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

and Applied Analysis 3 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 lnq n, q ∈ N \ {0}, as lnq n : ln ( lnq−1 n ) , 2.1 where ln0 n : n. We will write only lnn instead of ln1n. Further, for a fixed integer ≥ 0 define auxiliary functions: μ n : 1 8n2 1 8 n lnn 2 · · · 1 8 n lnn · · · ln n 2 , 2.2 p n : ( k k 1 )k · ( 1 k 1 kμ n ) , 2.3 ν n : ( k k 1 )n · √ n lnn ln2n · · · ln n, 2.4 α n : ( k k 1 )n · √ n lnn ln2n · · · ln n ln−σ 1n, 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 ∈ Za . Finally, let functions ψ,ω : Za → R satisfy for n ∈ Za the inequalities: ψ n ≤ ( k k 1 )k · δ n lnn · · · ln n 2ln 1n , 2.6 ω n ≤ ε ( k k 1 )k · k 2k − 1 16n3 , 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 n for some ≥ 0, then there exists a positive solution of 1.2 bounded above by the function ν n for n sufficiently large. Since limn→∞ν 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 n even if the coefficient p n satisfies a less restrictive inequality, namely, p n < p n ω n . Here we will prove that function α provides the lower estimate of the solution, supposing p n − ψ n ≤ p n ≤ p 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 → ∞. 4 Abstract and Applied Analysis Lemma 2.1. For fixed r ∈ R \ {0} and fixed q ∈ N, the asymptotic representation: lnq n − r lnq n − r n lnn · · · lnq−1n − r2 2n2 lnn · · · lnq−1n − r 2 2 n lnn ln2n · · · lnq−1n − · · · − r 2 2 ( n lnn · · · lnq−1n )2 o ( 1 n3 ) , 2.8


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 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 n − ψ n and p 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 n see 2.5 and from above by the function ν 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: R be a continuous function and let the inequalities: Then there exists a solution v v * n of 1.4 satisfying the inequalities 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 .

Auxiliary Functions and Lemmas
Define the expression ln q n, q ∈ N \ {0}, as ln q n : ln ln q−1 n , 2.1 where ln 0 n : n.We will write only ln n instead of ln 1 n.Further, for a fixed integer ≥ 0 define auxiliary functions: 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: 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 n for some ≥ 0, then there exists a positive solution of 1.2 bounded above by the function ν n for n sufficiently large.Since lim n → ∞ ν 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 n even if the coefficient p n satisfies a less restrictive inequality, namely, p n < p n ω n .Here we will prove that function α provides the lower estimate of the solution, supposing p n − ψ n ≤ p n ≤ p 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: 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: Proof.Both these relations are simple consequences of the asymptotic formula: 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 q n, we get

2.12
Abstract and Applied Analysis 5 Thus, putting and using 2.11 , we get 2.9 .
The following lemma is proved in 6 .

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 , ν , α , ψ, and ω were defined by 2.3 -2.6 and 2.7 , respectively.
Theorem 3.1.Suppose that there exist numbers a, ∈ N, and σ > 0, such that the function p in 1.2 satisfies the inequalities 2 such that for n sufficiently large the inequalities: Proof.Show that all the assumptions of Theorem 1.1 are fulfilled.For 1.2 , This is a continuous function.Put b n : α n , c n : ν n .

3.4
We have to prove that for every v 2 , . . ., v k 1 such that the inequalities 1.5 and 1.6 hold for n sufficiently large.Start with 1.5 .That gives that for which is equivalent to the inequality Denote the left-hand side of 3.8 as L 3.8 .As v k 1 > b n − k α n − k and as by 2.3 , 2.6 , and 3.1

3.10
Further, we can easily see that

3.11
Thus, to prove 3.8 , it suffices to show that for n sufficiently large, the following inequality holds:

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 and we will omit all the terms which are of order o 1/n 3 .Applying Lemma 2.4 with r k and q , we can write

3.14
Using Lemma 2.4 with r −1 and q , we get for R 3.12

3.15
Abstract and Applied Analysis 9 It is easy to see that the inequality 3.12 reduces to

3.16
This inequality is equivalent to 3.17 The last inequality holds for n sufficiently large because k ≥ 1, σ > 0, β > 2, and as n → ∞, tend to zero faster than does.Thus, we have proved that inequality 1.5 holds.Next, according to 1.6 , we have to prove that which is equivalent to the inequality:

3.23
Further, we can easily see that

3.24
Thus, to prove 3.21 , it suffices to show that for n sufficiently large, the following inequality holds:

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 , we can write

3.26
Using Lemma 2.3 with r −1 and q , we get for R 3.25

3.27
It is easy to see that the inequality 3.25 reduces to This inequality is equivalent to 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 .

3 . 21 Denote
the left-hand side of 3.21 as L 3.21 .As v k 1 < c n − k ν n − k and as by 2.3 , 3.1 , and 2.7