Non-local boundary value problems of arbitrary order

We give a new unified method of establishing the existence of multiple positive solutions for a large number of non-linear differential equations of arbitrary order with any allowed number of non-local boundary conditions (BCs). In particular, we are able to determine the Green's function for t...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2009-02, Vol.79 (1), p.238-258
Main Authors: Webb, J. R. L., Infante, Gennaro
Format: Article
Language:English
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Summary:We give a new unified method of establishing the existence of multiple positive solutions for a large number of non-linear differential equations of arbitrary order with any allowed number of non-local boundary conditions (BCs). In particular, we are able to determine the Green's function for these problems with very little explicit calculation, which shows that studying a more general version of a problem with appropriate notation can lead to a simplification in approach. We obtain existence and non-existence results, some of which are sharp, and give new results for both non-local and local BCs. We illustrate the theory with a detailed account of a fourth-order problem that models an elastic beam and also determine optimal values of constants that appear in the theory.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms/jdn066