Cauchy problems with non-unique solutions
1 Department of Higher Mathematics, National University of Water and Environmental Engineering, Rivne, 33028, Ukraine
Keywords:
Cauchy problems
Abstract
We prove the following theorem. Let $G$ be a domain in the space $\mathbb{R}^2$ and $f: G → \mathbb{R}$ be an arbitrary continuous map. For an arbitrary point $(t_0,x_0) ∈ G$ and a number $ε > 0$ there exists a continuous map $g: G → \mathbb{R}$ such that $\underset{(t,x) ∈ G} {sup} |g(t,x) - f(t,x)| ≤ ε$ and the Cauchy problem ${dz(t) \over dt} = g(t, z(t)), z(t_0) = x_0$ has more than one solution.
References
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[4] Slyusarchuk V. E. Density of the set of unsolvable Cauchy problems in the case of an infinite-dimensional Banach space // Nonlinear oscillations. - 2002. - 5, №1.P. 86-89.
Cite
- ACS Style
- Slyusarchuk , V.Y. Cauchy problems with non-unique solutions. Bukovinian Mathematical Journal. 2018, 1
- AMA Style
- Slyusarchuk VY. Cauchy problems with non-unique solutions. Bukovinian Mathematical Journal. 2018; 1(4).
- Chicago/Turabian Style
- Vasyl Yukhimovych Slyusarchuk . 2018. "Cauchy problems with non-unique solutions". Bukovinian Mathematical Journal. 1 no. 4.
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