<!DOCTYPE html>
<html lang="en">
<head>
<script>
  MathJax = { 
    tex: {
		    inlineMath: [['\\(','\\)']]
	} }
</script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<meta name="generator" content="plasTeX" />
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>On unique solvability <br />of the piecewise linear equation system: On unique solvability <br />of the piecewise linear equation system</title>
<link rel="stylesheet" href="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/styles/theme-white.css" />
</head>

<body>

<div class="wrapper">

<div class="content">
<div class="content-wrapper">


<div class="main-text">



<div class="titlepage">
<h1>On unique solvability <br />of the piecewise linear equation system</h1>
<p class="authors">
<span class="author">Shubham Kumar\(^\ast \) Deepmala\(^{\ast \ast }\)</span>
</p>
<p class="date">August 23, 2022; accepted: November 9, 2022; published online: December 31, 2022.</p>
</div>
<div class="abstract"><p> In this article, we take the piecewise linear equation system \(x-W|x|=b\), which is also known as the absolute value equation, where W \(\in \) \({\mathbb R}^{n\times n}\), b \(\in \) \({\mathbb R}^{n}\) are given and to determine the value of \(x\) \(\in \) \({\mathbb R}^{n}\). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future. </p>
<p><b class="bf">MSC.</b> 90C05, 90C30, 15A18. </p>
<p><b class="bf">Keywords.</b> Unique solvability, Absolute value equation, Linear complementarity problem. </p>
</div>
<p>\(^\ast \)This research work of the first author is supported by the Ministry of Education (Government of India) through GATE fellowship registration No. MA19S43033021. </p>
<p>\(^\ast \)Mathematics Discipline, PDPM-Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, M.P., India, e-mail: <span class="tt">shub.srma@gmail.com</span> </p>
<p>\(^{\ast \ast }\)Mathematics Discipline, PDPM-Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, M.P., India, e-mail: <span class="tt">dmrai23@gmail.com</span> </p>
<h1 id="a0000000002">1 Introduction</h1>
<p> Here we study the AVE (absolute value equation) of the type </p>
<div class="equation" id="Equ1">
<p>
  <div class="equation_content">
    \begin{equation} \label{Equ1} x-W|x| = b, \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where \(W \in {\mathbb R}^{n \times n}\), \(b\in {\mathbb R}^{n}\) are known, and \(x \in {\mathbb R}^{n}\) is unknown; \(|\cdot |\) denotes the absolute value operator. For a matrix \(W\in {\mathbb R}^{n \times n}\) and a vector \(x \in {\mathbb R}^{n}\), \(\vert W \vert \) and \(\vert x \vert \) denote the component-wise absolute value of the matrix and the vector, respectively. </p>
<p>The generalized absolute value equation (GAVE) is the standard form of (<a href="#Equ1">1</a>) defined as </p>
<div class="equation" id="Equ3i">
<p>
  <div class="equation_content">
    \begin{equation} \label{Equ3i} Vx-W|x| = b, \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> where \(V, W\in {\mathbb R}^{n \times n}\), \(b\in {\mathbb R}^{n}\) are given while \(x\in {\mathbb R}^{n}\) is to be determined. </p>
<p>In the literature, many authors discussed its unique solvability (see <span class="cite">
	[
	<a href="#M Achache 2019" >1</a>
	, 
	<a href="#Jiri Rohn 2009i" >16</a>
	, 
	<a href="#Jiri Rohn 2014i" >18</a>
	, 
	<a href="#Shi-Liang Wu 2019" >22</a>
	, 
	<a href="#Shi Liang Wu 2021" >23</a>
	]
</span> and the references therein) and its particular case, when \(W=I\), i.e., </p>
<div class="equation" id="Equ2">
<p>
  <div class="equation_content">
    \begin{equation} \label{Equ2} Vx-|x| = b, \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> is discussed in (<span class="cite">
	[
	<a href="#Sheng Long Hu 2010" >7</a>
	, 
	<a href="#Mangasarian 2006" >8</a>
	, 
	<a href="#O.L. Mangasarian 2017" >12</a>
	, 
	<a href="#Shi Liang Wu 2016" >20</a>
	, 
	<a href="#Shi Liang Wu 2018" >21</a>
	]
</span> and the references therein). </p>
<p>To the best of our knowledge, another particular case of Eq.(<a href="#Equ3i">2</a>) as \(V=I\) have not been discussed for the sufficient condition of unique solvability aspects in the literature in a strong way to date. In 2014, Jiri Rohn <span class="cite">
	[
	<a href="#Jiri Rohn 2014ii" >19</a>
	]
</span> took the particular case of (<a href="#Equ1">1</a>) as \(W\) \(\ge \) 0 and discussed the unique solution condition with restrictions on \(W\) and \(b\). In <span class="cite">
	[
	<a href="#Milan Hladik 2018" >6</a>
	]
</span>, authors provided some conditions for the bounding solution of AVE (<a href="#Equ1">1</a>). The study of the AVE (<a href="#Equ1">1</a>) is challenging and interesting, as it contains non-linear and non-differential terms \(W|x|\). Currently, many methods to solve AVE available in the literature, like the linear programming method <span class="cite">
	[
	<a href="#Mangasarian 2014" >11</a>
	]
</span>, globally and quadratically convergent method <span class="cite">
	[
	<a href="#L. Caccetta 2011" >3</a>
	]
</span>, matrix multi-splitting Picard-iterative method <span class="cite">
	[
	<a href="#Mehdi Dehghan 2020" >5</a>
	]
</span>, generalized Newton method <span class="cite">
	[
	<a href="#Mangasarian 2009" >10</a>
	]
</span>, iterative method <span class="cite">
	[
	<a href="#Jiri Rohn 2014i" >18</a>
	]
</span> and for more, one may refer to <span class="cite">
	[
	<a href="#N Anane 2020" >2</a>
	, 
	<a href="#Tangi 2018" >13</a>
	, 
	<a href="#Jiri Rohn 2009ii" >17</a>
	]
</span>. In recent years, some authors investigated the existence and uniqueness of different types of AVE (see <span class="cite">
	[
	<a href="#Mangasarian 2006" >8</a>
	, 
	<a href="#Mangasarian 2007i" >9</a>
	, 
	<a href="#Jiri Rohn 2004" >15</a>
	, 
	<a href="#Jiri Rohn 2009i" >16</a>
	, 
	<a href="#Shi Liang Wu 2016" >20</a>
	, 
	<a href="#Hongyu Zhou 2021" >26</a>
	]
</span> and references therein). </p>
<p>Throughout the paper, we will denote \( \hat{D}= \operatorname {diag}(\hat{d_i})\) with \(\hat{d_i} \in [{0,1}]\) is a diagonal matrix. For a matrix \(A\), \(\sigma _{1}(A)\) or \(\sigma _{\max }(A)\) and \(\sigma _{n}(A)\) or \(\sigma _{\min }(A)\) will denote the maximum singular value and the minimum singular value of A, respectively. </p>
<p>The structure of the paper is the following: in <a href="#section 2">section 2</a> we recall some necessary definitions and results that will be used throughout the paper. In <a href="#section 3">section 3</a>, with the help of LCP, some unique solution results are established for <a href="#Equ1" class="eqref">1</a>. Finally, the conclusions are discussed in <a href="#section 4">section 4</a>. </p>
<h1 id="section 2">2 Preliminaries</h1>

<p>Here, we recall some definitions and lemmas for additional results. <div class="definition_thmwrapper " id="a0000000003">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p>The LCP(q,M) is defined as <span class="cite">
	[
	<a href="#cottle 1992" >4</a>
	]
</span>: </p>
<div class="displaymath" id="Equ4">
  \begin{align} \label{Equ4} 0 \leq z \perp Mz + q \geq 0 \end{align}
</div>
<p> where \(q, z \in {\mathbb R}^n \), \(M \in {\mathbb R}^{n \times n}\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="01">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#KG Murty 1992" >14</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> A square matrix is called a P-matrix if all its principal minors are positive. Furthermore, every positive definite matrix is a P-matrix. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#cottle 1992" >4</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  A matrix M is a P-matrix if and only if the LCP(q,M) has a unique solution for any q. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#W.S Zhang 2006" >24</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  Let A \(\in C^{m \times m}\) satisfy the condition \(\| A\| _2 {\lt} 1\). Then A\(\pm \)I is non-singular. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">5</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#L. Caccetta 2011" >3</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(\| A^{-1}\| _2 {\lt} 1\). Then \(A+\bar{D}\) is non-singular for any diagonal matrix \(\bar{D} = \operatorname {diag}(\bar{d}_i)\) with \(-1 \leq \bar{d}_i \leq 1\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="4i">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Shi-Liang Wu 2019" >22</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> If the maximum singular value of the matrix W is strictly less than \(1\), then AVE <a href="#Equ1" class="eqref">1</a> has a unique solution. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="5">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">7</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#cottle 1992" >4</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> A matrix M is called a Q-matrix if, for any \(q\in \) \({\mathbb R}^{n}\), LCP(q,M) has a solution. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="7">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">8</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Shi Liang Wu 2021" >23</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> The AVE <a href="#Equ3i" class="eqref">2</a> has a unique solution if and only if matrix \(V+W-2W\hat{D}\) or \(V-W+2W\hat{D}\) is non-singular for any diagonal matrix \(\hat{D}\). </p>

  </div>
</div> </p>
<p><a href="#7">Lemma 8</a> is equivalent to the following lemma when W is a non-singular matrix. </p>
<p><div class="lemma_thmwrapper " id="8">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">9</span>
  </div>
  <div class="lemma_thmcontent">
  <p> The AVE <a href="#Equ3i" class="eqref">2</a> has a unique solution if and only if matrix </p>
<div class="displaymath" id="a0000000004">
  \[  W(W^{-1}V + I-2\hat{D})  \]
</div>
<p> or </p>
<div class="displaymath" id="a0000000005">
  \[  W(W^{-1}V-I+2\hat{D})  \]
</div>
<p> is non-singular for any diagonal matrix \(\hat{D}\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="9">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">10</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Fu Zhang 2011" >25</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> For square matrices \(A\) and \(B\), singular value inequality \(\sigma _{i}(A+B)\) \(\ge \) \(\sigma _{i}(A) \) \(-\) \(\sigma _{1}(B)\), \(i=1,2,\ldots ,n\) holds. </p>

  </div>
</div> </p>
<h1 id="section 3">3 Main Results</h1>
<p>  In this section, we introduce significant results for the unique solution of AVE <a href="#Equ1" class="eqref">1</a> and GAVE <a href="#Equ3i" class="eqref">2</a> and establish an equivalence relation between AVE and LCP. With the help of this relation, we get some results for the AVE <a href="#Equ1" class="eqref">1</a>. </p>
<p><div class="theorem_thmwrapper " id="Thm1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">11</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If matrix \((I-W)\) is non-singular, then the following results are equivalent: <br /><span class="rm">1.</span> for any b, the AVE <a href="#Equ1" class="eqref">1</a> has a unique solution; <br /><span class="rm">2.</span> matrix \((I-W)^{-1}(I+W)\) is a P-matrix; <br /><span class="rm">3.</span> matrix \((I-W)^{-1}(I+W)\) is a positive definite matrix; <br /><span class="rm">4.</span> all the principal minors of the matrix \((I-W)^{-1}(I+W)\) are positive. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000006">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(x_i^{+} = \max _i(0,x_i)\) and \(x_i^{-} = \max _i(0,-x_i) \). So </p>
<div class="equation" id="Equ5">
<p>
  <div class="equation_content">
    \begin{equation} \label{Equ5} x =x^{+}- x^{-} \quad \textrm{and} ~ ~  |x| =x^{+}+ x^{-} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>Now by <a href="#Equ5" class="eqref">5</a>, AVE <a href="#Equ1" class="eqref">1</a> is rewritten as </p>
<div class="equation" id="Equ6">
<p>
  <div class="equation_content">
    \begin{equation} \label{Equ6} x^{+}= (I-W)^{-1}(I+W)x^{-} + (I-W)^{-1}b. \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> <a href="#Equ6" class="eqref">6</a> is a LCP form, where M= \((I-W)^{-1}(I+W)\). </p>
<p>Then, by <a href="#1">lemma 3</a> and <a href="#01">definition 2</a>, our results hold. <div class="proof_wrapper" id="a0000000007">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Based on <a href="#5">lemma 7</a>, we get the existence result for <a href="#Equ1" class="eqref">1</a> in <a href="#Equ6i">theorem 12</a>. </p>
<p><div class="theorem_thmwrapper " id="Equ6i">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">12</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \((I-W)\) is non-singular matrix then for any \(b\), AVE <a href="#Equ1" class="eqref">1</a> has a solution, if \((I-W)^{-1}(I+W)\) is a \(Q\)-matrix. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> If matrix \((I-W)\) is non-singular, then <a href="#Equ1" class="eqref">1</a> can be written as the given below form: </p>
<div class="displaymath" id="a0000000009">
  \begin{equation*} \label{Equ6i} x^{+}= (I-W)^{-1}(I+W)x^{-} + (I-W)^{-1}b, \end{equation*}
</div>
<p> where \(x^{+}\) and \(x^{-}\) are defined in <a href="#Equ5" class="eqref">5</a>. Which is a linear complementarity problem form. Then, with the help of <a href="#5">lemma 7</a>, we get our result. <div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, by substituting \(V=I\) in <a href="#7">lemma 8</a>, we will get the necessary and sufficient conditions for AVE <a href="#Equ1" class="eqref">1</a> see <a href="#Thm8">theorem 13</a> and <a href="#Thm9">theorem 14</a>. </p>
<p><div class="theorem_thmwrapper " id="Thm8">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">13</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For any diagonal matrix \(\hat{D}\), the matrix </p>
<div class="displaymath" id="a0000000011">
  \[  (I-W+2W\hat{D})  \]
</div>
<p> is non-singular if and only if for any \(b\), AVE <a href="#Equ1" class="eqref">1</a> has a unique solution. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm9">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For any diagonal matrix \(\hat{D}\), the matrix \((I+W-2W\hat{D})\) is non-singular if and only if there exists a unique solution of AVE <a href="#Equ1" class="eqref">1</a> for any \(b\). </p>

  </div>
</div> </p>
<p>Based on the simple result “if the smallest singular value of a matrix is greater than \(0\), then the given matrix is non-singular", we get the following two results, see <a href="#Col04">theorem 15</a> and <a href="#Col004">theorem 16</a>. </p>
<p><div class="theorem_thmwrapper " id="Col04">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">15</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \(\sigma _{\min }(I+W){\gt}\) \(2\sigma _{\max }(W\hat{D})\) or \(\sigma _{\min }(I-W){\gt}\) \(2\sigma _{\max }(W\hat{D})\) for any diagonal matrix \(\hat{D}\), then there exists a unique solution of AVE <a href="#Equ1" class="eqref">1</a> for any \(b\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By <a href="#9">lemma 10</a>, we have \(\sigma _{\min }(I+W-2W\hat{D}) \ge \) \(\sigma _{\min }(I+W)-\) \(2\sigma _{\max }(W\hat{D}).\) </p>
<p>If \(\sigma _{\min }(I+W)\) \({\gt}\) \(2\sigma _{\max }(W\hat{D})\), then matrix \(I+W-2W\hat{D}\) is nonsingular for any diagonal matrix \(\hat{D}\). So by the previous result of the <a href="#Thm9">theorem 14</a>, AVE <a href="#Equ1" class="eqref">1</a> has a unique solution. By Similar way with the help of <a href="#Thm8">theorem 13</a> we can prove that if \(\sigma _{\min }(I-W)\) \({\gt}\) \(2\sigma _{\max }(W\hat{D})\) then AVE <a href="#Equ1" class="eqref">1</a> has a unique solution. <div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Moreover, the result of the <a href="#Col04">theorem 15</a> can be extended for the GAVE <a href="#Equ3i" class="eqref">2</a>. </p>
<p><div class="theorem_thmwrapper " id="Col004">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">16</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \(\sigma _{\min }(W(2\hat{D}-I)){\gt}1\) for any diagonal matrix \(\hat{D}\), then AVE <a href="#Equ1" class="eqref">1</a> has a unique solution for any \(b\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000014">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Based on <a href="#9">lemma 10</a>, we have </p>
<div class="displaymath" id="a0000000015">
  \[  \sigma _{\min }(I-W+2W\hat{D})\ge \sigma _{\min }(2W\hat{D}-W)-\sigma _{\max }(I).  \]
</div>
<p>Since \(\sigma _{\max }(I)=1\), so if \(\sigma _{\min }(2W\hat{D}-W){\gt}1\) then matrix \(I+W-2W\hat{D}\) is nonsingular for any diagonal matrix \(\hat{D}\), so the AVE <a href="#Equ1" class="eqref">1</a> has a unique solution for any \(b\). <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Based on <a href="#3">lemma 4</a>, we get the following result for GAVE <a href="#Equ3i" class="eqref">2</a>. </p>
<p><div class="theorem_thmwrapper " id="Thm9i">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">17</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For any diagonal matrix \(\hat{D}\), if \(\Vert W^{-1}V-2\hat{D}\Vert _2 {\lt} 1\) or \(\Vert W^{-1}V+2\hat{D}\Vert _2 {\lt} 1\) then for any b, the AVE <a href="#Equ3i" class="eqref">2</a> has a unique solution, where \(W\) is a non-singular matrix. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By the help of <a href="#3">lemma 4</a>, if \(\Vert W^{-1}V-2\hat{D}\Vert _2 {\lt} 1\), then \(W^{-1}V-2\hat{D}+I\) is non-singular, since W is non-singular, so \(W[W^{-1}V-2\hat{D}+I]\) is also non-singular, then by <a href="#8">lemma 9</a>, AVE <a href="#Equ3i" class="eqref">2</a> has a unique solution for any \(b\). </p>
<p>Or, when \(\Vert W^{-1}V+2\hat{D}\Vert _2 {\lt} 1\), then matrix \(W^{-1}V+2\hat{D}-I\) is non-singular, so \(W[W^{-1}V+2\hat{D}-I]\) is also non-singular, then by <a href="#8">lemma 9</a>, AVE <a href="#Equ3i" class="eqref">2</a> has a unique solution for any \(b\). <div class="proof_wrapper" id="a0000000018">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="Col4">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">18</span>
  </div>
  <div class="corollary_thmcontent">
  <p> There exists a unique solution of AVE <a href="#Equ2" class="eqref">3</a> if \(\Vert (V+2\hat{D})\Vert _2 {\lt} 1\) or \(\Vert (V-2\hat{D})\Vert _2 {\lt} 1\) for any diagonal matrix \(\hat{D}\). </p>

  </div>
</div> </p>
<p>By substituting \(W=I\) in <a href="#Thm9i">theorem 17</a>, then <a href="#Col4">corollary 18</a> will become the main result of the <span class="cite">
	[
	<a href="#Shi Liang Wu 2018" >21</a>
	, 
	Th.
	
	3.4
	]
</span>). </p>
<p>Based on <a href="#Thm8">theorem 13</a>, <a href="#Thm9">theorem 14</a> and <a href="#3">lemma 4</a>, we get the following result. </p>
<p><div class="theorem_thmwrapper " id="Thm10">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">19</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For any diagonal matrix \(\hat{D}\), matrix W satisfies \(\Vert W-2W\hat{D}\Vert _2 {\lt} 1\) or \(\Vert -W+2W\hat{D}\Vert _2 {\lt} 1\) then for any b, AVE <a href="#Equ1" class="eqref">1</a> has a unique solution. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> With the aid of <a href="#3">lemma 4</a>, if \(\Vert W-2W\hat{D}\Vert _2 {\lt} 1\), then \(W-2W\hat{D}+I\) is non-singular, then by <a href="#Thm9">theorem 14</a>, AVE <a href="#Equ1" class="eqref">1</a> has unique solution. </p>
<p>Or, when \(\Vert -W+2W\hat{D}\Vert _2 {\lt} 1\), then matrix \(-W+2W\hat{D}+I\) is non-singular, then by <a href="#Thm8">theorem 13</a>, AVE <a href="#Equ1" class="eqref">1</a> has unique solution. <div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Based on <a href="#4">lemma 5</a> and <a href="#7">lemma 8</a>, we get the following result. </p>
<p><div class="theorem_thmwrapper " id="Thm12">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">20</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If matrix V and non-singular matrix W will satisfy the conditions \(\Vert (W^{-1}V-I)^{-1}\Vert _2 {\lt} 2\) or \(\Vert (W^{-1}V+I)^{-1}\Vert _2 {\lt} 2\) then for any b, AVE <a href="#Equ3i" class="eqref">2</a> has a unique solution. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000021">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let </p>
<div class="equation" id="a0000000022">
<p>
  <div class="equation_content">
    \begin{equation} \begin{split} \label{Equ19 i} (i) ~  (V-W+2W\hat{D}) = 2W\left[\tfrac {1}{2}{W^{-1}{(V-W)}}+\hat{D}\right] \\ (ii)~  (V+W-2W\hat{D}) = 2W\left[\tfrac {1}{2}{W^{-1}{(V+W)}}-\hat{D}\right]. \end{split}\end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>With the aid of <a href="#4">lemma 5</a>, when \(\Vert \frac{1}{2}(W^{-1}V-I)^{-1}\Vert _2 {\lt} 1\), then \(\frac{1}{2}(W^{-1}V-I)+\hat{D} \) is non-singular. By <a href="#Equ19 i" class="eqref">8</a> and <a href="#7">lemma 8</a> our proof is complete. </p>
<p>Or, when \(\Vert \frac{1}{2}(W^{-1}V+I)^{-1}\Vert _2 {\lt} 1\), then matrix \(\frac{1}{2}(W^{-1}V+I)-\hat{D} \) is non-singular. Then, by <a href="#Equ19 i" class="eqref">8</a> and <a href="#7">lemma 8</a> our proof is completed. <div class="proof_wrapper" id="a0000000023">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="Col2">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">21</span>
  </div>
  <div class="corollary_thmcontent">
  <p> There exists a unique solution of AVE <a href="#Equ2" class="eqref">3</a> if &#160; \(\Vert (V-I)^{-1}\Vert _2 {\lt} 2\) or \(\Vert (V+I)^{-1}\Vert _2 {\lt} 2\). </p>

  </div>
</div> </p>
<p>By substituting \(W=I\) in <a href="#Thm12">theorem 20</a>, then <a href="#Col2">corollary 21</a> will become Theorem 3.5 as a main result in <span class="cite">
	[
	<a href="#Shi Liang Wu 2018" >21</a>
	]
</span>. </p>
<p><div class="corollary_thmwrapper " id="Col3">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">22</span>
  </div>
  <div class="corollary_thmcontent">
  <p> If \(\Vert (W^{-1}-I)^{-1}\Vert _2 {\lt} 2\) or \(\Vert (W^{-1}+I)^{-1}\Vert _2 {\lt} 2\), then there exists a unique solution of AVE <a href="#Equ1" class="eqref">1</a>. </p>

  </div>
</div> </p>
<p>We get our result by substituting \(V=I\) in <a href="#Thm12">theorem 20</a>. </p>
<p><div class="remark_thmwrapper " id="Remark 2">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">23</span>
  </div>
  <div class="remark_thmcontent">
  <p> Sometime the condition \(\Vert (V-I)^{-1}\Vert _2 {\lt} 2\) given in <a href="#Col2">corollary 21</a> is not satisfying even AVE \(Vx-|x|=b\) has a unique solution. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Let us consider the following example. </p>
<p><div class="example_thmwrapper " id="Ex1">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">24</span>
  </div>
  <div class="example_thmcontent">
  <p> The AVE \(Vx-|x|=(2,-2)^T\) (\(T\) denotes the transpose of the vector \((2,-2)\)) has a unique solution \(x=(20,-0.909091)^T\), for </p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  V= \begin{bmatrix}  1.1 

&  0.0 

\\ 0.0 

&  1.2 

\end{bmatrix} . \end{equation*}
</div>
<p> Clearly, \(\Vert (V-I)^{-1}\Vert _2 = 10 \nless 2.\) </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="Remark 3">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">25</span>
  </div>
  <div class="remark_thmcontent">
  <p> By the result \(\sigma _{\max }(W)\)\(\sigma _{\min }(W^{-1}) =1 \), the condition \(\sigma _{\max }(W) {\lt} 1\) of <a href="#4i">lemma 6</a> is equivalent to the condition \(\sigma _{\min }(W^{-1}) {\gt} 1.\) Sometimes, matrix \(W\) does not satisfy the conditions \(\sigma _{\max }(W) {\lt} 1\) or \(\sigma _{\min }(W^{-1}) {\gt} 1\) but AVE <a href="#Equ1" class="eqref">1</a> has a unique solution for some \(b.\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Let us consider the following example. </p>
<p><div class="example_thmwrapper " id="Ex2">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">26</span>
  </div>
  <div class="example_thmcontent">
  <p> Consider the matrix </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  W= \begin{bmatrix}  -0.9 

&  0.2 

\\ -0.4 

&  -0.8 

\end{bmatrix}. \end{equation*}
</div>
<p>Here \(\sigma _{\min }(W^{-1}) = 0.9870 \ngtr 1 \) and \(\sigma _{\max }(W) = 1.0132 \nless 1\), even AVE \(x-W|x|=b\) has a unique solution \(x=(2.66667,-15.33334)^{T}\) for some \(b=(2,-2)^T\) (\(T\) denotes the transpose). </p>
<p>Furthermore, the conditions \(\sigma _{\min }(W) {\lt} 1\) and \(\sigma _{\max }(W^{-1}) {\gt} 1\) maybe replace the existing conditions \(\sigma _{\max }(W) {\lt} 1\) and \(\sigma _{\min }(W^{-1}) {\gt} 1\) respectively, because of we have inequalities \(\sigma _{\min }(W) \le \sigma _{\max }(W) {\lt} 1\) and \(\sigma _{\max }(W^{-1}) \ge \sigma _{\min }(W^{-1}) {\gt} 1\). Also for <a href="#Ex2">example 26</a> we have \(\sigma _{\max }(W^{-1}) = 1.2665{\gt}1\) and \(\sigma _{\min }(W) =0.7896{\lt}1\). So existing singular value conditions may be revised in future. This needs further investigation. These conditions may be or may not be superior then the current singular value conditions. </p>

  </div>
</div> </p>
<h1 id="section 4">4 Conclusion</h1>

<p>This article discusses the existence of solution possibilities in the uniqueness of the GAVE \(Vx -W|x|=b\) and its particular cases. The conditions of <a href="#4i">lemma 6</a>, <a href="#Col2">corollary 21</a> and <a href="#Col3">corollary 22</a> may be revised in the future. With the help of the equivalence relation of LCP and AVE, we established some more conditions for the unique solvability of <a href="#Equ1" class="eqref">1</a>. The numerical results for the absolute value equations are also an interesting topic in the future. </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000026">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors are thankful to the anonymous referees for their insightful comments, which significantly improved the manuscript. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="M Achache 2019">1</a></dt>
  <dd><p><a href ="https://doi.org/10.33993/jnaat482-1182"> <i class="sc">M. Achache</i>, <i class="it">On the unique solvability and numerical study of absolute value equations</i>, J. Numer. Anal. Approx. Theory, <b class="bf">48</b> (2019) no. 2, pp.&#160;112–121. <a href="https://doi.org/10.33993/jnaat482-1182">https://doi.org/10.33993/jnaat482-1182</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="N Anane 2020">2</a></dt>
  <dd><p><a href ="https://doi.org/10.33993/jnaat491-1197"> <i class="sc">N. Anane, M. Achache</i>, <i class="it">Preconditioned conjugate gradient methods for absolute value equations</i>, J. Numer. Anal. Approx. Theory, <b class="bf">49</b> (2020) no. 1, pp.&#160;3–14. <a href="https://doi.org/10.33993/jnaat491-1197">https://doi.org/10.33993/jnaat491-1197</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="L. Caccetta 2011">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10589-009-9242-9"> <i class="sc">L. Caccetta, B. Qu, G.L. Zhou</i>, <i class="it">A globally and quadratically convergent method for absolute value equation</i>, Comput. Optim. Appl., <b class="bf">48</b> (2011), pp.&#160;45–58. <a href="https://doi.org/10.1007/s10589-009-9242-9">https://doi.org/10.1007/s10589-009-9242-9</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="cottle 1992">4</a></dt>
  <dd><p><i class="sc">R.W. Cottle, J.S. Pang, R.E. Stone</i>, <i class="it">The Linear Complementarity Problem</i>, Acad. Press, New York, 1992. </p>
</dd>
  <dt><a name="Mehdi Dehghan 2020">5</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.apnum.2020.08.001"> <i class="sc">M. Dehghan, A. Shirilord</i>, <i class="it">Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation</i>, Appl. Numer. Math., <b class="bf">158</b> (2020), pp.&#160;425–438. <a href="https://doi.org/10.1016/j.apnum.2020.08.001">https://doi.org/10.1016/j.apnum.2020.08.001</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Milan Hladik 2018">6</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10589-017-9939-0"> <i class="sc">M. Hladik</i>, <i class="it">Bounds for the solutions of absolute value equations</i>, Comput. Optim. Appl., <b class="bf">69</b> (2018), pp.&#160;243-–266. <a href="https://doi.org/10.1007/s10589-017-9939-0">https://doi.org/10.1007/s10589-017-9939-0</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Sheng Long Hu 2010">7</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11590-009-0169-y"> <i class="sc">S.L. Hu, Z.H. Huang</i>, <i class="it">A note on absolute value equations</i>, Optim. Lett., <b class="bf">4</b> (2010), pp.&#160;417–424. <a href="https://doi.org/10.1007/s11590-009-0169-y">https://doi.org/10.1007/s11590-009-0169-y</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Mangasarian 2006">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.laa.2006.05.004"> <i class="sc">O.L. Mangasarian, R.R. Meyer</i>, <i class="it">Absolute value equations</i>, Linear Algebra Appl., <b class="bf">419</b> (2006), pp.&#160;359–367. <a href="https://doi.org/10.1016/j.laa.2006.05.004">https://doi.org/10.1016/j.laa.2006.05.004</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Mangasarian 2007i">9</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10589-006-0395-5"> <i class="sc">O.L. Mangasarian</i>, <i class="it">Absolute value programming</i>, Comput. Optim. Appl., 36 (2007) no. 1, pp.&#160;43–53. <a href="https://doi.org/10.1007/s10589-006-0395-5">https://doi.org/10.1007/s10589-006-0395-5</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Mangasarian 2009">10</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11590-008-0094-5"> <i class="sc">O.L. Mangasarian</i>, <i class="it">A generalized Newton method for absolute value equations</i>, Optim. Lett., <b class="bf">3</b> (2009), pp.&#160;101–108. <a href="https://doi.org/10.1007/s11590-008-0094-5">https://doi.org/10.1007/s11590-008-0094-5</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Mangasarian 2014">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10957-013-0461-y"> <i class="sc">O.L. Mangasarian</i>, <i class="it">Absolute Value equation solution via linear programming</i>, J. Optim. Theory Appl., <b class="bf">161</b> (2014), pp.&#160;870–876. <a href="https://doi.org/10.1007/s10957-013-0461-y">https://doi.org/10.1007/s10957-013-0461-y</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="O.L. Mangasarian 2017">12</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11590-017-1115-z"> <i class="sc">O.L. Mangasarian</i>, <i class="it">Sufficient conditions for the unsolvability and solvability of the absolute value equation</i>. Optim. Lett., <b class="bf">11</b> (2017), pp.&#160;1469–1475. <a href="https://doi.org/10.1007/s11590-017-1115-z">https://doi.org/10.1007/s11590-017-1115-z</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Tangi 2018">13</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.cam.2017.06.019"> <i class="sc">L. Abdallah, M. Haddou, T. Migot</i>, <i class="it">Solving absolute value equation using complementarity and smoothing functions</i>, J. Comp. Appl. Math., <b class="bf">327</b> (2018), pp. 196–207. <a href="https://doi.org/10.1016/j.cam.2017.06.019">https://doi.org/10.1016/j.cam.2017.06.019</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="KG Murty 1992">14</a></dt>
  <dd><p><i class="sc">K.G. Murty</i>, <i class="it">Linear Complementarity, Linear and Nonlinear Programming</i>, Internet edition, 1997. </p>
</dd>
  <dt><a name="Jiri Rohn 2004">15</a></dt>
  <dd><p><a href ="https://doi.org/10.1080/0308108042000220686"> <i class="sc">J. Rohn</i>, <i class="it">A theorem of the alternatives for the equation \(Ax + B|x| = b,\)</i> Linear Multilinear Algebra, <b class="bf">52</b> (2004) no. 6, pp.&#160;421–426. <a href="https://doi.org/10.1080/0308108042000220686">https://doi.org/10.1080/0308108042000220686</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Jiri Rohn 2009i">16</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11590-009-0129-6"> <i class="sc">J. Rohn</i>, <i class="it">On unique solvability of the absolute value equation</i>, Optim. Lett., <b class="bf">3</b> (2009), pp.&#160;603–606. <a href="https://doi.org/10.1007/s11590-009-0129-6">https://doi.org/10.1007/s11590-009-0129-6</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Jiri Rohn 2009ii">17</a></dt>
  <dd><p><a href ="https://doi.org/10.13001/1081-3810.1332"> <i class="sc">J. Rohn</i>, <i class="it">An algorithm for solving the absolute value equations</i>, Electron. J. Linear Algebra, <b class="bf">18</b> (2009), pp. 589–599. <a href="https://doi.org/10.13001/1081-3810.1332">https://doi.org/10.13001/1081-3810.1332</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Jiri Rohn 2014i">18</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11590-012-0560-y"> <i class="sc">J. Rohn, V. Hooshyarbakhsh, R. Farhadsefat</i>, <i class="it">An iterative method for solving absolute value equations and sufficient conditions for unique solvability</i>, Optim. Lett., <b class="bf">8</b> (2014), pp.&#160;35–44. <a href="https://doi.org/10.1007/s11590-012-0560-y">https://doi.org/10.1007/s11590-012-0560-y</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Jiri Rohn 2014ii">19</a></dt>
  <dd><p><i class="sc">J. Rohn</i>, <i class="it">A class of explicitly solvable absolute value equations</i>, Technical report V-1202, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2014). </p>
</dd>
  <dt><a name="Shi Liang Wu 2016">20</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10957-015-0845-2"> <i class="sc">S.L. Wu, P. Guo, </i> <i class="it">On the unique solvability of the absolute value equation</i>, J. Optim. Theory Appl., <b class="bf">169</b> (2016), pp.&#160;705–712. <a href="https://doi.org/10.1007/s10957-015-0845-2">https://doi.org/10.1007/s10957-015-0845-2</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Shi Liang Wu 2018">21</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.aml.2017.08.012"> <i class="sc">S.L. Wu, C.X. Li</i>, <i class="it">The unique solution of the absolute value equations</i>, Appl. Math. Lett., <b class="bf">76</b> (2018), pp.&#160;195–200. <a href="https://doi.org/10.1016/j.aml.2017.08.012">https://doi.org/10.1016/j.aml.2017.08.012</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Shi-Liang Wu 2019">22</a></dt>
  <dd><p><i class="sc">S.L. Wu, C.X. Li</i>, <i class="it">A note on unique solvability of the absolute value equation</i>, Optim. Lett., <b class="bf">14</b> (2019), pp.&#160;1957–1960. </p>
</dd>
  <dt><a name="Shi Liang Wu 2021">23</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11590-020-01672-2"> <i class="sc">S.L. Wu, S. Shen</i>, <i class="it">On the unique solution of the generalized absolute value equation</i>, Optim. Lett., <b class="bf">15</b> (2021), pp.&#160;2017–2024. <a href="https://doi.org/10.1007/s11590-020-01672-2">https://doi.org/10.1007/s11590-020-01672-2</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="W.S Zhang 2006">24</a></dt>
  <dd><p><i class="sc">W.S. Zhang</i>, <i class="it">Finite Difference Methods for Partial Differential Equations in Science Computation</i>, Higher Education Press, 2006. </p>
</dd>
  <dt><a name="Fu Zhang 2011">25</a></dt>
  <dd><p><i class="sc">F. Zhang</i>, <i class="it">Matrix Theory: Basic results and techniques</i>, 2nd ed., Springer, New York, 2011. </p>
</dd>
  <dt><a name="Hongyu Zhou 2021">26</a></dt>
  <dd><p><a href ="https://doi.org/10.3934/math.2021517"> <i class="sc">H. Zhou, S. Wu</i>, <i class="it">On the unique solution of a class of absolute value equations \(Ax-B|Cx|=d\)</i>, AIMS Math., <b class="bf">6</b> (2021) no.&#160;8, pp.&#160;8912–8919. <a href="https://doi.org/10.3934/math.2021517">https://doi.org/10.3934/math.2021517</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
</div> <!-- content-wrapper -->
</div> <!-- content -->
</div> <!-- wrapper -->

<nav class="prev_up_next">
</nav>

<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/jquery.min.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/plastex.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/svgxuse.js"></script>
</body>
</html>