Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems - Slyusarchuk, V. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
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 existsa continuous mapping $g : G → E$ such that$$\sup_{(t,x)∈G}||g(t, x) − f(t, x)|| ≤ ε$$and the Cauchy problem$$\frac{dz(t)}{dt} = g(t, z(t)), z(t0) = x_0$$has more than one solution.