Doutoramento em Matemática
Problemas de valores na fronteira de ordem superior em domínios não limitados
Hugo Carrasco

Orientação: Feliz Manuel Barrão Minhós

The relative scarcity of results that guarantee the existence of solutions for BVP on unbounded domains, contrasts with the high applicability on real problems of differential equations defined on the half-line or on the whole real line. It is this gap the main reason that led to this work.

The differential equations studied vary from second order to higher orders and they can be discontinuous on time. Different types of boundary conditions will be discussed herein, for example, Sturm-Liouville, homoclinic, Lidstone and functional conditions.

The non-compactness of the time interval and the possibility of study unbounded functions will require the redefinition of the admissible Banach spaces. In fact the space considered and the functional framework assumed define the set of admissible solutions for each problem under a main goal: the functions must remain bounded for the space and the norm in consideration. This is achieved by defining some weight functions (polynomial or exponential) in the space or assuming some asymptotic behavior.

In addition to the existence, solutions will be localized in a strip. The lower and upper solutions method will play an important role, and combined with other tools like the one-sided Nagumo growth conditions, Green’s functions or Schauder’s fixed point theorem, provide the existence and location results for differential equations with various boundary conditions.

Different applications to real phenomena will be presented, most of them translated into classical equations as Duffing, Bernoulli-Eulerv. Karman, Fisher Kolmogorov, Swift-Hohenberg, Emden-Fowler or Falkner-Skan-type equations.

All these applications have a common denominator: they are defined in unbounded intervals and the existing results in the literature are scarce or proven only numerically in discrete problems.

Keywords : Unbounded intervals, Lower and upper solutions, Nagumo condition, Green’s function, Fixed point theory