Nonlinear Differential Equations on Bounded and Unbounded Domains

Differential equations represent one of the strongest connections between Mathematics and real life. This is due to the fact that almost all the physical phenomena, as well as many other in economy, biology or chemistry, are modelled by differential equations. This Thesis includes a detailed study of nonlinear differential equations, both on bounded and unbounded domains. In particular, we analyze the qualitative properties of the solutions of nonlinear differential equations, focusing on the study of constant sign solutions on the whole domain of definition or, at least, on some subset of it. The main technique is based on the construction of an abstract formulation included into functional analysis, in which the solutions of the differential equations coincide with the fixed points of certain operators

Lower and upper solutions for even order boundary value problems

In this paper, we prove the existence of solutions of nonlinear boundary value problems of arbitrary even order using the lower and upper solutions method. In particular, we point out the fact that the existence of a pair of lower and upper solutions of a considered problem could imply the existence of solution of another one with different boundary conditions. We consider Neumann, Dirichlet, mixed and periodic boundary conditionsThe authors were partially supported by Xunta de Galicia (Spain), project EM2014/032 and AIE, Spain and FEDER, grant MTM2016-75140-PS

Relationship of the Green's functions related to the Hill's equation coupled to different boundary value conditions

In this paper we will deduce several properties of the Green's functions related to the Hill's equation coupled to various boundary value conditions. In particular, the idea is to study the Green's functions of the second order differential operator coupled to Neumann, Dirichlet, Periodic and Mixed boundary conditions, by expressing the Green's function of a given problem as a linear combination of the Green's function of the other ones. This will allow us to compare different Green's functions when they have constant sign. Finally, such properties of the Green's function of the linear problem will be fundamental to deduce the existence of solutions to the nonlinear problem. The results are derived from the fixed point theory applied to related operators defined on suitable cones in Banach spaces

Physical therapy in unilateral and bilateral vestibular hypofunction

[ES] Introducción: La rehabilitación vestibular (RV) basada en la terapia física, tiene el objetivo, en el caso de patología vestibular, de inducir la compensación del sistema nervioso central (SNC) a nivel de núcleos vestibulares y de otros niveles del SNC. Incluye ejercicios de habituación, adaptación y sustitución vestibular, ejercicios para mejorar el equilibrio y el control postural dinámico y ejercicios para el acondicionamiento general. En este capítulo discutimos los recientes avances sobre el adiestramiento del equilibrio y de la marcha, la estabilidad de la mirada y la habituación, en el contexto de los trastornos vestibulares uni y bilaterales. Método: Revisión narrativa. Resultados: Los ejercicios se prescriben para mejorar la función; fortaleciendo, y favoreciendo la flexibilidad y la resistencia, a través de la adaptación del RVO, la habituación, la sustitución sensorial, la marcha y el equilibrio postural. Son más eficaces los programas personalizados que los genéricos. El cumplimiento mejora con la personalización y las visitas de seguimiento a un fisioterapeuta. Discusión/Conclusiones: La RV permite mejorar el déficit funcional y los síntomas subjetivos derivados de la hipofunción vestibular periférica uni y bilateral, así como las alteraciones del equilibrio de origen central. Los objetivos de la RV consisten en reducir los síntomas para mejorar la estabilidad postural y de la mirada (particularmente durante los movimientos de la cabeza) y devolver al individuo a sus actividades normales, incluyendo la actividad física, la conducción y el trabajo habitual. Los médicos deben ofrecer la RV a quienes muestren limitaciones funcionales relacionadas con un déficit vestibular, pues actualmente se considera el tratamiento estándar en la disfunción vestibular periférica

Existence and multiplicity results for some eigenvalue generalized Hammerstein equations

This paper considers the existence and multiplicity of fixed points for an integral operatorwith a positive parameter. The existence of solutions for these Hammerstein equations is obtained by fixed point index theory on new type of cones. Therefore some assumptions must hold only for, at least, one of the derivatives of the kernel or, even, for the kernel on a subset of the domain. Assuming some asymptotic conditions on the nonlinearity f, we get sufficient conditions for multiplicity of solutions. Two examples will illustrate the potentialities of the main results, namely the fact that the kernel function and/or some derivatives may only be positive on some subintervals, which can degenerate to a point. Moreover, an application of our method to general Lidstone problems improves the existent results in the literature in this field