Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions

J. R. L. Webb a1, Gennaro Infante a2 and Daniel Franco a3
a1 Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK (
a2 Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy (
a3 Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, c/ Juan del Rosal 12, Madrid 28040, Spain (

Article author query
webb jrl   [Google Scholar] 
infante g   [Google Scholar] 
franco d   [Google Scholar] 


We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.

(Published Online July 14 2008)
(Received October 20 2006)
(Accepted February 7 2007)