Ukrainian Mathematical Journal

, Volume 64, Issue 7, pp 1144–1150

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

  • V. Yu. Slyusarchuk

DOI: 10.1007/s11253-012-0705-2

Cite this article as:
Slyusarchuk, V.Y. Ukr Math J (2012) 64: 1144. doi:10.1007/s11253-012-0705-2
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 (t0, x0) ∈ 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.

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • V. Yu. Slyusarchuk
    • 1
  1. 1.National University of Water Management and Nature Resources UseRivneUkraine