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

Existence results for a clamped beam equation with integral boundary conditions

In this paper we investigate the existence of positive solutions of fourth-order non autonomous differential equations with integral boundary conditions, the nonlinearity is a continuous function that depends on the spatial variable and its the second-order derivative. The approach relies an extension of Krasnoselskii's fixed point theorem in a cone. Some examples are given to illustrate our results

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

New results on the sign of the Green function of a two-point n-th order linear boundary value problem

[EN] This paper provides conditions for determining the sign of all the partial derivatives of the Green functions of n-th order boundary value problems subject to a wide set of homogeneous two-point boundary conditions, removing restrictions of previous results about the distance between the two extremes that define the problem. To do so, it analyzes the sign of the derivatives of the solutions of related two-point n-th order boundary value problems subject to n ¿ 1 boundary conditions by introducing a new property denoted by `hyperdisfocality¿Almenar-Belenguer, P.; Jódar Sánchez, LA. (2022). New results on the sign of the Green function of a two-point n-th order linear boundary value problem. Boundary Value Problems. 1-22. https://doi.org/10.1186/s13661-022-01631-z12

Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).Comment: 80 pages. arXiv admin note: text overlap with arXiv:1105.375

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics