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

Biles, Daniel; Schechter, Eric

doi:10.1090/S0002-9939-00-05584-2

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations
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by Daniel C. Biles and Eric Schechter PDF

Proc. Amer. Math. Soc. 128 (2000), 3349-3360 Request permission

Abstract:

This paper proves the existence of solutions to the initial value problem \[ (\mathrm {IVP})\qquad \qquad \left \{\begin {array}{l} x’(t)=f(t,x(t))\qquad \quad (0\le t\le 1), x(0)=0,\end {array} \right .\] where $f:[0,1]\times \mathbb {R}^M\to \mathbb {R}^M$ may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set $M$ can be arbitrarily large (finite or infinite); our theorem is new even for $\mbox {card}(M)=2$. 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.

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