Proceedings of the American Mathematical Society

Existence theory for first order discontinuous functional differential equations

Author(s): Eduardo Liz; Rodrigo L. Pouso
Journal: Proc. Amer. Math. Soc. 130 (2002), 3301-3311.
MSC (1991): Primary 34A12, 34K07, 34K10
Posted: March 25, 2002
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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.

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 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

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34A12, 34K07, 34K10

Eduardo Liz
Affiliation: Departamento de Matemática Aplicada, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36280 Vigo, Spain
Email: eliz@dma.uvigo.es

Rodrigo L. Pouso
Affiliation: Departamento de Análise Matemática, Facultade de Matemáticas, Campus Sur, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Email: rodrigolp@correo.usc.es

DOI: 10.1090/S0002-9939-02-06480-8
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
Posted: 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
Copyright of article: Copyright 2002, American Mathematical Society

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