding to the order of the positive quadratic
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 20 of 21
Figure 1 The effect of increasing α on the number of iterations of Algorithm 4.1.
forms (which is the order of positive real numbers), which is total (since all its elements are comparable). So the elements of the cone of positive matrices P(n) have at
most one maximal and one minimal element. In Theorem ., we determined the in

∗
r
terval D = [βI, (I – m
i= Ai Ai ) ] that contains all positive definite solutions of Equation (.). In Theorem ., we proved the existence of maximal and minimal solutions

∗
r
in D = [βI, (I – m
i= Ai Ai ) ].
To compute the maximal solution, we constructed the decreasing sequence of positive
definite matrices (which are not solutions of Equation (.)) and proved that it converges
to the positive definite limit matrix, which is the maximal solution XL .
So, as α >  increases, the algorithm needs more iterations, as is explained in Table .
For example, since I > I, the algorithm needs more iterations at α =  than at α = .
Similarly, the same analysis holds for the minimal solution XS . See Figure .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Naglaa M. El-Shazly has made substantial contributions to conception and design, acquisition of data, and analysis and
interpretation of data. Abdel-Shakoor M. Sarhan has been involved in drafting the manuscript and revising it critically for
important intellectual content. Both authors read and approved the final manuscript.
Acknowledgements
The authors acknowledge the reviewers for reviewing the manuscript.
Received: 28 December 2015 Accepted: 9 May 2016
References
∗ δi
1. Duan, X, Liao, A, Tang, B: On the nonlinear matrix equation X – m
i=1 Ai X Ai = Q. Linear Algebra Appl. 429, 110-121
(2008)
2. Shi, X, Liu, F, Umoh, H, Gibson, F: Two kinds of nonlinear matrix equations and their corresponding matrix sequences.
Linear Multilinear Algebra 52, 1-15 (2004)
3. Cai, J, Chen, G: Some investigation on Hermitian positive definite solutions of the matrix equation X s + A∗ X –t A = Q.
Linear Algebra Appl. 430, 2448-2456 (2009)
4. Duan, X, Liao, A: On the existence of Hermitian positive definite solutions of the matrix equation X s + A∗ X –t A = Q.
Linear Algebra Appl. 429, 673-687 (2008)
Model. 49, 936-945 (2009)
5. Duan, X, Liao, A: On the nonlinear matrix equation X + A∗ X –q A = Q (q ≥ 1). Math. Comput.
∗ r
6. Duan, X, Liao, A: On Hermitian positive definite solution of the matrix equation X – m
i=1 Ai X Ai = Q. J. Comput. Appl.
Math. 229, 27-36 (2009)
√
2m –1
X A = I.
7. El-Sayed, SM, Ramadan, MA: On the existence of a positive definite solution of the matrix equation X – A∗
Int. J. Comput. Math. 76, 331-338 (2001)
m
8. Lim, Y: Solving the nonlinear matrix equation X = Q + i=1 Mi X δi M∗i via a contraction principle. Linear Algebra Appl.
430, 1380-1383 (2009)
9. Liu, XG, Gao, H: On the positive definite solutions of the matrix equations X s ± AT X –t A = In . Linear Algebra Appl. 368,
83-97 (2003)
Sarhan and El-Shazly Journal of Inequalities and Applications (2016) 2016:143
Page 21 of 21
10. Peng, ZY, El-Sayed, SM, Zhang, XL: Iterative methods for the extremal positive definite solution of the matrix equation
X + A∗ X –α A = Q. Appl. Math. Comput. 200, 520-527 (2007)
11. Ramadan, MA: On the existence of extremal positive definite solutions of a kind of matrix. Int. J. Nonlinear Sci. Numer.
Simul. 6, 115-126 (2005)
12. Ramadan, MA: Necessary and sufficient conditions to the existence of positive definite solution of the matrix
equation X + AT X –2 A = I. Int. J. Comput. Math. 82, 865-870 (2005)
√
2m –1
13. Ramadan, MA, El-Shazly, NM: On the matrix equation X + A∗
X A = I. Appl. Math. Comput. 173, 992-1013 (2006)
14. Yueting, Y: The iterative method for solving nonlinear matrix equation X s + A∗ X –t A = Q. Appl. Math. Comput. 188,
46-53 (2007)
15. Zhan, X: Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. Comput. 17, 1167-1174
(1996)
16. El-Sayed, SM, Ran, ACM: On an iteration method for solving a class of nonlinear matrix equation. SIAM J. Matrix Anal.
Appl. 23, 632-645 (2001)
17. Dehghan, M, Hajarian, M: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix
equations and its optimal approximation. Appl. Math. Comput. 202, 571-588 (2008)
18. Dehghan, M, Hajarian, M: The general coupled matrix equations over generalized bisymmetric matrices. Linear
Algebra Appl. 432, 1531-1552 (2010)
19. Hajarian, M: Matrix form of the CGS method for solving general coupled matrix equations. Appl. Math. Lett. 34, 37-42
(2014)
20. Hajarian, M: Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations.
J. Franklin Inst. 350, 3328-3341 (2013)
21. Sarhan, AM, El-Shazly,
NM, Shehata, EM: On the existence of extremal positive definite solutions of the nonlinear
∗ δi
matrix equation X r + m
i=1 Ai X Ai = I. Math. Comput. Model. 51, 1107-1117 (2010)
22. Bhatia, R: Matrix Analysis, vol. 169. Springer, Berlin (1997)
23. Choudhary, B, Nanda, S: Functional Analysis with Applications. Wiley, New Delhi (1989)