Existence of solutions for discontinuous differential equations. (English) Zbl 0824.34003

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 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.


34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems