The author considers the four-point boundary value problem (1)

${u}^{\text{'}\text{'}}=f(t,{u}^{\text{'}},{u}^{\text{'}\text{'}},s)$,

$u\in R$,

$t\in [a,b]$, (2)

$u\left(a\right)=u\left(c\right)$,

$u\left(d\right)=u\left(b\right)$,

$a<c\le d<b$, where

$s\in R$ is a bifurcation parameter and the continuous function

$f(t,x,y,s)$ is increasing in

$s$. Using the techniques of lower and upper solutions as well as the degree theory, the author proves an Ambrosetti-Prodi-like result for (1), (2) considered in some domain

$D=\{x\in {C}^{2}\left(J\right):-{\alpha}^{2}\ge x\left(t\right)\le {\beta}^{2}\forall t\in J\}$: namely, under certain conditions imposed on

$f$ there exists

${s}_{0}\in (p,{s}_{1})$ such that 1) for

$s<{s}_{0}$ BVP (1), (2) has no solution in

$D$; 2) for

$s={s}_{0}$ BVP (1), (2) has at least one solution; 3) for

$s\in (0;{s}_{1}]$ has at least two solutions.