The authors develop a new unifying theory, which allows to study the existence of multiple positive solutions for semi-positone problems for differential equations of arbitrary order with a mixture of local and nonlocal boundary conditions. The nonlocal boundary conditions are quite general, they involve positive linear functionals on the space \(C[0,1]\), given by Stieltjes integrals. These general BVPs are studied via a Hammerstein integral equation of the form
\[ u(t)=\int_0^1 k(t,s)g(s) f(s,u(s))ds, \]
where \(k\) is the corresponding Green’s function which is supposed to have certain positivity properties, and \(f:[0,1]\times [0,\infty)\to\mathbb R\) satisfies \(f(t,u)\geq -A\) for some \(A>0\). The proofs are based on fixed point index results. Examples of a second order and a fourth order problem are presented. Here, the authors determine explicit values of constants that appear in the theory.