Asymptotic Upper and Lower Estimates of a Class of Positive Solutions of a Discrete Linear Equation with a Single Delay

Abstract

We study a frequently investigated class of linear difference equations Δ𝑣(𝑛)=−𝑝(𝑛)𝑣(𝑛−𝑘) with a positive coefficient 𝑝(𝑛) and a single delay k. Recently, it was proved that if the function 𝑝(𝑛) is bounded above by a certain function, then there exists a positive vanishing solution of the considered equation, and the upper bound was found. Here we improve this result by finding even the lower bound for the positive solution, supposing the function 𝑝(𝑛) is bounded above and below by certain functions.

1. Introduction

Throughout this paper, we use the following notation: for an integer 𝑞, we define ℤ^∞_𝑞∶=𝑞,𝑞+1,…. (1.1) We investigate the asymptotic behavior as 𝑛→∞ of the solutions of the discrete delayed equation of the (𝑘+1)-th order Δ𝑣(𝑛)=−𝑝(𝑛)𝑣(𝑛−𝑘), (1.2) where 𝑛 is the independent variable assuming values from the set ℤ^∞_𝑎 with a fixed 𝑎∈ℕ={0,1,2,…}. The number 𝑘∈ℕ,𝑘≥1 is the fixed delay, Δ𝑣(𝑛)=𝑣(𝑛+1)−𝑣(𝑛), and 𝑝∶ℤ^∞_𝑎→ℝ⁺=(0,∞).

Along with (1.2), we consider 𝑘+1 initial conditions 𝑣(𝑎+𝑠−𝑘)=𝑣^{𝑎+𝑠−𝑘}∈ℝ,𝑠=0,1,…,𝑘. (1.3) Initial problem (1.2), (1.3) obviously has a unique solution, defined for every 𝑛∈ℤ^∞_𝑎−𝑘. Moreover, the solution of (1.2) continuously depends on initial conditions (1.3).

Equation (1.2) is investigated very frequently. It was analyzed, for example, in papers [1–3] (where the comparison method [4, 5] was used) and [6]. Similar problems for differential and dynamic equations are studied, for example, in [7–10].

In a recent work of the authors [6], it is proved that if the function 𝑝(𝑛) is bounded above by a certain function, then there exists a positive vanishing (i.e., tending to 0 as 𝑛→∞) solution of the considered equation. Moreover, its upper bound was found. Our aim is to improve this result and to show that if the coefficient 𝑝(𝑛) is between two functions 𝑝_ℓ(𝑛)−𝜓(𝑛) and 𝑝_ℓ(𝑛)+𝜔(𝑛) (see (2.3), (2.6), and (2.7) below) then (1.2) has a positive vanishing solution which is bounded from below by the function 𝛼_ℓ(𝑛) (see (2.5)) and from above by the function 𝜈_ℓ(𝑛) (see (2.4)). Due to the linearity of equation considered it becomes clear that a similar result holds for a one-parametric family of positive vanishing solutions of (1.2).

To prove this, we will use Theorem 1.1 which is one of the main results of [6]. This theorem is valid for any delayed difference equation of the form: Δ𝑣(𝑛)=𝑓(𝑛,𝑣(𝑛),𝑣(𝑛−1),…,𝑣(𝑛−𝑘)). (1.4)

Theorem 1.1. Let 𝑏(𝑛), 𝑐(𝑛), 𝑏(𝑛)<𝑐(𝑛) be real functions defined on ℤ^∞_𝑎−𝑘. Further, let 𝑓∶ℤ^∞_𝑎×ℝ^𝑘+1→ℝ be a continuous function and let the inequalities: 𝑏(𝑛)+𝑓𝑛,𝑏(𝑛),𝑣₂,…,𝑣_𝑘+1(),𝑐(𝑛)(𝑛)<𝑏𝑛+1+𝑓𝑛,𝑐,𝑣₂,…,𝑣_𝑘+1>𝑐(𝑛+1), (1.5)(1.6) hold for every 𝑛∈ℤ^∞_𝑎 and every 𝑣₂,…,𝑣_𝑘+1 such that 𝑏(𝑛−𝑖+1)<𝑣_𝑖<𝑐(𝑛−𝑖+1),𝑖=2,…,𝑘+1. (1.7) Then there exists a solution 𝑣=𝑣^∗(𝑛) of (1.4) satisfying the inequalities 𝑏(𝑛)<𝑣^∗(𝑛)<𝑐(𝑛), (1.8)
for every 𝑛∈ℤ^∞_𝑎−𝑘.

For related comparison theorems for solutions of difference equations as well as related methods and their applications, see, for example, [1, 11–21] and the related references therein. Investigation of positive solutions (and connected problems of oscillating solutions) attracted recently large attention. Except the references given above, one refers as well to [11, 22–33], and to the references therein. Existence of positive solutions of some classes of difference equations has been also studied in papers [12–16]. The existence of unbounded solutions by some comparison methods can be found, for example, in [17, 18].

2. Auxiliary Functions and Lemmas

Define the expression ln_𝑞𝑛, 𝑞∈ℕ⧵{0}, as ln_𝑞𝑛∶=lnln_𝑞−1𝑛, (2.1)where ln₀𝑛∶=𝑛. We will write only ln𝑛 instead of ln₁𝑛. Further, for a fixed integer ℓ≥0 define auxiliary functions: 𝜇_ℓ(𝑛)1∶=8𝑛²+18(𝑛ln𝑛)²1+⋯+8𝑛ln𝑛⋯ln_ℓ𝑛²,𝑝_ℓ(𝑛)𝑘∶=𝑘+1^𝑘⋅1𝑘+1+𝑘𝜇_ℓ(𝑛)𝜈_ℓ,(𝑛)𝑘∶=𝑘+1^𝑛⋅𝑛ln𝑛ln₂𝑛⋯ln_ℓ𝛼𝑛,_ℓ(𝑛)𝑘∶=𝑘+1^𝑛⋅𝑛ln𝑛ln₂𝑛⋯ln_ℓ𝑛ln^−𝜎_ℓ+1𝑛, (2.2)(2.3)(2.4)(2.5) where 𝜎∈ℝ,𝜎>0, is a constant. Notice that if 𝑎 is sufficiently large, all these functions are well defined for 𝑛∈ℤ^∞_𝑎.

Finally, let functions 𝜓,𝜔∶ℤ^∞_𝑎→ℝ⁺ satisfy for 𝑛∈ℤ^∞_𝑎 the inequalities: 𝜓(𝑛)≤𝑘𝑘+1^𝑘⋅𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛,𝜔(𝑛)𝑘≤𝜀𝑘+1^𝑘⋅𝑘(2𝑘−1)16𝑛³, (2.6)(2.7) for fixed 𝛿>0,𝛽>2 and 𝜀∈(0,1).

In [3], it was proved that if 𝑝(𝑛) in (1.2) is a positive function bounded above by 𝑝_ℓ(𝑛) for some ℓ≥0, then there exists a positive solution of (1.2) bounded above by the function 𝜈_ℓ(𝑛) for 𝑛 sufficiently large. Since lim_𝑛→∞𝜈_ℓ(𝑛)=0, such solution will vanish as 𝑛→∞. This result was further improved in [6], where it was shown that (1.2) has a positive solution bounded above by 𝜈_ℓ(𝑛) even if the coefficient 𝑝(𝑛) satisfies a less restrictive inequality, namely, 𝑝(𝑛)<𝑝_ℓ(𝑛)+𝜔(𝑛). Here we will prove that function 𝛼_ℓ provides the lower estimate of the solution, supposing 𝑝_ℓ(𝑛)−𝜓(𝑛)≤𝑝(𝑛)≤𝑝_ℓ(𝑛)+𝜔(𝑛). The proof of this statement will be based on the following four lemmas. The symbols “𝑜” and “𝑂” stand for the Landau order symbols and are used as 𝑛→∞.

Lemma 2.1. For fixed 𝑟∈ℝ⧵{0} and fixed 𝑞∈ℕ, the asymptotic representation: ln_𝑞(𝑛−𝑟)=ln_𝑞𝑟𝑛−𝑛ln𝑛⋯ln_𝑞−1𝑛−𝑟²2𝑛²ln𝑛⋯ln_𝑞−1𝑛−𝑟²2(𝑛ln𝑛)²ln₂𝑛⋯ln_𝑞−1𝑛𝑟−⋯−²2𝑛ln𝑛⋯ln_𝑞−1𝑛²1+𝑜𝑛³, (2.8) holds as 𝑛→∞.

Proof. Relation (2.8) can be proved by induction with respect to 𝑞, for details, see [6].

Lemma 2.2. For fixed 𝑟,𝑠∈ℝ⧵{0} and fixed 𝑞∈ℕ, the asymptotic representations: ln_𝑞(𝑛−𝑟)ln_𝑞𝑛^𝑠=1−𝑟𝑠𝑛ln𝑛⋯ln_𝑞𝑛−𝑟²𝑠2𝑛²ln𝑛⋯ln_𝑞𝑛−𝑟²𝑠2(𝑛ln𝑛)²ln₂𝑛⋯ln_𝑞𝑛−𝑟−⋯₂𝑠2𝑛ln𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛+𝑟²𝑠(𝑠−1)2𝑛ln𝑛⋯ln_𝑞𝑛²1+𝑜𝑛³,𝑛−𝑟𝑛𝑟=1−−𝑟2𝑛8𝑛²²−𝑟³16𝑛³1+𝑜𝑛³, (2.9)(2.10) hold as 𝑛→∞.

Proof. Both these relations are simple consequences of the asymptotic formula: (1−𝑥)^𝑠𝑠()=1−𝑠𝑥+𝑠−12𝑥²−𝑠(𝑠−1)(𝑠−2)6𝑥³+𝑜𝑥³as𝑥⟶0. (2.11) and of Lemma 2.1 (for formula (2.9)).
In the case of relation (2.10), we put 𝑥=𝑟/𝑛 and 𝑠=1/2.
To prove relation (2.9), first notice that dividing (2.8) by ln_𝑞𝑛, we getln_𝑞(𝑛−𝑟)ln_𝑞𝑛𝑟=1−𝑛ln𝑛⋯ln_𝑞−1𝑛ln_𝑞𝑛−𝑟²2𝑛²ln𝑛⋯ln_𝑞−1𝑛ln_𝑞𝑛−𝑟²2(𝑛ln𝑛)²ln₂𝑛⋯ln_𝑞−1𝑛ln_𝑞𝑛𝑟−⋯−²2𝑛ln𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛1+𝑜𝑛³. (2.12) Thus, putting 𝑥=1−ln_𝑞(𝑛−𝑟)ln_𝑞𝑛=𝑟𝑛ln𝑛⋯ln_𝑞𝑛+𝑟²2𝑛²ln𝑛⋯ln_𝑞𝑛𝑟+⋯+²2𝑛ln𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛1+𝑜𝑛³ (2.13) and using (2.11), we get (2.9).

The following lemma is proved in [6].

Lemma 2.3. For fixed 𝑟∈ℝ⧵{0} and fixed 𝑞∈ℕ, the asymptotic representation: (𝑛−𝑟)𝑛⋅ln(𝑛−𝑟)⋯ln𝑛ln_𝑞(𝑛−𝑟)ln_𝑞𝑛1=1−𝑟+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_𝑞𝑛−𝑟²𝜇_𝑞(𝑛)−𝑟³16𝑛³1+𝑜𝑛³, (2.14) holds as 𝑛→∞.

Lemma 2.4. For fixed 𝑟∈ℝ⧵{0}, 𝜎∈ℝ⁺ and 𝑞∈ℕ, the asymptotic representation: (𝑛−𝑟)𝑛⋅ln(𝑛−𝑟)⋯ln𝑛ln_𝑞(𝑛−𝑟)ln_𝑞𝑛⋅ln^−𝜎_𝑞+1(𝑛−𝑟)ln^−𝜎_𝑞+1𝑛1=1−𝑟+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_𝑞𝑛−𝜎2𝑛ln𝑛⋯ln_𝑞𝑛ln_𝑞+1𝑛−𝑟²𝜇_𝑞(𝑛)+𝜎(𝜎+2)8⋅𝑟²𝑛ln𝑛⋯ln_𝑞+1𝑛²−𝑟³16𝑛³1+𝑜𝑛³, (2.15) holds as 𝑛→∞.

Proof. Using Lemma 2.2 with 𝑠=−𝜎/2 and 𝑞+1 instead of 𝑞, we get for 𝑛→∞ln^−𝜎_𝑞+1(𝑛−𝑟)ln^−𝜎_𝑞+1𝑛=1+𝑟𝜎2𝑛ln𝑛⋯ln_𝑞+1𝑛+𝑟²𝜎4𝑛²ln𝑛⋯ln_𝑞+1𝑛+𝑟²𝜎4(𝑛ln𝑛)²ln₂𝑛⋯ln_𝑞+1𝑛+𝑟+⋯₂𝜎4𝑛ln𝑛⋯ln_𝑞𝑛²ln_𝑞+1𝑛+𝑟²𝜎(𝜎+2)8𝑛ln𝑛⋯ln_𝑞+1𝑛²1+𝑜𝑛³. (2.16) Multiplying the asymptotic representations (2.14) and (2.16), we get (2.15).

3. Main Result

Now we are ready to prove that there exists a positive solution of (1.2) which is bounded below and above. Remind the functions 𝑝_ℓ, 𝜈_ℓ, 𝛼_ℓ, 𝜓, and 𝜔 were defined by (2.3)–(2.6) and (2.7), respectively.

Theorem 3.1. Suppose that there exist numbers 𝑎,ℓ∈ℕ, and 𝜎>0, such that the function 𝑝 in (1.2) satisfies the inequalities 0<𝑝_ℓ(𝑛)−𝜓(𝑛)≤𝑝(𝑛)≤𝑝_ℓ(𝑛)+𝜔(𝑛), (3.1) for every 𝑛∈ℤ^∞_𝑎. Then there exists a solution 𝑣=𝑣^∗(𝑛),𝑛∈ℤ^∞_𝑎−𝑘 of (1.2) such that for 𝑛 sufficiently large the inequalities: 𝛼_ℓ(𝑛)<𝑣^∗(𝑛)<𝜈_ℓ(𝑛), (3.2) hold.

Proof. Show that all the assumptions of Theorem 1.1 are fulfilled. For (1.2), 𝑓𝑛,𝑣₁,…,𝑣_𝑘+1=−𝑝(𝑛)𝑣_𝑘+1. (3.3) This is a continuous function. Put 𝑏(𝑛)∶=𝛼_ℓ(𝑛),𝑐(𝑛)∶=𝜈_ℓ(𝑛). (3.4) We have to prove that for every 𝑣₂,…,𝑣_𝑘+1 such that 𝑏(𝑛−𝑖+1)<𝑣_𝑖<𝑐(𝑛−𝑖+1),𝑖=2,…,𝑘+1, (3.5) the inequalities (1.5) and (1.6) hold for 𝑛 sufficiently large. Start with (1.5). That gives that for 𝑏(𝑛−𝑘)<𝑣_𝑘+1<𝑐(𝑛−𝑘), (3.6) it has to be 𝛼_ℓ(𝑛)−𝑝(𝑛)⋅𝑣_𝑘+1<𝛼_ℓ(𝑛+1), (3.7) which is equivalent to the inequality −𝑝(𝑛)𝑣_𝑘+1<𝛼_ℓ(𝑛+1)−𝛼_ℓ(𝑛). (3.8) Denote the left-hand side of (3.8) as 𝐿_(3.8). As 𝑣_𝑘+1>𝑏(𝑛−𝑘)=𝛼_ℓ(𝑛−𝑘) and as by (2.3), (2.6), and (3.1) 𝑝(𝑛)≥𝑘𝑘+1^𝑘⋅1𝑘+1+𝑘𝜇_ℓ(𝑛)−𝑘𝑘+1^𝑘⋅𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛, (3.9) we have 𝐿_(3.8)𝑘<−𝑘+1^𝑘1𝑘+1+𝑘𝜇_ℓ(𝑛)−𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛×𝑘𝑘+1^𝑛−𝑘(𝑛−𝑘)ln(𝑛−𝑘)⋯ln_ℓ(𝑛−𝑘)ln^−𝜎_ℓ+1()𝑘𝑛−𝑘=−𝑘+1^𝑛1𝑘+1+𝑘𝜇_ℓ(𝑛)−𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛×(𝑛−𝑘)ln(𝑛−𝑘)⋯ln_ℓ(𝑛−𝑘)ln^−𝜎_ℓ+1(𝑛−𝑘). (3.10) Further, we can easily see that 𝛼_ℓ(𝑛+1)−𝛼_ℓ(𝑛)=𝑘𝑘+1^𝑛𝑛ln𝑛⋯ln_ℓ𝑛ln^−𝜎_ℓ+1𝑛×𝑘𝑘+1(𝑛+1)𝑛ln(𝑛+1)⋯ln𝑛ln_ℓ(𝑛+1)ln_ℓ𝑛ln^−𝜎_ℓ+1(𝑛+1)ln^−𝜎_ℓ+1𝑛−1. (3.11) Thus, to prove (3.8), it suffices to show that for 𝑛 sufficiently large, the following inequality holds: −1𝑘+1+𝑘𝜇_ℓ(𝑛)−𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛(𝑛−𝑘)𝑛ln(𝑛−𝑘)⋯ln𝑛ln_ℓ(𝑛−𝑘)ln_ℓ𝑛ln^−𝜎_ℓ+1(𝑛−𝑘)ln^−𝜎_ℓ+1𝑛<𝑘𝑘+1(𝑛+1)𝑛ln(𝑛+1)⋯ln𝑛ln_ℓ(𝑛+1)ln_ℓ𝑛ln^−𝜎_ℓ+1(𝑛+1)ln^−𝜎_ℓ+1𝑛−1. (3.12) Denote the left-hand side of inequality (3.12) as 𝐿_(3.12) and the right-hand side as 𝑅_(3.12). In the following computation we will use the fact that 𝛽>2 and 𝜇_ℓ(𝑛)=18𝑛²1+𝑂𝑛²ln²𝑛, (3.13) and we will omit all the terms which are of order 𝑜(1/𝑛³). Applying Lemma 2.4 with 𝑟=𝑘 and 𝑞=ℓ, we can write 𝐿_(3.12)1=−𝑘+1+𝑘𝜇_ℓ(𝑛)−𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛×11−𝑘+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝜎2𝑛ln𝑛⋯ln_ℓ𝑛ln_ℓ+1𝑛−𝑘²𝜇_ℓ(𝑛)+𝜎(𝜎+2)8⋅𝑘²𝑛ln𝑛⋯ln_ℓ+1𝑛²−𝑘³16𝑛³1+𝑜𝑛³1=−+𝑘𝑘+11𝑘+1+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝜎2𝑛ln𝑛⋯ln_ℓ𝑛ln_ℓ+1𝑛+𝑘²𝜇𝑘+1^ℓ(𝑛)−𝜎(𝜎+2)8()⋅𝑘𝑘+1𝑛ln𝑛⋯²ln_ℓ+1𝑛²+𝑘³16𝑛³(𝑘+1)−𝑘𝜇_ℓ(𝑛)+𝑘²16𝑛³+𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛1+𝑜𝑛³1=−+𝑘𝑘+11𝑘+1+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝜎2𝑛ln𝑛⋯ln_ℓ𝑛ln_ℓ+1𝑛−𝑘𝜇𝑘+1^ℓ(𝑛)−𝜎(𝜎+2)8()⋅𝑘𝑘+1𝑛ln𝑛⋯²ln_ℓ+1𝑛²+2𝑘³+𝑘²16𝑛³()+𝛿𝑘+1𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛1+𝑜𝑛³. (3.14) Using Lemma 2.4 with 𝑟=−1 and 𝑞=ℓ, we get for 𝑅_(3.12)𝑅_(3.12)=𝑘1𝑘+11++12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝜎2𝑛ln𝑛⋯ln_ℓ𝑛ln_ℓ+1𝑛−𝜇_ℓ(𝑛)+𝜎(𝜎+2)8⋅1𝑛ln𝑛⋯ln_ℓ+1𝑛²+116𝑛³1+𝑜𝑛³=−1−1+𝑘𝑘+11𝑘+1+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝜎2𝑛ln𝑛⋯ln_ℓ𝑛ln_ℓ+1𝑛−𝑘𝑘+1⋅𝜇_ℓ(𝑛)+𝜎(𝜎+2)𝑘8()⋅1𝑘+1𝑛ln𝑛⋯ln_ℓ+1𝑛²+𝑘16𝑛³()1𝑘+1+𝑜𝑛³. (3.15) It is easy to see that the inequality (3.12) reduces to −𝜎(𝜎+2)𝑘²8()⋅1𝑘+1𝑛ln𝑛⋯ln_ℓ+1𝑛²+2𝑘³+𝑘²16𝑛³()+𝛿𝑘+1𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛1+𝑜𝑛³<𝜎(𝜎+2)𝑘8()⋅1𝑘+1𝑛ln𝑛⋯ln_ℓ+1𝑛²+𝑘16𝑛³()1𝑘+1+𝑜𝑛³. (3.16) This inequality is equivalent to −𝜎(𝜎+2)𝑘8⋅1𝑛ln𝑛⋯ln_ℓ+1𝑛²+𝑘(2𝑘−1)16𝑛³+𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛1+𝑜𝑛³<0. (3.17) The last inequality holds for 𝑛 sufficiently large because 𝑘≥1, 𝜎>0, 𝛽>2, and as 𝑛→∞, 𝛿𝑛ln𝑛⋯ln_ℓ𝑛²ln^𝛽_ℓ+1𝑛,1𝑛³ (3.18) tend to zero faster than 1𝑛ln𝑛⋯ln_ℓ+1𝑛² (3.19) does.
Thus, we have proved that inequality (1.5) holds.
Next, according to (1.6), we have to prove that𝜈_ℓ(𝑛)−𝑝(𝑛)𝑣_𝑘+1>𝜈_ℓ(𝑛+1), (3.20) which is equivalent to the inequality: −𝑝(𝑛)𝑣_𝑘+1>𝜈_ℓ(𝑛+1)−𝜈_ℓ(𝑛). (3.21) Denote the left-hand side of (3.21) as 𝐿_(3.21). As 𝑣_𝑘+1<𝑐(𝑛−𝑘)=𝜈_ℓ(𝑛−𝑘) and as by (2.3), (3.1), and (2.7) 𝑝(𝑛)≤𝑘𝑘+1^𝑘⋅1𝑘+1+𝑘𝜇_ℓ(𝑛)𝑘+𝜀𝑘+1^𝑘⋅𝑘(2𝑘−1)16𝑛³, (3.22) we have 𝐿_(3.21)𝑘>−𝑘+1^𝑘1𝑘+1+𝑘𝜇_ℓ(𝑛)𝑘()+𝜀⋅2𝑘−116𝑛³×𝑘𝑘+1^𝑛−𝑘(𝑛−𝑘)ln(𝑛−𝑘)⋯ln_ℓ()𝑘𝑛−𝑘=−𝑘+1^𝑛1𝑘+1+𝑘𝜇_ℓ(𝑛)𝑘()+𝜀⋅2𝑘−116𝑛³⋅(𝑛−𝑘)ln(𝑛−𝑘)⋯ln_ℓ(𝑛−𝑘). (3.23) Further, we can easily see that 𝜈_ℓ(𝑛+1)−𝜈_ℓ(𝑛)=𝑘𝑘+1^𝑛𝑛ln𝑛⋯ln_ℓ𝑛𝑘𝑘+1(𝑛+1)𝑛⋅ln(𝑛+1)⋯ln𝑛ln_ℓ(𝑛+1)ln_ℓ𝑛−1. (3.24) Thus, to prove (3.21), it suffices to show that for 𝑛 sufficiently large, the following inequality holds: −1𝑘+1+𝑘𝜇_ℓ(𝑛)𝑘()+𝜀⋅2𝑘−116𝑛³(𝑛−𝑘)𝑛⋅ln(𝑛−𝑘)⋯ln𝑛ln_ℓ(𝑛−𝑘)ln_ℓ𝑛>𝑘𝑘+1(𝑛+1)𝑛⋅ln(𝑛+1)⋯ln𝑛ln_ℓ(𝑛+1)ln_ℓ𝑛−1. (3.25) Denote the left-hand side of inequality (3.25) as 𝐿_(3.25) and the right-hand side as 𝑅_(3.25). Using Lemma 2.3 with 𝑟=𝑘 and 𝑞=ℓ, we can write 𝐿_(3.25)1=−𝑘+1+𝑘𝜇_ℓ(𝑛)𝑘()+𝜀⋅2𝑘−116𝑛³×11−𝑘+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝑘²𝜇_ℓ(𝑛)−𝑘³16𝑛³1+𝑜𝑛³1=−+𝑘𝑘+11𝑘+1+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛+𝑘²𝜇𝑘+1^ℓ(𝑛)+𝑘³16𝑛³(𝑘+1)−𝑘𝜇_ℓ(𝑛)+𝑘²16𝑛³𝑘()−𝜀⋅2𝑘−116𝑛³1+𝑜𝑛³1=−+𝑘𝑘+11𝑘+1+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝑘𝜇𝑘+1^ℓ(𝑛)+𝑘³16𝑛³()+𝑘((𝑘+1𝑘−𝜀2𝑘−1))16𝑛³1+𝑜𝑛³. (3.26) Using Lemma 2.3 with 𝑟=−1 and 𝑞=ℓ, we get for 𝑅_(3.25)𝑅_(3.25)=𝑘1𝑘+11++12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝜇_ℓ(𝑛)+116𝑛³1+𝑜𝑛³=−1−1+𝑘𝑘+11𝑘+1+12𝑛12𝑛ln𝑛+⋯+2𝑛ln𝑛⋯ln_ℓ𝑛−𝑘𝑘+1⋅𝜇_ℓ(𝑛)+𝑘16𝑛³()1𝑘+1+𝑜𝑛³. (3.27) It is easy to see that the inequality (3.25) reduces to 𝑘³16𝑛³()+𝑘((𝑘+1𝑘−𝜀2𝑘−1))16𝑛³1+𝑜𝑛³>𝑘16𝑛³()1𝑘+1+𝑜𝑛³. (3.28) This inequality is equivalent to 𝑘(2𝑘−1)(1−𝜀)16𝑛³1+𝑜𝑛³>0. (3.29) The last inequality holds for 𝑛 sufficiently large because 𝑘≥1 and 1−𝜀∈(0,1). We have proved that all the assumptions of Theorem 1.1 are fulfilled and hence there exists a solution of (1.2) satisfying conditions (1.8), that is, in our case, conditions (3.2).

Acknowledgments

The research was supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30519 and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology.