Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems

Abstract

We prove the following theorem: Let $E$ be an arbitrary Banach space, $G$ be an open set in the space $R×E$, and $f : G → E$ be an arbitrary continuous mapping. Then, for an arbitrary point $(t_0, x_0) ∈ G$ and an arbitrary number $ε > 0$, there exists a continuous mapping $g : G → E$ such that $$\sup_{(t,x)∈G}||g(t, x) − f(t, x)|| \leq \varepsilon$$ and the Cauchy problem $$\frac{dz(t)}{dt} = g(t, z(t)), z(t0) = x_0$$ has more than one solution.