Hyers-Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient

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• 作者：Masakazu Onitsuka ; ^{onitsuka@xmath.ous.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Tomohiro Shoji
• 关键词：Hyers&ndash ; Ulam stability ; HUS constant ; Linear differential equation
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：63
• 期：Complete
• 页码：102-108
• 全文大小：562 K

文摘

This paper is concerned with the Hyers–Ulam stability of the first-order linear differential equation x^′−ax=0$x^{\prime }-ax=0$, where a$a$ is a non-zero real number. The main purpose is to find an explicit solution x(t)$x\left(t\right)$ of x^′−ax=0$x^{\prime }-ax=0$ satisfying db1cdc46b6913e" title="Click to view the MathML source">|ϕ(t)−x(t)|≤ε/|a|$|\varphi \left(t\right)-x\left(t\right)|\le \epsilon /\left|a\right|$ for all t∈R$t\in \mathbb{R}$ under the assumption that a differentiable function ϕ(t)$\varphi \left(t\right)$ satisfies |ϕ^′(t)−aϕ(t)|≤ε$|{\varphi }^{\prime }\left(t\right)-a\varphi \left(t\right)|\le \epsilon$ for all t∈R$t\in \mathbb{R}$. In addition, the precise behavior of the solutions of x^′−ax=0$x^{\prime }-ax=0$ near the function ϕ(t)$\varphi \left(t\right)$ is clarified on the semi-infinite interval. Finally, some applications to nonhomogeneous linear differential equations are included to illustrate the main result.