Scholarlink

<p>Motivated by prior studies on unique solvability of the absolute value equation (AVE), this paper further investigates that the AVE has only one solution. Firstly, from the special matrix point of view, a matrix-based unified sufficient condition for the AVE to have only one solution is obtained, which covers some existing results. Specifically, we show that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A-I\)</EquationSource> </InlineEquation> is a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P\)</EquationSource> </InlineEquation>-matrix, then the AVE has a unique solution for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(b \in \mathbb {R}^n\)</EquationSource> </InlineEquation>. Secondly, based on a published counterexample, two sufficient conditions based on matrix norm for the AVE to have only one solution are presented to take place of a false result published. Our approach leverages properties of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(P\)</EquationSource> </InlineEquation>-matrix and matrix norm inequalities, and the theoretical findings are supported by illustrative examples. These contributions extend and refine the existing uniqueness theory for the AVE.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Some New Sufficient Conditions for Only One Solution of Absolute Value Equation

  • 期刊论文

  • 发表日期：

    2026年4月17日

查看全文
  • Ji-Yu Wang,
  • Cui-Xia Li,
  • Shi-Liang Wu

摘要

Motivated by prior studies on unique solvability of the absolute value equation (AVE), this paper further investigates that the AVE has only one solution. Firstly, from the special matrix point of view, a matrix-based unified sufficient condition for the AVE to have only one solution is obtained, which covers some existing results. Specifically, we show that if \(A-I\) is a \(P\) -matrix, then the AVE has a unique solution for any \(b \in \mathbb {R}^n\) . Secondly, based on a published counterexample, two sufficient conditions based on matrix norm for the AVE to have only one solution are presented to take place of a false result published. Our approach leverages properties of \(P\) -matrix and matrix norm inequalities, and the theoretical findings are supported by illustrative examples. These contributions extend and refine the existing uniqueness theory for the AVE.