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On the solvability of first-order discontinuous scalar initial and boundary value problems

Published online by Cambridge University Press:  17 February 2009

R. P. Agarwal  and
S. Heikkilä
R. P. Agarwal
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA; e-mail: agarwal@fit.edu.
S. Heikkilä
Affiliation:
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland.
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Abstract

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In this paper we derive extremality and comparison results for explicit and implicit initial and boundary value problems of first-order differential equations. Both the differential equations and the boundary conditions may involve discontinuities.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 4 , April 2003 , pp. 513 - 538
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Biles, D. C., “Continuous dependence of nonmonotonic discontinuous differential equations”, Trans. Amer. Math. Soc. 339 (1993) 507–524.CrossRefGoogle Scholar
[2]Biles, D. C., “Existence of solutions for discontinuous differential equations”, Diff. Int. Eqns. 8 (1995) 1525–1532.Google Scholar
[3]Carathéodory, C., Vorlesungen über reelle Funktionen (Chelsea, New York, 1948).Google Scholar
[4]Carl, S., Heikkilä, S. and Kumpulainen, M., “On solvability of first-order discontinuous scalar differential equations”, Nonlinear Times and Digest 2 (1995) 11–24.Google Scholar
[5]Hassan, E. R. and Rzymowski, W., “Extremal solutions of a discontinuous scalar differential equation”, Nonlinear Analysis 37 (1999) 997–1017.CrossRefGoogle Scholar
[6]Heikkilä, S., On first-order discontinuous scalar differential equations, Pitman Research Notes in Mathematics Series 324 (Longman, 1995) 148–155.Google Scholar
[7]Heikkilä, S., “Notes on first-order discontinuous differential equations with nonlinear boundary conditions”, Proc. Dynamic Sys. Appl. 2 (1996) 255–260.Google Scholar
[8]Heikkilä, S., “On discontinuously perturbed Carathéodory type differential equations”, Nonlinear Anal. 26 (1996) 785–797.Google Scholar
[9]Heikkilä, S., “On first-order discontinuous differential equations with functional boundary conditions”, Adv. in Nonlinear Dynamics 5 (1997) 273–281.Google Scholar
[10]Heikkilä, S. and Cabada, A., “On first-order discontinuous differential equations with nonlinear boundary conditions”, Nonlinear World 3 (1996) 487–503.Google Scholar
[11]Heikkilä, S. and Lakshmikantham, V., “Extension of the method of upper and lower solutions for discontinuous differential equations”, Diff. Eqns. and Dyn. Syst. 1 (1993) 73–86.Google Scholar
[12]Heikkilä, S. and Lakshmikantham, V., Monotone iterative techniques for discontinuous nonlinear differential equations (Marcel Dekker, New York, 1994).Google Scholar
[13]Heikkilä, S. and Lakshmikantham, V., “A unified theory of first-order discontinuous scalar differential equations”, Nonlinear Anal. 26 (1996) 785–797.CrossRefGoogle Scholar
[14]Heikkilä, S., Lakshmikantham, V. and Leela, S., “Applications of monotone techniques to differential equations with discontinuous right hand side”, Diff. Int. Eqns. 1 (1988) 287–297.Google Scholar
[15]Kolmogorov, A. N. and Fomin, S. V., Introductory real analysis (Prentice-Hall, Englewood Cliffs, NJ, 1970).Google Scholar
[16]Kumpulainen, M., “On extremal and unique solutions of discontinuous ordinary differential equations and finite systems of ordinary differential equations”, Dissertation, Univ. of Oulu, Dept. of Math. Sci., 1996.Google Scholar
[17]McShane, E. J., Integration (Princeton Univ. Press, Princeton, NJ, 1974).Google Scholar
[18]Pouso, R. L., “Upper and lower solutions for first-order discontinuous ordinary differential equations”, J. Math. Anal. Appl. 244 (2000) 466–482.CrossRefGoogle Scholar
[19]Rzymowski, W. and Walachowski, D., “One-dimensional differential equation under weak assumptions”, J. Math. Anal. Appl. 198 (1996) 657–670.CrossRefGoogle Scholar