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On the Cauchy problem for ordinary differential equations with discontinuous right-hand sides. (English) Zbl 0724.34004

The authors are introducing the notion of “angular continuity” of \(f=f(t,x)\) and then they prove that this is sufficient for the existence of solutions of the Cauchy problem for \(x'(t)=f(t,x(t)).\)
The angular continuity is strictly more general than the continuity and therefore their results extend some well-known results, including Peano’s existence theorem.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A99 General theory for ordinary differential equations
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References:

[1] Binding, P., The differential equation \(x\)′ = \(f\) ° \(x\), J. Differential Equations, 31, 183-199 (1979) · Zbl 0363.34001
[2] Gelbaum, B.; Olmsted, J., Counterexamples in Analysis (1964), Holden-Day: Holden-Day San Francisco/London/Amsterdam · Zbl 0121.28902
[3] Giuntini, S.; Pianigiani, G., Equazioni differenziali ordinarie con seconde membro discontinue, (Atti Sem. Mat. Fis. Univ. Modena, 23 (1974)), 233-240 · Zbl 0343.34005
[4] Krasnosel’skii, M. A.; Pokrovskii, A. V., Systems with Hysteresis (1983), Nauka: Nauka Moscow · Zbl 1092.47508
[5] Peetre, J.; Persson, J., The Peano existence theorem under weaker assumptions, (Mimeograph (1971), University of Lund: University of Lund Lund)
[6] Szarski, J., Differential Inequalities (1965), PWN: PWN Warszawa · Zbl 0135.25804
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