Existence principle for BVPs with state-dependent impulses

Abstract

The paper provides an existence principle for the Sturm-Liouville boundary value problem with state-dependent impulses \begin{gather*} z''(t) = f(t,z(t),z'(t)) \quad \text{for a.e. } t \in [0,T] \subset \mathbb R, \\ z(0) - az'(0) = c_1, \quad z(T) + bz'(T) = c_2, \\ z(\tau_i+) - z(\tau_i) = J_i(\tau_i,z(\tau_i)), \quad z'(\tau_i+) - z'(\tau_i-) = \mathcal M_i(\tau_i,z(\tau_i)), \end{gather*} where the points $\tau_1, \ldots, \tau_p$ depend on $z$ through the equations \begin{equation*} \tau_i = \gamma(z(\tau_i)), \quad i = 1,\ldots,p, \ p \in \mathbb N. \end{equation*} Provided $a,b \in [0,\infty)$, $c_j \in \mathbb R$, $j = 1,2$, and the data functions $f,J_i,\mathcal M_i$, $i=1,\ldots,p$, are bounded, transversality conditions for barriers $\gamma_i$, $i = 1,\ldots,p$, which yield the solvability of the problem, are delivered. An application to the problem with unbounded data functions is demonstrated.