Continuous dependence of nonmonotonic discontinuous differential equations

Author:
Daniel C. Biles

Journal:
Trans. Amer. Math. Soc. **339** (1993), 507-524

MSC:
Primary 34A34

DOI:
https://doi.org/10.1090/S0002-9947-1993-1126212-0

MathSciNet review:
1126212

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Continuous dependence of solutions for a class of nonmonotonic, discontinuous differential equations is studied. First, a local existence theorem due to . Wu is extended to a larger class. Then, a result concerning continuous dependence for this larger class is proven. This employs a type of convergence similar to Gihman's Convergence Criterion, which is defined to be

**[1]**Z. Artstein,*Continuous dependence of solutions of Volterra integral equations*, SIAM J. Math. Anal.**6**(1975), 446-456. MR**0361656 (50:14101)****[2]**-,*Continuous dependence on parameters*:*On the best possible results*, J. Differential Equations**19**(1975), 214-225. MR**0390329 (52:11155)****[3]**E. A. Coddington and N. Levinson,*Theory of ordinary differential equations*, McGraw-Hill, New York, 1955. MR**0069338 (16:1022b)****[4]**A. F. Filippov,*Differential equations with discontinuous right-hand side*, Mat. Sb.**51**(1960), 99-128 (Russian); Amer. Math. Soc. Transl.**42**(1964), 199-231. MR**0114016 (22:4846)****[5]**I. I. Gihman,*Concerning a theorem of N. N. Bogolyubov*, Ukrainian Math. J.**4**(1952), 215-218. MR**0075386 (17:738g)****[6]**S. Gutman,*Evolutions governed by*-*accretive plus compact operators*, Nonlinear Anal.**7**(1983), 707-715. MR**707079 (84h:34094)****[7]**-,*Topological equivalence in the space of integrable vector-valued functions*, Proc. Amer. Math. Soc.**93**(1985), 40-42. MR**766523 (86b:46054)****[8]**L. Piccinini, -*convergence for ordinary differential equations*, Proc. Internat. Meeting on Recent Methods in Nonlinear Analysis (E. DeGiorgi et al., eds.), Pitagora Editrice, Bologna, 1979. MR**533171 (81a:34015c)****[9]**E. Schechter,*Existence and limits of Carathéodory-Martin evolutions*, Nonlinear Anal.**5**(1981), 897-930. MR**628110 (83m:34065)****[10]**-,*One-sided continuous dependence of maximal solutions*, J. Differential Equations**39**(1981), 413-425. MR**612595 (82e:45002)****[11]**-,*Perturbations of regularizing maximal monotone operators*, Israel J. Math.**43**(1982), 49-61. MR**728878 (85k:47110)****[12]**-,*Evolution generated by semilinear dissipative plus compact operators*, Amer. Math. Soc. Trans.**275**(1983), 297-308. MR**678351 (84g:34113)****[13]**-,*Necessary and sufficient conditions for convergence of temporally irregular evolutions*, Nonlinear Anal.**8**(1984), 133-153. MR**734447 (85h:34077)****[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), 432-449. MR**528904 (80c:93068)****[15]**Z. Wu,*The ordinary differential equations with discontinuous right members and the discontinuous solutions of the quasilinear partial differential equations*, Sci. Sinica**13**(1964), 1901-1917. MR**0178225 (31:2483)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34A34

Retrieve articles in all journals with MSC: 34A34

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1126212-0

Keywords:
Ordinary differential equations,
continuous dependence,
local existence

Article copyright:
© Copyright 1993
American Mathematical Society