Ukrainian Mathematical Journal

, Volume 64, Issue 7, pp 1144-1150

First online:

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

  • V. Yu. SlyusarchukAffiliated withNational University of Water Management and Nature Resources Use

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We prove the following theorem: Let E be an arbitrary Banach space, let G be an open set in the space \( \mathbb{R}\times E \), and let f: GE 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: GE such that
$$ \mathop{\sup}\limits_{{\left( {t,x} \right)\in G}}\left\| {g\left( {t,x} \right)-f\left( {t,x} \right)} \right\|\leqslant \varepsilon $$
and the Cauchy problem
$$ \frac{dz(t) }{dt }=g\left( {t,z(t)} \right),\quad z\left( {{t_0}} \right)={x_0} $$
has more than one solution.