×

Periodic boundary value problems for first-order impulsive ordinary differential equations. (English) Zbl 1015.34010

The author considers the following nonlinear impulsive periodic boundary value problem. Let \(T>0\), \(J=[0,T]\), and \(t_1, t_2,\dots,t_p\in J\) the impulse instants with \(0=t_0<t_1<\dots<t_p<t_{p+1}=T\), \(J'=J-\{t_1,\dots,t_p\}\), \[ u'(t)+F(t,u(t))=0 \text{ a.e. } t\in J', \quad \Delta u(t_j)=I_j(u(t_j)), \;j=1,\dots,p, \quad u(0)=u(T),\tag{1} \] where \[ \Delta u(t_j)=u(t_j^+)-u(t_j^-)=u(t_j^+)-u(t_j), \quad j=1,\dots,p, \] \(F:J\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function such that for every \(j=1,\dots, p\) and \(u\in \mathbb{R}\) there exist the limits \[ \lim_{t\to t_j^-}F(t,u)=F(t_j,u), \lim_{t\to t_j^+}F(t,u), \] and the impulses \(I_j:\mathbb{R}\to \mathbb{R}\), \(j=1,\dots, p\), are continuous. Sufficient conditions when problem (1) is solvable are obtained.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A37 Ordinary differential equations with impulses
34B37 Boundary value problems with impulses for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Akhmetov, M.U.; Zafer, A., Controllability of the vallée – poussin problem for impulsive differential systems, J. optim. theory appl., 102, 263-276, (1999) · Zbl 0941.93004
[2] Akhmetov, M.U.; Zafer, A., Stability of the zero solution of impulsive differential equations by the Lyapunov second method, J. math. anal. appl., 248, 69-82, (2000) · Zbl 0965.34007
[3] Cabada, A.; Nieto, J.J.; Franco, D.; Trofimchuk, S.I., A generalization of the monotone method for second order periodic boundary value problem with impulses at fixed points, Dynam. contin. discrete impuls. systems, 7, 145-158, (2000) · Zbl 0953.34020
[4] Eloe. J, P.W.; H.B. Thompson, Henderson., Extremal points for impulsiver lidstone boundary value problems, Math. comput. modelling, 32, 687-698, (2000) · Zbl 0963.34022
[5] Franco, D.; Liz, E.; Nieto, J.J.; Rogovchenko, Y.V., A contribution to the study of functional differential equations with impulses, Math. nachr., 218, 49-60, (2000) · Zbl 0966.34073
[6] Franco, D.; Nieto, J.J., A new maximum principle for impulsive first-order problems, Internat. J. theoret. phys., 37, 1607-1616, (1998) · Zbl 0946.34024
[7] Franco, D.; Nieto, J.J., First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear anal., 42, 163-173, (2000) · Zbl 0966.34025
[8] Fu, X.; Qi, J.; Liu, Y., General comparison principle for impulsive variable time differential equations with applications, Nonlinear anal., 42, 1421-1429, (2000) · Zbl 0985.34011
[9] George, R.K.; Nandakumaran, A.K.; Arapostathis, A., A note on contrabillity of impulsive systems, J. math. anal. appl., 241, 276-283, (2000) · Zbl 0965.93015
[10] Guan, Z.H.; Chen, G.; Ueta, T., On impulsive control of a periodically forced chaotic pendulum system, IEEE trans. automat. control, 45, 1724-1727, (2000) · Zbl 0990.93105
[11] Jinli, S.; Yihai, M., Initial value problems for the second order mixed monotone type impulsive differential equations in Banach spaces, J. math. anal. appl., 247, 506-516, (2000) · Zbl 0964.34008
[12] Lakmeche, A.; Arino, O., Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynam. contin. discrete impuls. systems, 7, 265-287, (2000) · Zbl 1011.34031
[13] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[14] Lenci, S.; Rega, G., Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos solitons fractals, 11, 2453-2472, (2000) · Zbl 0964.70018
[15] Nenov, S., Impulsive controllability and optimization problems in population dynamics, Nonlinear anal., 36, 881-890, (1999) · Zbl 0941.49021
[16] Nieto, J.J., Basic theory for nonresonance impulsive periodic problems of fisrt order, J. math. anal. appl., 205, 423-433, (1997) · Zbl 0870.34009
[17] J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett., in press. · Zbl 1022.34025
[18] O’Regan, D., Existence theory for nonlinear ordinary differential equations, (1997), Kluwer Dordrecht · Zbl 1077.34505
[19] Rogovchenko, Y.V., Impulsive evolution systems: main results and new trends, Dynam. contin. discrete impuls. systems, 3, 57-88, (1997) · Zbl 0879.34014
[20] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[21] Smart, D.R., Fixed point theorems, (1974), Cambridge University Press Cambridge · Zbl 0297.47042
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.