Impulsive problems on the half-line with infinite impulse moments (Q2627899)

The author considers the impulsive differential equation of the second order with impulses at fixed times on the half-line \[ \begin{aligned} & u''(t) = f(t,u(t),u'(t)) \quad \text{a.e.}\;t \in (0,\infty), \\ & \triangle u(t_k) = I_{0k}(t_k,u(t_k),u'(t_k)), \quad \triangle u'(t_k) = I_{1k}(t_k,u(t_k),u'(t_k)), \quad k = 1,2,\ldots, \\ & u(0) = A, \lim_{t\to\infty} u'(t) = B, \end{aligned} \] where \(I_{ik}\) are continuous, \(f\) satisfies some kind of Carathéodory conditions, \(0< t_1< t_2< \ldots\), \(\lim_{k\to\infty} t_k = \infty\), \(A,B\) are real numbers. Sufficient conditions ensuring the existence of at least one solution to this boundary value problem are obtained. The proofs are based on Schauder's fixed point theorem. Illustrating example is given as well.