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

Abstract

We consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville eigenvalue problems { 1 p ( t ) ( p ( t ) u ‴ ( t ) ) ′ − λ g ( t ) F ( t , u , u ″ ) = 0 , 0 < t < 1 , α 1 u ( 0 ) − β 1 u ′ ( 0 ) = 0 , γ 1 u ( 1 ) + δ 1 u ′ ( 1 ) = 0 , α 2 u ″ ( 0 ) − β 2 lim t → 0 + p ( t ) u ‴ ( t ) = 0 , γ 2 u ″ ( 1 ) + δ 2 lim t → 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.