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

Abstract

The aim of this paper is to give an Ulam stability result for the linear differential operator \(D:C^{1}(\mathbb {R}, X)\rightarrow C(\mathbb {R}, X)\) defined by \(Dy=y^{\prime }+f\cdot y,\) where X is a real or complex Banach space. Moreover, if there exists \(\inf \limits _{x\in \mathbb {R}}|\Re f(x)|=m\) and \(m\ne 0,\) we prove that \(K=\frac{1}{m}\) 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.