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

Non-Atkinson perturbations of nonautonomous linear Hamiltonian systems: exponential dichotomy and nonoscillation

Producción CientíficaWe analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of $M^+$, then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of $M^+$ are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.MINECO/FEDER, MTM2015-66330-PEuropean Commission, H2020-MSCA-ITN-201

Friedrichs extensions for a class of singular discrete linear Hamiltonian systems

This paper is concerned with the characterizations of the Friedrichs extension for a class of singular discrete linear Hamiltonian systems. The existence of recessive solutions and the existence of the Friedrichs extension are proved under some conditions. The self-adjoint boundary conditions are obtained by applying the recessive solutions and then the characterization of the Friedrichs extension is obtained in terms of boundary conditions via linear independently recessive solutions

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

Constant sign solution for a simply supported beam equation

The aim of this paper is to ensure the existence of constant sign solutions for the fourth order boundary value problem: $$\begin{cases} u^{(4)}(t)-p\,u''(t)+c(t)\,u(t)=h(t)(\geq 0)\,,&t\in I\equiv[a,b]\,,\\ u(a)=u''(a)=u(b)=u''(b)=0\,, \end{cases}$$ where $c,\ h\in C(I)$ and $p\geq 0$. This problem models the behavior of a suspension bridge assuming that the vertical displacement is small enough. By using variational methods, we weaken the previously known sufficient conditions on $c$ to ensure that the obtained solution is of constant sign