Multi-point boundary value problems on an unbounded domain at resonance

We consider the second-order nonlinear differential equation (p(t)u′(t))′=f(t,u(t),u′(t)),a.e. in (0,∞), satisfying two sets of boundary conditions: u′(0)=0,∑i=1nκiu(Ti)=limt→∞u(t) and u(0)=0,∑i=1nκiu(Ti)=limt→∞u(t), where n≥1n≥1, f:[0,∞)×R2→Rf:[0,∞)×R2→R is Carathéodory with respect to L1[0,∞)L1[0,∞). The parameters in the multi-point boundary conditions are such that the corresponding differential operator is non-invertible but nevertheless is a Fredholm map of index zero. As a result the coincidence degree theory can be applied to establish existence theorems.