Non-local boundary value problems of arbitrary order

We give a new unified method of establishing the existence ofmultiple positive solutions for a large number of non-lineardifferential equations of arbitrary order with any allowed numberof non-local boundary conditions (BCs). In particular, we areable to determine the Green's function for these problems withvery little explicit calculation, which shows that studyinga more general version of a problem with appropriate notationcan lead to a simplification in approach. We obtain existenceand non-existence results, some of which are sharp, and givenew results for both non-local and local BCs. We illustratethe theory with a detailed account of a fourth-order problemthat models an elastic beam and also determine optimal valuesof constants that appear in the theory.