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 w the MathML source">x^′−ax=0, where w the MathML source">a is a non-zero real number. The main purpose is to find an explicit solution w the MathML source">x(t) of w the MathML source">x^′−ax=0 satisfying w the MathML source">|ϕ(t)−x(t)|≤ε/|a| for all w the MathML source">t∈R under the assumption that a differentiable function w the MathML source">ϕ(t) satisfies w the MathML source">|ϕ^′(t)−aϕ(t)|≤ε for all w the MathML source">t∈R. In addition, the precise behavior of the solutions of w the MathML source">x^′−ax=0 near the function w the MathML source">ϕ(t) is clarified on the semi-infinite interval. Finally, some applications to nonhomogeneous linear differential equations are included to illustrate the main result.