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.
English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 7, pp 1144-1150.
Citation Example: Slyusarchuk V. Yu. Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems // Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 1001-1006.
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