<p>The aim of this paper is to give an Ulam stability result for the linear differential operator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$D:C^{1}(\mathbb {R}, X)\rightarrow C(\mathbb {R}, X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>:</mo> <msup> <mi>C</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Dy=y^{\prime }+f\cdot y,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mi>y</mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>f</mi> <mo>·</mo> <mi>y</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <i>X</i> is a real or complex Banach space. Moreover, if there exists <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\inf \limits _{x\in \mathbb {R}}|\Re f(x)|=m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">inf</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </munder> <mrow> <mo stretchy="false">|</mo> <mi>ℜ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m\ne 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K=\frac{1}{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is the best Ulam constant of the operator. As applications, we provide some stability results for Hill’s operator and for the n-th order linear differential operator.</p>

The best Ulam constant of a first-order linear differential operator

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发表日期：

2026年3月16日

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Alina Ramona Baias,
Alexandra Paicu,
Dorian Popa

摘要

The aim of this paper is to give an Ulam stability result for the linear differential operator $D:C^{1}(\mathbb {R}, X)\rightarrow C(\mathbb {R}, X)$ $D : C^{1} (R, X) \to C (R, X)$ defined by $Dy=y^{\prime }+f\cdot y,$ $D y = y^{'} + f \cdot y,$ where X is a real or complex Banach space. Moreover, if there exists $\inf \limits _{x\in \mathbb {R}}|\Re f(x)|=m$ $inf_{x \in R} | ℜ f (x) | = m$ and $m\ne 0,$ $m \neq 0,$ we prove that $K=\frac{1}{m}$ $K = \frac{1}{m}$ is the best Ulam constant of the operator. As applications, we provide some stability results for Hill’s operator and for the n-th order linear differential operator.