We consider a differentiable map $f$ from an open interval to a real Banach space of all bounded continuous real-valued functions on a topological space. We show that $f$ can be approximated by the solution to the differential equation $x'(t)=\lambda x(t)$, if $||f'(t)-\lambda f(t)||_\infty\leq\varepsilon$ holds.
"On the Hyers-Ulam Stability of Real Continuous Function Valued Differentiable Map." Tokyo J. Math. 24 (2) 467 - 476, December 2001. https://doi.org/10.3836/tjm/1255958187