multiple solutions of sturm–liouville boundary value problem via lower and upper solutions and variational methods

in this paper, we prove the existence of multiple solutions for second order sturm–liouville

boundary value problem

 {−lu = f (x, u), x ∈ [0, 1]

r₁(u) = 0, r₂(u) = 0,

where lu = (p(x)u′)′ − q(x)u is a sturm–liouville operator, r₁(u) = αu′(0) − βu(0), r₂(u) = γ u′(1)+σu(1). the technical approach is fully based on lower and upper solutions and variational methods. the interesting point is that the existence of four solutions and seven solutions is given.