Convex (α, β)-Generalized Contraction and Its Applications in Matrix Equations

Abstract

This paper investigates the existence and convergence of solutions for linear and nonlinear matrix equations. This study explores the potential of convex ($\alpha ,\beta$)-generalized contraction mappings in geodesic spaces, ensuring the existence of solutions for both linear and nonlinear matrix equations. This paper extends the concept to partially ordered geodesic spaces and establishes new existence and convergence results. Illustrative examples are provided to demonstrate the practical relevance of the findings. Overall, this research contributes a novel approach to the field of matrix equations.

1. Introduction

Matrix equations arise as essential mathematical tools in various fields, such as ladder networks, control theory, stochastic filtering, dynamic programming, and statistics. Understanding the solutions of linear and nonlinear matrix equations is crucial for advancing research and practical applications in these areas [1]. This paper aims to investigate the existence and convergence of solutions for such matrix equations.

We commence our study by focusing on a linear matrix equation of the form:

$$U-{\Gamma }_1^{*}U{\Gamma }_1-\cdots -{\Gamma }_{m}U{\Gamma }_m=Q,$$

(1)

where ${\Gamma }_1,\cdots ,{\Gamma }_m$ are arbitrary matrices of order $n\times n$, and Q is a matrix of order $n\times n$. Our objective is to ensure the existence of solutions for Equation (1), which also possess the property of being positive definite. To achieve this, we employ the Krasnosel’skiĭ iterative method, a proven technique that converges to solutions of linear matrix equations [2].

Moving on, we delve into the realm of nonlinear matrix equations and consider the following equation:

$$U=Q\pm \sum _{j=1}^m{\Gamma }_j^{*}F\left(U\right){\Gamma }_j,$$

(2)

where F is a continuous mapping from the set of all positive definite matrices to itself. We impose certain assumptions on F, such as order-preserving or order-reversing properties. Our objective is to determine solutions to Equation (2) under these assumptions.

To tackle the challenge of solving both linear and nonlinear matrix equations, we draw inspiration from the concept of Picard–Banach contractions. Picard operators, which are a specific type of mapping, have been widely recognized as valuable tools in nonlinear analysis, pioneered by Banach [3]. The fundamental idea behind the Banach contraction principle (BCP) is that within a complete metric space $(X,\Omega )$, any mapping $\Phi :X\to X$ satisfying the contraction condition $\Omega (\Phi (\xi ),\Phi (\varrho \left)\right)\le \beta \Omega (\xi ,\varrho )$ for all $\xi ,\varrho \in X$, possesses a unique fixed point.

Building upon the foundations laid by previous researchers, such as Ran and Reurings [2], who utilized the Banach contraction principle in partially ordered sets to obtain positive definite solutions for matrix equations, and Nieto and Rodríguez-López [4], who applied similar fixed point theorems to find solutions of differential equations, we explore the potential of convex ($\alpha ,\beta$)-generalized contraction mappings in geodesic spaces. This perspective enables us to ensure the existence of solutions for both linear and nonlinear matrix equations under appropriate assumptions.

Over the past nine decades, the Picard–Banach fixed point theorem and its various extensions have become invaluable tools in solving a wide range of nonlinear problems, including differential equations, integral equations, integro-differential equations, optimization problems, and variational inequalities. This is evident from the extensive body of literature surrounding these topics, including monographs by authors such as Zeidler, Khamsi, and Granas, as well as numerous references [5,6,7]. The following results are related to integral type contractive conditions [8,9,10]. See also [11,12,13].

Recent research by Petruşel and Petruşel [14] and Popescu [15] has focused on convex orbital $\beta$-Lipschitz mappings and convex orbital ($\alpha ,\beta$)-Lipschitz mappings, respectively, in Hilbert spaces (see also [16]). They demonstrated the existence of fixed points within these classes of mappings and established connections with the admissible perturbations approach. Motivated by their work, we consider convex ($\alpha ,\beta$)-generalized contraction mappings in the setting of geodesic spaces. Our aim is to ensure the solution of Equation (2) under appropriate assumptions. Specifically, we utilize fixed-point techniques and introduce the concept of monotone convex ($\alpha ,\beta$)-generalized contraction mappings in ordered spaces, where the condition on the mapping only needs to hold for comparable elements.

The primary objective of this research is to investigate the existence and convergence of solutions for matrix equations. To accomplish this, we employ the concept of monotone convex ($\alpha ,\beta$)-generalized contraction mappings in partially ordered Banach spaces. We extend the concept of convex ($\alpha ,\beta$)-generalized contraction mappings to partially ordered geodesic spaces and establish new existence and convergence results. Furthermore, we utilize these findings to solve the linear matrix Equation (1) and the nonlinear matrix Equation (2). We provide illustrative examples to demonstrate the applicability and effectiveness of our results. In conclusion, this research contributes to the field of matrix equations by introducing a novel approach based on monotone convex ($\alpha ,\beta$)-generalized contractions in partially ordered Banach spaces. Our theoretical results pave the way for further investigations and applications in various areas of mathematics and related disciplines.

2. Preliminaries

Consider a pair of points, $\xi$ and $\varrho$, within the metric space $(X,\Omega )$. We define a path, denoted as $\zeta :[0,1]\to X$, to connect $\xi$ and $\varrho$ if the conditions are met:

$$\zeta \left(0\right)=\xi \mathrm{and} \zeta \left(1\right)=\varrho .$$

A path $\zeta$ is categorized as a geodesic if the following criteria are satisfied for all s and t in the interval $[0,1]$:

$$\Omega \left(\zeta \right(s),\zeta (t\left)\right)=\Omega \left(\zeta \right(0),\zeta (1\left)\right)|s-t|.$$

When each pair of points $\xi$ and $\varrho$ in X can be linked by a geodesic, the metric space $(X,\Omega )$ obtains the classification of a geodesic space. Further, if the geodesics in a geodesic space are unique, the space falls under the classification of a Busemann space, according to [17]. Notable instances of these spaces include normed spaces, CAT(0)-spaces, Hadamard manifolds, and the Hilbert open unit ball equipped with the hyperbolic metric, as shown in References [18,19]. Kohlenbach [19] introduced a precise formulation of hyperbolic spaces, which is presented below.

Definition 1.

If a function $W:X\times X\times [0,1]\to X$ exists such that $(X,\Omega )$ is a metric space and $(X,\Omega ,W)$ satisfies the following conditions, then it is referred to as a hyperbolic metric space:

 (i) 

$\Omega (z,W(\xi ,\varrho ,\theta \left)\right)\le (1-\theta )\Omega (z,\xi )+\theta \Omega (z,\varrho );$

 (ii) 

$\Omega (W(\xi ,\varrho ,\theta ),W(\xi ,\varrho ,\overline{\theta }))=|\theta -\overline{\theta }|\Omega (\xi ,\varrho );$

 (iii) 

$W(\xi ,\varrho ,\theta )=W(\varrho ,\xi ,1-\theta );$

 (iv) 

$\Omega \left(W\right(\xi ,z,\theta ),W(\varrho ,\zeta ,\theta \left)\right)\le (1-\theta )\Omega (\xi ,\varrho )+\theta \Omega (z,\zeta ),$

for all $\xi ,\varrho ,z,\zeta \in X$ and $\theta ,\overline{\theta }\in [0,1].$

Remark 1.

If $W(\xi ,\varrho ,\theta )=(1-\theta )\xi +\theta \varrho$ for all $\xi ,\varrho \in X,\theta \in [0,1],$ then these spaces include all normed linear spaces.

For $\xi ,\varrho \in X,$

$$[\xi ,\varrho ]=\left\{\right(1-\theta )\xi \oplus \theta \varrho :\theta \in [0,1\left]\right\}$$

denotes geodesic segments.

A map $x:[a,b]\to X$ is an affinely reparametrized geodesic if there exist an interval $[c,d]$ and a geodesic $x^{\prime }:[c,d]\to X$ such that $x=x^{\prime } o \psi ,$ where $\psi :[a,b]\to [c,d]$ is the unique affine homeomorphism between the intervals $[a,b]$ and $[c,d]$, or x is a constant path. A geodesic space $(X,\Omega )$ is a Busemann space if for any two affinely reparametrized geodesics $x:[a,b]\to X$ and $x^{\prime }:[c,d]\to X,$ and the map $D_{x,x^{\prime }}:[a,b]\times [c,d]\to \mathbb{R}$ is defined as

$$D_{x,x^{\prime }}(s,t)=\Omega (x\left(s\right),x^{\prime }\left(t\right))$$

is convex, see [17].

A unique mapping for convexity, denoted as W, exists such that the triple $(X,\Omega ,W)$ forms a uniquely geodesic W-hyperbolic space when $(X,\Omega )$ is a Busemann space. This implies that for any distinct pair of points $\xi$ and $\varrho$ in X, and for any value of $\theta$ within the interval $[0,1]$, there exists a sole element $\zeta \in X$ (specifically denoted as $\zeta =W(\xi ,\varrho ,\theta )$) that satisfies the following conditions:

$$\Omega (\xi ,\zeta )=\theta \Omega (\xi ,\varrho ) \mathrm{and} \Omega (\varrho ,\zeta )=(1-\theta )\Omega (\xi ,\varrho ).$$

3. Convex ($\mathit{\alpha },\mathit{\beta }$)-Generalized Contraction Mapping

In this section, we delve into the following concept:

Definition 2.

Let $(X,\Omega ,W)$ be a W-hyperbolic space, and $\Phi :X\to X$ a mapping. The mapping $\Phi$ is called convex orbital $(\alpha ,\beta )$-contraction mapping if there exists $\alpha ,\beta \in (0,1)$ such that

$$\Omega \left(\Phi \left(W(\xi ,\Phi (\xi ),\alpha ),\Phi \left(\varrho \right)\right)\right)\le (1-\alpha )\beta d(\xi ,\varrho )+\alpha \beta \Omega (\Phi \left(\xi \right),\varrho )$$

(3)

for all $\xi ,\varrho \in X$.

If we consider $\alpha =0$, then convex $(\alpha ,\beta )$-generalized contraction is contraction mapping. Thus, we take $\alpha \in (0,1).$

Theorem 1.

Let X be a complete Busemann space, and $\Phi :X\to X$ be a convex $(\alpha ,\beta )$-generalized contraction mapping. Then, $\Phi$ has a unique fixed point in $X.$

Proof. 

Let ${\xi }_0\in X$ and the following sequence be defined as follows:

$${\xi }_{n+1}=W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\mathrm{for} \mathrm{all} n\in \mathbb{N}\cup \left\{0\right\}$$

(4)

From the condition on space, we have

$$\Omega ({\xi }_n,{\xi }_{n+1})=\alpha \Omega ({\xi }_n,\Phi \left({\xi }_n\right)).$$

(5)

Using the definition of mapping $\Phi$

$$\begin{array}{ccc}\hfill \Omega (\Phi \left({\xi }_n\right),\Phi \left({\xi }_{n+1}\right))& =& \Omega (\Phi \left({\xi }_n\right),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & \le & (1-\alpha )\beta d({\xi }_n,{\xi }_n)+\alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))\hfill \\ & =& \beta \Omega ({\xi }_n,{\xi }_{n+1}).\hfill \end{array}$$

(6)

Using the Definition 1,

$$\begin{array}{ccc}\hfill \Omega ({\xi }_{n+2},{\xi }_{n+1})& =& \Omega (W({\xi }_{n+1},\Phi \left({\xi }_{n+1}\right),\alpha ),W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ))\hfill \\ & \le & (1-\alpha )\Omega ({\xi }_n,{\xi }_{n+1})+\alpha \Omega (\Phi \left({\xi }_{n+1}\right),\Phi \left({\xi }_n\right)).\hfill \end{array}$$

From (6), we have

$$\begin{array}{ccc}\hfill \Omega ({\xi }_{n+2},{\xi }_{n+1})& \le & (1-\alpha )\Omega ({\xi }_n,{\xi }_{n+1})+\alpha \beta \Omega ({\xi }_n,{\xi }_{n+1})\hfill \\ & \le & (1-\alpha +\alpha \beta )\Omega ({\xi }_n,{\xi }_{n+1}).\hfill \end{array}$$

By the successive approximation,

$$\Omega ({\xi }_{n+1},{\xi }_n)\le {(1-\alpha +\alpha \beta )}^n\Omega ({\xi }_1,{\xi }_0).$$

(7)

Take $c=(1-\alpha +\alpha \beta )<1$. Thus, let $m,n\in \mathbb{N}$ with $n<m$. From (7) and by the triangle inequality,

$$\begin{array}{cc}\hfill \Omega \left({\xi }_m,{\xi }_n\right)& \le \Omega \left({\xi }_m,{\xi }_{m-1}\right)+\Omega \left({\xi }_{m-1},{\xi }_{m-2}\right)+\cdots +\Omega \left({\xi }_{n+1},{\xi }_n\right)\hfill \\ & \le \left(c^{m-1}+c^{m-2}+\cdots +c^n\right)\Omega \left({\xi }_1,{\xi }_0\right)\hfill \\ & \le c^n\left(c^{m-n-1}+c^{m-n-2}+\cdots +1\right)\Omega \left({\xi }_1,{\xi }_0\right)\hfill \\ & \le \frac{c^n}{1-c}\Omega \left({\xi }_1,{\xi }_0\right).\hfill \end{array}$$

Since $\underset{n\to \infty }{lim}{c}^n=0$ and $\Omega \left({\xi }_1,{\xi }_0\right)$ are fixed, the sequence $\left\{{\xi }_n\right\}$ is a Cauchy sequence in X. Since X is complete, there exists ${\xi }^{†}\in X$ such that ${\xi }_n\to {\xi }^{†}$ as $n\to \infty$. We prove that ${\xi }^{†}$ is a fixed point of $\Phi$. Since ${\xi }_n\to {\xi }^{†}$, $\underset{n\to \infty }{lim}\Omega ({\xi }_{n+1},{\xi }_n)=0.$ From (5), we have

$$\underset{n\to \infty }{lim}\Omega ({\xi }_n,\Phi \left({\xi }_n\right))=0.$$

(8)

In view of the triangle inequality and the condition on mapping $\Phi$, we have

$$\begin{array}{ccc}\hfill \Omega ({\xi }_n,\Phi \left({\xi }^{†}\right))& \le & \Omega ({\xi }_n,\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right))+\Omega (\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right),\Phi \left({\xi }^{†}\right))\hfill \\ & \le & \Omega ({\xi }_n,W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ))+\Omega (W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & & +\Omega (\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right),\Phi \left({\xi }^{†}\right))\hfill \\ & =& \alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega (W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & & +\Omega (\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right),\Phi \left({\xi }^{†}\right)).\hfill \end{array}$$

Again, by the condition on the mapping $\Phi$, we obtain

$$\begin{array}{ccc}\hfill \Omega ({\xi }_n,\Phi \left({\xi }^{†}\right))& \le & \alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega (W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & & +(1-\alpha )\beta \Omega ({\xi }_n,{\xi }^{†})+\alpha \beta \Omega (\Phi \left({\xi }_n\right),{\xi }^{†})\hfill \\ & \le & \alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega ({\xi }_{n+1},\Phi \left({\xi }_{n+1}\right))\hfill \\ & & +(1-\alpha )\beta \Omega ({\xi }_n,{\xi }^{†})+\alpha \beta \Omega ({\xi }_n,{\xi }^{†})+\alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))\hfill \\ & \le & (\alpha \beta +\alpha \beta )\Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega ({\xi }_{n+1},\Phi \left({\xi }_{n+1}\right))\hfill \\ & & +\beta \Omega ({\xi }_n,{\xi }^{†}).\hfill \end{array}$$

From (8) $\underset{n\to \infty }{lim}\Omega ({\xi }_n,\Phi \left({\xi }^{†}\right))=0.$ Therefore, ${\xi }^{†}$ is a fixed point of $\Phi .$ To prove the uniqueness, let q be another fixed point of $\Phi$. Then,

$$\begin{array}{ccc}\hfill 0<\Omega ({\xi }^{†},q)& =& \Omega (\Phi (W({\xi }^{†},\Phi \left({\xi }^{†}\right),\alpha ){\xi }^{†},\Phi \left(q\right))\le (1-\alpha )\beta \Omega ({\xi }^{†},q)+\alpha \beta \Omega (\Phi \left({\xi }^{†}\right),q)\hfill \\ & =& \beta \Omega ({\xi }^{†},q)<\Omega ({\xi }^{†},q)\hfill \end{array}$$

a contradiction unless ${\xi }^{†}=q$. Therefore, $\Phi$ has a unique fixed point. □

Corollary 1.

Let X be a Banach space, and $\Phi :X\to X$ be a convex $(\alpha ,\beta )$-generalized contraction mapping. Then, $\Phi$ has a unique fixed point in $X.$

Example 1.

Let $X=[0,4]\subset \mathbb{R}$ with the usual norm. Define $\Psi :X\to X$ by

$$\Psi \left(\xi \right)=\left\{\begin{array}{cc}0,\hfill & if\xi \ne 4\hfill \\ 1,\hfill & if\xi =4.\hfill \end{array}\right.$$

First, we show that $\Psi$ is convex $(\alpha ,\beta )$-generalized contraction mapping for $\alpha ={\displaystyle \frac{1}{2}}$ and $\beta ={\displaystyle \frac{2}{3}}$. We consider the following cases:

 (1) 

If $\xi \ne 4$ and $\varrho \ne 4$, then the condition is trivially satisfied.

 (2) 

If $\xi \ne 4$, and $\varrho =4$, then

$$\begin{array}{ccc}\hfill \parallel \Psi \left((1-\alpha )\xi +\alpha \Psi \left(\xi \right)\right)-\Psi \left(\varrho \right)\parallel & =& ∥\Psi \left(\left(1-\frac{1}{2}\right)\times \xi +\frac{1}{2}\Psi \left(\xi \right)\right)-\Psi \left(4\right)∥\hfill \\ & =& ∥\Psi \left(\frac{\xi }{2}\right)-1∥=1<\frac{1}{3}\parallel 4-\xi \parallel +\frac{4}{3}\hfill \\ & =& (1-\alpha )\beta \parallel 4-\xi \parallel +4\alpha \beta \hfill \\ & =& (1-\alpha )\beta \parallel \xi -\varrho \parallel +\alpha \beta \parallel \Psi \left(\xi \right)-\varrho \parallel .\hfill \end{array}$$

 (3) 

If $\xi =4$, and $\varrho \ne 4$, then

$$\begin{array}{ccc}\hfill \parallel \Psi \left((1-\alpha )\xi +\alpha \Psi \left(\xi \right)\right)-\Psi \left(\varrho \right)\parallel & =& ∥\Psi \left(\left(1-\frac{1}{2}\right)\times 4+\frac{1}{2}\Psi \left(4\right)\right)-\Psi \left(\varrho \right)∥\hfill \\ & =& ∥\Psi \left(\frac{5}{2}\right)∥=0<(1-\alpha )\beta \parallel \xi -\varrho \parallel +\alpha \beta \parallel \Psi \left(\xi \right)-\varrho \parallel .\hfill \end{array}$$

 (4) 

If $\xi =4$, and $\varrho =4$, then

$$\begin{array}{ccc}\hfill \parallel \Psi \left((1-\alpha )\xi +\alpha \Psi \left(\xi \right)\right)-\Psi \left(\varrho \right)\parallel & =& ∥\Psi \left(\left(1-\frac{1}{2}\right)\times 4+\frac{1}{2}\Psi \left(4\right)\right)-\Psi \left(4\right)∥\hfill \\ & =& ∥\Psi \left(\frac{5}{2}\right)-1∥=1\hfill \\ & =& (1-\alpha )\beta \parallel \xi -\varrho \parallel +\alpha \beta \parallel \Psi \left(\xi \right)-\varrho \parallel .\hfill \end{array}$$

On the other hand, $\Psi$ is not continuous mapping, thus $\Psi$ is not contraction mapping (even nonexpansive mapping).

Example 2

([20], Example 1.3.1). Suppose $X=\mathbb{R}$ and $\Phi :X\to X$ is defined by

$$\Psi \left(\xi \right)=\left\{\begin{array}{cc}0,\hfill & if\xi \in (-\infty ,2]\hfill \\ -\frac{1}{3},\hfill & if\xi \in (2,+\infty ).\hfill \end{array}\right.$$

Although mapping $\Phi$ is discontinuous and does not satisfy the contraction mapping property, mapping ${\Phi }^2$ can be considered a contraction mapping.

If the Picard–Banach contraction mapping principle cannot be used, we may find the following fixed-point theorem to be a useful option. This theorem is discussed in many places.

Theorem 2

([20], Theorem 1.3.2). Let X be a complete metric space and $\Phi :X\to X$ a mapping. If $\exists a N\in \mathbb{N}$ such that ${\Phi }^N$ is a contraction, then $F(\Phi )=\left\{{\xi }^{*}\right\}$.

Theorem 3.

Let X be a complete Busemann space, $G:X\to X$ be a mapping and $\exists a N\in \mathbb{N}$ such that $G^N$ is a convex ($\alpha ,\beta$)-generalized contraction mapping. For given ${\xi }_0\in X$, define a sequence

$${\xi }_{n+1}=W({\xi }_n,G^N\left({\xi }_n\right),\alpha ),n\ge 0.$$

(9)

Then, the sequence $\left\{{\xi }_n\right\}$ converges to a unique fixed point of G.

Proof. 

Suppose that $\Phi =G^N$, then the sequence (9) becomes

$${\xi }_{n+1}=W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ).$$

By applying Theorem 1, we can conclude that the sequence $\left\{{\xi }_n\right\}$ converges to a unique fixed point of $G^N$, say p, and $F(\Phi )=F\left(G^N\right)=\left\{p\right\}$. We obtain

$$G^N\left(G\left(p\right)\right)=G^{N+1}\left(p\right)=G\left(G^N\left(p\right)\right)=G\left(p\right),$$

Hence, $G\left(p\right)$ is a fixed point of $G^N$. But $F\left(G^N\right)=\left\{p\right\}$; therefore, $G\left(p\right)=p$. □

4. Monotone Convex ($\mathit{\alpha },\mathit{\beta }$)-Generalized Contraction Mapping

In this section, we take this idea further and extend the concept of convex $(\alpha ,\beta )$-generalized contraction to partially ordered Busemann spaces.

Let X be a partially ordered Busemann space. A subset K of X is said to be convex if $[\xi ,\varrho ]\subset K$ whenever $\xi ,\varrho \in K.$ In this section, we denote the order intervals in X by

$$[\xi ,\to ):=\{z\in X:\xi ⪯z\} \mathrm{and} (\to ,\varrho ]:=\{z\in X:z⪯\varrho \},$$

we also assume the following hypothesis in the framework of partially ordered Busemann spaces

$$\left(H\right):\mathrm{For} \mathrm{any} \xi \in X, \mathrm{the} \mathrm{order} \mathrm{interval} [\xi ,\to )\mathrm{is} \mathrm{a} \mathrm{closed} \mathrm{and} \mathrm{convex} \mathrm{subset}.$$

Definition 3.

Let X be the same as above and K a convex subset of X. A mapping $\Phi :K\to K$ is said to be monotone if

$$\xi ⪯\varrho \mathrm{implies} \Phi \left(\xi \right)⪯\Phi \left(\varrho \right),\forall \xi ,\varrho \in K.$$

Definition 4.

Let X and K be the same as in Definition 3. The mapping $\Phi :K\to K$ is called monotone convex $(\alpha ,\beta )$-generalized contraction mapping if $\Phi$ is monotone and there exist $\alpha ,\beta \in (0,1)$ such that

$$\Omega \left(\Phi \left(W(\xi ,\Phi (\xi ),\alpha ),\Phi \left(\varrho \right)\right)\right)\le (1-\alpha )\beta d(x,\varrho )+\alpha \beta \Omega (\Phi \left(\xi \right),\varrho )$$

(10)

for all $\xi ,\varrho \in K$ with $\xi ⪯\varrho .$

Theorem 4.

Let X be a complete Busemann space, K a convex closed subset of X, and $\Phi :K\to K$ a monotone convex $(\alpha ,\beta )$-generalized contraction mapping. Suppose that ${\xi }_0\in K$ such that ${\xi }_0⪯\Phi \left({\xi }_0\right).$ Then, $\Phi$ has a fixed point in $K.$

Proof. 

Let ${\xi }_0\in X$ and the following sequence is defined as follows:

$${\xi }_{n+1}=W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\mathrm{for} \mathrm{all} n\in \mathbb{N}\cup \left\{0\right\}$$

(11)

from the condition on space,

$$\Omega ({\xi }_n,{\xi }_{n+1})=\alpha \Omega ({\xi }_n,\Phi \left({\xi }_n\right)).$$

(12)

We prove the following:

$${\xi }_n⪯{\xi }_{n+1}⪯\Phi \left({\xi }_n\right)\mathrm{for} \mathrm{all} n\in \mathbb{N}\cup \left\{0\right\}.$$

(13)

We will prove the above claim by induction. Since ${\xi }_0⪯\Phi \left({\xi }_0\right)$, in view of the convexity of order interval,

$${\xi }_0⪯{\xi }_1⪯\Phi \left({\xi }_0\right).$$

Since $\Phi$ is monotone, $\Phi \left({\xi }_0\right)⪯\Phi \left({\xi }_1\right)$ and

$${\xi }_1⪯\Phi \left({\xi }_0\right)⪯\Phi \left({\xi }_1\right).$$

in view the convexity of order interval

$${\xi }_1⪯{\xi }_2⪯\Phi \left({\xi }_1\right).$$

Hence, (13) is true for $n=1.$ Assume it is true for a fixed $k\in \mathbb{N}$, that is, ${\xi }_k⪯{\xi }_{k+1}⪯\Phi \left({\xi }_k\right)$. Again, $\Phi$ is monotone, $\Phi \left({\xi }_k\right)⪯\Phi \left({\xi }_{k+1}\right)$ and using convexity

$${\xi }_{k+1}⪯{\xi }_{k+2}⪯\Phi \left({\xi }_{k+1}\right).$$

This verifies the claim. Based on the definition of mapping $\Phi$

$$\begin{array}{ccc}\hfill \Omega (\Phi \left({\xi }_n\right),\Phi \left({\xi }_{n+1}\right))& =& \Omega (\Phi \left({\xi }_n\right),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right))\hfill \\ & \le & (1-\alpha )\beta \Omega ({\xi }_n,{\xi }_n)+\alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))\hfill \\ & =& \beta \Omega ({\xi }_n,{\xi }_{n+1}).\hfill \end{array}$$

(14)

In view of the Definition 1,

$$\begin{array}{ccc}\hfill \Omega ({\xi }_{n+2},{\xi }_{n+1})& =& \Omega (W({\xi }_{n+1},\Phi \left({\xi }_{n+1}\right),\alpha ),W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ))\hfill \\ & \le & (1-\alpha )\Omega ({\xi }_n,{\xi }_{n+1})+\alpha \Omega (\Phi \left({\xi }_{n+1}\right),\Phi \left({\xi }_n\right)).\hfill \end{array}$$

By (14), we have

$$\begin{array}{ccc}\hfill \Omega ({\xi }_{n+2},{\xi }_{n+1})& \le & (1-\alpha )\Omega ({\xi }_n,{\xi }_{n+1})+\alpha \beta \Omega ({\xi }_n,{\xi }_{n+1})\hfill \\ & \le & (1-\alpha +\alpha \beta )\Omega ({\xi }_n,{\xi }_{n+1}).\hfill \end{array}$$

By the successive approximation,

$$\Omega ({\xi }_{n+1},{\xi }_n)\le {(1-\alpha +\alpha \beta )}^n\Omega ({\xi }_1,{\xi }_0).$$

(15)

Suppose $c=(1-\alpha +\alpha \beta )<1$. From the proof of Theorem 1, one can show that $\left\{{\xi }_n\right\}$ is a Cauchy sequence. Since K is closed, the sequence ${\xi }_n\to {\xi }^{†}\in K$ as $n\to \infty$ with ${\xi }_n⪯{\xi }^{†}$∀$n\in \mathbb{N}$ and $\underset{n\to \infty }{lim}\Omega ({\xi }_{n+1},{\xi }_n)=0.$ From (13), we obtain

$$\underset{n\to \infty }{lim}\Omega ({\xi }_n,\Phi \left({\xi }_n\right))=0.$$

(16)

In view of the triangle inequality and the condition on mapping $\Phi$, we have

$$\begin{array}{ccc}\hfill \Omega ({\xi }_n,\Phi \left({\xi }^{†}\right))& \le & \Omega ({\xi }_n,\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right))+\Omega (\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right),\Phi \left({\xi }^{†}\right))\hfill \\ & \le & \Omega ({\xi }_n,W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ))+\Omega (W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & & +\Omega (\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right),\Phi \left({\xi }^{†}\right))\hfill \\ & =& \alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega (W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & & +\Omega (\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right),\Phi \left({\xi }^{†}\right)).\hfill \end{array}$$

Again, by the condition on the mapping $\Phi$, we obtain

$$\begin{array}{ccc}\hfill \Omega ({\xi }_n,\Phi \left({\xi }^{†}\right))& \le & \alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega (W({\xi }_n,\Phi \left({\xi }_n\right),\alpha ),\Phi \left(W({\xi }_n,\Phi \left({\xi }_n\right),\alpha )\right)\hfill \\ & & +(1-\alpha )\beta \Omega ({\xi }_n,{\xi }^{†})+\alpha \beta \Omega (\Phi \left({\xi }_n\right),{\xi }^{†})\hfill \\ & \le & \alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega ({\xi }_{n+1},\Phi \left({\xi }_{n+1}\right))\hfill \\ & & +(1-\alpha )\beta \Omega ({\xi }_n,{\xi }^{†})+\alpha \beta \Omega ({\xi }_n,{\xi }^{†})+\alpha \beta \Omega ({\xi }_n,\Phi \left({\xi }_n\right))\hfill \\ & \le & (\alpha \beta +\alpha \beta )\Omega ({\xi }_n,\Phi \left({\xi }_n\right))+\Omega ({\xi }_{n+1},\Phi \left({\xi }_{n+1}\right))\hfill \\ & & +\beta \Omega ({\xi }_n,{\xi }^{†}).\hfill \end{array}$$

From (16) $\underset{n\to \infty }{lim}\Omega ({\xi }_n,\Phi \left({\xi }^{†}\right))=0.$ Therefore, ${\xi }^{†}$ is a fixed point of $\Phi .$ □

Theorem 5.

Let X, K, and $\Phi$ be the same as in Theorem 4. Then, the fixed point of $\Phi$ is unique, if

$$\mathrm{for} \mathrm{all} \xi ,\varrho \in K,\exists w\in K \mathrm{such} \mathrm{that} w⪯\Phi \left(w\right),\xi ⪯w \mathrm{and} \varrho ⪯w.$$

Proof. 

Given two fixed points u and v of $\Phi$ such that $u\ne v,$ in light of the assumption, there exists $w\in K$ such that $u⪯w$ and $v⪯w$. Let $w_0=w\in K$, and consider the following sequence

$$w_{n+1}=W(w_n,\Phi \left(w_n\right),\alpha )\mathrm{for} \mathrm{all} n\in \mathbb{N}\cup \left\{0\right\}$$

(17)

and

$$\Omega (w_n,w_{n+1})=\alpha \Omega (w_n,\Phi \left(w_n\right)).$$

(18)

By largely following the proof of Theorem 4, it can be shown that

$$\underset{n\to \infty }{lim}\Omega (w_n,\Phi \left(w_n\right))=0.$$

(19)

Since $v⪯w=w_0$ and $w_0⪯\Phi \left(w_0\right)$, it can be seen that $v⪯w_n$ and $u⪯w_n$∀$n\in \mathbb{N}$.

Case 1. If $v=w_{n_0}$ for some $n_0\ge 0,$ then $v=\Phi \left(v\right)=\Phi \left(w_{n_0}\right)$ and

$$\begin{array}{ccc}\hfill w_{n_0+1}& =& (1-\alpha )w_{n_0}+\alpha \Phi \left(w_{n_0}\right)\hfill \\ & =& (1-\alpha )v+\alpha v=v.\hfill \end{array}$$

Therefore, $w_n=v$∀$n\ge n_0.$

Case 2. If $v⪯w_n$ and $v\ne w_n$∀$n\ge 0,$ then

$$\begin{array}{ccc}\hfill \Omega (v,w_n)& =& \Omega (W(v,\Phi \left(v\right),\alpha ),w_n)\le \Omega (W(v,\Phi \left(v\right),\alpha ),\Phi \left(w_n\right))+\Omega (w_n,\Phi \left(w_n\right))\hfill \\ & \le & (1-\alpha )\beta \Omega (v,w_n)+\alpha \beta \Omega (\Phi \left(v\right),w_n)+\Omega (w_n,\Phi \left(w_n\right))\hfill \\ & =& \beta \Omega (v,w_n)+\Omega (w_n,\Phi \left(w_n\right)).\hfill \end{array}$$

Thus, from (19)

$$(1-\beta )\Omega (v,w_n)\le \Omega (w_n,\Phi \left(w_n\right))$$

and $\Omega (v,\Phi \left(w_n\right))\to 0$ as $n\to \infty .$ Therefore, the sequence $\left\{w_n\right\}$ converges to v. Similarly, $\left\{w_n\right\}$ converges to u, by the uniqueness of the limit, we have $u=v.$ Hence, $\Phi$ admits a unique fixed point. □

Corollary 2.

Let X be a Banach space and K a closed convex subset of X. Let $\Phi :K\to K$ be a monotone convex $(\alpha ,\beta )$-generalized contraction mapping. Suppose that ${\xi }_0\in K$ such that ${\xi }_0⪯\Phi \left({\xi }_0\right).$ Then, $\Phi$ has a fixed point in $X.$ If

$$\mathrm{for} \mathrm{all} \xi ,\varrho \in K,\exists w\in K \mathrm{such} \mathrm{that} w⪯\Phi \left(w\right),\xi ⪯w \mathrm{and} \varrho ⪯w.$$

Then, the fixed point of $\Phi$ is unique.

5. Solutions to Linear Matrix Equation

In the following section, we are primarily interested in the positive definite solution of matrix equation:

$$U-{\Gamma }_1^{*}U{\Gamma }_1-\cdots -{\Gamma }_{m}U{\Gamma }_m=Q$$

(20)

where ${\Gamma }_1,\cdots ,{\Gamma }_m$ are arbitrary matrices of order $n\times n$ and Q is a matrix of order $n\times n$. We can define a mapping G on the set $\mathcal{H}\left(n\right)$, which comprises all Hermitian matrices of size $n\times n$, as follows:

$$G\left(U\right)=Q+\sum _{j=1}^m{\Gamma }_m^{*}U{\Gamma }_m$$

(21)

The fixed points of G correspond to the solutions of Equation (20). Let $\Gamma$ belong to the set $\mathcal{M}\left(n\right)$, which comprises all matrices of size $n\times n$. In this context, the norm of $\Gamma$, denoted as ${\parallel \Gamma \parallel }_1$, can be expressed as the sum of its singular values $s_j(\Gamma )$ for $j=1,2,\cdots ,n$, that is, ${\parallel \Gamma \parallel }_1={\displaystyle \sum _{j=1}^n}{s}_j(\Gamma ).$ Now, consider a given matrix $Q_{+}$ from the set $\mathcal{P}\left(n\right)$, which consists of all positive definite matrices of size $n\times n$. We can define the following norm:

$${\parallel \Gamma \parallel }_{1,Q_{+}}={\parallel Q_{+}^{\frac{1}{2}}\Gamma Q_{+}^{\frac{1}{2}}\parallel }_1.$$

It can be observed that, for any $Q_{+}\in \mathcal{P}\left(n\right)$, the set $\mathcal{H}\left(n\right)$, equipped with the norm ${\parallel .\parallel }_{1,Q_{+}}$, forms a Banach space. Additionally, $\mathcal{H}\left(n\right)$ possesses a partial order. To indicate this partial order, we use the notation $U\ge V$ (or $U>V$) when $U-V\ge 0$ (or $U-V>0$), respectively. Furthermore, we define the spectral norm, denoted by $\parallel .\parallel$, as follows: $\parallel \Gamma \parallel =\sqrt{{\lambda }_{+}({\Gamma }^{*}\Gamma )}$, where ${\lambda }_{+}({\Gamma }^{*}\Gamma )$ represents the largest eigenvalue of the matrix ${\Gamma }^{*}\Gamma$. Also, let $I_n$ represent the identity matrix of size $n\times n$.

Lemma 1

([2]). Consider matrices Γ and B, both of size $n\times n$, such that $\Gamma ,B\ge 0$. Then

$$tr(\Gamma B)\le \parallel \Gamma \parallel tr\left(B\right).$$

Theorem 6.

Let $Q\in \mathcal{P}\left(n\right)$, additionally, there exists a matrix $Q_{+}\in \mathcal{P}\left(n\right)$ such that

$$∥\sum _{j=1}^m{Q}_{+}^{\frac{1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{1}{2}}∥<1$$

Then,

 (1) 

The mapping G has a unique fixed point in $\mathcal{H}\left(n\right).$

 (2) 

Given an initial matrix $U_1\in \mathcal{H}\left(n\right)$, the sequence $U_k$ is defined as follows:

$$U_{k+1}=\frac{1}{2}{U}_k+\frac{1}{2}G\left(U_k\right)\forall k\in \mathbb{N}$$

converges to the solution of (20), which lies within $\mathcal{P}\left(n\right)$.

Proof. 

It can be observed that, for all $U,V\in \mathcal{H}\left(n\right)$, there exists either a lower bound or an upper bound. When considering the case where $U_1=0$, we find that $G\left(0\right)=L\left(0\right)=Q>0$. Therefore, we have $U_1\le G\left(U_1\right)$ and $U_1\le L\left(U_1\right)$. Now, we shall demonstrate that the mapping G satisfies condition (10). For any matrix $U,V\in \mathcal{H}\left(n\right)$ with $U\le V$, it follows that $G\left(U\right)\le G\left(V\right)$. Therefore,

$$\begin{array}{ccc}& & {∥G\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-G\left(V\right)∥}_{1,Q_{+}}={∥\sum _{j=1}^m{\Gamma }_j\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)-V\right){\Gamma }_j^{*}∥}_{1,Q_{+}}\hfill \\ & =& tr\left\{\sum _{j=1}^m{Q}_{+}^{\frac{1}{2}}{\Gamma }_j^{*}\left[\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)\right]{\Gamma }_j{Q}_{+}^{\frac{1}{2}}\right\}\hfill \\ & =& \sum _{j=1}^{m}tr\left\{Q_{+}^{\frac{1}{2}}{\Gamma }_j^{*}\left[\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)\right]{\Gamma }_j{Q}_{+}^{\frac{1}{2}}\right\}\hfill \\ & =& \sum _{j=1}^{m}tr\left\{{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}\left[\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)\right]\right\}\hfill \\ & =& \sum _{j=1}^{m}tr\left\{{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}{Q}_{+}^{\frac{1}{2}}\left[\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)\right]Q_{+}^{\frac{1}{2}}{Q}_{+}^{\frac{-1}{2}}\right\}\hfill \\ & =& \sum _{j=1}^{m}tr\left\{Q_{+}^{\frac{-1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}{Q}_{+}^{\frac{1}{2}}\left[\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)\right]Q_{+}^{\frac{1}{2}}\right\}\hfill \\ & =& tr\left\{\sum _{j=1}^m\left(Q_{+}^{\frac{-1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}\right)\left(Q_{+}^{\frac{1}{2}}\left[\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)\right]Q_{+}^{\frac{1}{2}}\right)\right\}.\hfill \end{array}$$

Thus, from Lemma 1, we have

$$\begin{array}{ccc}\hfill {∥G\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-G\left(V\right)∥}_{1,Q_{+}}& \le & ∥\sum _{j=1}^m{Q}_{+}^{\frac{-1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}∥\hfill \\ & & {∥\frac{1}{2}(U-V)+\frac{1}{2}\left(Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V\right)∥}_{1,Q_{+}}\hfill \\ & \le & \frac{1}{2}∥\sum _{j=1}^m{Q}_{+}^{\frac{-1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}∥{\parallel U-V\parallel }_{1,Q_{+}}\hfill \\ & +& \frac{1}{2}∥\sum _{j=1}^m{Q}_{+}^{\frac{-1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}∥{∥Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V∥}_{1,Q_{+}}\hfill \end{array}$$

Take $\beta =∥{\displaystyle \sum _{j=1}^m}{Q}_{+}^{\frac{-1}{2}}{\Gamma }_j{Q}_{+}{\Gamma }_j^{*}{Q}_{+}^{\frac{-1}{2}}∥<1$. Thus

$${∥G\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-G\left(V\right)∥}_{1,Q_{+}}\le \frac{1}{2}\beta {\parallel U-V\parallel }_{1,Q_{+}}+\frac{1}{2}\beta {∥Q+\sum _{i=1}^m{\Gamma }_i^{*}U{\Gamma }_i-V∥}_{1,Q_{+}}$$

and

$${∥G\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-G\left(V\right)∥}_{1,Q_{+}}\le \frac{1}{2}\beta {\parallel U-V\parallel }_{1,Q_{+}}+\frac{1}{2}\beta {∥G\left(U\right)-V∥}_{1,Q_{+}}$$

Therefore, the mapping G possesses a unique fixed point, and the sequence $\left\{U_k\right\}$ converges to the solution of (20). It is apparent that G maps $\mathcal{P}\left(n\right)$ into the set $\{U\in \mathcal{H}(n)\mid Q\le U\}$, thereby indicating that the solution lies within this set and is positive definite. □

6. Solutions to Nonlinear Matrix Equations

In this section, our focus is on investigating the following nonlinear matrix equations, given as follows:

$$U=Q\pm \sum _{j=1}^m{\Gamma }_j^{*}F\left(U\right){\Gamma }_j$$

(22)

Here, $F:\mathcal{P}\left(n\right)\to \mathcal{P}\left(n\right)$ represents a continuous mapping. For a comprehensive understanding of these equation classes, refer to [21]. We explore various scenarios based on different conditions on the mapping F, which are discussed as follows:

Case 1. Assuming that F is order-preserving, we examine the following equation:

$$U=Q+\sum _{j=1}^m{\Gamma }_j^{*}F\left(U\right){\Gamma }_j.$$

(23)

To simplify our analysis, we define the mapping $G\left(U\right)$ as:

$$G\left(U\right)=Q+\sum _{j=1}^m{\Gamma }_j^{*}F\left(U\right){\Gamma }_j.$$

The mapping G is well-defined on $\mathcal{P}\left(n\right)$ and preserves the order of matrices. For any matrix $U\in \mathcal{P}\left(n\right)$, it holds that $Q\le G\left(U\right)$. Particularly, $Q\le G\left(Q\right)$ since Q is positive definite. Since G is order-preserving, we have the following inequality chain:

$$Q\le G\left(Q\right)\le G^2\left(Q\right)\le G^3\left(Q\right)\le \cdots$$

Consequently, the sequence $\left\{G^j\left(Q\right)\right\}$ is increasing.

Proposition 1.

Assume the existence of a matrix $U_0$ satisfying $G\left(U_0\right)\le U_0$. In this case, the mapping G preserves the set $\{U:Q\le U\le U_0\}$ within itself. The sequence $\left\{G^j\left(Q\right)\right\}$ gradually approaches a certain point denoted as $U_{-}$, which represents the smallest solution of Equation (23). Moreover, the sequence $\left\{G^j\left(U_0\right)\right\}$ exhibits a decreasing pattern, constituting the largest solution within the interval $[Q,U_0]$.

The subsequent theorem guarantees the uniqueness of the solution to Equation (23).

Theorem 7.

Suppose there is a positive number M for which ${\displaystyle \sum _{j=1}^m}{\Gamma }_j{\Gamma }_j^{*}<M{I}_n.$ Assume that for all $U,V\in \mathcal{H}\left(n\right)$ with $U\le V$, we have

$$\begin{array}{ccc}\hfill \left|tr\left\{F\left(\frac{1}{2}U+\frac{1}{2}Q+\frac{1}{2}\sum _{i=1}^m{\Gamma }_i^{*}F\left(U\right){\Gamma }_i\right)-F\left(V\right)\right\}\right|& \le & \frac{1}{2M}|tr(U-V)|\hfill \\ & +& \frac{1}{2M}\left|tr\left\{Q+\sum _{i=1}^m{\Gamma }_i^{*}F\left(U\right){\Gamma }_i-F\left(V\right)\right\}\right|\hfill \end{array}$$

Then, (23) has a unique solution which is positive definite. Moreover, for a given $U_1\in \mathcal{H}\left(n\right)$ with $U_1\le Q+{\displaystyle \sum _{i=1}^m}{\Gamma }_i^{*}F\left(U_1\right){\Gamma }_i$ the sequence $\left\{U_k\right\}$ defined as

$$U_{k+1}=\frac{1}{2}{U}_k+\frac{1}{2}G\left(U_k\right)\mathrm{for} \mathrm{all} k\in \mathbb{N}$$

converges (in sense of norm ${\parallel .\parallel }_1$) to the solution of (23).

Proof. 

Let $U,V\in \mathcal{H}\left(n\right)$ with $U\le V$

$$\begin{array}{ccc}\hfill \parallel G\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right){-G\left(V\right)\parallel }_1& =& tr\left\{\sum _{j=1}^m{\Gamma }_j^{*}\left(F\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-F\left(V\right)\right){\Gamma }_j\right\}\hfill \\ & =& \sum _{j=1}^{m}tr\left\{{\Gamma }_j^{*}\left(F\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-F\left(V\right)\right){\Gamma }_j\right\}\hfill \\ & =& \sum _{j=1}^{m}tr\left\{{\Gamma }_j{\Gamma }_j^{*}\left(F\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-F\left(V\right)\right)\right\}\hfill \\ & =& tr\left\{\sum _{j=1}^m{\Gamma }_j{\Gamma }_j^{*}\left(F\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-F\left(V\right)\right)\right\}\hfill \\ & =& tr\left\{\left(\sum _{j=1}^m{\Gamma }_j{\Gamma }_j^{*}\right)\left(F\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-F\left(V\right)\right)\right\}\hfill \\ & \le & ∥\sum _{j=1}^m{\Gamma }_j{\Gamma }_j^{*}∥tr\left\{F\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right)-F\left(V\right)\right\}\hfill \\ & \le & ∥\sum _{j=1}^m{\Gamma }_j{\Gamma }_j^{*}∥tr\left\{F\left(\frac{1}{2}U+\frac{1}{2}Q+\frac{1}{2}\sum _{i=1}^m{\Gamma }_i^{*}F\left(U\right){\Gamma }_i\right)-F\left(V\right)\right\}\hfill \end{array}$$

From the assumptions in the theorem,

$$\begin{array}{cc}\hfill \parallel G\left(\frac{1}{2}U+\frac{1}{2}G\left(U\right)\right){-G\left(V\right)\parallel }_1& \le M\left(\frac{1}{2M}|tr(U-v)|+\frac{1}{2M}|tr\{Q+\sum _{i=1}^m{\Gamma }_i^{*}F\left(U\right){\Gamma }_i-F\left(V\right)\}|\right).\end{array}$$

Thus, all the assumptions of Corollary 2 are satisfied. Hence, (23) has a unique solution which is positive definite. □

Case 2. Consider the following equation

$$U=Q-\sum _{j=1}^m{\Gamma }_j^{*}F\left(U\right){\Gamma }_j.$$

(24)

We can define

$$G\left(U\right)=Q-\sum _{j=1}^m{\Gamma }_j^{*}F\left(U\right){\Gamma }_j.$$

If we assume that F is order-reversing in (24), then the mapping G becomes order-preserving. Furthermore, we consider the existence of a matrix $U_0$ that satisfies $U_0\le Q$ and $U_0\le G\left(U_0\right)$. Then

$$U_0\le G\left(U_0\right)\le G\left(Q\right)\le G\left(Q\right).$$

It is evident that the interval $[U_0,Q]$ is mapped onto itself.

Proposition 2.

Assume the existence of a matrix $U_0$ such that $U_0\le G\left(U_0\right)$. In this case, the mapping G preserves the set $\{U:U_0\le U\le Q\}$ within itself. The sequence $\left\{G^j\left(Q\right)\right\}$ approaches a certain point, denoted as $U_{+}$, which represents the largest solution of Equation (24). Moreover, the sequence $\left\{G^j\left(U_0\right)\right\}$ is increasing and converges to a point $U_{-}$, which represents the smallest solution within the interval $[U_0,Q]$.

Theorem 8.

Let $Q\in \mathcal{P}\left(n\right)$ and M be the same as in Theorem 7. Assume the existence of a matrix $U_0$ such that $U_0\le Q$ and $U_0\le G\left(U_0\right)$. Furthermore, suppose that for any matrix U and V satisfying $U_0\le U\le V\le Q$, we have the following:

$$\begin{array}{ccc}\hfill \left|tr\left\{F\left(\frac{1}{2}U+\frac{1}{2}Q+\frac{1}{2}\sum _{i=1}^m{\Gamma }_i^{*}F\left(U\right){\Gamma }_i\right)-F\left(V\right)\right\}\right|& \le & \frac{1}{2M}|tr(U-V)|\hfill \\ & +& \frac{1}{2M}\left|tr\left\{Q+\sum _{i=1}^m{\Gamma }_i^{*}F\left(U\right){\Gamma }_i-F\left(V\right)\right\}\right|\hfill \end{array}$$

Then, (24) has a unique solution which is positive definite. Moreover, for given $U_1\in \mathcal{H}\left(n\right)$ with $U_1\le Q-{\displaystyle \sum _{i=1}^m}{\Gamma }_i^{*}F\left(U_0\right){\Gamma }_i$, the sequence $\left\{U_k\right\}$ defined as

$$U_{k+1}=\frac{1}{2}{U}_k+\frac{1}{2}G\left(U_k\right) \mathrm{for} \mathrm{all} k\in \mathbb{N}$$

converges (in sense of norm ${\parallel .\parallel }_1$) to the solution of (24).