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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