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.
\[ \begin{aligned} -x''(t) = f(t,x(t)),&\quad t \in [0,1]\setminus\{t_1,\dots,t_n\}, \\ -\triangle x'| _{t = t_k} = I_k(x(t_k)), &\quad k = 1,\dots,n,\\ x(0) = \sum_{i=1}^{m-2} a_i x(\xi_i), &\quad x(1) = \sum_{i=1}^{m-2} b_ix(\xi_i). \end{aligned} \]
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.
Reviewer: Jan Tomeček (Olomouc)
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 |
Keywords:
multi-point boundary value problem; impulse effect; positive solution; fixed-point theory; existenceReferences:
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