Maximum principles for periodic impulsive first order problems. (English) Zbl 0898.34010
The authors demonstrate some comparison principles for first-order ordinary differential equations with impulses at fixed moments. The results are applied to prove the existence of solutions using the method of upper and lower solutions to periodic boundary value problems for nonlinear differential equations with impulses:
\[
u'(t)=f(t,u(t)),\quad t\neq t_{k};\quad u(t_{k}^{+})=I_{k}(u(t_{k})), \quad k=1,\dots , p;\quad u(0)=u(T),
\]
where \(f:I\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function, \(I=[0,T]\), \(0<t_{1}<\cdots <t_{p}<T\), and \(I_{k}:\mathbb{R}\to \mathbb{R}\) are continuous functions, \(k=1,\dots, p\).
To establish the existence result, new definitions of upper and lower solutions, involving the usual concepts, are considered and the function \(f\) is assumed to satisfy a one-sided Lipschitz condition.
Finally, the validity of the monotone iterative technique to approximate extremal solutions to the considered problems in a sector is proved.
To establish the existence result, new definitions of upper and lower solutions, involving the usual concepts, are considered and the function \(f\) is assumed to satisfy a one-sided Lipschitz condition.
Finally, the validity of the monotone iterative technique to approximate extremal solutions to the considered problems in a sector is proved.
Reviewer: Eduardo Liz (Vigo)
MSC:
34A37 | Ordinary differential equations with impulses |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
34C25 | Periodic solutions to ordinary differential equations |
Keywords:
differential equations with impulses; maximum principles; upper and lower solutions; monotone iterative techniqueReferences:
[1] | Cabada, A., The monotone method for first order problems with linear and nonlinear boundary conditions, Appl. Math. Comput., 63, 163-186 (1994) · Zbl 0807.34022 |
[2] | Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman Boston · Zbl 0658.35003 |
[3] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 |
[4] | Nieto, J. J., Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl., 205, 423-433 (1997) · Zbl 0870.34009 |
[5] | Nieto, J. J.; Alvarez-Noriega, N., Periodic boundary value problems for nonlinear first order ordinary differential equations, Acta Math. Hungar., 71, 49-58 (1996) · Zbl 0853.34023 |
[6] | Pierson-Gorez, C., Impulsive differential equations of first order with periodic boundary conditions, Diff. Equations Dyn. Systems, 185-196 (1993) · Zbl 0868.34007 |
[7] | Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003 |
[8] | Smart, D. R., Fixed Points Theorems (1980), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0427.47036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.