Second order boundary value problems on an unbounded domain

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,limt→∞u(t)=0 and u(0)=0,limt→∞u(t)=0, where f:[0,∞)×R2→Rf:[0,∞)×R2→R is Carathéodory with respect to L1[0,∞)L1[0,∞), p∈C[0,∞)∩C1(0,∞)p∈C[0,∞)∩C1(0,∞) and p(t)>0p(t)>0 for all t≥0t≥0. We obtain the existence of at least one solution using the Leray–Schauder Continuation Principle.