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Open Access Research

Positive solutions for Sturm-Liouville BVPs on time scales via sub-supersolution and variational methods

Quan-Guo Zhang1, Xi-Ping He2 and Hong-Rui Sun1*

Author Affiliations

1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China

2 Center of Teaching Guidance, Gansu Radio and TV University, Lanzhou, Gansu, 730000, People’s Republic of China

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Boundary Value Problems 2013, 2013:123  doi:10.1186/1687-2770-2013-123

Published: 13 May 2013

Abstract

This paper is concerned with the existence of one and two positive solutions for the following Sturm-Liouville boundary value problem on time scales

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M1">View MathML</a>

Under a locally nonnegative assumption on the nonlinearity f and some other suitable hypotheses, positive solutions are sought by considering the corresponding truncated problem, constructing the variational framework and combining the sub-supersolution method with the mountain pass lemma.

MSC: 34B10, 34B18.

Keywords:
positive solution; Sturm-Liouville; time scales; sub-supersolution; variational methods