Open Access Research

Bifurcation from interval and positive solutions for a class of fourth-order two-point boundary value problem

Wenguo Shen* and Tao He

Author Affiliations

Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou, 730050, People’s Republic of China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:170  doi:10.1186/1687-2770-2013-170

Published: 22 July 2013

Abstract

We consider the fourth-order two-point boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M3">View MathML</a>, which is not necessarily linearizable. We give conditions on the parameters k, l and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M4">View MathML</a> that guarantee the existence of positive solutions. The proof of our main result is based upon topological degree theory and global bifurcation techniques.

MSC: 34B15.

Keywords:
topological degree; fourth-order ordinary differential equation; bifurcation; positive solution; eigenvalue