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Unique solutions for a class of discontinuous differential equations

Author: Alberto Bressan
Journal: Proc. Amer. Math. Soc. 104 (1988), 772-778
MSC: Primary 34A10; Secondary 34A60
MathSciNet review: 964856
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Abstract: This paper is concerned with the Cauchy Problem

$\displaystyle \dot x\left( t \right) = f\left( {t,x\left( t \right)} \right),\quad x\left( {{t_0}} \right) = {x_0} \in {\mathbb{R}^n},$

where the vector field $ f$ may be discontinuous with respect to both variables $ t,x$. If the total variation of $ f$ along certain directions is locally finite, we prove the existence of a unique solution, depending continuously on the initial data.

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  • [1] Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330
  • [2] Alberto Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31 (1988), no. 3, 459–470. MR 987798
  • [3] Alberto Bressan, Upper and lower semicontinuous differential inclusions: a unified approach, Nonlinear controllability and optimal control, Monogr. Textbooks Pure Appl. Math., vol. 133, Dekker, New York, 1990, pp. 21–31. MR 1061382
  • [4] Alberto Cambini and Sauro Querci, Equazioni differenziali del primo ordine con secondo membro discontinuo rispetto all’incognita, Rend. Ist. Mat. Univ. Trieste 1 (1969), 89–97 (Italian, with English summary). MR 0261064
  • [5] A. F. Filippov, Differential equations with discontinuous right-hand sides, Trans. Amer. Math. Soc. 42 (1964), 199-231.
  • [6] Otomar Hájek, Discontinuous differential equations. I, II, J. Differential Equations 32 (1979), no. 2, 149–170, 171–185. MR 534546, 10.1016/0022-0396(79)90056-1
  • [7] Alfredo Pucci, Sistemi di equazioni differenziali con secondo membro discontinuo rispetto all’incognita, Rend. Ist. Mat. Univ. Trieste 3 (1971), 75–80 (Italian, with English summary). MR 0291528
  • [8] Alfredo Pucci, Traiettorie di campi di vettori discontinui, Rend. Ist. Mat. Univ. Trieste 8 (1976), no. 1, 84–93. MR 0430369
  • [9] Rémi Sentis, Équations différentielles à second membre mesurable, Boll. Un. Mat. Ital. B (5) 15 (1978), no. 3, 724–742 (French, with English summary). MR 524095

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Article copyright: © Copyright 1988 American Mathematical Society