Abstract
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, α1u(0) − β1u′(0) = 0, γ1u(1) + δ1u′(1) = 0, α2u′′(0) − β2u′′′(0) = 0, γ2u′′(1) + δ2u′′′(1) = 0, with αi, βi, γi, δi ≥ 0 and βiγi + αiγi + αiδi > 0, i = 1,2, where L denotes the linear operator Lu : = (ru′′′)′ − qu′′, r ∈ C1([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 Carathéodory 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.
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