Antonio J. Ureña

A Hartman-Nagumo inequality for the vector ordinary p-Laplacian and Applications to Nonlinear Boundary Value Problems

A generalization of the well-known Hartman-Nagumo inequality to the case of the vector ordinary p-Laplacian and classical degree theory provide existence results for some associated nonlinear boundary value problems.

Dynamics of periodic second-order equations between an ordered pair of lower and upper solutions

Abstract We consider periodic second-order equations having an ordered pair of lower and upper solutions and show the existence of asymptotic trajectories heading towards the maximal and minimal periodic solutions which lie between them.

The Spectrum of Reversible Minimizers

Poincaré and, later on, Carathéodory, showed that the Floquet multipliers of 1-dimensional periodic curves minimizing the Lagrangian action are real and positive. Even though Carathéodory himself observed that this result loses its validity in the general higherdimensional case, we shall show that it remains true for systems which are reversible in time. In this way, we also generalize a previous result by Offin on the hyperbolicity of nondegenerate symmetric minimizers. Our arguments rely on the higher-dimensional generalizations of the Sturm theory which were developed during the second half of the twentieth century by several authors, including Hartman, Morse or Arnol’d.

A Counterexample for Singular Equations with Indefinite Weight

Abstract We construct a second-order equation x ¨ = h ⁢ ( t ) / x p {\ddot{x}=h(t)/x^{p}} , with p > 1 {p>1} and the sign-changing, periodic weight function h having negative mean, which does not have periodic solutions. This contrasts with earlier results which state that, in many cases, such periodic problems are solvable.

THE REGION OF SOLVABILITY OF A PARAMETERIZED BOUNDARY VALUE PROBLEM CAN BE DISCONNECTED

A celebrated result by Amann, Ambrosetti and Mancini [1] implies the connectedness of the region of existence for some parameter-depending boundary value problems which are resonant at the first eigenvalue. The analogous thing does not hold for problems which are resonant at the second eigenvalue.