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Nonresonance conditions and extremal solutions for first-order impulsive problems under weak assumptions

Published online by Cambridge University Press:  17 February 2009

Daniel Franco  and
Rodrigo L. Pouso
Daniel Franco
Affiliation:
Departamento de Matemática Aplicada 1, ETSI Industriales, UNED, Apartado de Correos 60149, 28080 Madrid, Spain; e-mail: [email protected].
Rodrigo L. Pouso
Affiliation:
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Campus Sur s/n, 15782 Santiago de Compostela, Spain.
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Abstract

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In this work we shall study the existence of extremal solutions for an impulsive problem with functional-boundary conditions and weak regularity assumptions, not only on the right-hand side of the equation and on the functions that define the boundary conditions, but also on the impulse functions, which will be required to be nondecreasing, but not continuous as well, as is customary in the literature.

Moreover, in order to prove one of our results we shall study a general impulsive linearproblem, giving a complete characterisation of resonance for it.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 3 , January 2003 , pp. 393 - 407
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Appell, J. and Zabrejko, P. P., Nonlinear superposition operators (Cambridge University Press, 1990).CrossRefGoogle Scholar
[2]Cabada, A. and Liz, E., “Discontinuous impulsive differential equations with nonlinear boundary conditions”, Nonlinear Anal. 28 (1997) 1491–1497.CrossRefGoogle Scholar
[3]Cabada, A. and Liz, E., “Boundary value problems for higher order ODEs with impulses”, Nonlinear Anal. 32 (1998) 775–786.CrossRefGoogle Scholar
[4]Cabada, A. and Pouso, R. L., “On first-order discontinuous scalar differential equations”, Nonlinear Studies 6 (1999) 161–170.Google Scholar
[5]Erbe, L. H., Freedman, H. I., Liu, X. Z. and Wu, J. H., “Comparison principles for impulsive parabolic equations with applications to models of single-species growth”, J. Austral. Math. Soc. Ser. B 32 (1991) 382–400.CrossRefGoogle Scholar
[6]Franco, D. and Nieto, J. J., “A new maximum principle for impulsive first-order problems”, Internat. J. Theoret. Phys. 37 (1998) 1607–1616.CrossRefGoogle Scholar
[7]Frigon, M. and O'Regan, D., “Existence results for first-order impulsive differential equations”, J. Math. Anal. Appl. 193 (1995) 96–113.CrossRefGoogle Scholar
[8]Hassan, E. R. and Rzymowski, W., “Extremal solutions of a discontinuous differential equation”. Nonlinear Anal. 37 (1999) 997–1017.CrossRefGoogle Scholar
[9]Heikkilä, S., “First-order discontinuous differential equations with functional boundary conditions”, in Advances in nonlinear dynamics, Stability Control Theory Methods Appl. 5, (Gordon and Breach, Amsterdam, 1997) 273–281.Google Scholar
[10]Heikkilä, S. and Lakshmikantham, V., Monotone iterative techniques for discontinuous nonlinear differential equations (Marcel Dekker, New York, 1994).Google Scholar
[11]Kruger-Thiemer, E., “Formal theory of drug dosage regiments, I”, J. Theoret. Biol. 13 (1966) 212–235.CrossRefGoogle Scholar
[12]Kruger-Thiemer, E., “Formal theory of drug dosage regiments, II”, J. Theoret. Biol. 23 (1969) 169–190.CrossRefGoogle Scholar
[13]Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations (World Scientific, Singapore, 1989).CrossRefGoogle Scholar
[14]Liu, X. and Willms, A., “Impulsive stabilizability of autonomous systems”, J. Math. Anal. Appl. 187 (1994) 17–39.CrossRefGoogle Scholar
[15]Liz, E. and Nieto, J. J., “Positive solutions of linear impulsive differential equations”, Commun. Appl. Anal. 2 (1998) 565–571.Google Scholar
[16]Nieto, J. J., “Basic theory for nonresonance impulsive periodic problems of first order”, J. Math. Anal. Appl. 205 (1997) 423–433.CrossRefGoogle Scholar
[17]Rogovchenko, Y. V., “Nonlinear impulse evolution systems and applications to population models”, J. Math. Anal. Appl. 207 (1997) 300–315.CrossRefGoogle Scholar