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Multiplicity of positive solutions to period boundary value problems for second order impulsive differential equations. (English) Zbl 1193.34058

The authors consider the impulsive boundary value problem
\[ \begin{aligned} &-x''(t) + \rho^2x = f(t,x), \quad t\neq t_k, t \in [0,2\pi],\\ &\Delta x(t_k) = I_k(x(t_k)),\;-\Delta x'(t_k) = J_k(x(t_k)),\;k = 1,\dots,p, \\ &x(0) = x(2\pi),\quad x'(0) = x'(2\pi), \end{aligned} \]
where \(0 < t_1 < \dots < t_p < 2\pi\), \(\rho > 0\); the functions \(f\), \(I_k\), \(J_k\) are continuous. Existence and multiplicity results for positive solutions are obtained. As a main tool the fixed point theorem in cones is used.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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