Rigoberto Medina

The Freezing Method for Linear Difference Equations

The freezing method for ordinary differential systems is extended to difference systems. By virtue of that method, new stability criteria and solution estimates for linear difference systems are derived.

Asymptotic behavior of solutions of second order nonlinear difference equations

~0 + 2) + @)J@ + 1) + b(n)u(n) = g(n, Y(~),_Y(H + 1)) (1) or, equivalently, E’Y + a0 + by = g(n, y, EY), where E is the operator defined by Ey(n) = y(n + 1). (2) Let (z, , z2) be a fundamental system of solutions of the linear difference equation Lz := E2z f aEz + bz = 0 (3) satisfying z,(O) = 1, z,(l) = 0; zz(O) = 0, 22(l) = 1. (4) We will prove that there exists a ball B(0, p) c R2 such that if (y(O), y(1)) E B(0, p), then the solution y of equation (1) is defined on all N, and for n -+ co it satisfies

Asymptotic Representation of Solutions of Linear Second Order Difference Equations

Abstract Using dichotomy conditions, we obtain asymptotic formulae for the solutions of perturbed linear second order difference equations.

Exponential stabilization of nonlinear discrete-time systems

In this paper, we give sufficient conditions for the exponential stabilizability of a class of perturbed non-autonomous difference equations with slowly varying coefficients. Under appropriate growth conditions on the perturbations, we establish explicit results concerning the feedback exponential stabilizability.

ESTIMATES FOR THE NORMS OF SOLUTIONS OF DELAY DIFFERENCE SYSTEMS

We derive explicit stability conditions for delay difference equations in ℂn (the set of n complex vectors) and estimates for the size of the solutions are derived. Applications to partial difference equations, which model diffusion and reaction processes, are given.

Dichotomies and asymptotic equivalence of nonlinear difference systems

Some asymptotic formulae for the solutions of quasilinear systems are obtained. Several dichotomies for the linear part are considered. Moreover, one result for constant multiple eigenvalues is presented.

Linear differential systems with conditionally integrable coefficients

Abstract For perturbations conditionally integrable. Levinsons theorems are obtained and applied to almost constant systems.

Delay difference equations in infinite-dimensional spaces

We present explicit stability conditions for the solutions of nonlinear delay difference equations defined in infinite-dimensional spaces , where X is a Banach space; are linear bounded operators and the nonlinearities satisfy a local comparison condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators as well as the Freezing Method for difference systems, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.

Delay difference equations in Banach spaces

We derive explicit stability conditions for semilinear delay difference equations in a Banach space. It is assumed that the nonlinearities of the considered equations satisfy the local Lipschitz condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.

Asymptotic constancy of solutions of systems of difference equations

Assuming only conditional summability we study the convergence of the solutions of difference systems.