The authors consider semilinear boundary value problems
where , denotes either the Dirichlet or the Neumann or the periodic boundary conditions, respectively, and is the first eigenvalue of the corresponding linear problem , , , . The nonlinear function is supposed to be bounded and, in some cases, satisfies additional differentiability assumptions and asymptotic conditions. The authors emphasize the dependence of on the derivative of the solution in order to show the qualitative difference of this case and the Landesman-Lazer-type problem in which the nonlinearity depends only on the solution . The authors establish the solvability of the problem (1), (2).