#### Abstract

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

#### 1. Introduction

Throughout this paper, we use the following notation: for an integer , we define We investigate the asymptotic behavior for of the solutions of the discrete delayed equation of the -th order where is the independent variable assuming values from the set with a fixed . The number , is the fixed delay, , and .

A function is a solution of (1.2) if it satisfies (1.2) for every .

We will study (1.2) together with initial conditions Initial problem (1.2), (1.3) obviously has a unique solution, defined for every . If the function is continuous with respect to its last arguments, then the solution of (1.2) continuously depends on initial conditions (1.3).

Now we give a general description of the problem solved in this paper.

*Problem 1. *Let be functions such that for every . The problem under consideration is to find sufficient conditions for the right-hand side of (1.2) that will guarantee the existence of a solution of initial problem (1.2), (1.3) such that

This problem can be solved with help of a result which is valid for systems of first-order difference equations and which will be presented in the next section. This is possible because the considered equation (1.2) can be rewritten as a system of first-order difference equations, similarly as a differential equation of a higher order can be transformed to a special system of first-order differential equations. Although the process of transforming a -st order difference equation to a system of first order equations is simple and well-known (it is described in Section 3), the determination of the asymptotic properties of the solutions of the resulting system using either Liapunov approach or retract-type method is not trivial. These analogies of classical approaches, known from the qualitative theory of differential equations, were developed for difference systems in [1] (where an approach based on Liapunov method was formulated) and in [2–5] (where retract-type analysis was modified for discrete equations). It occurs that for the mentioned analysis of asymptotic problems of system (1.2), neither the ideas of Liapunov, nor the retract-type technique can be applied directly. However, in spite of the fact that each of the two mentioned methods fails when used independently, it appears that the combination of both these techniques works for this type of systems. Therefore, in Section 2 we prove the relevant result suitable for the asymptotic analysis of systems arising by transformation of (1.2) to a system of first-order differential equations (Theorem 2.1), where the assumptions put to the right-hand side of the system are of both types: those caused by the application of the Liapunov approach and those which are typical for the retract-type technique. Such an idea was applied in a particular case of investigation of asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation in [6, 7]. The approach is demonstrated in Section 3 where, moreover, its usefulness is illustrated on the problem of detecting the existence of positive solutions of linear equations with a single delay (in Section 3.4) and asymptotic estimation of solutions (in Section 3.3).

Advantages of our approach can be summarized as follows. We give a general method of analysis which is different from the well-known comparison method (see, e.g., [8, 9]). Comparing our approach with the scheme of investigation in [10, 11] which is based on a result from [12], we can see that the presented method is more general because it unifies the investigation of systems of discrete equations and delayed discrete equations thanks to the Liapunov-retract-type technique.

For related results concerning positive solutions and the asymptotics of solutions of discrete equations, the reader is referred also to [13–25].

#### 2. The Result for Systems of First-Order Equations

Consider the system of difference equations
where , , and , . The solution of system (2.1) is defined as a vector function such that for every , (2.1) is fulfilled. Again, if we prescribe initial conditions
the initial problem (2.1), (2.2) has a unique solution. Let us define a set as
where
with , , being auxiliary functions such that for each . Such set is called a *polyfacial set*.

Our aim (in this part) is to solve, in correspondence with formulated Problem 1, the following similar problem for systems of difference equations.

*Problem 2. *Derive sufficient conditions with respect to the right-hand sides of system (2.1) which guarantee the existence of at least one solution , , satisfying
for every .

As we mentioned above, in [1] the above described problem is solved via a Liapunov-type technique. Here we will combine this technique with the retract-type technique which was used in [2–5] so as the result can be applied easily to the system arising after transformation of (1.2). This brings a significant increase in the range of systems we are able to investigate. Before we start, we recall some basic notions that will be used.

##### 2.1. Consequent Point

Define the mapping as
For any point , the point is called the *first consequent point* of the point . The geometrical meaning is that if a point lies on the graph of some solution of system (2.1), then its first consequent point is the next point on this graph.

##### 2.2. Liapunov-Type Polyfacial Set

We say that a polyfacial set is *Liapunov-type* with respect to discrete system (2.1) if
for every and every . The geometrical meaning of this property is this: if a point lies inside the set , then its first consequent point stays inside .

In this contribution we will deal with sets that need not be of Liapunov-type, but they will have, in a certain sense, a similar property. We say that a polyfacial set is *Liapunov-type with respect to the **th variable* () and to discrete system (2.1) if
The geometrical meaning is that if , then the -coordinate of its first consequent point stays between and , meanwhile the other coordinates of may be arbitrary.

##### 2.3. Points of Strict Egress and Their Geometrical Sense

An important role in the application of the retract-type technique is played by the so called strict egress points. Before we define these points, let us describe the boundaries of the sets , , in detail. As one can easily see,
with
In accordance with [3, Lemmas 1 and 2], a point is a *point of the type of strict egress* for the polyfacial set with respect to discrete system (2.1) if and only if for some
or
Geometrically these inequalities mean the following: if a point is a point of the type of strict egress, then the first consequent point .

##### 2.4. Retract and Retraction

If are any two sets in a topological space and is a continuous mapping from onto such that for every , then is said to be a *retraction* of onto . If there exists a retraction of onto , then is called a *retract* of .

##### 2.5. The Existence Theorem for the System of First-Order Equations (Solution of Problem 2)

The following result, solving Problem 2, gives sufficient conditions with respect to the right-hand sides of (2.1) which guarantee the existence of at least one solution satisfying (2.5) for every .

Theorem 2.1. *Let , , be real functions defined on and let , , be continuous functions. Suppose that for one fixed all the points of the sets , are points of strict egress, that is, if , then
**
and if , then
**
Further suppose that the set is of Liapunov-type with respect to the th variable for every , , that is, that for every **
Then there exists a solution of system (2.1) satisfying the inequalities
**
for every .*

*Proof. *The proof will be by contradiction. We will suppose that there exists no solution satisfying inequalities (2.16) for every . Under this supposition we prove that there exists a continuous mapping (a retraction) of a closed interval onto both its endpoints which is, by the intermediate value theorem of calculus, impossible.

Without the loss of generality we may suppose that the index in Theorem 2.1 is equal to 1, that is, all the points of the sets and are strict egress points. Each solution of system (2.1) is uniquely determined by the chosen initial condition
For the following considerations, let with , , be chosen arbitrarily but fixed. Now the solution of (2.1) is given just by the choice of , we can write
Define the closed interval . Hereafter we show that, under the supposition that there exists no solution satisfying inequalities (2.16), there exists a retraction (which will be a composition of two auxiliary mappings and defined below) of the set onto the set . This contradiction will prove our result. To arrive at such a contradiction, we divide the remaining part of the proof into several steps.*Construction of the Leaving Value *

Let a point be fixed. The initial condition defines a solution . According to our supposition, this solution does not satisfy inequalities (2.16) for every . We will study the moment the solution leaves the domain for the first time. The first value of for which inequalities (2.16) are not valid will be denoted as .

(I) First consider the case . Then there exists a value in such that
while
As the set is of the Liapunov-type with respect to all variables except the first one and , then
Because was assumed, and is of Liapunov-type for each variable , , then the validity of inequalities (2.16) has to be violated in the -coordinate. The geometrical meaning was explained in Section 2.2.

Now, two cases are possible: either or . In the first case or . In the second case or and, due to (2.13) and (2.14), or , respectively.

(II) If , then . Thus, for this case, we could put . Further, because of the strict egress property of and , either (if ) or (if ) and thus .

Unfortunately, for the next consideration the value (the first value of the independent variable for which the graph of the solution is out of ) would be of little use. What we will need is the last value for which the graph of the solution stays in . We will denote this value as and will call it the *leaving value*. We can define as
As the value of depends on the chosen initial point , we could write but we will mostly omit the argument , unless it is necessary. From the above considerations it follows that
*Auxiliary Mapping *

Now we construct the auxiliary mapping . First extend the discrete functions onto the whole interval :
being the integer part of (the floor function). Note that are now piecewise linear continuous functions of a real variable such that for every and that the original values of for are preserved. This means that the graphs of these functions connect the points or for , respectively. Denote the set
The boundary of consists of three mutually disjoint parts , , and :
where
Define the mapping as follows: let be the point of intersection of the line segment defined by its end points , with (see Figure 1). The mapping is obviously well defined on and , .

Prove that the mapping is continuous. The point lies either on or on . Without the loss of generality, consider the second case (the first one is analogical). The relevant boundary line segment for , which is a part of , is described by (see (2.24))
and the line segment joining the points , by the equation
The coordinates of the point , which is the intersection of both these line segments, can be obtained as the solution of the system consisting of (2.28) and (2.29). Solving this system with respect to and , we get
Let be any sequence with such that . We will show that . Because of the continuity of the functions , ,
We have to consider two cases: (I), that is, , (II), that is, .

Recall that (due to our agreement) in both cases .

(I)In this case also and for sufficiently large. That means that the leaving value is the same as given by and thus the point is given by
The desired convergence is implied by equations (2.30) to (2.34).(II)Suppose . Then , for all and as , . A minor edit of the text in the case (I) proof provides the continuity proof. Suppose . In this case there can be for some members of the sequence and for the others. Without the loss of generality, we can suppose that splits into two infinite subsequences and such that For the subsequence , the text of the proof of (I) can be subjected to a minor edit to provide the proof of continuity. As for the subsequence , the leaving value is different from given by because is already out of . For sufficiently large,because and thus, as , .Hence, the value of the mapping for is (in (2.33), (2.34) we replace by ) Due to (2.32), and thus We have shown that and and thus .*Auxiliary Mapping *

Define as
The mapping is obviously continuous.*Resulting Mapping and Its Properties*

Define . Due to construction we have
and . The mapping is continuous because of the continuity of the two mappings and . Hence, it is the sought retraction of onto . But such a retraction cannot exist and thus we get a contradiction and the proof is complete.

#### 3. Application of Theorem 2.1 to the Delayed Discrete Equation

Now, let us return to the original delayed discrete equation (1.2), that is, As it was said in Section 1, this equation will be transformed to a system of first-order discrete equations. Then we will apply Theorem 2.1 to this system and prove that under certain conditions there exists a solution of delayed equation (1.2) that stays in the prescribed domain. In the end, we will study a special case of (1.2).

##### 3.1. Transformation of (1.2) to the System of First-Order Equations

We will proceed in accordance with the well-known scheme similarly as when constructing the system of first-order differential equations from a differential equation of a higher order. Put where are new unknown functions. From (1.2) we get . Obviously . Rewriting these equalities in terms of differences, we have . Altogether, we get the system which is equivalent to (1.2).

##### 3.2. The Existence Theorem for the Delayed Equation (1.2) (Solution of Problem 1)

The following theorem is a consequence of Theorem 2.1. In fact, this theorem has been already proved in [12]. There, the proof is based upon a modification of the retract method for delayed equations. Our method (rearranging a delayed equation to a system of first-order equations) is, by its principle, more general than that used in [12].

Theorem 3.1. *Let , , , be real functions defined on . Further, let be a continuous function and let the inequalities
**
hold for every and every such that
**
Then there exists a solution of (1.2) satisfying the inequalities
**
for every .*

*Proof. *We have shown that (1.2) is equivalent to system (3.3) which can be seen as a special case of system (2.1) with and where
Define the polyfacial set as
with
We will show that for system (3.3) and the set , all the assumptions of Theorem 2.1 are satisfied.

As the function is supposed to be continuous, the mapping is continuous, too. Put the index from Theorem 2.1, characterizing the points of egress, equal to 1. We will verify that the set is of Liapunov-type with respect to the th variable for any , that is, (see (2.8)) that for every
First, we compute
Thus we have to show that for
Because , then for any , and therefore
But, by (3.10), we have
meanwhile
and thus . Analogously we get that . Thus inequalities (3.11) are fulfilled.

Further we will show that all the boundary points are points of strict egress for the set with respect to system (3.3). According to (2.11), we have to show that if and for , then
that is,
Notice that the condition for is equivalent with condition (see (3.10)). Looking at the supposed inequality (3.4) and realizing that and , we can see that inequality (3.18) is fulfilled.

Analogously, according to (2.12), we have to prove that for and for the inequality
that is,
holds.

Again, considering (3.5) and the fact that and , we can see that this inequality really holds.

Thus, by the assertion of Theorem 2.1, there exists a solution of system (3.3) such that for every

In our case, is the solution of the original equation (1.2). Further, and , and thus the existence of a solution of the delayed equation (1.2) such that inequalities (3.7) are satisfied is guaranteed.

##### 3.3. Asymptotic Solution Estimates for Delayed Difference Equations

Let us suppose that two functions are given such that Consider the problem of whether there exists a solution , of (1.2) such that The following corollary of Theorem 3.1 presents sufficient conditions for the existence of a solution of this problem.

Corollary 3.2. *Let functions satisfy inequalities (3.22)–(3.24). Let be a continuous function such that
**
for every and every such that
**
Then there exists a solution of (1.2) satisfying inequalities (3.25) for every .*

*Proof. *This assertion is an easy consequence of Theorem 3.1.

Put , . Considering inequalities (3.23) and (3.26), we can see that
for every and every such that
Similarly,
for every and every , .

Obviously, inequalities (3.29) and (3.31) are equivalent with inequalities (3.4) and (3.5), respectively. Thus, all the assumptions of Theorem 3.1 are satisfied and there exists a solution of (1.2) satisfying inequalities (3.25) for every .

*Example 3.3. *Consider the equation
for which is a second-order delayed discrete equation with delay . We will show that there exists a solution of (3.32) that satisfies the inequalities
for .

We will prove that for the functions all the assumptions of Corollary 3.2 are satisfied. Inequality (3.22) is obviously fulfilled for . Inequality (3.23) can be also proved very easily: and thus for every , .

As for inequality (3.24), we get and thus for .

Finally, the functions are decreasing with respect to . Therefore, conditions (3.26) and (3.27) are satisfied, too. Hence, due to Corollary 3.2, there exists a solution of (3.32) satisfying (3.33).

##### 3.4. Positive Solutions of a Linear Equation with a Single Delay

We will apply the result of Theorem 3.1 to the investigation of a simple linear difference equation of the -st order with only one delay, namely, the equation where, again, is the independent variable and , , is the fixed delay. The function is assumed to be positive. Our goal is to give sharp sufficient conditions for the existence of positive solutions. The existence of such solutions is very often substantial for a concrete model considered. For example, in biology, when a model of population dynamics is described by an equation, the positivity of a solution may mean that the studied biological species can survive in the supposed environment.

For its simple form, (3.38) often serves for testing new results and is very frequently investigated. It was analyzed, for example, in papers [10, 11, 26]. A sharp result on existence of positive solutions given in [26] is proved by a comparison method [8, 9]. Here we will use Theorem 3.1 to generalize this result.

For the purposes of this section, define the expression , where , as We will write only instead of . Further, for a fixed integer define auxiliary functions In [26], it was proved that if in (3.38) is a positive function bounded by for some , then there exists a positive solution of (3.38) bounded by the function for sufficiently large. Since , such solution will vanish for . Here we show that (3.38) has a positive solution bounded by even if the coefficient satisfies a less restrictive inequality (see inequality (3.58) below). The proof of this statement will be based on the following four lemmas. The symbols “” and “” stand for the Landau order symbols and are used for .

Lemma 3.4. *The formula
**
holds for any numbers such that and .*

*Proof. *The assertion is a simple consequence of the well-known Maclaurin expansion
As , substituting we get
and adding to both sides of this equality, we get (3.42).

Lemma 3.5. *For fixed and fixed , the asymptotic representation
**
holds for .*

*Proof. *We will prove relation (3.45) by induction with respect to . For , (3.45) reduces to
which holds due to Lemma 3.4. Suppose that relation (3.45) holds for some . We can write with and
Now we will show that (3.45) holds for . Notice that in our case, the condition from Lemma 3.4 is fulfilled for sufficiently large because for , meanwhile for . Thus we are justified to use Lemma 3.4 and doing so, we get
Thus, formula (3.45) holds for , too, which ends the proof.

Lemma 3.6. *For fixed and fixed , the asymptotic representations
**
hold for .*

*Proof. *Both these relations are simple consequences of the asymptotic formula
and of Lemma 3.5 (for formula (3.49)). In the case of relation (3.49), we put
and in the case of relation (3.50), we put .

Lemma 3.7. *For fixed and fixed , the asymptotic representation
**
holds for .*

*Proof. *We will prove relation (3.53) by induction with respect to . For , (3.53) reduces to
On the other hand, using Lemma 3.6, we get
Thus, for , relation (3.53) holds. Now suppose that (3.53) holds for some and prove that it holds for . In the following calculations, we use Lemma 3.6 and we skip some tedious expressions handling.
We can see that formula (3.53) holds for , too, which ends the proof.

Now we are ready to prove that there exists a bounded positive solution of (3.38). Remind that functions and were defined by (3.40) and (3.41), respectively.

Theorem 3.8. *Let satisfy the inequality
**
for a fixed . Suppose that there exists an integer such that the function satisfies the inequalities
**
for every . Then there exists a solution , of (3.38) such that for sufficiently large the inequalities
**
hold.*

*Proof. *Show that all the assumptions of Theorem 3.1 are fulfilled. For (3.38), . This is a continuous function. Put
We have to prove that for every such that , , the inequalities (3.4) and (3.5) hold for sufficiently large. Start with (3.4). That gives that for , it has to be
This certainly holds, because the function is positive and so is .

Next, according to (3.5), we have to prove that
which is equivalent to the inequality
Denote the left-hand side of (3.63) as . As and as by (3.40), (3.58), and (3.57)
we have
Further, we can easily see that
Thus, to prove (3.63), it suffices to show that for sufficiently large,
Denote the left-hand side of inequality (3.67) as and the right-hand side as . Using Lemma 3.7 with and , we can write
Using Lemma 3.7 with and , we get for