Existence of solutions for discontinuous differential equations (Q1891584)

The author investigates the initial value problem for the first order nonlinear differential equation. The existence of Carathéodory solutions is proved under conditions weaker than those due to \textit{C. Carathéodory} [``Vorlesungen über reelle Funktionen'', Leipzig (1927)], and the main difference is the assumption of nondecreasing of \(f(t, x)\) in \(x\) for almost all \(t\in [0, T]\) instead of traditional continuity in \(x\) for all \(t\in [0,T]\). The discussion of the hypotheses and comparison to known results are also given.