Solvability conditions for the first-order Cauchy problem in a Banach space (Q801458)

Roughly speaking the author gives (without proof) necessary conditions (on g) for the inequality \(\| f(x)-f(y)\| \leq g(\| x-y\|),\) x,y\(\in X\), \(g: R_+\to R_+\) to imply the local existence of the solution to the Cauchy problem \(\dot x=f(x)\), \(x(0)=x_ 0\in X\), where \(f: X\to X\) is continuous and X is a Banach space of infinite dimension. The author recalls that \textit{F. E. Browder} [Proc. Sympos. Pure Math. 18 (1976; Zbl 0327.47022)] has given sufficient conditions in this direction.