J. Ángel Cid

On comparison principles for the periodic Hill's equation

In this work we make an exhaustive study of the properties of the Green’s function related to the periodic boundary value problem La x ≡ x′′ + a(t) x = 0, x(0) = x(T ), x′(0) = x′(T ), with a sign-changing potential a(t). Moreover, we obtain new explicit criteria that ensures the maximum or antimaximum principle holds for this equation. The given criteria complement previous results in the literature.

Ordinary differential equations and systems with time-dependent discontinuity sets

In this paper we prove new existence results for nonautonomous 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.

On first-order ordinary differential equations with nonnegative right-hand sides

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.

On uniqueness criteria for systems of ordinary differential equations

Abstract We present some new uniqueness criteria for the Cauchy problem x′(t)=f t,x(t) , x(t 0 )=x 0 , based on the local equivalence with another initial value problem.

Existence of Solutions for Elliptic Systems with Nonlocal Terms in One Dimension

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.

Existence of positive solutions for second-order ODEs with reverse Carathéodory functions

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

Integration by Parts and by Substitution Unified with Applications to Green's Theorem and Uniqueness for ODEs

Abstract We present a rather unknown version of the change of variables formula for nonautonomous functions. We will show that this formula is equivalent to Greens theorem for regions of the plane bounded by the graphs of two continuously differentiable functions. Moreover, the formula has interesting applications in the uniqueness of solutions of ordinary differential equations.