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 $f:[0,1]\times [0,\infty)\to\Bbb R$ satisfies $f(t,u)\ge -A$ 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.