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.

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