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We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

We study the existence of positive solutions for equations of the form where 0 < ω < π, subject to various non-local boundary conditions defined in terms of the Riemann–Stieltjes integrals. We prove the existence and multiplicity of positive solutions for these boundary value problems in both resonant and non-resonant cases. We discuss the resonant case by making a shift and considering an equivalent non-resonant problem.

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Non-trivial solutions of local and non-local Neumann boundary-value problems

MULTIPLE POSITIVE SOLUTIONS OF RESONANT AND NON-RESONANT NON-LOCAL FOURTH-ORDER BOUNDARY VALUE PROBLEMS

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Non-trivial solutions of local and non-local Neumann boundary-value problems

MULTIPLE POSITIVE SOLUTIONS OF RESONANT AND NON-RESONANT NON-LOCAL FOURTH-ORDER BOUNDARY VALUE PROBLEMS