Multiple Positive Solutions of Singular Nonlinear Sturm‐Liouville Problems with Carathéodory Perturbed Term

Yuefeng Han,Lishan Liu,Yonghong Wu,Xinguang Zhang

doi:10.1155/2012/160891

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.

Highlights

In this paper, we consider the existence of multiple positive solutions for the following fourthorder singular Sturm-Liouville boundary value problem involving a perturbed termLu 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, 1.1Journal of Applied Mathematics where αi, βi, γi, δi ≥ 0 and βiγi αiγi αiδi > 0, i 1, 2, and L denotes the linear operator Lu : ru − qu, r ∈ C1 0, 1, 0, ∞ and q ∈ C 0, 1, 0, ∞ and q ∈ C 0, 1, 0, ∞
It mainly describes the deformation of an elastic beam for g t, u, u ≡ 0; for example, under the Lidstone boundary condition, u 0 u 1 u 0 u 1 0, 1.2 problem 1.1 is used to model such phenomena as the deflection of an elastic beam supported at the endpoints; see 1, 3, 5, 7–11
If the boundary condition of 1.1 is a Focal boundary condition, it describes the deflection of an elastic beam having both end-points fixed, or having one end supported and the other end clamped with sliding clamps

Summary

Introduction

We consider the existence of multiple positive solutions for the following fourthorder singular Sturm-Liouville boundary value problem involving a perturbed termLu 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, 1.1Journal of Applied Mathematics where αi, βi, γi, δi ≥ 0 and βiγi αiγi αiδi > 0, i 1, 2, and L denotes the linear operator Lu : ru − qu , r ∈ C1 0, 1 , 0, ∞ and q ∈ C 0, 1 , 0, ∞ and q ∈ C 0, 1 , 0, ∞. The perturbed term, g : 0, 1 × 0, ∞ × −∞, ∞ → 0, ∞ , satisfies global Caratheodory’s conditions.

Results

Conclusion