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

• V. Yu. Slyusarchuk

### 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.
Published
25.07.2012
How to Cite
SlyusarchukV. 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, http://umj.imath.kiev.ua/index.php/umj/article/view/2635.
Issue
Section
Short communications
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