We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.

In many phenomena of the real world, not only does their evolution prove to be dependent on the present state, but it is essentially specified by the entire previous history. These processes are encountered in the theory of viscoelasticity [

The aim of this article is to develop a technique for investigating stability and boundedness of nonlinear implicit Volterra difference systems described by Volterra operator equations. Only a few papers deal with the theory of general Volterra equations and most of them are devoted to the stability analysis of explicit Volterra linear difference equations with constant coefficients or of convolution type. For example, in Minh [

One of the basic methods in the theory of stability and boundedness of Volterra difference equations is the direct Lyapunov method (see [

In this paper, to establish boundedness conditions of solutions, we will interpret the Volterra difference equations with nonlinear kernels as operator equations in appropriate spaces. Such an approach for linear Volterra difference equations has been used by Myshkis [

Existence and uniqueness problems for the Volterra difference equations were discussed by some authors. Usually the solutions were sought in the phase space

Our results compare favorably with the above-mentioned works in the following sense:

Sufficient conditions for the existence and uniqueness of solutions of implicit nonlinear Volterra difference equations are obtained.

We established a theory on the asymptotic behavior of implicit nonlinear Volterra difference systems which are described by Volterra operator equations.

Explicit estimates for the solutions of nonlinear Volterra operator equations in Hilbert spaces are derived.

The remainder of this article is organized as follows: in Section

Let

Denote

Equation (

We ask when the solutions of this equation belong to the space

The operator

The causal property of the operator

If the operator

Finally, we will determine sufficient conditions on the coefficients of (

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.

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

An operator

The norm

A bounded linear operator

Let

Let

Denote

Assume that

Assume that

From (

By Hölder’s inequality, we have

Assume that

From (

Now, we will complete the Proof of Theorem

By Urysohn’s Lemma [

Consider the Volterra difference equation of the form

Assume that

Proceeding in a similar way to the proof of Theorem

Consider

New conditions for the existence, uniqueness, and boundedness of solutions of infinite-dimensional nonlinear Volterra difference systems are derived. Unlike the classic method of stability analysis, we do not use the technique of the Lyapunov functions in the process of construction of the estimates for the solutions. The proofs are carried out using estimates for the norm of powers of quasi-nilpotent operators. That is, we interpreted the Volterra difference equations, with nonlinear kernels, as operator equations defined in the Banach spaces

In connection with the above investigations, some open problems arise. The richness of the spectral properties of operators acting on infinite-dimensional Hilbert spaces will need new stability formulations. Consequently, natural directions for future research is the generalization of our results to local exponential stabilizability of nonlinear Volterra difference equations or investigating the feedback stabilization of implicit nonlinear Volterra systems defined by operator Volterra equations.

The author declares that there is no conflict of interests regarding the publication of this paper.

This research was supported by Dirección de Investigación under Grant NU06/16.