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

Author(s): Daniel C. Biles; Eric Schechter
Journal: Proc. Amer. Math. Soc. 128 (2000), 3349-3360.
MSC (2000): Primary 34A12, 34A40; Secondary 45G15
Posted: May 18, 2000
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Abstract:

This paper proves the existence of solutions to the initial value problem

\begin{displaymath}(\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.\end{displaymath}

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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Additional Information:

Daniel C. Biles
Affiliation: Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576
Email: Daniel.Biles@wku.edu

Eric Schechter
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240-0001
Email: schectex@math.vanderbilt.edu

DOI: 10.1090/S0002-9939-00-05584-2
PII: S 0002-9939(00)05584-2
Keywords: Quasimonotone, semicontinuous, quasisemicontinuous, supmeasurable, subsolution
Received by editor(s): January 13, 1999
Posted: May 18, 2000
Communicated by: Hal L. Smith
Copyright of article: Copyright 2000, American Mathematical Society

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