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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Existence theory for first order discontinuous functional differential equations

Authors: Eduardo Liz and Rodrigo L. Pouso
Journal: Proc. Amer. Math. Soc. 130 (2002), 3301-3311
MSC (1991): Primary 34A12, 34K07, 34K10
Published electronically: March 25, 2002
Corrigendum: Proc. Amer. Math. Soc. 132 (2004), 3135-3136.
MathSciNet review: 1913010
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of extremal solutions for a first order functional differential equation subject to nonlinear boundary conditions of functional type. Moreover, the functions that define our problem are allowed to be discontinuous. The proof of our main result is based on a generalized iterative technique.

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  • 1. S. Carl and S. Heikkilä, Extremality results for first-order discontinuous functional differential equations, Comput. Math. Appl. 40 (2000), 1217-1232. MR 2001g:34095
  • 2. A. Fryszkowski, Existence of solutions of functional-differential inclusions in nonconvex case, Ann. Polon. Math. 45 (1985), 121-124. MR 86i:34018
  • 3. J. R. Haddock and M. N. Nkashama, Periodic boundary value problems and monotone iterative methods for functional differential equations, Nonlinear Anal. 22, 3 (1994), 267-276. MR 95h:34102
  • 4. A. Halanay, ``Differential equations: Stability, oscillations, time lags", Academic Press, New York (1966). MR 35:6938
  • 5. J. K. Hale and S. M. Verduyn Lunel ``Introduction to functional-differential equations", Springer-Verlag, New York (1993). MR 94m:34169
  • 6. E. R. Hassan and W. Rzymowski, Extremal solutions of a discontinuous differential equation, Nonlinear Anal. 37 (1999), 997-1017. MR 2000d:34015
  • 7. S. Heikkilä and V. Lakshmikantham, ``Monotone iterative techniques for discontinuous nonlinear differential equations", Marcel Dekker, New York (1994). MR 95d:34002
  • 8. A. Ivanov, E. Liz and S. Trofimchuk, Halanay inequality, Yorke $3/2$ stability criterion, and differential equations with maxima, Tohoku Math.J., to appear.
  • 9. V. Kolmanovskii and A. Myshkis, ``Introduction to the theory and applications of functional-differential equations", Kluwer Academic Publishers, Dordrecht (1999). MR 2000c:34164
  • 10. J. W. Lee and D. O'Regan, Existence results for differential delay equations, I. J. Differential Equations 102 (1993), 342-359. MR 94c:34096
  • 11. S. Leela and M.N. Oguztöreli, Periodic boundary value problem for differential equations with delay and monotone iterative method. J. Math. Anal. Appl. 122 (1987), 301-307. MR 88d:34095
  • 12. X. Liu, Periodic boundary value problems for differential equations with finite delay. Dynam. Systems Appl. 3 (1994), 357-367. MR 95f:34096
  • 13. E. Liz and J. J. Nieto, Periodic boundary value problems for a class of functional-differential equations, J. Math. Anal. Appl. 200 (1996), 680-686. MR 97k:34097
  • 14. M. Pinto and S. Trofimchuk, Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima. Proc. Roy. Soc. Edimburgh Sect. A 130 (2000), 1103-1118. MR 2001j:34101
  • 15. E. Stepanov, On solvability of some boundary value problems for differential equations with ``maxima", Topol. Methods Nonlinear Anal. 8 (1996), 315-326. MR 99a:34178
  • 16. H. K. Xu and E. Liz, Boundary value problems for differential equations with maxima, Nonlinear Stud. 3, 2 (1996), 231-241.
  • 17. H. K. Xu and E. Liz, Boundary value problems for functional differential equations, Nonlinear Anal. 41 (2000), 971-988. MR 2001d:34111

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Additional Information

Eduardo Liz
Affiliation: Departamento de Matemática Aplicada, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36280 Vigo, Spain

Rodrigo L. Pouso
Affiliation: Departamento de Análise Matemática, Facultade de Matemáticas, Campus Sur, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

PII: S 0002-9939(02)06480-8
Keywords: Discontinuous functional differential equations, extremal solutions, existence results
Received by editor(s): July 10, 2000
Received by editor(s) in revised form: June 13, 2001
Published electronically: March 25, 2002
Additional Notes: This research was partially supported by D.G.E.S. (Spain), projects PB97 – 0552 and HP1999-0026.
Communicated by: Carmen Chicone
Article copyright: © Copyright 2002 American Mathematical Society