Best constant in Hyers–Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient

Abstract

This paper is concerned with Hyers–Ulam stability of the first-order homogeneous linear differential equation x′−a(t)x=0 on R, where a:R→R is a continuous periodic function. It is known that if a(t)≡0 then the above equation does not have Hyers–Ulam stability on R. However, sufficient conditions for Hyers–Ulam stability are presented in spite of a(t) has infinitely many zeros and changes sign. Furthermore, the best constant in Hyers–Ulam stability is clarified. To illustrate the obtained results, some examples are included.