A strongly nonlinear Neumann problem at resonance with restrictions on the nonlinearity just in one direction. (English) Zbl 1018.34016

Summary: Using topological degree techniques, the authors state and prove new sufficient conditions for the existence of a solution to the Neumann boundary value problem \[ (|x'|^{p-2} x')'+ f(t, x)+ h(t, x)= 0,\quad x'(0)= x'(1)= 0, \] where \(h\) is bounded, \(f\) satisfies a one-sided growth condition, \(f+h\) some sign condition, and the solutions to some associated homogeneous problem are not oscillatory. A generalization of the Lyapunov inequality is proved for a \(p\)-Laplacian equation. Similar results are given for the periodic problem.


34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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