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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.

34B37Boundary value problems for ODE with impulses
34B18Positive solutions of nonlinear boundary value problems for ODE
Full Text: DOI
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