Existence of solutions for a cantilever beam problem

We are concerned with the fourth-order nonuniform cantilever beam problem I (x)W ∇(x) ∇ = f x,W(x) , W(a) = W (a) = 0, I (b)W ∇(b) = (I (x)W ∇(x)) |x=b = 0. Under various assumptions on f we prove the existence of at least one, two, or three positive solutions for this measure chain boundary value problem using the Green function approach. © 2005 Elsevier Inc. All rights reserved