Existence of one-signed solutions of nonlinear four-point boundary value problems

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$ - u'' + Mu = rg(t)f(u),u(0) = u(\varepsilon ),u(1) = u(1 - \varepsilon ) $$ and $$u'' + Mu = rg(t)f(u),u(0) = u(\varepsilon ),u(1) = u(1 - \varepsilon ) $$ , where ε ∈ (0, 1/2), M ∈ (0,∞) is a constant and r > 0 is a parameter, g ∈ C(0, 1], (0,+∞)), f ∈ C(?,?) with sf(s) > 0 for s ≠ 0. The proof of the main results is based upon bifurcation techniques.