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Positivity and variational methods in second and fourth order boundary value problems
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Orientador(es)
Resumo(s)
We study the existence of solutions for a nonlocal singular second order ordinary differential
equation. We obtain results through Krasnoselskii’s fixed point Theorem and using
some properties of the eigenvalues of the underlying singular linear problem and, on a
different approach, through the monotone method associated with well-ordered lower and
upper solutions.
We deal with second and fourth order problems in infinite intervals, where we prove
the existence of an homoclinic or an heteroclinic solution. For the second order we consider
both superlinear and bounded nonlinearities, and prove existence results through
variational methods. A non-variational approach was made for a second order problem
with a dissipative term and a p-laplacian problem was also adressed. Simpler fourth order
bvp’s were also tackled from a variational point of view.
We also analyse fourth order boundary value problems related to beam deflection
theory, generalizing some well known results for the second order. We analysed two types
of problems: the case where the correspondent fourth order operator can be decomposed
in two positive second order operators and the case where that cannot be done. The
results are obtained through topological arguments in association with lower and upper
solutions.
Descrição
Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2010
Palavras-chave
Problemas de valores na fronteira Singularidades (Matemática) Métodos variacionais Análise matemática Teses de doutoramento - 2010