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

Jean Mawhin, David Ruiz

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.

Keywords


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

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