# Jose Angel CidUniversity of Vigo | UVIGO · Department of Mathematics

Jose Angel Cid

## About

67

Publications

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669

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Introduction

**Skills and Expertise**

## Publications

Publications (67)

Motivated by the Brillouin equation we deal with the existence and Lyapunov stability of periodic solutions for a more general kind of equations. Our approach is based on the third order approximation in combination with some location information obtained by the averaging method. We will show that our main results apply to some singular models not...

This book is devoted to Prof. Juan J. Nieto, on the occasion of his 60th birthday. Juan José Nieto Roig (born 1958, A Coruña) is a Spanish mathematician, who has been a Professor of Mathematical Analysis at the University of Santiago de Compostela since 1991. His most influential contributions to date are in the area of differential equations. Niet...

We show that in the existing bibliography there are many contradictory claims about the exact date of death of Hermann Weyl and, after a detailed exposition of the way we got our evidence, we demonstrate that Weyl died on 8 December 1955. The paper also shows up the fact that an authoritative intellectual biography on Weyl is yet to be written.

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the ex...

We prove the following result: if a continuous vector field $F$ is Lipschitz when restricted to the hypersurfaces determined by a suitable foliation and a transversal condition is satisfied at the initial condition, then $F$ determines a locally unique integral curve. We also present some illustrative examples and sufficient conditions in order to...

Maximum Principles for the Hill's Equation focuses on the application of these methods to nonlinear equations with singularities (e.g. Brillouin-bem focusing equation, Ermakov-Pinney,...) and for problems with parametric dependence. The authors discuss the properties of the related Green's functions coupled with different boundary value conditions....

We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are presented to illustrate the theoretical results.

We prove new results regarding the existence of positive solutions for a nonlin-ear periodic boundary value problem related to the Liebau phenomenon. As a consequence we obtain new sufficient conditions for the existence of a pump in a simple model. Our methodology relies on the use of classical fixed point index. Some examples are provided to illu...

We present a visual proof of an identity for three consecutive terms of a recurrence relation satisfied by the Fibonacci and Lucas numbers.

The aim of this paper is to show certain properties of the Green's functions related to the Hill's equation coupled with various two point boundary value conditions. We will obtain the expression of the Green's function of Neumann, Dirichlet, Mixed and anti-periodic problems as a combination of the Green's function related to periodic ones. As a co...

The aim of this paper is to show certain properties of the Green's functions
related to the Hill's equation coupled with different two point boundary value
conditions. We will obtain the expression of the Green's function of Neumann,
Dirichlet, Mixed and anti-periodic problems as a combination of the Green's
function related to periodic ones.
As a...

We present a new uniqueness result for first order systems of ordinary differential equations which contains a generalization of Montel–Tonelli's Uniqueness Theorem as a particular case. An example is given to illustrate its applicability.

We present a rather unknown version of the change of variables formula for
non-autonomous functions. We will show that this formula is equivalent to
Green's Theorem for regions of the plane bounded by the graphs of two
continuously differentiable functions. Besides, the formula has interesting
applications in the uniqueness of solution of ordinary...

We give some sufficient conditions for existence, non-existence and
localization of positive solutions for a periodic boundary value problem
related to the Liebau phenomenon. Our approach is of topological nature and
relies on the Krasnosel'ski\u\i{}-Guo theorem on cone expansion and
compression. Our results improve and complement earlier ones in t...

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the...

We give new criteria for the existence of nontrivial fixed points on cones assuming some monotonicity of the operator on a suitable conical shell. Moreover, we give an application to the existence of multiple solutions for a nonlocal boundary value problem that models the displacement of a beam subject to some feedback controllers.
MSC:
47H10, 34B...

In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation ( (x0)) 0 +cx0 +g(x) = e(t) +s.

We study the existence of heteroclinics connecting the two equi-libria ±1 of the third order differential equation u = f (u) + p(t)u 1 Partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2010-15314. 2 Supported by Fundação para a Ciência e a Tecnologia, Financiamento Base ISL-209 (2010) and PEst-OE/MAT/UI0209/2011 . 1 where...

a b s t r a c t This paper is devoted to construct an algorithm that allows us to calculate the explicit expression of the Green's function related to a nth-order linear ordinary differential equa-tion, with constant coefficients, coupled with two-point linear boundary conditions. We develop this algorithm by making a Mathematica package.

In this work, we make an exhaustive study of the properties of the Green's function related to the periodic boundary value
problem
with a sign-changing potential a(t).
Moreover, we obtain new explicit criteria that ensures that the maximum or anti-maximum principle holds for this equation.
The given criteria complement previous results in the lit...

We prove the existence of infinitely many solutions for a second-order singular initial value problem between given lower and upper solutions. Our study is motivated by a singular problem which arises in the field of nonlinear massive gravity. Moreover, we also discuss the global behavior of solutions of the motivating problem. Our arguments lean a...

Keeping in mind the singular model for the periodic oscillations of the axis of a satellite in the plane of the elliptic orbit around its center of mass, we give sufficient conditions for the solvability of a class of singular Sturm-Liouville equations with peri-odic boundary value conditions. To this end, under a suitable change of variables, we p...

We deal with the existence and multiplicity of solutions for the
periodic boundary value problem , where is a positive parameter. The function is allowed to be singular, and the related Green's function is nonnegative
and can vanish at some points.

We study the solvability of a system of second-order differential equations with Dirichlet boundary conditions and non-local terms depending upon a parameter. The main tools used are a dual variational method and the topological degree.

It is known that the anti-maximum principle holds for the quasilinear periodic problem (| u′ |p - 2 u′)′ + μ (t) (| u |p - 2 u) = h (t), u (0) = u (T), u′ (0) = u′ (T), if μ ≥ 0 in [0, T] and 0 < {norm of matrix} μ {norm of matrix}∞ ≤ (πp / T)p, where πp = 2 (p - 1)1 / p ∫01 (1 - sp)- 1 / p d s, or p = 2 and 0 < {norm of matrix} μ {norm of matrix}α...

In this paper we use Krasnoselskii’s fixed point theorem on cone expansions to prove a new fixed point theorem for nondecreasing operators on ordered Banach spaces. Moreover we apply this abstract result to prove the existence of a positive periodic solution for a nonlinear boundary value problem.

We present an alternative method to that of [5] to teach the students how to discover if a differential equation y = f (x, y) is separable or not when the nonlinearity f (x, y) is not explicitly factorized. Our approach is completely elementary and provides a necessary and suffi-cient condition for separability, as well as simple formulas for the e...

This work presents sufficient conditions for the existence of at least one positive solution for a nonlinear fourth-order beam equation under Lidstone boundary conditions. The main tool used is a fixed point theorem that essentially combines the monotone iterative technique with fixed point theorems of cone expansion or compression type. The last s...

Most uniqueness tests for differential equations need assumptions about how the nonlinear part depends on the unknown. A paradigm for this is the classical Lipschitz Theorem. We show that the theorem on differentiation of inverse functions yields, in a significant situation, an elementary technique for turning such uniqueness results into alternati...

In this note we give a L(p)- criterium for the positiveness of the Green's function of the periodic boundary value problem: x" + a(t)x = 0, x(0) = x(T), x'(0) = x'(T) with and indefinite potential a(t). Moreover, we prove that such Green's function is negative provided a(t) belongs to the image of a suitable periodic Ricatti type operator.

We consider the existence of unique absolutely continuous solutions for x′ = p(t)f(x) + p(t)h(t), t ≥ 0, x(0) = 0, where p, f, and h are positive almost everywhere, but none of them needs be continuous or monotone. Moreover, p and f can be unbounded around zero. Our uniqueness results are not based on assumptions on the differences f(x) - f(y), as...

We study the existence of solution for the one-dimensional φ-laplacian equation (φ(u)) = λf (t, u, u) with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate λ 0 is given such that the problem possesses a solution for any |λ| < λ 0 .

This paper is devoted to the study the boundary value problem We prove the existence of at least one, two or three solutions in the presence of a pair of, not necessarily ordered, lower and upper solutions.The proof follows from maximum principles related to the operator u(4)+Mu and Amann’s three solutions theorem.

We study the existence of W2,1 solutions for singular and nonsmooth initial value problems of the type
whereT > 0 is a priori fixed, x0, x1 ∈ ℝ, and F: [0, T ] × ℝ → (ℝ) \ {∅} is a multivalued mapping. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We prove general existence results for
x¢¢ = f(x)g(x¢), x(0) = x0, x¢(0) = x1,x^{\prime\prime} = f(x)g(x^{\prime}), x(0) = x_{0}, x^{\prime}(0) = x_{1},
where f and g need not be continuous or monotone. Moreover neither f nor g need be bounded around, respectively, x
0 and x
1, thus allowing singularities in the equation. Several other basic topic...

In this paper we deal with some boundary value problems related with diffusion processes in the presence of lower and upper solutions. Singularities as well as non local boundary conditions are allowed. We also prove the existence of extremal solutions and the uniqueness of solution for a particular case.

We study the existence of extremal solutions for an infinite system of first-order discontinuous functional differential equations in the Banach space of the bounded functions I∞(M).

We establish some uniqueness and existence results for first-order ordinary differential equations with constant-signed discontinuous nonlinear parts. Several examples are given to illustrate the applicability of our work.

The aim of this paper is to ensure the existence of extremal solutions lying between a pair of lower and upper solutions for a second order dif-ferential equation that includes both the φ–laplacian and the diffusion equations. The results hold from a suitable change of variables. An ex-ample is given to expose the applicability of the obtained resu...

In this paper we use Tarski's fixed point theorem to extend in a systematic way the existence of extremal solutions from scalar initial value problems to boundary value problems for infinite quasimonotone functional systems of differential equations.

We derive sufficient conditions for the existence of extremal solutions for a second order singular ordinary differential equation subject to initial data. The type of equations that we study here can be regarded as stationary and one dimensional models for diffusion processes in which the diffusion coefficient is not a constant. We have also tried...

We deal with the existence of Carathéodory solutions for initial value problems for first-order ordinary differential equations. Our approach consists in passing from the differential equation to a solvable differential inclusion, and then, we look for Carathéodory solutions of the former equation among those of the inclusion.

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

Abstract In this work we present a necessary and sucient,condition for a decreasing map to have at most one fixed point. Some applications to dierential,equations are also given. Keywords: Uniqueness of fixed point; Decreasing operators; Directed sets.

This paper contains two contributions of the theory of T-monotone operators introduced by Chen. First, we prove a new fixed-point theorem for a discontinuous T-monotone mapping. After, we use this theory to obtain the solution of a classical continuous problem, for which the usual iterative methods fail.

We derive necessary and sufficient conditions for the existence of a least and a greatest fixed point of an operator which satisfies the hypothesis of Schauder's theorem. The so obtained results are applied to prove existence of extremal solutions for some initial and boundary value problems.

This paper contains two contributions of the theory of T –monotone operators introduced by Chen. First, we prove a new fixed point theorem for a discontinuous T –monotone mapping. After, we use this theory to obtain the solution of a classical continuous problem, for which the usual iterative methods fail.

We deal with the existence of periodic solutions for problems with a jump discontinuity. We use an approximation procedure and the method of the lower and upper solutions.

In this work, we use Schauder's fixed point theorem to prove the existence of positive solutions for second-order problems. We call the nonlinearities considered reverse Carathéodory functions, because they are continuous in the independent variable and measurable in the dependent one.

We derive existence and approximation results for a type of implicit second order differential equations related with diffusion processes. Singularities, discontinuities and functional dependence are allowed.

We present some new uniqueness criteria for the Cauchy problem based on the local equivalence with another initial value problem.

Existence of extremal solutions for scalar initial value problems is investigated. We shall concentrate upon nonlinearities having constant sign, which lead to new existence results.

We derive sufficient conditions for the existence of extremal solutions for a second-order singular functional differential equation subject to initial data. The type of equations that we study here can be regarded as stationary and one-dimensional models for diffusion processes in which the diffusion coefficient is not a constant. We have also tri...

In this paper we prove new existence results for nonautonomous systems of first order ordinary dierential,equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown,are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition. Keywords. Di...

## Projects

Projects (5)

Boundary Value Problems welcomes submissions to an article collection titled 'Differential Equations with Nonlocal and Functional Terms'.
Differential equations with nonlocal and functional terms have become an active area of research; these terms may occur in the differential equation and/or in the initial or boundary conditions.
Their study is driven not only by theoretical interest but also to the fact that these type of problems occur naturally when modeling real-world applications.
This special issue will focus on all aspects of nonlocal and functional terms including discrete and continuous equations, regular, singular and resonant problems and their applications.
Potential topics include, but are not limited to:
Development of new theoretical tools that can be used to study problems with nonlocal and functional terms
Existence, uniqueness and multiplicity results
Qualitative properties of the solutions, for example, positivity, oscillation, symmetry, bifurcation, asymptotic behavior, regularity, and stability
Approximation of the solutions
Eigenvalue problems
Applications to real-world phenomena

Provide new criteria for the existence of fixed points in cones with topological methods.