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 
Similar Articles 
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.''
 1.
N. Azbelev, Z. Calyuk, Uniqueness of the solution of an integral equation, Soviet Math. Dokl. 5 (1964) 611614.
 2.
Daniel
C. Biles, Continuous dependence of nonmonotonic
discontinuous differential equations, Trans.
Amer. Math. Soc. 339 (1993), no. 2, 507–524. MR 1126212
(93m:34011), http://dx.doi.org/10.1090/S00029947199311262120
 3.
Daniel
C. Biles, Existence of solutions for discontinuous differential
equations, Differential Integral Equations 8 (1995),
no. 6, 1525–1532. MR 1329854
(96d:34003)
 4.
Paul
Binding, The differential equation
𝑥=𝑓∘𝑥, J. Differential Equations
31 (1979), no. 2, 183–199. MR 525443
(81c:34002), http://dx.doi.org/10.1016/00220396(79)901438
 5.
Alberto
Bressan, Directionally continuous selections and differential
inclusions, Funkcial. Ekvac. 31 (1988), no. 3,
459–470. MR
987798 (90d:34038)
 6.
C. Carathéodory, Vorlesungen über reelle Funktionen, Teubner, 1918.
 7.
Gerald
S. Goodman, Subfunctions and intitialvalue problem for
differential equations satisfying Carathéodory’s
hypotheses, J. Differential Equations 7 (1970),
232–242. MR 0255880
(41 #540)
 8.
Seppo
Heikkilä, V.
Lakshmikantham, and S.
Leela, Applications of monotone techniques to differential
equations with discontinuous righthand side, Differential Integral
Equations 1 (1988), no. 3, 287–297. MR 929916
(89g:34002)
 9.
D.
L. Hanson and Paul
Waltman, A note on a functional equation, J. Math. Anal. Appl.
10 (1965), 330–333. MR 0173872
(30 #4079)
 10.
J.
S. Muldowney and D.
Willett, An intermediate value property for operators with
applications to integral and differential equations, Canad. J. Math.
26 (1974), 27–41. MR 0338522
(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.
Alfred
Tarski, A latticetheoretical fixpoint theorem and its
applications, Pacific J. Math. 5 (1955),
285–309. MR 0074376
(17,574d)
 15.
David
V. V. Wend, Existence and uniqueness of solutions
of ordinary differential equations, Proc. Amer.
Math. Soc. 23
(1969), 27–33. MR 0245879
(39 #7185), http://dx.doi.org/10.1090/S00029939196902458794
 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.
 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
 5.
 A. Bressan, Directionally continuous selections and differential inclusions, Func. Ekvac. 31 (1988) 459470. MR 90d:34038
 6.
 C. Carathéodory, Vorlesungen über reelle Funktionen, Teubner, 1918.
 7.
 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.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
34A12,
34A40
Retrieve articles in all journals
with MSC (1991):
34A12,
34A40
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
