The paper deals with the second order three-point nonlinear boundary value problem (1)

${u}^{\text{'}\text{'}}=f(x,u\left(x\right),{u}^{\text{'}}\left(x\right))-e\left(x\right)$,

$0<x<1$,

$u\left(0\right)=0$,

$u\left(\eta \right)=u\left(1\right)$, where

$f$ is a Carathéodory function and

$e$ is a Lebesgue integrable function. Provided

$f$ has at most linear growth in its phase variables, the author establishes conditions for the existence of solutions to (1) and for the uniqueness of problem (1). The proofs are based on the topological degree theory and the Leray-Schauder continuation theorem. A priori estimates are obtained by Wirtinger-type inequalities.