A. Cañada

Matrix Lyapunov inequalities for ordinary and elliptic partial differential equations

This paper is devoted to the study of

Lyapunov-type Inequalities and Applications to PDE

L_p

Nonselfadjoint semilinear elliptic boundary value problems

Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant

Topological methods in the study of positive solutions for some nonlinear delay integral equations

p \geq 1

Nonlinear boundary value problems for elliptic systems

. We consider ordinary and elliptic problems. The results obtained in the linear case are combined with Schauder fixed point theorem to provide new results about the existence and uniqueness of solutions for resonant nonlinear problems. The proof uses in a fundamental way the nontrivial relation between the best Lyapunov constants and the minimum value of some especial minimization problems.

Systems of Equations

This work is devoted to the study of resonant nonlinear boundary problems with Neumann boundary conditions. First, we consider the linear case doing a careful analysis which involves Lyapunov-type inequalities with the Lp— norms of the coefficient function. After this end, combining these results with Schauder fixed point theorem, we obtain some new results about the existence and uniqueness of solutions for resonant nonlinear problems.

Partial Differential Equations

SummaryIn this paper we study the existence of solutions of nonselfadjoint semilinear elliptic boundary value problems with a bounded nonlinear term. We emphasize that this nonlinear term may depend on the derivatives of the function in a nontrivial way. In the proof of our main result we use the Leray-Schauder degree theory.

A Variational Characterization of the Best Lyapunov Constants

In this work we give an extension of the results obtained in [2], we are interested in producing sufficient conditions for the existence of positive periodic solution to x(t) = ∫ τ(t) 0 h(t, s, x(t− s− l)) ds, in the special case where h(t, s, x) = f(t, s, x)g(t, s, x). For it, we use topological methods, more precisely, the fixed point index. AMS Subject Classification: 37C25, 47J10, 47B07

An applied mathematical excursion through Lyapunov inequalities, classical analysis and Differential Equations

The purpose of this paper is to discuss non-linear boundary value problems for elliptic systems of the type where A k is a second order uniformly elliptic operator and is such that the problem has a one-dimensional space of solutions that is generated by a non-negative function. The boundary ∂ G is supposed to be smooth and the functions g k , 1≦ k ≦ m are defined on Ḡ× R m and are continuously differentiate (usually, B k represents Dirichlet or Neumann conditions and is the first eigenvalue associated with A k and such boundary conditions).

Lyapunov inequalities for partial differential equations

This chapter is devoted to the study of L p Lyapunov-type inequalities for linear systems of ordinary differential equations with different boundary conditions (which include the case of Neumann, Dirichlet, periodic, and antiperiodic boundary value problems) and for any constant p ≥ 1. Elliptic problems are also considered. As in the scalar case, the results obtained in the linear case are combined with Schauder fixed point theorem to provide several results about the existence and uniqueness of solutions for resonant nonlinear systems. In addition, we study the stable boundedness of linear periodic conservative systems. The proof uses in a fundamental way the nontrivial relation (proved in Chap. 2) between the best Lyapunov constants and the minimum value of some especial constrained or unconstrained minimization problems (depending on the considered problems are resonant or nonresonant, respectively).