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Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. (English) Zbl 1159.34022

The authors consider the impulsive boundary value problem for ordinary differential equation of second order
\[ \begin{aligned} -x''(t) = f(t,x(t)),&\quad t \in [0,1]\setminus\{t_1,\dots,t_n\}, \\ -\triangle x'| _{t = t_k} = I_k(x(t_k)), &\quad k = 1,\dots,n,\\ x(0) = \sum_{i=1}^{m-2} a_i x(\xi_i), &\quad x(1) = \sum_{i=1}^{m-2} b_ix(\xi_i). \end{aligned} \]
where \(0 < t_1 < \dots < t_n < 1\), \(f \in C([0,1]\times {\mathbb R}^+,{\mathbb R}^+)\), \(I_k \in C({\mathbb R}^+,{\mathbb R}^+)\) for \(k = 1,\dots,n\), \(a_i\), \(b_i \in {\mathbb R}\) are positive, \(\xi_i \neq t_k\) for \(i=1,\dots,m-2\), \(j = 1,\dots,n\). Sufficient conditions to ensure the existence of at least one or two positive solutions are given. The proofs are based on a fixed-point theorem in cones.

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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