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<p>A recent study employing general integral transform techniques presents an incomplete treatment of the Hyers–Ulam stability of linear second-order Cauchy–Euler differential equations. That paper asserts that such second-order equations are universally Hyers–Ulam stable. This paper aims to clarify that the Hyers–Ulam stability of these second-order equations is critically dependent on whether the characteristic roots have non-zero real parts. We discuss and illustrate instability cases to provide a more complete stability and instability analysis. Next, we prove that the <i>n</i>th-order Cauchy–Euler equation is Hyers–Ulam stable on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if the real parts of the roots of the corresponding characteristic equation are all non-zero. Further, we prove that the <i>n</i>th-order Cauchy–Euler equation is Hyers–Ulam stable on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if a related <i>n</i>th-order linear constant coefficient differential equation is Hyers–Ulam stable on the real line, where the coefficients of the two equations are related via Stirling numbers of the first kind, and the Hyers–Ulam stability constant is the minimum constant for one if and only if it is the minimum constant for the other.</p>

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Hyers–Ulam Instability and Stability in Second and Higher-Order Cauchy–Euler Equations

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Douglas R. Anderson,
  • Masakazu Onitsuka

摘要

A recent study employing general integral transform techniques presents an incomplete treatment of the Hyers–Ulam stability of linear second-order Cauchy–Euler differential equations. That paper asserts that such second-order equations are universally Hyers–Ulam stable. This paper aims to clarify that the Hyers–Ulam stability of these second-order equations is critically dependent on whether the characteristic roots have non-zero real parts. We discuss and illustrate instability cases to provide a more complete stability and instability analysis. Next, we prove that the nth-order Cauchy–Euler equation is Hyers–Ulam stable on \((0,\infty )\) ( 0 , ∞ ) if and only if the real parts of the roots of the corresponding characteristic equation are all non-zero. Further, we prove that the nth-order Cauchy–Euler equation is Hyers–Ulam stable on \((0,\infty )\) ( 0 , ∞ ) if and only if a related nth-order linear constant coefficient differential equation is Hyers–Ulam stable on the real line, where the coefficients of the two equations are related via Stirling numbers of the first kind, and the Hyers–Ulam stability constant is the minimum constant for one if and only if it is the minimum constant for the other.