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Zbl 0532.34045
Szufla, Stanisław
On the existence of solutions of differential equations in Banach spaces.
(English)
[J] Bull. Acad. Pol. Sci., Sér. Sci. Math. 30, 507-515 (1982). ISSN 0137-639X

This note is concerned with the local existence of solutions to the initial value problem (*) $x'=f(t,x)$, $x(0)=\theta$, where $f$ maps $[0,a]\times B$ into a Banach space E and B is a ball in E centered at the origin. It is assumed that $t\to f(t,x)$ is measurable, $x\to f(t,x)$ is continuous, and $\Vert f(t,x)\Vert \le m(t)$ for $(t,x)\in [0,a]\times B$ where $\int\sp{a}\sb{0}m(s)ds<\infty$. The main assumption on f involves the Kuratowski (or ball) measure of noncompactness and has the form that for each $\epsilon>0$ and $X\subset B$, there is a closed subset $I\sb{\epsilon}$ of [0,a] such that $\mu [I- I\sb{\epsilon}]<\epsilon$ ($\mu$ denotes Lebesgue measure) and $\alpha(f(T\times X))\le \sup\sb{t\in T}h(t,\alpha(X))$ for each compact subset T of $I\sb{\epsilon}$ (It seems that there is a misprint in the statement of this theorem since it is stated in the paper that T is a compact subset of I rather than $I\sb{\epsilon})$. The function h satisfies the Caratheodory conditions on $[0,a]\times [0,\infty)$ and $u(t)\equiv 0$ is the maximal solution to $u'=h(t,u)$, $u(0)=0$. Under these hypotheses it is shown that the initial value problem (*) has a solution. The result is an improvement of a preceding one of {\it G. Pianigiani} [ibid. 23, 853-857 (1975; Zbl 0317.34050)].
[R.H.Martin]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces
34A12 Initial value problems for ODE

Keywords: local existence of solutions; Caratheodory conditions

Citations: Zbl 0317.34050

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