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Open Access Research

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

Wenguo Shen

Author affiliations

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

Citation and License

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

Published: 20 March 2012

Abstract

In this article, we give conditions on parameters k, l that the generalized eigenvalue problem x″″ + kx″ + lx = λh(t)x, 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0 possesses an infinite number of simple positive eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/31/mathml/M1">View MathML</a> and to each eigenvalue there corresponds an essential unique eigenfunction ψk 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 limx→-∞ f(t,x)/x = 0 and limx→+∞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