A strongly nonlinear Neumann problem at resonance with restrictions on the nonlinearity just in one direction

Using topological degree techniques, we state and prove new sufficientconditions for the existence of a solution of the Neumann boundary valueproblem$$(|x'|^{p-2} x')' +f(t, x)+ h(t, x) =0,\quadx'(0) = x'(1)=0,$$when $h$ is bounded, $f$ satisfies a one-sided growth condition, $f + h$ somesign condition, and the solutions of some associated homogeneous problem arenot oscillatory. A generalization of Lyapunov inequality is proved for a $p$-Laplacian equation. Similar results are given for the periodic problem.