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Continuous dependence of nonmonotonic discontinuous differential equations. (English) Zbl 0787.34009

The author obtains a result on continuous dependence of solutions of a differential equation \(u'=f(t,u)\) on the function \(f\) if \(f\) is not continuous or monotone but is bounded, Lebesgue measurable in \(t\) for each fixed \(u\), and satisfies \(\limsup_{y \to u-} f(t,y) \leq f(t,u)=\lim_{y \to u+} f(t,y)\).
Reviewer: F.Brauer (Madison)

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A37 Ordinary differential equations with impulses
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[1] Zvi Artstein, Continuous dependence of solutions of Volterra integral equations, SIAM J. Math. Anal. 6 (1975), 446 – 456. · Zbl 0341.45005 · doi:10.1137/0506039
[2] Zvi Artstein, Continuous dependence on parameters: on the best possible results, J. Differential Equations 19 (1975), no. 2, 214 – 225. · Zbl 0342.34001 · doi:10.1016/0022-0396(75)90002-9
[3] Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. · Zbl 0064.33002
[4] A. F. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.) 51 (93) (1960), 99 – 128 (Russian). · Zbl 0138.32204
[5] I. I. Gihman, Concerning a theorem of N. N. Bogolyubov, Ukrain. Mat. Ž. 4 (1952), 215 – 219 (Russian).
[6] S. Gutman, Evolutions governed by \?-accretive plus compact operators, Nonlinear Anal. 7 (1983), no. 7, 707 – 715. · Zbl 0518.34055 · doi:10.1016/0362-546X(83)90027-5
[7] Semion Gutman, Topological equivalence in the space of integrable vector-valued functions, Proc. Amer. Math. Soc. 93 (1985), no. 1, 40 – 42. · Zbl 0529.46027
[8] Livio C. Piccinini, Homogeneization problems for ordinary differential equations, Rend. Circ. Mat. Palermo (2) 27 (1978), no. 1, 95 – 112. , https://doi.org/10.1007/BF02843869 Livio Clemente Piccinini, \?-convergence for ordinary differential equations with Peano phenomenon, Rend. Sem. Mat. Univ. Padova 58 (1977), 65 – 86 (1978). L. Piccinini, \?-convergence for ordinary differential equations, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 257 – 279.
[9] Eric Schechter, Existence and limits of Carathéodory-Martin evolutions, Nonlinear Anal. 5 (1981), no. 8, 897 – 930. · Zbl 0465.34036 · doi:10.1016/0362-546X(81)90093-6
[10] Eric Schechter, One-sided continuous dependence of maximal solutions, J. Differential Equations 39 (1981), no. 3, 413 – 425. · Zbl 0427.34007 · doi:10.1016/0022-0396(81)90066-8
[11] Eric Schechter, Perturbations of regularizing maximal monotone operators, Israel J. Math. 43 (1982), no. 1, 49 – 61. · Zbl 0516.34060 · doi:10.1007/BF02761684
[12] Eric Schechter, Evolution generated by semilinear dissipative plus compact operators, Trans. Amer. Math. Soc. 275 (1983), no. 1, 297 – 308. · Zbl 0516.34061
[13] Eric Schechter, Necessary and sufficient conditions for convergence of temporally irregular evolutions, Nonlinear Anal. 8 (1984), no. 2, 133 – 153. · Zbl 0546.35007 · doi:10.1016/0362-546X(84)90065-8
[14] G. I. Stassinopoulos and R. B. Vinter, Continuous dependence of solutions of a differential inclusion on the right hand side with applications to stability of optimal control problems, SIAM J. Control Optim. 17 (1979), no. 3, 432 – 449. · Zbl 0442.49025 · doi:10.1137/0317031
[15] Wu Zhuo-qun, The ordinay differential equations with discontinuous right members and the discontinuous solutions of the quasilinear partial differential equations, Sci. Sinica 13 (1964), 1901 – 1917. · Zbl 0154.10901
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