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Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations. (English) Zbl 1156.34019
The authors consider the impulsive boundary value problem $$\aligned &-x'' + Mx = f(t,x), \quad 0 < t < 2\pi,\ t \not= t_k\\ &\triangle x\vert _{t=t_k} = I_k(x(t_k)), \quad -\triangle x'\vert _{t=t_k} = J_k(x(t_k)), \quad k = 1,2,\dots,l,\\ &x(0) = x(2\pi),\quad x'(0) = x'(2\pi), \endaligned$$ where $0 < t_1 < \dots < t_l < 2\pi$, $M > 0$, $f: [0,2\pi] \times [0,\infty) \to [0,\infty)$, $I_k : [0,\infty) \to R$, $J_k : [0,\infty) \to [0,\infty)$ are continuous functions. Sufficient conditions for the existence of at least two positive solutions are found. The arguments are based on fixed point index theory in cones. An example illustrating the results is included.

MSC:
34B37Boundary value problems for ODE with impulses
34B18Positive solutions of nonlinear boundary value problems for ODE
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References:
[1] Agarwal, R. P.; O’regan, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. math. Comput. 114, 51-59 (2000) · Zbl 1047.34008
[2] Cong, Fuzhong: Periodic solutions for second order differential equations. Appl. math. Lett. 18, 957-961 (2005) · Zbl 1094.34523
[3] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045
[4] Jiang, Daqing: On the existence of positive solutions to second order periodic BVPs. Acta math. Sci. 18, 31-35 (1998)
[5] Jiang, Daqing; Wei, Junjie: Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations. Nonlinear anal. 50, 885-898 (2002) · Zbl 1014.34049
[6] Hristova, S. G.; Bainov, D. D.: Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential-difference equations. J. math. Anal. appl. 1997, 1-13 (1996) · Zbl 0849.34051
[7] Lin, Xiaoning; Jiang, Daqing: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. math. Anal. appl. 321, 501-514 (2006) · Zbl 1103.34015
[8] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[9] Lee, E. K.; Lee, Y. H.: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation. Appl. math. Comput. 158, 745-759 (2004) · Zbl 1069.34035
[10] Zhang, Zhongxin; Wang, Junyu: On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations. J. math. Anal. appl. 281, 99-107 (2003) · Zbl 1030.34024
[11] Wei, Zhongli: Periodic boundary value problem for second order impulsive integrodifferential equations of mixed type in Banach space. J. math. Appl. anal. 195, 214-229 (1995) · Zbl 0849.45006