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On the existence of solutions of differential equations in Banach spaces. (English) Zbl 0858.34047

Using the abstract concepts of the measure of noncompactness, the regularity of mappings and the Kamke comparison function, the author proves some general versions of well-known results concerning the existence and uniqueness of solutions of the Cauchy problem \(x'=f(t,x)\), \(x(0)=x_0\) in Banach spaces. Such an approach permits to unify the classical results involving compactness or dissipative conditions.

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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