Portugaliae Mathematica


Full-Text PDF (217 KB) | Table of Contents | PM summary
Volume 64, Issue 3, 2007, pp. 255–279
DOI: 10.4171/PM/1786

A monotone method for fourth order boundary value problems involving a factorizable linear operator

P. Habets[1] and Margarita Ramalho[2]

(1) Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, Chemin du Cylcotron, 2, B-1348, LOUVAIN-LA-NEUVE, BELGIUM
(2) Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C1, Piso 3, 1749-016, LISBOA, PORTUGAL

We consider the nonlinear fourth order beam equation $$ u^{\iv}=f(t,u,u''), $$ with boundary conditions corresponding to the periodic or the hinged beam problem. In presence of upper and lower solutions, we consider a monotone method to obtain solutions. The main idea is to write the equation in the form $$ u^{\iv}-cu''+du=g(t,u,u''), $$ where $c$, $d$ are adequate constants, and use maximum principles and a suitable decomposition of the operator appearing in the left-hand side.

Keywords: Beam equation, fourth order boundary value problem, periodic solutions, maximum principle, monotone method, upper and lower solutions

Habets P, Ramalho M. A monotone method for fourth order boundary value problems involving a factorizable linear operator. Port. Math. 64 (2007), 255-279. doi: 10.4171/PM/1786