# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the existence of solutions of differential equations in Banach spaces. (English) Zbl 0532.34045
This note is concerned with the local existence of solutions to the initial value problem (*) ${x}^{\text{'}}=f\left(t,x\right)$, $x\left(0\right)=\theta$, where $f$ maps $\left[0,a\right]×B$ into a Banach space E and B is a ball in E centered at the origin. It is assumed that $t\to f\left(t,x\right)$ is measurable, $x\to f\left(t,x\right)$ is continuous, and $\parallel f\left(t,x\right)\parallel \le m\left(t\right)$ for $\left(t,x\right)\in \left[0,a\right]×B$ where ${\int }_{0}^{a}m\left(s\right)ds<\infty$. The main assumption on f involves the Kuratowski (or ball) measure of noncompactness and has the form that for each $ϵ>0$ and $X\subset B$, there is a closed subset ${I}_{ϵ}$ of [0,a] such that $\mu \left[I-{I}_{ϵ}\right]<ϵ$ ($\mu$ denotes Lebesgue measure) and $\alpha \left(f\left(T×X\right)\right)\le {sup}_{t\in T}h\left(t,\alpha \left(X\right)\right)$ for each compact subset T of ${I}_{ϵ}$ (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 ${I}_{ϵ}\right)$. The function h satisfies the Caratheodory conditions on $\left[0,a\right]×\left[0,\infty \right)$ and $u\left(t\right)\equiv 0$ is the maximal solution to ${u}^{\text{'}}=h\left(t,u\right)$, $u\left(0\right)=0$. Under these hypotheses it is shown that the initial value problem (*) has a solution. The result is an improvement of a preceding one of G. Pianigiani [ibid. 23, 853-857 (1975; Zbl 0317.34050)].
Reviewer: R.H.Martin
##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions
##### Keywords:
local existence of solutions; Caratheodory conditions