Existence of solutions for non-linear boundary value problems

This Thesis contains a detailed collection of the different results proved by the author in her predoctoral stage. The interest of the non-linear differential equations is well-known. This is due to their applications in different fields, such as physics, economy, medicine, biology or chemistry. It is very important to make a precise study of the existence of solutions for this kind of problems, as well as their uniqueness or multiplicity. We focus on the qualitative analysis of diverse boundary value problems, both linear and non-linear ones. Indeed, in most of the cases, our aim is to prove the existence of constant sign solutions in their definition interval. This interest comes from the constant sign of many of the magnitudes which are modelled by this kind of problems

The eigenvalue characterization for the constant sign Green’s functions of (k,n−k)(k,n−k) problems

This paper is devoted to the study of the sign of the Green’s function related to a general linear nth-order operator, depending on a real parameter, Tn[M]Tn[M], coupled with the (k,n−k)(k,n−k) boundary value conditions. If the operator Tn[M¯]Tn[M¯] is disconjugate for a given M̄, we describe the interval of values on the real parameter M for which the Green’s function has constant sign. One of the extremes of the interval is given by the first eigenvalue of the operator Tn[M¯]Tn[M¯] satisfying (k,n−k)(k,n−k) conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of (k−1,n−k+1)(k−1,n−k+1) and (k+1,n−k−1)(k+1,n−k−1) problems. Moreover, if n−kn−k is even (odd) the Green’s function cannot be nonpositive (nonnegative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green’s functions for particular operators. Our method avoids the necessity of calculating the expression of the Green’s function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator Tn[M¯]Tn[M¯] for a given M̄ cannot be eliminatedAlberto Cabada was partially supported by Ministerio de Educación, Cultura y Deporte, Spain, and FEDER, project MTM2013-43014-P. Lorena Saavedra was partially supported by Ministerio de Educación, Cultura y Deporte, Spain, and FEDER, project MTM2013-43014-P, and Plan I2C scholarship, Consellería de Educación, Cultura e O.U., Xunta de Galicia, and FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain. The authors would also like to express their special thanks to the reviewer of the paper for his/her remarks, which considerably improved the content of this paperS

Characterization of constant sign Green's function for a two-point boundary-value problem by means of spectral theory

This article is devoted to the study of the parameter’s set where the Green’s function related to a general linear nth-order operator, depending on a real parameter, Tn[M], coupled with many different two point boundary value conditions, is of constant sign. This constant sign is equivalent to the strongly inverse positive (negative) character of the related operator on suitable spaces related to the boundary conditions. This characterization is based on spectral theory, in fact the extremes of the obtained interval are given by suitable eigenvalues of the differential operator with different boundary conditions. Also, we obtain a characterization of the strongly inverse positive (negative) character on some sets, where non homogeneous boundary conditions are considered. To show the applicability of the results, we give some examples. Note that this method avoids the explicit calculation of the related Green’s function.This research was Partially supported by AIE Spain and FEDER, grants MTM2013-43014-P, MTM2016-75140-P. The second author was supported by FPU scholarship, Ministerio de Educaci´on, Cultura y Deporte, SpS

Diabetes Mellitus tipo 2 como factor de riesgo de catarata en mayores de 40 años atendido en el Hospital II-2 Tarapoto 2012 - 2014

Esta investigación buscó establecer a diabetes mellitus tipo 2 como factor de riesgo de catarata en mayores de 40 años atendidos en el Hospital II-2 Tarapoto, se seleccionó a la población de forma aleatoria simple, la investigación se realizó mediante la revisión de historias clínicas del servicio de oftalmología. Método: todos los casos de cataratas, de manera aleatoria, aplicando el programa Epidat 3.1 se escogió la muestra de 105 casos. El mismo procedimiento se realizó para los controles hasta obtener los 315 controles. Resultados: al establecer el riesgo de desarrollo de cataratas en la diabetes mellitus observamos el valor de OR es menor que la unidad, pero estadísticamente el valor de x2 tiene un valor p< 0.05 por consiguiente la Diabetes Mellitus 2 si es un factor de riesgo de catarata. Por otro lado la frecuencia de diabetes mellitus tipo 2 en pacientes con catarata es de 83 casos y sin cataratas 183 de un total de 360