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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 ' =f(t,x), x(0)=θ, where f maps [0,a]×B into a Banach space E and B is a ball in E centered at the origin. It is assumed that tf(t,x) is measurable, xf(t,x) is continuous, and f(t,x)m(t) for (t,x)[0,a]×B where 0 a m(s)ds<. The main assumption on f involves the Kuratowski (or ball) measure of noncompactness and has the form that for each ϵ>0 and XB, there is a closed subset I ϵ of [0,a] such that μ[I-I ϵ ]<ϵ (μ denotes Lebesgue measure) and α(f(T×X))sup tT h(t,α(X)) 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 ). The function h satisfies the Caratheodory conditions on [0,a]×[0,) and u(t)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 G. Pianigiani [ibid. 23, 853-857 (1975; Zbl 0317.34050)].
Reviewer: R.H.Martin
34G20Nonlinear ODE in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions