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

DOI: http://dx.doi.org/10.12775/TMNA.2002.021

#### Abstract

Using topological degree techniques, we state and prove new sufficient

conditions for the existence of a solution of the Neumann boundary value

problem

$$

(|x'|^{p-2} x')' +f(t, x)+ h(t, x) =0,

\quad

x'(0) = x'(1)=0,

$$

when $h$ is bounded, $f$ satisfies a one-sided growth condition, $f + h$ some

sign condition, and the solutions of some associated homogeneous problem are

not oscillatory. A generalization of Lyapunov inequality is proved for

a $p$-Laplacian equation. Similar results are given for the periodic problem.

conditions for the existence of a solution of the Neumann boundary value

problem

$$

(|x'|^{p-2} x')' +f(t, x)+ h(t, x) =0,

\quad

x'(0) = x'(1)=0,

$$

when $h$ is bounded, $f$ satisfies a one-sided growth condition, $f + h$ some

sign condition, and the solutions of some associated homogeneous problem are

not oscillatory. A generalization of Lyapunov inequality is proved for

a $p$-Laplacian equation. Similar results are given for the periodic problem.

#### Keywords

Neuman problem; periodic solutions; p-Laplacian equation; topological degree; Lyapunov inequality

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