^{1}

^{2}

^{2}

^{2}

^{1}

^{2}

A discrete equation

We use the following notation: for integers

In this paper we study a discrete equation with two delays

The results concern the asymptotic convergence of all solutions of (

Such a critical case is characterized by the constant value

Consider (

Let

Let

The case

Then the relevant solutions

Let

Gathering all the cases considered, we have the following:

if

if

The above analysis is not applicable in the case of a nonconstant function

The proofs of the results are based on comparing the solutions of (

The problem concerning the asymptotic convergence of solutions in the continuous case, that is, in the case of delayed differential equations or other classes of equations, is a classical one and has attracted much attention recently. The problem of the asymptotic convergence of solutions of discrete and difference equations with delay has not yet received much attention. We mention some papers from both of these fields (in most of them, equations and systems with a structure similar to the discrete equation (

Arino and Pituk [

Bereketoğlu and Huseynov [

Comparing the known investigations with the results presented, we can see that our results can be applied to the critical case giving strong sufficient conditions of asymptotic convergence of solutions for this case. Nevertheless, we are not concerned with computing the limits of the solutions as

The paper is organized as follows. In Section

Throughout the paper we adopt the customary notation

Let

The auxiliary inequality

We give some properties of solutions of inequalities of the type (

Let

If

This follows directly from (

Let

Let

Now we consider an inequality of the type (

Let

Let

We will construct a solution of inequality (

Let there exist a discrete function

For

It is well known that every absolutely continuous function is representable as the difference of two increasing absolutely continuous functions [

Every function

Let constants

The following lemma can be proved easily by induction. The symbol

For fixed

This part deals with the problem of detecting the existence of asymptotically convergent solutions. The results shown below provide sufficient conditions for the existence of an asymptotically convergent solution of (

Let

From Lemma

Let there exist a function

Assuming that the coefficient

Let there exist a

In the proof, we assume (without loss of generality) that

Comparing

In this part we present results concerning the convergence of all solutions of (

Let there exist a

First we prove that every solution defined by a monotone initial function is convergent. We will assume that a strictly monotone initial function

By Theorem

Summarizing the previous section, we state that every monotone solution is convergent. It remains to consider a class of all nonmonotone initial functions. For the behavior of a solution

Now we use the statement of Lemma

We will finish the paper with two obvious results. Inequality (

If (

Combining the statements of Theorems

The following three statements are equivalent.

Equation (

All solutions of (

Inequality (

We will demonstrate the sharpness of the criterion (

The research was supported by the Project APVV-0700-07 of the Slovak Research and Development Agency and by the Grant no. 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).