This paper investigates the solvability of the second-order boundary value problems with the one-dimensional p-Laplacian at resonance on a half-line
⎩⎨⎧(c(t)ϕp(x′(t)))′=f(t,x(t),x′(t)),0<t<∞,x(0)=i=1∑nμix(ξi),t→+∞limc(t)ϕp(x′(t))=0
and
{(c(t)ϕp(x′(t)))′+g(t)h(t,x(t),x′(t))=0,0<t<∞,x(0)=∫0∞g(s)x(s)ds,t→+∞limc(t)ϕp(x′(t))=0
with multi-point and integral boundary conditions, respectively, where ϕp(s)=∣s∣p−2s, p>1. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results