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Solvability of a finite or infinite system of discontinuous quasimonotone differential equations. (English) Zbl 0959.34012
The authors prove the existence of solutions to the initial value problem $$x'(t)=f(t,x(t)), \quad t\in[0,1] \text{ and }x(0)=0.$$ The function $f: [0,1]\times {\Bbb R}^M \to {\Bbb R}^M$ may be discontinuous but has to satisfy superposition-measurability (also called sup-measurability), quasimonotonicity and integrability conditions. The set $M$ can be arbitrarily large. The proof is based on a more general theorem on the existence and certain properties of subsolutions to the same problem. A brief discussion of the assumptions and further generalizations is presented; the theorem contains results given earlier by {\it D. C. Biles} [Differ. Integral Equ. 8, No. 6, 1525-1532 (1995; Zbl 0824.34003)] and {\it D. C. Biles} and {\it P. A. Binding} [Proc. Am. Math. Soc. 125, No. 5, 1371-1376 (1997; Zbl 0869.34004)] as special cases.

MSC:
34A35ODE of infinite order
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
45G15Systems of nonlinear integral equations
34A40Differential inequalities (ODE)
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