Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

This paper proves the existence of solutions to the initial value problem \[ ( I V P ) { x ′ ( t ) = f ( t , x ( t ) ) ( 0 ≤ t ≤ 1 ) , x ( 0 ) = 0 , (\mathrm {IVP})\qquad \qquad \left \{\begin {array}{l} x’(t)=f(t,x(t))\qquad \quad (0\le t\le 1), x(0)=0,\end {array} \right . \] where f : [ 0 , 1 ] × R M → R M f:[0,1]\times \mathbb {R}^M\to \mathbb {R}^M may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set M M can be arbitrarily large (finite or infinite); our theorem is new even for card ( M ) = 2 \mbox {card}(M)=2 . The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.