Multiple positive solutions of singular nonlinear Sturm-Liouville problems with Carathéodory perturbed term

By employing a well-known fixed point theorem,we establish the existence of multiple positive solutions for the following fourth-order singular differential equation lu = p(t)f(t,u(t),u″(t)) - g(t,u(t),u″(t)),0 < t < 1,α 1u(0) - β 1u′(0) = 0,γ 1u(1) + δ 1u′(1) = 0,α 2u″(0) - β 2u‴(0) = 0,γ 2u″(1) + δ 2u‴(1) = 0,with α i,β i,γ i,δ i ≥ 0 and β iγ i + α iγ i + α iδ i > 0,i = 1,2,where l denotes the linear operator lu := (ru‴)′ - qu″,r ∈ c 1([0,1],(0,+∞)),and q ∈ c([0,1],[0,+∞)). this equation is viewed as a perturbation of the fourth-order sturm-liouville problem,where the perturbed term g : (0,1) × [0,+∞) × (-∞,+∞) → (-∞,+∞) only satisfies the global carath́eodory conditions,which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points. copyright © 2012 yuefeng han et al.