For part I see [the authors, ibid. 1, No. 2, 295-302 (1993; Zbl 0787.34022)]. The authors study the boundary value problem \[ u''+ k^2u+g(x,u)=h(x), \quad u(0)= u(\pi)=0, \] where \(k\geq 1\) is an integer, the Carathéodory function \(g\) for almost all \(x\) satisfies the growth restriction \(\limsup_{| u|\to \infty}| g(x,u)/u |\leq \Gamma(x)\) with \(\| \Gamma \|_{L^1}\leq 2k\). Conditions of Landesman-Lazer type are imposed, which originate from \textit{E. M. Landesman} and \textit{A. C. Lazer} [J. Math. Mech. 19, 609-623 (1970; Zbl 0193.39203)]. The case \(k=1\) is treated separately under the weaker assumption \(\|\Gamma \|_{L^1}\leq 4\).