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Impulsive resonance periodic problems of first order. (English) Zbl 1022.34025
A theorem on the existence of solutions to the nonlinear periodic boundary value problem $$u'(t)+ F(t, u(t))= 0,\quad t\ne t_1,\dots, t_p,\quad u(t^+_j)= u(t^-_j)+ I_j(u(t^-_j)),\quad j= 1,\dots, p,$$ is proved.

34B37Boundary value problems for ODE with impulses
34C25Periodic solutions of ODE
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[2] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations. (1995) · Zbl 0837.34003
[3] O’regan, D.: Existence theory for nonlinear ordinary differential equations. (1997)
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[5] Rogovchenko, Y. V.: Impulsive evolution systems: Main results and new trends. Dynam. contin. Discrete impuls. Systems 3, 57-88 (1997) · Zbl 0879.34014
[6] Akhmetov, M. U.; Zafer, A.: Successive approximation method for quasilinear impulsive differential equations with control. Appl. math. Lett. 13, No. 5, 99-105 (2000) · Zbl 1125.93349
[7] 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
[8] Liu, X.; Shen, J.: Asymptotic behavior of solutions of impulsive neutral differential equations. Appl. math. Lett. 12, No. 7, 51-58 (1999) · Zbl 0943.34065
[9] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order. J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009
[10] Mawhin, J.: Topological degree methods in nonlinear boundary value problems. 40 (1979) · Zbl 0414.34025