A four-point boundary value problem with singular ϕ-Laplacian

We prove that the four-point boundary value problem -[ϕ(u′)]′=f(t,u,u′),u(0)=αu(ξ),u(T)=βu(η),where f: [ 0 , T] × R 2 → R is continuous, α,β∈[0,1), 0 < ξ< η< T, and ϕ: (- a, a) → R (0 < a< ∞) is an increasing homeomorphism, which is always solvable. When instead of f is some g: [ 0 , T] × [ 0 , ∞) → [ 0 , ∞) , we obtain existence, localization, and multiplicity of positive solutions. Our approach relies on Schauder and Krasnoselskii’s fixed point theorems, combined with a Harnack-type inequality.