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 exists a continuous mapping g : G → E such that _(t,x)∈G||g(t, x) − f(t, x)|| ≤ ε and the Cauchy problem dz(t)dt = g(t, z(t)), z(t0) = x_0 has more than one solution.