A second-order boundary value problem with impulsive effects on an unbounded domain (Q949717)

The authors consider the impulsive boundary value problem \[ \begin{aligned} &(p(t)u'(t))' = f(t,u(t)),\quad t \in (0,\infty)\setminus\{t_1,\ldots,t_n\},\\ &\triangle u'(t_k) = I_k(u(t_k)), \quad k = 1,\ldots,n,\\ &\alpha u(0) - \beta\lim_{t\to 0^+} p(t)u'(t) = 0,\\ &\gamma\lim_{t\to\infty} u(t) + \delta\lim_{t\to\infty} p(t)u'(t) = 0, \end{aligned} \] where \(f: [0,\infty) \times {\mathbb R} \to {\mathbb R}\) is continuous, \(p \in C[0,\infty) \cap C^1(0,\infty)\) is positive on \((0,\infty)\), \(I_k : {\mathbb R} \to {\mathbb R}\) are Lipschitz continuous for \(k = 1,\dots,n\). The non-resonant case is considered. Sufficient conditions ensuring the existence of at least one solution to this problem are found. The proofs of these results are based on the Krasnoselskii fixed point theorem.