Ger type stability of the first order linear differential equation y'(t) = h(t)y(t)

Let 0 ∈ I [??] R be an open interval and let [C.sup.1](I,[C.sup.x) be the set of all continuously differentiable functions from I to [C.sup.x], where [C.sup.x] is the set of all non-zero complex numbers. If h: I → [C.sup.x] is a continuous function with M = [sup.sub.t∈I |[∫.sub.0.sup.t]|h(s)|ds| < ∞, then for each ζ ≥ 0 and f ∈ [C.sup.1](I, [C.sup.x]) satisfying |[[f'(t)]/[h(t)f(t)]] - 1|≤ζ (∀t∈I) there exists [f.sub.0] ∈ [C.sup.1](I, [C.sup.x]) such that [f'.sub.0](t) = h(t)[f.sub.0](t) and that max{|[[f(t)]/[[f.sub.0](t)]] - 1|,|[[[f.sub.0](t)]/[f(t)]] - 1|}≤[e.sup.Mζ] - 1 for all t ∈ I. We give an example that the constant [e.sup.Mζ] - 1 can not be improved in general. We also prove that the assumption [sup.sub.t∈I] |[t.∫.0]|h(s)|ds| < ∞ is essential for Ger type stability. Keywords and Phrases: Exponential functions, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Ger type stability
1. Introduction It seems that the stability problem of functional equations had been first raised by S. M. Ulam (cf. [(24), Chapter VI]). 'For what metric groups G is it [...]