# Existence of nodal solutions of a nonlinear fourth-order two-point boundary value problem

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People's Republic of China

Department of Basic Courses, Lanzhou Polytechnic College, Lanzhou 730050, People's Republic of China

*Boundary Value Problems* 2012, **2012**:31
doi:10.1186/1687-2770-2012-31

### Abstract

In this article, we give conditions on parameters *k, l *that the generalized eigenvalue problem *x″″ *+ *kx″ + lx *= λ*h*(*t*)*x*, 0 *< t < *1, *x*(*0*) = *x*(1) = *x*′(0) = *x*′(1) = 0 possesses an infinite number of simple positive eigenvalues and to each eigenvalue there corresponds an essential unique eigenfunction *ψ _{k }*which has exactly

*k -*1 simple zeros in (0,1) and is positive near 0. It follows that we consider the fourth-order two-point boundary value problem

*x″″*+

*kx″*+

*lx = f*(

*t,x*), 0

*< t <*1,

*x*(

*0*) =

*x*(1) =

*x*′(0) =

*x′*(1) = 0, where

*f*(

*t, x*) ∈

*C*([0,1]

*×*ℝ, ℝ) satisfies

*f*(

*t, x*)

*x >*0 for all

*x ≠*0,

*t*∈ [0,1] and lim

*(*

_{|x|→0 }f*t,x*)/

*x = a*(

*t*), lim

*(*

_{|x|→+∞ }f*t,x*)/

*x = b*(

*t*) or lim

*(*

_{x→-∞ }f*t,x*)/

*x*= 0 and lim

*(*

_{x→+∞}f*t,x*)/

*x = c*(

*t*) for some

*a*(

*t*),

*b*(

*t*),

*c*(

*t*) ∈

*C*([0,1], (0,+∞)) and

*t*∈ [0,1]. Furthermore, we obtain the existence and multiplicity results of nodal solutions for the above problem. The proofs of our main results are based upon disconjugate operator theory and the global bifurcation techniques.

**MSC (2000): **34B15.

##### Keywords:

disconjugacy theory; bifurcation; nodal solutions; eigenvalue*Boundary Value Problems*