ON THE HYERS-ULAM STABILITY OF THE BANACH SPACE-VALUED DIFFERENTIAL EQUATION y'=λy

Let I be an open interval and X a complex Banach space. Let<TEX>ε≥0\;and\;λ</TEX> a non-zero complex number with Re <TEX>λ≠0</TEX>. If <TEX>φ</TEX> is a strongly differentiable map from I to X with <TEX>∥φ^'(t)-λφ(t)∥≤ε\;for\;all\;t∈\;I</TEX>, then we show that the distance between <TEX>φ</TEX> and the set of all solutions to the differential equation y'=<TEX>λ</TEX>y is at most <TEX>ε/<TEX></TEX>Reλ<TEX>$</TEX>$</TEX>.