Positive solutions of fourth-order nonlinear singular Sturm–Liouville eigenvalue problems

We consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville eigenvalue problems ⎧⎪ ⎪⎪⎪⎪⎪⎪⎨⎪ ⎪⎪⎪⎪⎪⎪⎩ 1 p(t) p(t)u (t) −λg(t)F (t, u, u ) = 0, 0 < t <1, α1u(0) − β1u (0) = 0, γ1u(1) +δ1u (1) = 0, α2u (0)− β2 limt→0+ p(t)u (t) = 0, γ2u (1) +δ2 limt→1− p(t)u (t) = 0, whereλ>0, g,p may be singular at t = 0 and/or 1. Moreover, F(t, x, y) may also have singularity at x = 0 and/or y = 0. By using fixed point theory in cones, an explicit interval for λ is derived such that for any λ in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and nonsingular cases. © 2006 Elsevier Inc. All rights reserved.