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

  • V. Yu. Slyusarchuk


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.
How to Cite
Slyusarchuk, V. Y. “Denseness of the Set of Cauchy Problems With Nonunique Solutions in the Set of All Cauchy Problems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 7, July 2012, pp. 1001-6,
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