In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$ -u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon) $$ and $$ u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ), $$ where $\varepsilon \in (0,{1}/{2})$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C([0,1],(0,+\infty ))$, $f\in C(\mathbb {R},\mathbb {R})$ with $sf(s)>0$ for $s\neq 0$. The proof of the main results is based upon bifurcation techniques.
We prove the existence of solutions of four-point boundary value problems under the assumption that $f$ fulfils various combinations of sign conditions and no growth restrictions are imposed on $f$. In contrast to earlier works all our results are proved for the Carathéodory case.