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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.

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