Existence of positive solutions for a fourth-order multi-point beam problem on measure chains

This article concerns the fourth-order multi-point beam problem $$displaylines{ (EIW^{Delta abla }) ^{ abla Delta }(x)=m(x)f(x,W(x)),quad xin [x_{1},x_{n}]_{mathbb{X}} cr W( ho ^2(x_{1}))=sum_{i=2}^{n-1}a_iW(x_i),quad W^{Delta}( ho ^2(x_{1}))=0, cr (EIW^{Delta abla }) (sigma (x_{n}))=0,quad (EIW^{Delta abla })^{ abla }(sigma(x_{n})) =sum_{i=2}^{n-1}b_i(EIW^{Delta abla })^{ abla}(x_i). }$$ Under various assumptions on the functions $f$ and $m$ and the coefficients $a_i$ and $b_i$ we establish the existence of one or two positive solutions for this measure chain boundary value problem using the Green's function approach.