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Positive solutions for elastic beam equations with nonlinear boundary conditions and a parameter

Wenxia Wang*, Yanping Zheng, Hui Yang and Junxia Wang

Author Affiliations

Department of Mathematics, Taiyuan Normal University, Taiyuan, 030012, P.R. China

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Boundary Value Problems 2014, 2014:80  doi:10.1186/1687-2770-2014-80

Published: 9 April 2014

Abstract

This paper is concerned with the existence, nonexistence, and uniqueness of convex monotone positive solutions of elastic beam equations with a parameter λ. The boundary conditions mean that the beam is fixed at one end and attached to a bearing device or freed at the other end. By using fixed point theorem of cone expansion, we show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1">View MathML</a> such that the beam equation has at least two, one, and no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M4">View MathML</a>, respectively; furthermore, by using cone theory we establish some uniqueness criteria for positive solutions for the beam and show that such solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> depends continuously on the parameter λ. In particular, we give an estimate for critical value of parameter λ.

MSC: 34B18, 34B15.

Keywords:
elastic beam equation; positive solution; fixed point; cone