On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space

Authors:

Slyusarchuk V E

Institution:

(1) Rivne State Technical University, Rivne

Abstract:

We prove the following statement: Theorem 1. Let E and $f:\mathbb{R} \times E \to E$ be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every $(t_0 ,z_0 ) \in \mathbb{R} \times E$ and epsi

> 0, there exists a continuous mapping $g:\mathbb{R} \times E \to E$ such that

$\mathop {\sup }\limits_{(t,x) \in \mathbb{R} \times E} \left\| {f(t,x) - g(t,x)} \right\| \leqslant \varepsilon$

and the Cauchy problem

$z'(t) = g(t,z(t)),{\text{ z(}}t_0 ) = z_0 ,{\text{ }}t \in (t_0 - \delta ,t_0 + \delta ),$

does not have a solution for every delta

> 0.

Keywords:

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