The paper contains some existence and multiplicity results of Ambrosetti-Prodi type on nonlinear boundary value problems the \[ x''(t)+x(t)+f(t,x(t),x'(t))=s\varphi (t), \quad t \in [0,\pi], \;x(0)=x(\pi)=\gamma, \] where \(f\) (a Carathéodory function) satisfies a condition which includes Bernstein-Nagumo and Wintner growth conditions, \(s\) and \(\gamma\) are real constants and \(\varphi\) is the principal positive eigenfunction of the corresponding homogeneous linear problem. In the proofs, the author uses lower and upper solutions and Leray-Schauder degree theory.