We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.

The problem of stability and robustness of difference systems has been extensively studied in the last years [

Martynyuk [

Our results compare favorably with the abovementioned works in the following sense:

We established a local stability theory of discrete evolution equations.

The Lipschitz assumptions are local in the state and general in the time.

Explicit estimates to the norm of the associated evolution semigroups are established.

In this paper we consider evolution difference systems defined in infinite-dimensional Banach spaces, with bounded operators on the right-hand side represented in the pseudolinear form. New estimates for the norms of solutions are derived giving us explicit stability and boundedness conditions. Our approach is based on the generalization of the freezing method to abstract difference systems. The equations will be represented as a perturbation about a fixed value of the coefficient operator. Thus, applying norm estimates for the involved operator-valued functions, new stability results are established.

Although the freezing method appears to be often utilized in practice in the control of linear time-varying systems, not much is currently known regarding the stability or asymptotic behavior of pseudolinear difference systems of the form considered here. In fact, we will develop a local stability theory to evolution pseudolinear difference systems.

The remainder of this paper is organized as follows: In Section

Let

Consider in a Banach space

Denote by

Here we will consider system (

In order to establish the stability properties of (

Consider in

The zero solution of system (

Under conditions (

Now, returning to system (

Put

For a positive

Let us introduce the equation

Consequently

Theorem

As the previous theorems show, the extension of the freezing method to evolution equations is based on norm estimates for relevant semigroups. However, obtaining these estimates is usually not an easy task. Because of this, we restrict ourselves by equations in a separable Hilbert space with Hilbert-Schmidt coefficient operators.

In the finite-dimensional case, the spectrum of a linear operator consists of its eigenvalues. The spectral theory of bounded linear operators on infinite-dimensional spaces is an important but challenging area of research. For example, an operator may have a continuous spectrum in addition to, or instead of, a point spectrum of eigenvalues. A particularly simple and important case is that of compact, self-adjoint operators. Compact operators may be approximated by finite-dimensional operators, and their spectral theory is close to that of finite-dimensional operators. We will assume that coefficient operators of (

To formulate the next results, let us introduce the following notations and definitions: Let

An operator

The norm

A bounded linear operator

Let

Denote

Under conditions (

Now, in order to apply Theorem

Under conditions (

By Theorem

Consider a perturbed system of autonomous pseudolinear equations

Let suppose that

Under conditions (

The leading part of (

We present an example that illustrates Theorem

Hence, using the inequality

If the inequality

Let

There exists a nonnegative

where

New conditions for the exponential stability of a class of infinite-dimensional nonlinear difference systems are derived. We establish the robustness of the exponential stability, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. Unlike the classic method of stability analysis, we do not use the technique of Lyapunov function in the process of construction of the stability results. The proofs are carried out using the semigroup theory combined with the freezing method. That is, the equation is represented as a perturbation about a fixed value of the operator and then applying norm estimates for operator-valued functions the results follow. We have presented two examples which show how this approach brings out different aspects of the stability problem of pseudolinear equations.

The author declares that there are no competing interests regarding the publication of this paper.

This research was supported by Fondecyt Chile under Grant no. 1130112.