Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case

Abstract

A discrete equation Δ𝑦(𝑛)=𝛽(𝑛)[𝑦(𝑛−𝑗)−𝑦(𝑛−𝑘)] with two integer delays 𝑘 and 𝑗,𝑘>𝑗≥0 is considered for 𝑛→∞. We assume 𝛽∶ℤ^∞_𝑛0−𝑘→(0,∞), where ℤ^∞_𝑛0={𝑛₀,𝑛₀+1,…},𝑛₀∈ℕ and 𝑛∈ℤ^∞_𝑛0. Criteria for the existence of strictly monotone and asymptotically convergent solutions for 𝑛→∞ are presented in terms of inequalities for the function 𝛽. Results are sharp in the sense that the criteria are valid even for some functions 𝛽 with a behavior near the so-called critical value, defined by the constant (𝑘−𝑗)⁻¹. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

1. Introduction

We use the following notation: for integers 𝑠,𝑞,𝑠≤𝑞, we define ℤ^𝑞_𝑠∶={𝑠,𝑠+1,…,𝑞}, where the cases 𝑠=−∞ and 𝑞=∞ are admitted too. Throughout this paper, using the notation ℤ^𝑞_𝑠 or another one with a pair of integers 𝑠, 𝑞, we assume 𝑠≤𝑞.

In this paper we study a discrete equation with two delaysΔ𝑦(𝑛)=𝛽(𝑛)𝑦𝑛−𝑗−𝑦(𝑛−𝑘) (1.1) as 𝑛→∞. Integers 𝑘 and 𝑗 in (1.1) satisfy the inequality 𝑘>𝑗≥0 and 𝛽∶ℤ^∞_𝑛0−𝑘→ℝ⁺∶=(0,∞), where 𝑛₀∈ℕ and 𝑛∈ℤ^∞_𝑛0. Without loss of generality, we assume 𝑛₀−𝑘>0 throughout the paper (this is a technical detail, necessary for some expressions to be well defined).

The results concern the asymptotic convergence of all solutions of (1.1). We focus on what is called the critical case (with respect to the function 𝛽) which separates the case when all solutions are convergent from the case when there exist divergent solutions.

Such a critical case is characterized by the constant value𝛽(𝑛)≡𝛽_cr∶=𝑘−𝑗⁻¹,𝑛∈ℤ^∞_𝑛0−𝑘, (1.2) and below we explain its meaning and importance by an analysis of the asymptotic behavior of solutions of (1.1).

Consider (1.1) with 𝛽(𝑛)=𝛽₀, where 𝛽₀ is a positive constant; that is, we consider the following equation:Δ𝑦(𝑛)=𝛽₀⋅𝑦𝑛−𝑗−𝑦(𝑛−𝑘). (1.3) Looking for a solution of (1.3) in the form 𝑦(𝑛)=𝜆^𝑛, 𝜆∈ℂ⧵{0} using the usual procedure, we get the characteristic equation𝜆^𝑘+1−𝜆^𝑘=𝛽₀⋅𝜆^𝑘−𝑗−1. (1.4) Denote its roots by 𝜆_𝑖, 𝑖=1,…,𝑘+1. Then characteristic equation (1.4) has a root 𝜆_𝑘+1=1. Related solution of (1.3) is 𝑦_𝑘+1(𝑛)=1. Then there exists a one-parametric family of constant solutions of (1.3) 𝑦(𝑛)=𝑐_𝑘+1𝑦_𝑘+1(𝑛)=𝑐_𝑘+1, where 𝑐_𝑘+1 is an arbitrary constant. Equation (1.4) can be rewritten as 𝜆^𝑘(𝜆−1)=𝛽₀⋅(𝜆−1)𝜆^{𝑘−𝑗−1}+𝜆^{𝑘−𝑗−2}+⋯+1, (1.5) and, instead of (1.4), we can consider the following equation:𝑓(𝜆)∶=𝜆^𝑘−𝛽₀⋅𝜆^{𝑘−𝑗−1}+𝜆^{𝑘−𝑗−2}+⋯+1=0. (1.6)

Let 𝛽₀=𝛽_cr. Then (1.6) has a root 𝜆_𝑘=1 which is a double root of (1.4). By the theory of linear difference equations, (1.3) has a solution 𝑦_𝑘(𝑛)=𝑛, linearly independent with 𝑦_𝑘+1(𝑛). There exists a two-parametric family of solutions of (1.3) 𝑦(𝑛)=𝑐_𝑘𝑦_𝑘(𝑛)+𝑐_𝑘+1𝑦_𝑘+1(𝑛)=𝑐_𝑘𝑛+𝑐_𝑘+1, (1.7) where 𝑐_𝑘, 𝑐_𝑘+1 are arbitrary constants. Then lim_𝑛→∞𝑦(𝑛)=∞ if 𝑐_𝑘≠0. This means that solutions with 𝑐_𝑘≠0 are divergent.

Let 𝛽₀<𝛽_cr and 𝑘−𝑗>1. We define two functions of a complex variable 𝜆𝐹(𝜆)∶=𝜆^𝑘,Ψ(𝜆)∶=𝛽₀⋅𝜆^{𝑘−𝑗−1}+𝜆^{𝑘−𝑗−2}+⋯+1, (1.8) and (1.6) can be written as 𝐹(𝜆)−Ψ(𝜆)=0. (1.9) By Rouche's theorem, all roots 𝜆_𝑖, 𝑖=1,2,…,𝑘 of (1.6) satisfy |𝜆_𝑖|<1 because, on the boundary 𝐶 of a unit circle |𝜆|<1, we have |Ψ(𝜆)|_𝐶=𝛽₀⋅𝜆^{𝑘−𝑗−1}+𝜆^{𝑘−𝑗−2}<1+⋯+1|𝐹(𝜆𝑘−𝑗𝑘−𝑗=1=)|_𝐶, (1.10) and the functions 𝐹(𝜆), 𝐹(𝜆)−Ψ(𝜆) have the same number of zeros in the domain |𝜆|<1.

The case 𝛽₀<𝛽_cr and 𝑘−𝑗=1 is trivial because (1.6) turns into 𝜆^𝑘−𝛽₀=0 (1.11) and, due to inequality |𝜆|^𝑘=𝛽₀<𝛽_cr=1, has all its roots in the domain |𝜆|<1.

Then the relevant solutions 𝑦_𝑖(𝑛), 𝑖=1,2,…,𝑘 satisfy lim_𝑛→∞𝑦_𝑖(𝑛)=0, and the limit of the general solution of (1.3), 𝑦(𝑛)=lim_𝑛→∞∑^𝑘+1_𝑖=1𝑐_𝑖𝑦_𝑖(𝑛) where 𝑐_𝑖 are arbitrary constants, is finite because lim_𝑛→∞𝑦(𝑛)=lim_𝑛→∞^𝑘+1_𝑖=1𝑐_𝑖𝑦_𝑖(𝑛)=𝑐_𝑘+1. (1.12)

Let 𝛽₀>𝛽_cr. Since 𝑓(1)=1−𝛽₀⋅(𝑘−𝑗)<0 and 𝑓(+∞)=+∞, there exists a root 𝜆=𝜆_∗>1 of (1.6) and a solution 𝑦_∗(𝑛)=(𝜆_∗)^𝑛 of (1.3) satisfying lim_𝑛→∞𝑦_∗(𝑛)=∞. This means that solution 𝑦_∗(𝑛) is divergent.

Gathering all the cases considered, we have the following: (i)if 0<𝛽₀<𝛽_cr, then all solutions of (1.3) have a finite limit as 𝑛→∞, (ii)if 𝛽₀≥𝛽_cr, then there exists a divergent solution of (1.3) when 𝑛→∞.

The above analysis is not applicable in the case of a nonconstant function 𝛽(𝑛) in (1.1). To overcome some difficulties, the method of auxiliary inequalities is applied to investigate (1.1). From our results it follows that, for example, all solutions of (1.1) have a finite limit for 𝑛→∞ (or, in accordance with the below definition, are asymptotically convergent) if there exists a 𝑝>1 such that the inequality𝛽(𝑛)≤1−𝑝𝑘−𝑗𝑘+𝑗+12𝑛𝑘−𝑗 (1.13) holds for all 𝑛∈ℤ^∞_𝑛0−𝑘, where 𝑛₀ is a sufficiently large natural number. The limit of the right-hand side of (1.13) as 𝑛→∞ equals the critical value 𝛽_cr: lim_𝑛→∞1−𝑝𝑘−𝑗𝑘+𝑗+1=12𝑛𝑘−𝑗𝑘−𝑗=𝛽_cr. (1.14) It means that the function 𝛽(𝑛) in (1.1) can be sufficiently close to the critical value 𝛽_cr but such that all solutions of (1.1) are convergent as 𝑛→∞.

The proofs of the results are based on comparing the solutions of (1.1) with those of an auxiliary inequality that formally copies (1.1). First, we prove that, under certain conditions, (1.1) has an increasing and convergent solution 𝑦=𝑦(𝑛) (i.e., there exists a finite limit lim_𝑛→∞𝑦(𝑛)). Then we extend this statement to all the solutions of (1.1). It is an interesting fact that, in the general case, the asymptotic convergence of all solutions is characterized by the existence of a strictly increasing and bounded solution.

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 (1.1) are considered).

Arino and Pituk [1], for example, investigate linear and nonlinear perturbations of a linear autonomous functional-differential equation which has infinitely many equilibria. Bereketoğlu and Karakoç [2] derive sufficient conditions for the asymptotic constancy and estimates of the limits of solutions for an impulsive system, and Györi et al. give sufficient conditions for the convergence of solutions of a nonhomogeneous linear system of impulsive delay differential equations and a limit formula in [3]. Bereketoğlu and Pituk [4] give sufficient conditions for the asymptotic constancy of solutions of nonhomogeneous linear delay differential equations with unbounded delay. The limits of the solutions can be computed in terms of the initial conditions and a special matrix solution of the corresponding adjoint equation. In [5] Diblík studies the scalar equation under the assumption that every constant is its solution. Criteria and sufficient conditions for the convergence of solutions are found. The paper by Diblík and Růžičková [6] deals with the asymptotic behavior of a first-order linear homogeneous differential equation with double delay. The convergence of solutions of the delay Volterra equation in the critical case is studied by Messina et al. in [7]. Berezansky and Braverman study a behavior of solutions of a food-limited population model with time delay in [8].

Bereketoğlu and Huseynov [9] give sufficient conditions for the asymptotic constancy of the solutions of a system of linear difference equations with delays. The limits of the solutions, as 𝑡→∞, can be computed in terms of the initial function and a special matrix solution of the corresponding adjoint equation. Dehghan and Douraki [10] study the global behavior of a certain difference equation and show, for example, that zero is always an equilibrium point which satisfies a necessary and suffient condition for its local asymptotic stability. Györi and Horváth [11] study a system of linear delay difference equations such that every solution has a finite limit at infinity. The stability of difference equations is studied intensively in papers by Stević [12, 13]. In [12], for example, he proves the global asymptotic stability of a class of difference equations. Baštinec and Diblík [14] study a class of positive and vanishing at infinity solutions of a linear difference equation with delay. Nonoscillatory solutions of second-order difference equations of the Poincaré type are investigated by Medina and Pituk in [15].

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 𝑛→∞.

The paper is organized as follows. In Section 2 auxiliary results are collected, an auxiliary inequality is studied, and the relationship of its solutions with the solutions of (1.1) is derived. The existence of a strictly increasing and convergent solution of (1.1) is established in Section 3. Section 4 contains results concerning the convergence of all solutions of (1.1). An example illustrating the sharpness of the results derived is given as well.

Throughout the paper we adopt the customary notation ∑^𝑘_{𝑖=𝑘+𝑠}ℬ(𝑖)=0, where 𝑘 is an integer, 𝑠 is a positive integer, and ℬ denotes the function under consideration regardless of whether it is defined for the arguments indicated or not.

2. Auxiliary Results

Let 𝒞∶=𝒞(ℤ⁰_−𝑘,ℝ) be the space of discrete functions mapping the discrete interval ℤ⁰_−𝑘 into ℝ. Let 𝑣∈ℤ^∞_𝑛0 be given. The function 𝑦∶ℤ^∞_𝑣−𝑘→ℝ is said to be a solution of (1.1) on ℤ^∞_𝑣−𝑘 if it satisfies (1.1) for every 𝑛∈ℤ^∞_𝑣. A solution 𝑦 of (1.1) on ℤ^∞_𝑣−𝑘 is asymptotically convergent if the limit lim_𝑛→∞𝑦(𝑛) exists and is finite. For a given 𝑣∈ℤ^∞_𝑛0 and 𝜑∈𝒞, we say that 𝑦=𝑦_(𝑣,𝜑) is a solution of (1.1) defined by the initial conditions (𝑣,𝜑) if 𝑦_(𝑣,𝜑) is a solution of (1.1) on ℤ^∞_𝑣−𝑘 and 𝑦_(𝑣,𝜑)(𝑣+𝑚)=𝜑(𝑚) for 𝑚∈ℤ⁰_−𝑘.

2.1. Auxiliary Inequality

The auxiliary inequalityΔ𝜔(𝑛)≥𝛽(𝑛)𝜔𝑛−𝑗−𝜔(𝑛−𝑘) (2.1) will serve as a helpful tool in the analysis of (1.1). Let 𝑣∈ℤ^∞_𝑛0. The function 𝜔∶ℤ^∞_𝑣−𝑘→ℝ is said to be a solution of (2.1) on ℤ^∞_𝑣−𝑘 if 𝜔 satisfies inequality (2.1) for 𝑛∈ℤ^∞_𝑣. A solution 𝜔 of (2.1) on ℤ^∞_𝑣−𝑘 is asymptotically convergent if the limit lim_𝑛→∞𝜔(𝑛) exists and is finite.

We give some properties of solutions of inequalities of the type (2.1), which will be utilized later on. We will also compare the solutions of (1.1) with the solutions of inequality (2.1).

Lemma 2.1. Let 𝜑∈𝒞 be strictly increasing (nondecreasing, strictly decreasing, nonincreasing) on ℤ⁰_−𝑘. Then the corresponding solution 𝑦_{(𝑛∗,𝜑)}(𝑛) of (1.1) with 𝑛^∗∈ℤ^∞_𝑛0 is strictly increasing (nondecreasing, strictly decreasing, nonincreasing) on ℤ^∞_{𝑛∗−𝑘} too.
If 𝜑 is strictly increasing (nondecreasing) and 𝜔∶ℤ^∞_𝑛0−𝑘→ℝ is a solution of inequality (2.1) with 𝜔(𝑛₀+𝑚)=𝜑(𝑚), 𝑚∈ℤ^𝑛0_𝑛0−𝑘, then 𝜔 is strictly increasing (nondecreasing) on ℤ^∞_𝑛0−𝑘.

Proof. This follows directly from (1.1), inequality (2.1), and from the properties 𝛽(𝑛)>0, 𝑛∈ℤ^∞_𝑛0−𝑘, 𝑘>𝑗≥0.

Theorem 2.2. Let 𝜔(𝑛) be a solution of inequality (2.1) on ℤ^∞_𝑛0−𝑘. Then there exists a solution 𝑦(𝑛) of (1.1) on ℤ^∞_𝑛0−𝑘 such that the inequality 𝑦(𝑛)≤𝜔(𝑛) (2.2) holds on ℤ^∞_𝑛0−𝑘. In particular, a solution 𝑦(𝑛₀,𝜙) of (1.1) with 𝜙∈𝒞 defined by the equation 𝜙(𝑚)∶=𝜔𝑛₀+𝑚,𝑚∈ℤ⁰_−𝑘 (2.3) is such a solution.

Proof. Let 𝜔(𝑛) be a solution of inequality (2.1) on ℤ^∞_𝑛0−𝑘. We will show that the solution 𝑦(𝑛)∶=𝑦_(𝑛0,𝜙)(𝑛) of (1.1) satisfies inequality (2.2), that is, 𝑦_(𝑛0,𝜙)(𝑛)≤𝜔(𝑛) (2.4) on ℤ^∞_𝑛0−𝑘. Let 𝑊∶ℤ^∞_𝑛0−𝑘→ℝ be defined by 𝑊(𝑛)=𝜔(𝑛)−𝑦(𝑛). Then 𝑊=0 on ℤ^𝑛0_𝑛0−𝑘, and, in addition, 𝑊 is a solution of (2.1) on ℤ^∞_𝑛0−𝑘. Lemma 2.1 implies that 𝑊 is nondecreasing. Consequently, 𝜔(𝑛)−𝑦(𝑛)≥𝜔(𝑛₀)−𝑦(𝑛₀)=0 for all 𝑛≥𝑛₀.

2.2. Comparison Lemma

Now we consider an inequality of the type  (2.1)Δ𝜔^∗(𝑛)≥𝛽₁(𝑛)𝜔^∗𝑛−𝑗−𝜔^∗(𝑛−𝑘), (2.5) where 𝛽₁∶ℤ^∞_𝑛0−𝑘→ℝ⁺ is a discrete function satisfying 𝛽₁(𝑛)≥𝛽(𝑛) on ℤ^∞_𝑛0−𝑘. The following comparison lemma holds.

Lemma 2.3. Let 𝜔^∗∶ℤ^∞_𝑛0−𝑘→ℝ⁺ be a nondecreasing positive solution of inequality (2.5) on ℤ^∞_𝑛0−𝑘. Then 𝜔^∗ is a solution of inequality (2.1) on ℤ^∞_𝑛0−𝑘 too.

Proof. Let 𝜔^∗ be a nondecreasing solution of (2.5) on ℤ^∞_𝑛0−𝑘. We have 𝜔^∗𝑛−𝑗−𝜔^∗(𝑛−𝑘)≥0 (2.6) because 𝑛−𝑘<𝑛−𝑗. Then Δ𝜔^∗(𝑛)≥𝛽₁(𝑛)𝜔^∗𝑛−𝑗−𝜔^∗(𝑛−𝑘)≥𝛽(𝑛)𝜔^∗𝑛−𝑗−𝜔^∗(𝑛−𝑘) (2.7) on ℤ^∞_𝑛0. Consequently, the function 𝜔∶=𝜔^∗ solves inequality (2.1) on ℤ^∞_𝑛0, too.

2.3. A Solution of Inequality (2.1)

We will construct a solution of inequality (2.1). In the following lemma, we obtain a solution of inequality (2.1) in the form of a sum. This auxiliary result will help us derive sufficient conditions for the existence of a strictly increasing and asymptotically convergent solution of (1.1) (see Theorem 3.2 below).

Lemma 2.4. Let there exist a discrete function 𝜀∶ℤ^∞_𝑛0−𝑘→ℝ⁺ such that 𝜀(𝑛+1)≥^𝑛−𝑗_{𝑖=𝑛−𝑘+1}𝛽(𝑖−1)𝜀(𝑖) (2.8) on ℤ^∞_𝑛0. Then there exists a solution 𝜔(𝑛)=𝜔_𝜀(𝑛) of inequality (2.1) defined on ℤ^∞_𝑛0−𝑘 having the form 𝜔_𝜀(𝑛)∶=^𝑛_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖). (2.9)

Proof. For 𝑛∈ℤ^∞_𝑛0, we get Δ𝜔_𝜀(𝑛)=𝜔_𝜀(𝑛+1)−𝜔_𝜀(𝑛)=^𝑛+1_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖)−^𝑛_{𝑖=𝑛0−𝑘+1}𝛽()𝜀(𝑖)(𝑛)𝜀(),𝜔𝑖−1=𝛽𝑛+1_𝜀𝑛−𝑗−𝜔_𝜀(𝑛−𝑘)=^𝑛−𝑗_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖)−^𝑛−𝑘_{𝑖=𝑛0−𝑘+1}𝛽()𝜀(𝑖)=𝑖−1_{𝑛−𝑗𝑖=𝑛−𝑘+1}𝛽(𝑖−1)𝜀(𝑖). (2.10) We substitute 𝜔_𝜀 for 𝜔 in (2.1). Using (2.10), we get 𝛽(𝑛)𝜀(𝑛+1)≥𝛽(𝑛)^𝑛−𝑗_{𝑛−𝑘+1}𝛽(𝑖−1)𝜀(𝑖). (2.11) This inequality will be satisfied if inequality (2.8) holds. Indeed, reducing the last inequality by 𝛽(𝑛), we obtain 𝜀(𝑛+1)≥^𝑛−𝑗_{𝑛−𝑘+1}𝛽(𝑖−1)𝜀(𝑖), (2.12) which is inequality (2.8).

2.4. Decomposition of a Function into the Difference of Two Strictly Increasing Functions

It is well known that every absolutely continuous function is representable as the difference of two increasing absolutely continuous functions [16, page 318]. We will need a simple discrete analogue of this result.

Lemma 2.5. Every function 𝜑∈𝒞 can be decomposed into the difference of two strictly increasing functions 𝜑_𝑗∈𝒞, 𝑗=1,2, that is, 𝜑(𝑛)=𝜑₁(𝑛)−𝜑₂(𝑛),𝑛∈ℤ⁰_−𝑘. (2.13)

Proof. Let constants 𝑀_𝑛>0, 𝑛∈ℤ⁰_−𝑘 be such that inequalities 𝑀_𝑛+1>𝑀_𝑛+max0,𝜑(𝑛)−𝜑(𝑛+1) (2.14) are valid for 𝑛∈ℤ⁻¹_−𝑘. We set 𝜑₁(𝑛)∶=𝜑(𝑛)+𝑀_𝑛,𝑛∈ℤ⁰_−𝑘,𝜑₂(𝑛)∶=𝑀_𝑛,𝑛∈ℤ⁰_−𝑘. (2.15) It is obvious that (2.13) holds. Now we verify that both functions 𝜑_𝑗, 𝑗=1,2 are strictly increasing. The first one should satisfy 𝜑₁(𝑛+1)>𝜑₁(𝑛) for 𝑛∈ℤ⁻¹_−𝑘, which means that 𝜑(𝑛+1)+𝑀_𝑛+1>𝜑(𝑛)+𝑀_𝑛 (2.16) or 𝑀_𝑛+1>𝑀_𝑛+𝜑(𝑛)−𝜑(𝑛+1). (2.17) We conclude that the last inequality holds because, due to (2.14), we have 𝑀_𝑛+1>𝑀_𝑛+max0,𝜑(𝑛)−𝜑(𝑛+1)≥𝑀_𝑛+𝜑(𝑛)−𝜑(𝑛+1). (2.18) The inequality 𝜑₂(𝑛+1)>𝜑₂(𝑛) obviously holds for 𝑛∈ℤ⁻¹_−𝑘 due to (2.14) as well.

2.5. Auxiliary Asymptotic Decomposition

The following lemma can be proved easily by induction. The symbol 𝒪 stands for the Landau order symbol.

Lemma 2.6. For fixed 𝑟, 𝜎∈ℝ⧵{0}, the asymptotic representation (𝑛−𝑟)^𝜎=𝑛^𝜎1−𝜎𝑟𝑛1+𝒪𝑛² (2.19) holds for 𝑛→∞.

3. Convergent Solutions of (1.1)

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 (1.1). First we present two obvious statements concerning asymptotic convergence. From Lemma 2.1 and Theorem 2.2, we immediately get the following.

Theorem 3.1. Let 𝜔(𝑛) be a strictly increasing and bounded solution of (2.1) on ℤ^∞_𝑛0−𝑘. Then there exists a strictly increasing and asymptotically convergent solution 𝑦(𝑛) of (1.1) on ℤ^∞_𝑛0−𝑘.

From Lemma 2.1, Theorem 2.2, and Lemma 2.4, we get the following.

Theorem 3.2. Let there exist a function 𝜀∶ℤ^∞_𝑛0−𝑘→ℝ⁺ satisfying ^∞_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖)<∞ (3.1) and inequality (2.8) on ℤ^∞_𝑛0. Then the initial function 𝜑(𝑛)=^𝑛0+𝑛_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖),𝑛∈ℤ⁰_−𝑘 (3.2) defines a strictly increasing and asymptotically convergent solution 𝑦_(𝑛0,𝜑)(𝑛) of (1.1) on ℤ^∞_𝑛0−𝑘 satisfying the inequality 𝑦(𝑛)≤^𝑛_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖) (3.3) on ℤ^∞_𝑛0.

Assuming that the coefficient 𝛽(𝑛) in (1.1) can be estimated by a suitable function, we can prove that (1.1) has a convergent solution.

Theorem 3.3. Let there exist a 𝑝>1 such that the inequality 𝛽(𝑛)≤1−𝑝𝑘−𝑗𝑘+𝑗+12𝑛𝑘−𝑗 (3.4) holds for all 𝑛∈ℤ^∞_𝑛0−𝑘. Then there exists a strictly increasing and asymptotically convergent solution 𝑦(𝑛) of (1.1) as 𝑛→∞.

Proof. In the proof, we assume (without loss of generality) that 𝑛₀ is sufficiently large for asymptotic computations to be valid. Let us verify that inequality (2.8) has a solution 𝜀 such that ^∞_{𝑖=𝑛0−𝑘+1}𝛽(𝑖−1)𝜀(𝑖)<∞. (3.5) We put 𝛽(𝑛)=𝛽^∗(𝑛)1∶=−𝑝𝑘−𝑗^∗(𝑛)12𝑛,𝜀∶=𝑛^𝛼 (3.6) in inequality (2.8), where 𝑝^∗>0 and 𝛼>1 are constants. Then, for the right-hand side ℛ(𝑛) of (2.8), we have ℛ(𝑛)=^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1−𝑝𝑘−𝑗2(^∗)1𝑖−1𝑖^𝛼=1𝑘−𝑗^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1𝑖^𝛼−𝑝^∗2^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1(𝑖−1)𝑖^𝛼=11𝑘−𝑗(𝑛−𝑘+1)^𝛼+1(𝑛−𝑘+2)^𝛼1+⋯+𝑛−𝑗^𝛼−𝑝^∗21(𝑛−𝑘)(𝑛−𝑘+1)^𝛼+1(𝑛−𝑘+1)(𝑛−𝑘+2)^𝛼1+⋯+𝑛−𝑗−1𝑛−𝑗^𝛼. (3.7) We asymptotically decompose ℛ(𝑛) as 𝑛→∞ using decomposition formula (2.19) in Lemma 2.6. We apply this formula to each term in the first square bracket with 𝜎=−𝛼 and with 𝑟=𝑘−1 for the first term, 𝑟=𝑘−2 for the second term, and so forth, and, finally, 𝑟=𝑗 for the last term. To estimate the terms in the second square bracket, we need only the first terms of the decomposition and the order of accuracy, which can be computed easily without using Lemma 2.6. We get ℛ(𝑛)=1𝑘−𝑗𝑛^𝛼𝛼()1+𝑘−1𝑛𝛼()+1+𝑘−2𝑛+⋯+1+𝛼𝑗𝑛1+𝒪𝑛²−𝑝^∗2𝑛^𝛼+111+1+⋯+1+𝒪𝑛=1𝑘−𝑗𝑛^𝛼+1()()1𝑘−𝑗𝑛+𝛼𝑘−1+𝛼𝑘−2+⋯+𝛼𝑗+𝒪𝑛−𝑝^∗2𝑛^𝛼+11𝑘−𝑗+𝒪𝑛=1𝑛^𝛼+𝛼𝑘−𝑗𝑛^𝛼+1𝑘+𝑗−1𝑘−𝑗2−𝑝^∗2𝑛^𝛼+11𝑘−𝑗+𝒪𝑛^𝛼+2, (3.8) and, finally, ℛ(𝑛)=1𝑛^𝛼+𝛼2𝑛^𝛼+1−𝑝𝑘+𝑗−1^∗2𝑛^𝛼+11𝑘−𝑗+𝒪𝑛^𝛼+2. (3.9) A similar decomposition of the left-hand side ℒ(𝑛)=𝜀(𝑛+1)=(𝑛+1)^−𝛼 in inequality (2.8) leads to ℒ(𝑛)≡1(𝑛+1)^𝛼=1𝑛^𝛼𝛼1−𝑛1+𝒪𝑛²=1𝑛^𝛼−𝛼𝑛^𝛼+11+𝒪𝑛^𝛼+2 (3.10) (we use decomposition formula (2.19) in Lemma 2.6 with 𝜎=−𝛼 and 𝑟=−1).
Comparing ℒ(𝑛) and ℛ(𝑛), we see that, for ℒ(𝑛)≥ℛ(𝑛), it is necessary to match the coefficients of the terms 𝑛^−𝛼−1 because the coefficients of the terms 𝑛^−𝛼 are equal. It means that we need the inequality 𝛼−𝛼>𝑘+𝑗−12−𝑝^∗2𝑘−𝑗. (3.11) Simplifying this inequality, we get 𝑝^∗2𝛼𝑘−𝑗>𝛼+𝑘+𝑗−12,𝑝^∗𝑘−𝑗>𝛼𝑘+𝑗+1, (3.12) and, finally, 𝑝^∗>𝛼𝑘+𝑗+1.𝑘−𝑗 (3.13) We set 𝑝^∗∶=𝑝𝑘+𝑗+1𝑘−𝑗, (3.14) where 𝑝=const. Then the previous inequality holds for 𝑝>𝛼, that is, for 𝑝>1. Consequently, the function 𝛽^∗ defined by (3.6) has the form 𝛽^∗(𝑛)=1−𝑝𝑘−𝑗𝑘+𝑗+12𝑘−𝑗𝑛 (3.15) with 𝑝>1, and, for the function 𝜔_𝜀 defined by formula (2.9), we have 𝜔_𝜀(𝑛)=^𝑛_{𝑖=𝑛0−𝑘+1}1−𝑝𝑘−𝑗𝑘+𝑗+12()1𝑘−𝑗𝑖−1𝑖^𝛼. (3.16) Function 𝜔_𝜀(𝑛) is a positive solution of inequality (2.1), and, moreover, it is easy to verify that 𝜔_𝜀(∞)<∞ since 𝛼>1. This is a solution to every inequality of the type (2.1) if the function 𝛽^∗ fixed by formula (3.15) is changed by an arbitrary function 𝛽 satisfying inequality (3.4). This is a straightforward consequence of Lemma 2.3 if, in its formulation, we set 𝛽₁(𝑛)∶=𝛽^∗(𝑛)=1−𝑝𝑘−𝑗𝑘+𝑗+12𝑘−𝑗𝑛 (3.17) with 𝑝>1 since 𝜔^∗≡𝜔_𝜀 is the desired solution of inequality (2.5). Finally, by Theorem 3.1 with 𝜔∶=𝜔_𝜀 as defined by (3.16), we conclude that there exists a strictly increasing and convergent solution 𝑦(𝑛) of (1.1) as 𝑛→∞ satisfying the inequality 𝑦(𝑛)<𝜔_𝜀(𝑛) (3.18) on ℤ^∞_𝑛0−𝑘.

4. Convergence of All Solutions

In this part we present results concerning the convergence of all solutions of (1.1). First we use inequality (3.4) to state the convergence of all the solutions.

Theorem 4.1. Let there exist a 𝑝>1 such that inequality (3.4) holds for all 𝑛∈ℤ^∞_𝑛0−𝑘. Then all solutions of (1.1) are convergent as 𝑛→∞.

Proof. First we prove that every solution defined by a monotone initial function is convergent. We will assume that a strictly monotone initial function 𝜑∈𝒞 is given. For definiteness, let 𝜑 be strictly increasing or nondecreasing (the case when it is strictly decreasing or nonincreasing can be considered in much the same way). By Lemma 2.1, the solution 𝑦_(𝑛0,𝜑) is monotone; that is, it is either strictly increasing or nondecreasing. We prove that 𝑦_(𝑛0,𝜑) is convergent.
By Theorem 3.3 there exists a strictly increasing and asymptotically convergent solution 𝑦=𝑌(𝑛) of (1.1) on ℤ^∞_𝑛0−𝑘. Without loss of generality we assume 𝑦_(𝑛0,𝜑)≢𝑌(𝑛) on ℤ^∞_𝑛0−𝑘 since, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume Δ𝑌(𝑛)>0,𝑛∈ℤ^𝑛0_𝑛⁻¹_0−𝑘. (4.1) Hence, there is a constant 𝛾>0 such that Δ𝑌(𝑛)−𝛾Δ𝑦(𝑛)>0,𝑛∈ℤ^𝑛0_𝑛⁻¹_0−𝑘 (4.2) or Δ𝑌(𝑛)−𝛾𝑦(𝑛)>0,𝑛∈ℤ^𝑛0_𝑛⁻¹_0−𝑘, (4.3) and the function 𝑌(𝑛)−𝛾𝑦(𝑛) is strictly increasing on ℤ^𝑛0_𝑛⁻¹_0−𝑘. Then Lemma 2.1 implies that 𝑌(𝑛)−𝛾𝑦(𝑛) is strictly increasing on ℤ^∞_𝑛0−𝑘. Thus 𝑌(𝑛)−𝛾𝑦(𝑛)>𝑌𝑛₀−𝛾𝑦𝑛₀,𝑛∈ℤ^∞_𝑛0 (4.4) or 𝑦(𝑛)<1𝛾𝑌(𝑛)−𝑌𝑛₀+𝑦𝑛₀,𝑛∈ℤ^∞_𝑛0, (4.5) and, consequently, 𝑦(𝑛) is a bounded function on ℤ^∞_𝑛0−𝑘 because of the boundedness of 𝑌(𝑛). Obviously, in such a case, 𝑦(𝑛) is asymptotically convergent and has a finite limit.
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 𝑦_(𝑛0,𝜑) generated by a nonmonotone initial function 𝜑∈𝒞, there are two possibilities: 𝑦_(𝑛0,𝜑) is either eventually monotone and, consequently, convergent, or 𝑦_(𝑛0,𝜑) is eventually nonmonotone.
Now we use the statement of Lemma 2.5 that every discrete function 𝜑∈𝒞 can be decomposed into the difference of two strictly increasing discrete functions 𝜑_𝑗∈𝒞, 𝑗=1,2. In accordance with the previous part of the proof, every function 𝜑_𝑗∈𝒞, 𝑗=1,2 defines a strictly increasing and asymptotically convergent solution 𝑦_{(𝑛0,𝜑𝑗)}. Now it is clear that the solution 𝑦_(𝑛0,𝜑) is asymptotically convergent.

We will finish the paper with two obvious results. Inequality (3.4) in Theorem 3.3 was necessary only for the proof of the existence of an asymptotically convergent solution. If we assume the existence of an asymptotically convergent solution rather than inequality (3.4), we can formulate the following result, the proof of which is an elementary modification of the proof of Theorem 4.1.

Theorem 4.2. If (1.1) has a strictly monotone and asymptotically convergent solution on ℤ^∞_𝑛0−𝑘, then all the solutions of (1.1) defined on ℤ^∞_𝑛0−𝑘 are asymptotically convergent.

Combining the statements of Theorems 2.2, 3.1, and 4.2, we get a series of equivalent statements below.

Theorem 4.3. The following three statements are equivalent. (a)Equation (1.1) has a strictly monotone and asymptotically convergent solution on ℤ^∞_𝑛0−𝑘.(b)All solutions of (1.1) defined on ℤ^∞_𝑛0−𝑘 are asymptotically convergent.(c)Inequality (2.1) has a strictly monotone and asymptotically convergent solution on ℤ^∞_𝑛0−𝑘.

Example 4.4. We will demonstrate the sharpness of the criterion (3.4) by the following example. Let 𝑘=1, 𝑗=0, 𝛽(𝑛)=1−1/𝑛, 𝑛∈ℤ^∞_𝑛0−1, 𝑛₀=2 in (1.1); that is, we consider the equation (𝑛)=1Δ𝑦1−𝑛𝑦(𝑛)−𝑦(𝑛−1). (4.6) By Theorems 3.3 and 4.3, all solutions are asymptotically convergent if 𝛽(𝑛)≤1−𝑝𝑘−𝑗𝑘+𝑗+1𝑝2𝑛𝑘−𝑗=1−𝑛, (4.7) where a constant 𝑝>1. In our case the inequality (4.7) does not hold since inequality 𝛽(𝑛)1=1−𝑛𝑝≤1−𝑛 (4.8) is valid only for 𝑝≤1. Inequality (4.7) is sharp because there exists a solution 𝑦=𝑦^∗(𝑛) of (4.6) having the form of an 𝑛th partial sum of harmonic series, that is, 𝑦^∗(𝑛)=^𝑛_𝑖=11𝑖 (4.9) with the obvious property lim_𝑛→∞𝑦^∗(𝑛)=+∞. Then (by Theorem 4.3), all strictly monotone (increasing or decreasing) solutions of (4.6) tend to infinity.

Acknowledgments

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).