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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.

MSC:

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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