On Carathéodory's conditions for the initial value problem
Authors:
D. C. Biles and P. A. Binding
Journal:
Proc. Amer. Math. Soc. 125 (1997), 13711376
MSC (1991):
Primary 34A12; Secondary 34A40
MathSciNet review:
1403114
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We prove a local existence theorem of CarathéodoryGoodman type for where instead of being continuous in we require only that it have no ``downward discontinuities.''
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 1.
 N. Azbelev, Z. Calyuk, Uniqueness of the solution of an integral equation, Soviet Math. Dokl. 5 (1964) 611614.
 2.
 D.C. Biles, Continuous dependence of nonmonotonic discontinuous differential equations, Trans. Amer. Math. Soc. 339 (1993) 507524. MR 93m:34011
 3.
 D.C. Biles, Existence of solutions for discontinuous differential equations, Diff. Int. Eqn. 8 (1995) 15251532. MR 96d:34003
 4.
 P.A. Binding, The differential equation , J. Diff. Eqn. 31 (1979) 183199. MR 81c:34002
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 A. Bressan, Directionally continuous selections and differential inclusions, Func. Ekvac. 31 (1988) 459470. MR 90d:34038
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 C. Carathéodory, Vorlesungen über reelle Funktionen, Teubner, 1918.
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 G.S. Goodman, Subfunctions and the initial value problem for differential equations satisfying Carathéodory's hypotheses, J. Diff. Eqn. 7 (1970) 232242. MR 41:540
 8.
 S. Heikkilä, V. Lakshmikantham, S. Leela, Applications of monotone techniques to differential equations with discontinuous right hand side, Diff. Int. Eqn. 1 (1988) 287297. MR 89g:34002
 9.
 D. Hanson, P. Waltman, A note on a functional equation, J. Math. Anal. Appl. 10 (1965) 330333. MR 30:4079
 10.
 J.S. Muldowney, D. Willett, An intermediate value property for operators with applications to integral and differential equations, Canad. J. Math. 26 (1974) 2741. MR 49:3286
 11.
 M. Nagumo, Über das Verfahren der sukzessiven Approximationen zur Integration gewöhnlicher Differentialgleichungen, Japan. J. Math. 7 (1930) 143160.
 12.
 G. Peano, Sull'integrabilitá delle equazione differenziali di primo ordine, Atti Acad. Torino 21 (1885/6) 677685.
 13.
 W. Rzymowski, D. Walachowski, Onedimensional differential equation under weak assumptions, J. Math. Anal. Appl., 198 (1996), no. 3, 657670. CMP 96:09
 14.
 A. Tarski, A latticetheoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955) 285309. MR 17:574d
 15.
 D.V.V. Wend, Existence and uniqueness of solutions of ordinary differential equations, Proc. Amer. Math. Soc. 23 (1969) 2733. MR 39:7185
 16.
 Z. Wu, The ordinary differential equations with discontinuous right members and the discontinuous solutions of the quasilinear partial differential equations, Sci. Sinica 13 (1964) 19011917.
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Additional Information
D. C. Biles
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101
Email:
Daniel.Biles@wku.edu
P. A. Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Alberta, Canada T2N 1N4
Email:
binding@acs.ucalgary.ca
DOI:
http://dx.doi.org/10.1090/S0002993997039427
PII:
S 00029939(97)039427
Received by editor(s):
November 8, 1995
Additional Notes:
The second author’s research was supported by NSERC of Canada and the I. W. Killam Foundation.
Communicated by:
Hal L. Smith
Article copyright:
© Copyright 1997 American Mathematical Society
