On certain singular ordinary differential equations of the first order in Banach spaces (Q1203057)

One denotes by \(B\) a real Banach space, \(I_ 0\) the zero element of \(L(B)\); \((0,T)\), \(T\leq\infty\) an interval from \(R\); \(\alpha:[0,T)\to L(B)\) with \(\alpha(0)=I_ 0\) and \(f:[0,T)\times B\to B\). Sufficient conditions are given on \(\alpha\) and \(f\) in order that the equation \({d\over dt}(\alpha(t)x(t))=f(t,x(t))\), \(t\in(0,T)\) has a unique solution in the set \(C([0,T),B)\cap C^ 1([0,T),B)\) satisfying \(x(0)=x_ 0\) where \(x_ 0\) is the unique solution of the equation \(x_ 0=(\alpha'(0))^{-1}f(0,x_ 0)\). These results are also formulated in the particular case of a linear equation. Let's underline the fact that the results of the paper are proved with the aid of classical methods of mathematical analysis and using a generalization of the del'Hospital rules for \(f/g\) to the case where \(g\) is a real function and \(f\) takes values in a normed space.