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Remarks on the theorem of M. and S. Rădulescu about an initial value problem for the differential equation \(x^{(n)}=f(t,x)\). (English) Zbl 0757.34049
It is well-known that in general, the classical theorem of Peano can not be generalized to initial value problems in infinite-dimensional Banach spaces. M. and S. Rádulescu proved a global existence and uniqueness result for the initial value problem \(x^{(n)}(t)=f(t,x(t))\), \(x^{(i)}(a)=x_ i\in F\) for \(i=0,1,\dots,n-1\), where the Fréchet derivative \(f_ x\) satisfies suitable growth restrictions. The goal of the present paper is to prove a corresponding result under weaker conditions for \(f\), namely, under growth restrictions with regard to \(x\) for \(f\) itself.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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