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

AbstractWe consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville eigenvalue problems{1p(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)−β2limt→0+p(t)u‴(t)=0,γ2u″(1)+δ2limt→1−p(t)u‴(t)=0, where λ>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