Existence of positive solutions for singular fourth-order three-point boundary value problems

• Published in: Advances in difference equations Vol. 2013; no. 1; p. 51
• Main Authors: Sun, Yan, Zhu, Cun
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 08.03.2013; BioMed Central Ltd
• Subjects: Analysis; Difference and Functional Equations; Functional Analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; existence; positive solution; cone; boundary value problems
• ISSN: 1687-1847; 1687-1847
• DOI: 10.1186/1687-1847-2013-51

In this article, we consider the boundary value problem u ( 4 ) ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , subject to the boundary conditions u ( 0 ) = u ′ ( 0 ) = u ″ ( 0 ) = 0 and u ″ ( 1 ) − α u ″ ( η ) = λ . In this setting, 0 < η < 1 and α ∈ [ 0 , 1 η ) are constants and λ ∈ [ 0 , + ∞ ) is a parameter. By imposing a sufficient structure on the nonlinearity f ( t , u ) , we deduce the existence of at least one positive solution to the problem. The novelty in our setting lies in the fact that f ( t , u ) may be singular at t = 0 and t = 1 . Our results here are achieved by making use of the Krasnosel’skii fixed point theorem. We conclude with examples illustrating our results and the improvements that they present. MSC: 34B15, 34B25, 34B18.