×

zbMATH — the first resource for mathematics

Unique solutions for a class of discontinuous differential equations. (English) Zbl 0692.34004
Summary: This paper is concerned with the Cauchy problem \(\dot x(t)=f(t,x(t)),\) \(x(t_ 0)=x_ 0\in {\mathbb{R}}^ n\), 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.

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. · Zbl 0538.34007
[2] Alberto Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31 (1988), no. 3, 459 – 470. · Zbl 0676.34014
[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. · Zbl 0704.49011
[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). · Zbl 0193.04203
[5] A. F. Filippov, Differential equations with discontinuous right-hand sides, Trans. Amer. Math. Soc. 42 (1964), 199-231. · Zbl 0148.33002
[6] Otomar Hájek, Discontinuous differential equations. I, II, J. Differential Equations 32 (1979), no. 2, 149 – 170, 171 – 185. · Zbl 0365.34017
[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). · Zbl 0238.34008
[8] Alfredo Pucci, Traiettorie di campi di vettori discontinui, Rend. Ist. Mat. Univ. Trieste 8 (1976), no. 1, 84 – 93. · Zbl 0343.57010
[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). · Zbl 0429.34014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.