Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type

Josef Diblík; Evgeniya Korobko

doi:10.1515/anona-2023-0105

Article Open Access

Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type

Josef Diblík and Evgeniya Korobko

Published/Copyright: October 5, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

The article investigates a second-order nonlinear difference equation of Emden-Fowler type

Δ 2 u ( k ) ± k α u m ( k ) = 0 ,

where k is the independent variable with values k = k 0 , k 0 + 1 , … , u : { k 0 , k 0 + 1 , … } → R is the dependent variable, k 0 is a fixed integer, and Δ 2 u ( k ) is its second-order forward difference. New conditions with respect to parameters α ∈ R and m ∈ R , m ≠ 1 , are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation

y ″ ( x ) ± x α y m ( x ) = 0 .

Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.

Keywords: discrete equation; Emden-Fowler equation; equation of the second-order; asymptotic behavior; nonlinearity

MSC 2010: 39A22; 39A12

1 Introduction

Let k 0 be an integer. Define a set N ( k 0 ) ≔ { k 0 , k 0 + 1 , … } . In the article, we consider a second-order nonlinear difference equation of Emden-Fowler type

(1) Δ 2 u ( k ) ± k α u m ( k ) = 0 ,

where k ∈ N ( k 0 ) is the independent variable, u : N ( k 0 ) → R is the dependent variable, Δ 2 u ( k ) is the second-order forward difference of u ( k ) defined, as customary, by using the first-order forward difference Δ u ( k ) = u ( k + 1 ) − u ( k ) and Δ 2 u ( k ) = Δ ( Δ u ( k ) ) = u ( k + 2 ) − 2 u ( k + 1 ) + u ( k ) . A solution to equation (1) is defined as a function u : N ( k 0 ) → R satisfying equation (1) for every k ∈ N ( k 0 ) . In equation (1), it is assumed that α ∈ R and m ∈ R , m ≠ 1 (i.e., the linear variant is excluded) are fixed constants.

Equation (1) was considered in several papers. In [4], for some values of the parameters α and m are discussed solutions with asymptotic behavior characterized by a power function. This form is deduced from the exact solution

(2) y ( x ) = a ± x s

of continuous second-order differential Emden-Fowler equation having the form used in [6],

(3) y ″ ( x ) ± x α y m ( x ) = 0 ,

where

(4) a ± ≔ [ ∓ s ( s + 1 ) ] 1 ⁄ ( m − 1 )

and

(5) s ≔ α + 2 m − 1 .

Below, we assume that the coefficient a ± defined by formula (4) exists and is nonzero. This necessitates s ≠ 0 and s + 1 ≠ 0 , i.e., α ≠ − 2 and α ≠ − m − 1 in terms of the coefficients α and m . Then, the existence of the power 1 ⁄ ( m − 1 ) in equation (4) is a sufficient condition to guarantee this, and we assume the following throughout the article. If, in the right-hand side of equation (4), either the upper sign, i.e., “ − ,” is in force while s ( s + 1 ) > 0 or the lower sign, i.e., “ + ,” is in force while s ( s + 1 ) < 0 , then we assume that the constant m is of the form m 1 ⁄ m 2 , where m 1 and m 2 are relatively prime, and if m 2 is odd, then the difference m 1 − m 2 is odd as well. Such a convention guarantees that the formula (4) defines two or at least one value. As equation (1) splits in two, when formulating the results, we assume that a concrete variant is fixed (either with the sign + or with the sign − ).

For other investigations of discrete equations of Emden-Fowler-type, we refer, e.g., to a few studies [3,8,9,13–15,18–24]. Differential equation (3) itself and its various modifications and generalizations are more widely investigated; we refer at least to [5,6] and the references therein.

The discrete equation (1), treated in the article, is derived by discretizing equation (3) using the formulas

(6) x ∼ k , y ( x ) ∼ u ( k ) , y ″ ( x ) ∼ Δ 2 u ( k ) .

The goal of the article is to derive new results for the existence of a solution u = u ( k ) to discrete equation (1) satisfying the inequalities specified in the article. In particular, these inequalities imply an asymptotic behavior of u ( k ) described by the formula

(7) u ( k ) ∼ a ± k s , k → ∞

which well coincides with equation (2), giving the exact solution to differential equation (3). The results derived are then compared with those published previously. A little more general difference equation than equation (1),

(8) Δ 2 v ( k ) ± p k α v m ( k ) = 0 ,

where p is a positive constant, can obviously be transformed to the form equation (1) by the transformation v ( k ) = q u ( k ) , where q is a positive number defined by q = p 1 ⁄ ( 1 − m ) .

The article is structured as follows. In Section 2, auxiliary material used in the article is given. The existence of solutions u = u ( k ) to equation (1) with asymptotic behavior implying formula (7) is proved in Section 3, where two variants are considered. In Section 3.1, the existence of solutions is proved under the assumptions s + 1 < 0 and m s < 0 , whereas in Part 3.2, it is proved under under the assumptions s + 1 < 0 and m s > 0 . Comparisons with previously known results are formulated in Section 4 along with possible generalizations, further problems, and comments.

2 Preliminaries

Below, we formulate the necessary preliminaries, reproducing them (with possible minor modifications) from [4,10,12]. In [4] the following transformation, transforming equation (1) into a system of two discrete equations, is applied. Let us define auxiliary transformations

(9) u ( k ) = a ± k s + b ± k s + 1 ( 1 + Y 0 ( k ) ) ,

(10) Δ u ( k ) = Δ a ± k s + Δ b ± k s + 1 ( 1 + Y 1 ( k ) ) ,

(11) Δ 2 u ( k ) = Δ 2 a ± k s + Δ 2 b ± k s + 1 ( 1 + Y 2 ( k ) ) ,

where k ∈ N ( k 0 ) , a ± , and s are defined by equations (4) and (5) as follows:

(12) b ± ≔ a ± s ( s + 2 ) s + 2 − m s ,

and Y i ( k ) , i = 0 , 1 , 2 , are new dependent variables. In formula (12), we assume (in addition to other restrictions formulated above) s + 2 ≠ 0 and s + 2 − m s ≠ 0 , i.e., α ≠ − 2 m and α ≠ 0 . Then, from equation (1), provided that

(13) Y 0 ( k ) = O ( 1 ) ,

an equivalence

(14) Y 2 ( k ) = m s s + 2 Y 0 ( k ) + O 1 k

between Y 2 ( k ) and Y 0 ( k ) is derived. Using equations (9)–(11) and (14), equation (1) is transformed into a system

(15) Δ Y 0 ( k ) = F 1 ( k , Y 0 ( k ) , Y 1 ( k ) ) ,

(16) Δ Y 1 ( k ) = F 2 ( k , Y 0 ( k ) , Y 1 ( k ) ) ,

where the functions F i ( k , Y 0 , Y 1 ) : N ( k 0 ) × R × R → R , i = 1 , 2 , are continuous with respect to Y 0 and Y 1 and can be expressed by the following formulas:

(17) F 1 ( k , Y 0 , Y 1 ) ≔ − s + 1 k + O 1 k 2 ( − Y 0 + Y 1 ) ,

(18) F 2 ( k , Y 0 , Y 1 ) ≔ − s + 2 k + O 1 k 2 m s s + 2 Y 0 − Y 1 + O 1 k .

For details, we refer to the study by Astashova et al. [4]. The equations (14) and (18) are the consequences of equations (29) and (30) in the study by Astashova et al. [4] being valid, as mentioned above, if equation (13) holds. Note that, in the below investigation, the assumption (13) is fulfilled, and this fact will not be mentioned each time explicitly.

Assume that the functions b i , and c i , i = 1 , 2 which map N ( k 0 ) to R satisfy b i ( k ) < c i ( k ) , k ∈ N ( k 0 ) , i = 1 , 2 . For i = 1 , 2 , define the sets

Ω b i ≔ { ( k , Y 0 , Y 1 ) : k ∈ N ( k 0 ) , b j ( k ) ≤ Y j − 1 ≤ c j ( k ) , j = 1 , 2 , j ≠ i , Y i − 1 = b i ( k ) } , Ω c i ≔ { ( k , Y 0 , Y 1 ) : k ∈ N ( k 0 ) , b j ( k ) ≤ Y j − 1 ≤ c j ( k ) , j = 1 , 2 , j ≠ i , Y i − 1 = c i ( k ) }

used in the below lemma as a particular case of [12, Theorem 1] and [10, Theorem 2].

Lemma 1

Assume that the inequality

(19) F i ( k , Y 0 , Y 1 ) < b i ( k + 1 ) − b i ( k )

holds for every ( k , Y 0 , Y 1 ) ∈ Ω b i and the inequality

(20) F i ( k , Y 0 , Y 1 ) > c i ( k + 1 ) − c i ( k )

holds for every ( k , Y 0 , Y 1 ) ∈ Ω c i , i = 1 , 2 . Then, there exists a solution ( Y 0 , Y 1 ) = ( Y 0 ( k ) , Y 1 ( k ) ) , k ∈ N ( k 0 ) , to the system of equations (15) and (16) such that

(21) b i ( k ) < Y i − 1 ( k ) < c i ( k ) ,

where k ∈ N ( k 0 ) and i = 1 , 2 ,

3 Main results

In this part, the main results are formulated in Theorems 1 and 2. In both theorems, to guarantee the existence of a nonzero value b ± defined by formula (12), we assume α ∉ { 0 , − 2 m } . Both theorems basically concern the existence and properties of two different solutions u ± with the following exception. If the coefficient a ± , defined by formula (4), reduces to a single value (i.e., if a + = a − ), then the coefficient b ± reduces to a single value as well, and consequently, u + = u − . Then, both theorems describe the behavior of only one solution to equation (1). Moreover, the value k 0 in the set N ( k 0 ) is assumed to be sufficiently large for the asymptotic computations to be correct.

Let functions b i and c i , i = 1 , 2 , be defined by the following formulas:

(22) b 1 ( k ) ≔ − ε 1 k γ , c 1 ( k ) ≔ ε 2 k γ , b 2 ( k ) ≔ − ε 3 k β , c 2 ( k ) ≔ ε 4 k β ,

where β , γ , and ε i , i = 1 , 2 , 3 , 4 , are positive constants. If, for such a choice, all hypotheses of Lemma 1 hold, then we obtain from equation (21) a characterization of the behavior of a solution ( Y 0 , Y 1 ) T = ( Y 0 ( k ) , Y 1 ( k ) ) T , k ∈ N ( k 0 ) , to the system of equations (15), (16), and consequently, using equations (9)–(11), (13), and (14), we obtain a characterization of the behavior of a solution u = u ( k ) , k ∈ N ( k 0 ) , to equation (1).

To apply Lemma 1, we need auxiliary computations concerning the estimates of the right-hand sides and the left-hand sides in inequalities (19) and (20). We start by estimating the differences

b i ( k + 1 ) − b i ( k ) , c i ( k + 1 ) − c i ( k ) , i = 1 , 2 ,

in the right-hand sides of inequalities (19) and (20) for k → ∞ . These can be derived easily by developing an asymptotic decomposition using the binomial formula. We obtain

(23) b 1 ( k + 1 ) − b 1 ( k ) = − ε 1 ( k + 1 ) γ + ε 1 k γ = − ε 1 k − γ 1 + 1 k − γ − 1 = − ε 1 k γ 1 − γ k + O 1 k 2 − 1 = ε 1 γ k γ + 1 1 + O 1 k .

Proceeding similarly, we derive

(24) c 1 ( k + 1 ) − c 1 ( k ) = − ε 2 γ k γ + 1 1 + O 1 k ,

(25) b 2 ( k + 1 ) − b 2 ( k ) = ε 3 β k β + 1 1 + O 1 k ,

(26) c 2 ( k + 1 ) − c 2 ( k ) = − ε 4 β k β + 1 1 + O 1 k .

Furthermore, we need to estimate the left-hand sides of inequalities (19) and (20). As these estimates are different, we will consider them separately provided that

(27) s + 1 = α + m + 1 m − 1 < 0

and the expression

m s = m α + 2 m − 1

is such that either

(28) m s < 0

(29) m s > 0 .

3.1 Analysis of the cases (27) and (28)

If i = 1 , for the left-hand sides of inequalities (19) and (20), we derive

(30) F 1 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω b 1 = F 1 ( k , b 1 ( k ) , Y 1 ) ∣ b 2 ( k ) ≤ Y 1 ≤ c 2 ( k ) = − s + 1 k + O 1 k 2 ε 1 k γ + Y 1 b 2 ( k ) ≤ Y 1 ≤ c 2 ( k ) ≤ − s + 1 k + O 1 k 2 ε 1 k γ + ε 4 k β

and

(31) F 1 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω c 1 = F 1 ( k , c 1 ( k ) , Y 1 ) ∣ b 2 ( k ) ≤ Y 1 ≤ c 2 ( k ) = − s + 1 k + O 1 k 2 − ε 2 k γ + Y 1 b 2 ( k ) ≤ Y 1 ≤ c 2 ( k ) ≥ − s + 1 k + O 1 k 2 − ε 2 k γ − ε 3 k β .

If i = 2 , for the left-hand sides of inequalities (19) and (20), we derive

(32) F 2 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω b 2 = F 2 ( k , Y 0 , b 2 ( k ) ) ∣ b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) = − s + 2 k + O 1 k 2 m s s + 2 Y 0 + ε 3 k β + O 1 k b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) ≤ − s + 2 k + O 1 k 2 m s s + 2 ε 2 k γ + ε 3 k β + O 1 k

and

(33) F 2 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω c 2 = F 2 ( k , Y 0 , c 2 ( k ) ) ∣ b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) = − s + 2 k + O 1 k 2 m s s + 2 Y 0 − ε 4 k β + O 1 k b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) ≥ − s + 2 k + O 1 k 2 − m s s + 2 ε 1 k γ − ε 4 k β + O 1 k .

Equations (23)–(26) and (30)–(33) will be used in the proof of the following theorem. In their formulation, the system of inequalities (37) has an important role. The proof of the theorem verifies its solvability.

Theorem 1

Let α ∉ { 0 , − 2 m } and let inequalities (27) and (28) hold. Let, moreover,

(34) ( m − 1 ) ( 2 α + 5 m − 1 ) > 0 ,

(35) ( m − 1 ) ( α 2 + α m − α − 4 m ) < 0 ,

and

(36) γ + α + m + 1 m − 1 > 0 ,

where γ is a fixed number such that γ ∈ ( γ * , 1 ) ⊂ ( 0 , 1 ) ,

γ * ≔ 1 2 − 2 α + 3 m + 1 m − 1 + 4 m α + 2 m − 1 ⋅ α + m + 1 m − 1 + 1 .

If fixed positive constants δ and ε satisfy the system of inequalities

(37) ε < − δ γ + s + 1 s + 1 , δ < − ε γ + s + 2 m s ,

then equation (1) has solutions u ± = u ± ( k ) , k ∈ N ( k 0 ) , such that

(38) ∣ u ± ( k ) − a ± k − s − b ± k − s − 1 ∣ < δ ∣ b ± ∣ k s + γ + 1 ,

(39) ∣ Δ u ± ( k ) − a ± Δ k − s − b ± Δ k − s − 1 ∣ < Δ b ± k s + 1 ε k γ ,

and

(40) ∣ Δ 2 u ± ( k ) − a ± Δ 2 k − s − b ± Δ 2 k − s − 1 ∣ < Δ 2 b ± k s + 1 δ ∣ m s ∣ k γ ∣ s + 2 ∣ + O 1 k .

Proof

The change of u ( k ) , Δ u ( k ) , and Δ 2 u ( k ) with the new variables Y 0 ( k ) , Y 1 ( k ) , and Y 2 ( k ) using formulas (9)–(11) and equation (1) leads to the system of equations (15) and (16) with right-hand sides defined by formulas (17) and (18) provided that Y 2 ( k ) is replaced by (14) and Y 0 ( k ) = O ( 1 ) . The system of equations (15) and (16) is analyzed below using Lemma 1. By formulas (23)–(26), the right-hand sides of the system of equations (15) and (16) are estimated by formulas (30)–(33). Referring to equations (30) and (23), inequality (19) for i = 1 will hold if

(41) − s + 1 k + O 1 k 2 ε 1 k γ + ε 4 k β < ε 1 γ k γ + 1 1 + O 1 k ;

referring to equations (31) and (24), inequality (20) for i = 1 will hold if

(42) − s + 1 k + O 1 k 2 − ε 2 k γ − ε 3 k β > − ε 2 γ k γ + 1 1 + O 1 k ;

referring to equations (32) and (25), inequality (19) for i = 2 will hold if

(43) − s + 2 k + O 1 k 2 m s s + 2 ε 2 k γ + ε 3 k β + O 1 k < ε 3 β k β + 1 1 + O 1 k ;

and referring to equations (33) and (26), inequality (20) for i = 2 will hold if

(44) − s + 2 k + O 1 k 2 − m s s + 2 ε 1 k γ − ε 4 k β + O 1 k > − ε 4 β k β + 1 1 + O 1 k .

Lemma 1 will be applicable provided that inequalities (41)–(44) hold. Below, we assume that these inequalities are considered for all sufficiently large k , i.e., fixed k 0 should be sufficiently large. Analyzing the above inequalities, it is easy to conclude that β = γ ∈ ( 0 , 1 ) is a necessary condition for them to hold . Then, inequality (41) will hold if

(45) ε 4 < − ε 1 γ + s + 1 s + 1 ,

inequality (42) will hold if

(46) ε 3 < − ε 2 γ + s + 1 s + 1 ,

inequality (43) will hold if

(47) ε 2 < − ε 3 γ + s + 2 m s

and inequality (44) will hold if

(48) ε 1 < − ε 4 γ + s + 2 m s .

Note the following. First, for the system of inequalities (45)–(48) to be solvable, the inequality

(49) γ + s + 1 > 0

is necessary as, in the opposite case, inequalities (45) and (46) cannot be satisfied due to the positivity of ε i , i = 1 , 2 , 3 , 4 , and the property s + 1 < 0 . Second, due to a symmetry between the sub-system of inequalities (45) and (48) and the sub-system of inequalities (46) and (47) with the first sub-system being independent of the second one and vice versa, it is sufficient to analyze the solvability of only one of these two sub-systems. Below, the sub-system of inequalities (45) and (48) is considered where we set ε = ε 4 and δ = ε 1 . This subsystem coincides with equation (37). We obtain

ε < − δ γ + s + 1 s + 1 < ε γ + s + 2 m s ⋅ γ + s + 1 s + 1

(50) ( γ + s + 1 ) ( γ + s + 2 ) − m s ( s + 1 ) > 0 .

We rewrite equation (50) as a quadratic inequality with respect to γ as follows:

(51) Γ ( γ ) ≔ γ 2 + γ ( 2 s + 3 ) + ( s + 1 ) ( − m s + s + 2 ) > 0

with the discriminant of the quadratic equation Γ ( γ ) = 0 being

D = ( 2 s + 3 ) 2 − 4 ( s + 1 ) ( − m s + s + 2 ) = 4 m s ( s + 1 ) + 1 > 0 .

The two reals roots γ 1 , γ 2 , γ 1 < γ 2 of equation Γ ( γ ) = 0 are

(52) γ 1 , 2 = 1 2 ( − ( 2 s + 3 ) ∓ 4 m s ( s + 1 ) + 1 ) .

Therefore, the system of inequalities (45)–(48) (i.e., the system (37)) will be solvable if γ 1 > 0 or γ 2 < 1 . Then, it will be possible to fix a γ ∈ ( 0 , 1 ) within the interval ( 0 , γ 1 ) if γ 1 > 0 or the interval ( γ 2 , 1 ) if γ 2 < 1 . Below, both cases are discussed. We will show that the first case is not possible, whereas in the second case, the system of inequalities (45)–(48) is solvable.

The case γ 1 > 0 . If γ 1 > 0 , then, as it follows from equation (52),

(53) 4 m s ( s + 1 ) + 1 < − ( 2 s + 3 ) ,

and consequently, inequality

(54) 2 s + 3 < 0

must be fulfilled. Replacing in equation (53) the value s by the formula (5), we obtain an inequality

4 m α + 2 m − 1 α + m + 1 m − 1 + 1 < ( 2 ( α + 2 ) + 3 ( m − 1 ) ) 2 ( m − 1 ) 2 ,

which can be reduced to

α ( α + m + 1 ) ( m − 1 ) < 0

or to

(55) α ( s + 1 ) < 0 .

Inequality (55) can only hold if α > 0 , and from equation (5), we deduce that m ∈ ( 0 , 1 ) . Therefore, based on all the assumptions, we conclude that Lemma 1 is applicable if

(56) α > 0 , 0 < m < 1 .

It is easy to verify that equation (56) implies the validity of inequality (54). However, inequality (49) is not satisfied if γ ∈ ( 0 , γ 1 ) because

γ 1 + s + 1 = 1 2 ( − ( 2 s + 3 ) − 4 m s ( s + 1 ) + 1 ) + s + 1 = − 1 2 ( 1 + 4 m s ( s + 1 ) + 1 ) < 0 .

The case γ 2 < 1 . The root γ 2 is positive, see Lemma 2 for further details. Then, by formula (52), we see that the inequality

(57) 4 m s ( s + 1 ) + 1 < 2 s + 5

holds only if the inequality

2 s + 5 > 0 ,

equivalent with equation (34), holds. Inequality (57) is equivalent with

(58) 4 m s ( s + 1 ) + 1 < ( 2 s + 5 ) 2

and, substituting equation (5) for s in equation (58), with

(59) 4 m α + 2 m − 1 ⋅ α + m + 1 m − 1 + 1 < 2 α + 2 m − 1 + 5 2 .

Simplifying inequality (59), we obtain its equivalent form

( m − 1 ) ( α 2 + α m − α − 4 m ) < 0 .

Considering all assumptions, we see that Lemma 1 is applicable if inequalities (27), and (28), (34)–(36) hold and γ * = γ 2 .

Note that, with a proper choice of the functions b 1 ( k ) = − ε 1 ⁄ k γ and c 1 ( k ) = ε 2 ⁄ k γ , where γ > 0 , the assumption Y 0 ( k ) = O ( 1 ) is fulfilled as a consequence of the definitions of domains Ω b i and Ω c i , i = 1 , 2 . Then Lemma 1 is applicable and by formula (21), there exists a solution ( Y 0 , Y 1 ) = ( Y 0 ( k ) , Y 1 ( k ) ) , k ∈ N ( k 0 ) , to the system of equations (15) and (16) such that

− δ k γ < Y 0 ( k ) < δ k γ , − ε k γ < Y 1 ( k ) < ε k γ

∣ Y 0 ( k ) ∣ < δ k γ , ∣ Y 1 ( k ) ∣ < ε k γ .

Inequalities (38)–(40) are the consequences of formulas (9)–(11) and (14).□

The following lemma shows that the root γ 2 , given by formula (52), is positive.

Lemma 2

Let all hypotheses of Theorem 1 hold. Then, the root γ 2 , defined by formula (52), is positive.

Proof

Below we exclude the case γ 2 ≤ 0 . First, assume γ 2 = 0 . From inequality (51), we have

(60) Γ ( 0 ) = ( s + 1 ) ( − m s + s + 2 ) = 0 .

Because s + 1 < 0 , equation (60) implies

− m s + s + 2 = 0

and by formula (5),

− m s + s + 2 = − m α + 2 m − 1 + α + 2 m − 1 + 2 = − α = 0 ,

and we obtain a contradiction with α ≠ 0 in Theorem 1. Therefore, γ 2 ≠ 0 . Second, let γ 2 < 0 . Then, equation (52) implies 2 s + 3 > 0 , and the inequality

4 m s ( s + 1 ) + 1 < 2 s + 3

yields

m s ( s + 1 ) < ( s + 1 ) ( s + 2 ) .

As s + 1 < 0 , the last inequality is equivalent to

s ( m − 1 ) > 2 .

Substituting equation (5) for s , we obtain α > 0 . Let us show that α cannot be positive. If m > 1 , then s + 1 < 0 implying that α < − 2 . If m < 1 , the conditions s + 1 < 0 and m s < 0 imply 0 < m < 1 and 2 s + 3 > 0 can be transformed into 2 α + 3 m + 1 < 0 . This inequality does not hold as α > 0 and m > 0 .□

Remark 1

The range of the admissible values of parameters m and α in equation (1) is defined by inequalities (27), (28), (34), and (35), and the condition α ∉ { 0 , − 2 m } in Theorem 1. This is visualized in the ( m , α ) -plane in Figure 1 (omitting the lines playing no role in its specification). The set of points satisfying all these restrictions splits into two open domains. The first one, shown in blue, is bounded. The second one, in green, is unbounded. In the following two examples, the parameters of particular cases of equation (1) lie either within the blue domain (in Example 1) or within the green domain (in Example 2).

Figure 1

To Remark 1.

Example 1

Consider equation (1) with parameters m = 1 ⁄ 2 and α = − 27 ⁄ 20 (taken from the blue subdomain in Figure 1), i.e., the equation

(61) Δ 2 u ( k ) ± k − 27 ⁄ 20 u 1 ⁄ 2 ( k ) = 0 .

We will show that all hypotheses of Theorem 1 are satisfied. Inequality (27) holds since

s = α + 2 m − 1 = − 13 10 , s + 1 = α + m + 1 m − 1 = − 3 10 < 0 ,

inequality (28) holds since

m s = m α + 2 m − 1 = − 13 20 < 0 ,

inequality (34) holds since

( m − 1 ) ( 2 α + 5 m − 1 ) = 3 5 > 0 ,

and inequality (35) holds since

( m − 1 ) ( α 2 + α m − α − 4 m ) = − 199 800 < 0 .

Moreover, γ * ≐ 0.467 and inequality (36) holds since

γ + α + m + 1 m − 1 = γ − 3 10 > 0 ,

if γ is a fixed number such that γ ∈ ( γ * , 1 ) . Let, e.g., γ = 0.8 . Then, the system of inequalities (37) is

ε < 5 3 δ , δ < 30 13 ε ,

and the choice, e.g., δ = ε = 1 is one of its solutions. Theorem 1 with δ = ε = 1 and γ = 0.8 is applicable. Coefficients a ± computed by equation (4) give only one value a * since

a * = a ± = [ ∓ s ( s + 1 ) ] 1 ⁄ ( m − 1 ) ≐ 6.5746 .

Then, expressions b ± take on only a single value of b * , while formula (12) gives

b * = b ± = a ± s ( s + 2 ) s + 2 − m s = a * s ( s + 2 ) s + 2 − m s ≐ − 4.43178 .

We conclude that there exists a solution u = u ( k ) , k ∈ N ( k 0 ) , to equation (61) such that, by formulas (38)–(40),

∣ u ( k ) − a * k 13 ⁄ 10 − b * k 3 ⁄ 10 ∣ < ∣ b * ∣ k ,

∣ Δ u ( k ) − a * Δ k 13 ⁄ 10 − b * Δ k 3 ⁄ 10 ∣ < Δ k 3 ⁄ 10 ⋅ ∣ b * ∣ k 8 ⁄ 10 ,

and

∣ Δ 2 u ( k ) − a * Δ 2 k 13 ⁄ 10 − b * Δ 2 k 3 ⁄ 10 ∣ < ∣ Δ 2 k 3 ⁄ 10 ∣ ⋅ 13 ∣ b * ∣ 14 k 8 ⁄ 10 + O 1 k .

Example 2

Consider equation (1) with parameters m = 2 and α = − 3.1 (taken from the green subdomain in Figure 1), i.e., the equation

(62) Δ 2 u ( k ) ± k − 3.1 u 2 ( k ) = 0 .

We will show that all hypotheses of Theorem 1 are satisfied. Since

s = α + 2 m − 1 = − 1.1 , s + 1 = α + m + 1 m − 1 = − 0.1 < 0 , m s = m α + 2 m − 1 = − 2.2 < 0 ,

( m − 1 ) ( 2 α + 5 m − 1 ) = 2.8 > 0 , ( m − 1 ) ( α 2 + α m − α − 4 m ) = − 1.49 < 0 , γ * ≐ 0.286 ,

and

γ + α + m + 1 m − 1 = 0.4 > 0 for γ = 0.5 ∈ ( γ * , 1 ) ,

inequalities (27), (28), and (34)–(36) are satisfied. System of inequalities (37)

ε < 4 δ , δ < 7 11 ε

is solvable with one of the solutions being, e.g., δ = 1 and ε = 2 . For a ± computed by equation (4), we obtain

a ± = [ ∓ s ( s + 1 ) ] 1 ⁄ ( m − 1 ) = ∓ 0.11 ,

and for b ± computed by formula (12), we have

b ± = a ± s ( s + 2 ) s + 2 − m s ≐ ± 0.0351 .

By Theorem 1, equation (62) has two solutions u ± = u ± ( k ) , k ∈ N ( k 0 ) , satisfying inequalities (38)–(40), i.e.,

∣ u ± ( k ) ± 0.11 k 1.1 ∓ 0.0351 k 0.1 ∣ < 0.0351 k 0.4 ,

∣ Δ u ± ( k ) ± 0.11 Δ k 1.1 ∓ 0.0351 Δ k 0.1 ∣ < Δ ( 0.0351 k 0.1 ) 2 k 0.5 ,

and

∣ Δ 2 u ± ( k ) ± 0.11 Δ 2 k 1.1 ∓ 0.0351 Δ 2 k 0.1 ∣ < ∣ Δ 2 ( 0.0351 k 0.1 ) ∣ 22 9 k 0.5 + O 1 k .

3.2 Analysis of cases (27) and (29)

If i = 1 , only the condition s + 1 < 0 is used so that the left-hand sides of inequalities (19) and (20) are estimated by inequalities (30) and (31). For i = 2 , in particular, we have

(63) F 2 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω b 2 = F 2 ( k , Y 0 , b 2 ( k ) ) ∣ b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) = − s + 2 k + O 1 k 2 m s s + 2 Y 0 + ε 3 k β + O 1 k b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) ≤ − s + 2 k + O 1 k 2 m s s + 2 − ε 1 k γ + ε 3 k β + O 1 k

and

(64) F 2 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω c 2 = F 2 ( k , Y 0 , c 2 ( k ) ) ∣ b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) = − s + 2 k + O 1 k 2 m s s + 2 Y 0 − ε 4 k β + O 1 k b 1 ( k ) ≤ Y 0 ≤ c 1 ( k ) ≥ − s + 2 k + O 1 k 2 m s s + 2 ε 2 k γ − ε 4 k β + O 1 k .

Being unchanged, inequalities (30) and (31) will hold if inequalities (41) and (42) do. Inequalities (63) and (64) will hold if (we refer to similar computations leading to (43) and (44))

(65) − s + 2 k + O 1 k 2 m s s + 2 − ε 1 k γ + ε 3 k β + O 1 k < ε 3 β k β + 1 1 + O 1 k

and

(66) − s + 2 k + O 1 k 2 m s s + 2 ε 2 k γ − ε 4 k β + O 1 k > − ε 4 β k β + 1 1 + O 1 k .

The following theorem includes a hypothesis consisting of a system of inequalities (69), which is solvable as shown in the proof.

Theorem 2

Let α ∉ { 0 , − 2 m } and let inequalities (27), (29), and (34) hold. Let, moreover,

(67) α 2 + 8 m 2 + 8 m α − α + m α 2 + m 2 α > 0 ,

and

(68) γ + α + m + 1 m − 1 > 0 ,

where γ is a fixed number such that for γ ∈ ( γ * , 1 ) ⊂ ( 0 , 1 ) ,

γ * = 1 2 − 2 α + 3 m + 1 m − 1 + 1 − 4 m α + 2 m − 1 ⋅ α + m + 1 m − 1 .

If fixed positive constants ε i , i = 1 , 2 , 3 , 4 , satisfy the system of inequalities

(69) ε 4 < − ε 1 γ + s + 1 s + 1 , ε 3 < − ε 2 γ + s + 1 s + 1 , ε 1 < ε 3 γ + s + 2 m s , ε 2 < ε 4 γ + s + 2 m s ,

then equation (1) has solutions u ± = u ± ( k ) , k ∈ N ( k 0 ) , such that

(70) ∣ u ± ( k ) − a ± k − s − b ± k − s − 1 ∣ < max { ε 1 , ε 2 } ∣ b ± ∣ k s + γ + 1 ,

(71) ∣ Δ u ± ( k ) − a ± Δ k − s − b ± Δ k − s − 1 ∣ < Δ b ± k s + 1 max { ε 3 , ε 4 } k γ ,

(72) ∣ Δ 2 u ± ( k ) − a ± Δ 2 k − s − b ± Δ 2 k − s − 1 ∣ < Δ 2 b ± k s + 1 max { ε 1 , ε 2 } m s k γ ∣ s + 2 ∣ + O 1 k .

Proof

The proof can be done in much the same way as that of Theorem 1. We point out only some of the differences. The validity of inequalities (41), (42), (65), and (66) is necessitated by the condition β = γ ∈ ( 0 , 1 ) . The following system of inequalities (the first two of them coincide with equations (45) and (46)) forms a sufficient condition. Inequality (41) will hold if

(73) ε 4 < − ε 1 γ + s + 1 s + 1 ,

inequality (42) will hold if

(74) ε 3 < − ε 2 γ + s + 1 s + 1 ,

inequality (65) will hold if

(75) ε 1 < ε 3 γ + s + 2 m s ,

and inequality (66) will hold if

(76) ε 2 < ε 4 γ + s + 2 m s .

Note that the system of inequalities (73)–(76) coincides with the system (69). It is easy to see that the inequality

(77) γ + s + 1 > 0

is necessary for the system of inequalities (73)–(76) to have a solution. To solve the system of inequalities (73)–(76), provided that equation (77) holds, we write down the chain of inequalities as follows:

ε 4 < − ε 1 γ + s + 1 s + 1 < − ε 3 γ + s + 1 s + 1 γ + s + 2 m s < ε 2 ( γ + s + 1 ) 2 ( γ + s + 2 ) ( s + 1 ) 2 m s < ε 4 ( γ + s + 1 ) 2 ( γ + s + 2 ) 2 ( s + 1 ) 2 ( m s ) 2 .

As ε 4 > 0 , we have

1 < ( γ + s + 1 ) 2 ( γ + s + 2 ) 2 ( s + 1 ) 2 ( m s ) 2

or, simplifying this inequality,

(78) G 1 ( γ ) G 2 ( γ ) > 0 ,

where

G 1 ( γ ) ≔ ( γ + s + 1 ) ( γ + s + 2 ) − ( s + 1 ) m s , G 2 ( γ ) ≔ ( γ + s + 1 ) ( γ + s + 2 ) + ( s + 1 ) m s .

Inequality (78) will hold if either

(79) G 1 ( γ ) > 0 , G 2 ( γ ) > 0

(80) G 1 ( γ ) < 0 , G 2 ( γ ) < 0 .

Below, both possibilities are analyzed.

The case (79). Consider a system of inequalities (79). Because s + 1 < 0 and m s > 0 , we have G 1 ( γ ) > 0 . Consequently, it is just sufficient to find out when G 2 ( γ ) > 0 . Rewrite the last inequality as a quadratic one with respect to γ ,

G 2 ( γ ) = γ 2 + γ ( 2 s + 3 ) + ( s + 1 ) ( s + m s + 2 ) > 0 ,

with discriminant D of a quadratic equation G 2 ( γ ) = 0 defined as follows:

D = ( 2 s + 3 ) 2 − 4 ( s + 1 ) ( s + m s + 2 ) = 1 − 4 m s ( s + 1 ) > 0 .

The two real roots γ 1 , γ 2 , γ 1 < γ 2 of the equation G 2 ( γ ) = 0 are

(81) γ 1 , 2 = − ( 2 s + 3 ) ∓ 1 − 4 m s ( s + 1 ) 2 = 1 2 − 2 α + 3 m + 1 m − 1 ∓ 1 − 4 m α + 2 m − 1 α + m + 1 m − 1 .

System (73)–(76) will be solvable (i.e., a suitable ε i , i = 1 , 2 , 3 , 4 , will exist) if γ 1 > 0 or if γ 2 < 1 . Note that the root γ 2 is positive, see Lemma 3. Consider the case γ 1 > 0 . Then, the necessary condition (77) does not hold as, for γ ≤ γ 1 ,

γ + s + 1 ≤ γ 1 + s + 1 = − ( 2 s + 3 ) − 1 − 4 m s ( s + 1 ) 2 + s + 1 = − 1 − 1 − 4 m s ( s + 1 ) 2 < 0 .

Consider the case γ 2 < 1 . This inequality is equivalent with the following one:

(82) 1 − 4 m s ( s + 1 ) < 2 s + 5 .

The condition necessary for its solvability is represented by the inequality 2 s + 5 > 0 equivalent with equation (34). If it holds, then inequality (82) is equivalent to

1 − 4 m s ( s + 1 ) < ( 2 s + 5 ) 2 ,

and replacing s by formula (5), we obtain the condition (67),

α 2 + 8 m 2 + 8 m α − α + m α 2 + m 2 α > 0 .

Since inequalities (27), (29), (34), (67), and (68) hold and γ is a fixed number such that γ ∈ ( γ * , 1 ) , where γ * ≔ γ 2 , we see that Lemma 1 is applicable, and, therefore, there exists a solution ( Y 0 , Y 1 ) = ( Y 0 ( k ) , Y 1 ( k ) ) , k ∈ N ( k 0 ) , to the system of equations (15), (16) such that

− ε 1 k γ < Y 0 ( k ) < ε 2 k γ , − ε 3 k γ < Y 1 ( k ) < ε 4 k γ

∣ Y 0 ( k ) ∣ < max { ε 1 , ε 2 } k γ , ∣ Y 1 ( k ) ∣ < max { ε 3 , ε 4 } k γ .

Inequalities (70)–(72) are the consequences of formulas (9)–(11), and (14).

The case (80). We show that this case is not possible. Because ( s + 1 ) m s < 0 , and by (77), γ + s + 1 > 0 , we have

G 1 ( γ ) = ( γ + s + 1 ) ( γ + s + 2 ) − ( s + 1 ) m s > 0 .

This contradicts the inequality G 1 ( γ ) < 0 . Thus, the case G 1 ( γ ) < 0 , G 2 ( γ ) < 0 is not possible.□

Lemma 3

Let inequalities (27), (29), (34), and (67) hold. Then, the root γ 2 defined by formula (81) is positive.

Proof

Assume that γ 2 ≤ 0 . Then,

(83) 1 − 4 m s ( s + 1 ) ≤ 2 s + 3 .

The condition necessary for equation (83) to hold is the inequality

(84) 2 s + 3 ≥ 0 .

Then, inequality equation (83) is equivalent with

(85) − m s ≥ s + 2 .

From equations (84) and (85), we derive a chain of inequalities

− m s ≥ s + 2 ≥ − 3 2 + 2 = 1 2 > 0 .

We have arrived at a contradiction with the assumption m s > 0 . Therefore, γ 2 > 0 .□

Remark 2

The range of the admissible values of parameters m and α in equation (1) is defined by inequalities (27), (29), (34), and (67) and the condition α ∉ { 0 , − 2 m } in Theorem 2. It is visualized in the ( m , α ) -plane in Figure 2 (omitting the lines not relevant for its specification). The set of points satisfying all these restrictions splits into two open domains. The first one, in pink, is unbounded. The second one, in violet, is bounded. In Example 3, the parameters of a particular case of equation (1) lie within the pink domain.

Figure 2

To Remark 2.

Example 3

Consider equation (1) with parameters m = − 2 and α = 3 ⁄ 2 (taken from the pink domain in Figure 2), i.e., the equation

(86) Δ 2 u ( k ) ± k k u 2 ( k ) = 0 .

We show that all hypotheses of Theorem 2 hold. Let us show that inequalities (27), (29), (34), and (67) are satisfied. Indeed, inequality (27) holds since

s = α + 2 m − 1 = − 7 6 , s + 1 = α + m + 1 m − 1 = − 1 6 < 0 ,

inequality (29) holds since

m s = m α + 2 m − 1 = 7 3 > 0 ,

inequality (34) holds, since

2 α + 5 m − 1 = − 8 < 0

and inequality (67) holds since

α 2 + 8 m 2 + 8 m α − α + m α 2 + m 2 α = 41 4 > 0 .

Moreover, γ * ≐ 0.46597 , and for γ = 5 ⁄ 6 ∈ ( γ * , 1 ) , inequality (68) holds since

γ + α + m + 1 m − 1 = 2 3 > 0 .

Then, system (69) can be written as follows:

ε 4 < 4 ε 1 , ε 3 < 4 ε 2 , ε 1 < 5 7 ε 3 , ε 2 < 5 7 ε 4 .

The values ε 1 = ε 2 = 1 and ε 3 = ε 4 = 2 solve this system. By formula (4), we obtain

a ± = [ ∓ s ( s + 1 ) ] 1 ⁄ ( m − 1 ) = ∓ 36 7 3 ≐ ∓ 1.7261 ,

and by formula (12), we have

b ± = a ± s ( s + 2 ) s + 2 − m s ≐ ∓ 1.1188 .

Theorem 2 is applicable, and equation (86) has two solutions u = u ± ( k ) , k ∈ N ( k 0 ) , satisfying inequalities (70)–(72), i.e.,

∣ u ± ( k ) ± 1.7261 k 7 ⁄ 6 ± 1.1188 k 1 ⁄ 6 ∣ < 1.1188 k 2 ⁄ 3 ,

∣ Δ u ± ( k ) ± 1.7261 Δ k 7 ⁄ 6 ± 1.1188 Δ k 1 ⁄ 6 ∣ < Δ ( 1.1188 k 1 ⁄ 6 ) 2 k 5 ⁄ 6 ,

and

∣ Δ 2 u ± ( k ) ± 1.7261 Δ 2 k 7 ⁄ 6 ± 1.1188 Δ 2 k 1 ⁄ 6 ∣ < ∣ Δ 2 ( 1.1188 k 1 ⁄ 6 ) ∣ 14 5 k 5 ⁄ 6 + O 1 k .

4 Final remarks

In the article, two-term asymptotic representations are derived for the solution of equation (1) and for its first- and second-order forward differences. This section is divided into three parts: discussing possible generalizations and extensions; comments on the method used and on its applicability; and comparisons with previously known results.

4.1 Generalizations

In the introduction, we pointed out that the results of the article formulated for equation (1) can be, using a simple transformation, reformulated for its variant – equation (8). The method used makes it also possible to generalize all results to some kinds of perturbed equations. Consider, instead of equation (1), an equation

(87) Δ 2 u ( k ) ± k α u m ( k ) = ω ( k , u ( k ) , Δ u ( k ) ) ,

where the assumptions for α and m are not changed and ω : N ( k 0 ) × R × R → R is a function continuous with respect to the second and third arguments. Tracing carefully all steps of the method used, we can find assumptions for the perturbation ω such that the theorems of the article remain valid. After some additional computation similar to that in the study by Astashova et al. [4], formula (14) can be written as follows:

Y 2 ( k ) = m s s + 2 Y 0 ( k ) + ω ± * ( k , Y 0 , Y 1 ) + O 1 k ,

where

ω ± * ( k , Y 0 , Y 1 ) ≔ 1 b ± ( s + 1 ) ( s + 2 ) ω k , a ± k s + b ± k s + 1 ( 1 + Y 0 ( k ) ) , Δ a ± k s + Δ b ± k s + 1 ( 1 + Y 1 ( k ) ) .

The first equation in the auxiliary system of equations (15) and (16), with the right-hand side defined by formula (17), is unchanged. In the second one, the function F 2 , defined by formula (18), must be modified as follows:

F 2 ( k , Y 0 , Y 1 ) ≔ − s + 2 k + O 1 k 2 m s s + 2 Y 0 − Y 1 + ω ± * ( k , Y 0 , Y 1 ) + O 1 k .

Obviously, if function ω ± * is sufficiently small, then the perturbation ω in equation (87) will have no impact on the results derived (except for to the value of k 0 ). Let us formulate one of the possible assumptions for ω ± * , leading to such a small value of ω and covering all the cases considered in the article. Assume

(88) ω ± * ( k , Y 0 , Y 1 ) = O 1 k when k ∈ N ( k 0 ) and ∣ Y i ∣ < 1 , i = 0 , 1 .

Then, assuming that Theorem 1 (or Theorem 2) can be applied and assumption (88) holds, the conclusion of Theorem 1 (or Theorem 2) is applicable to equation (87) as well.

By the suggested method, other classes can be considered of nonlinear difference equations as well. It seems that, e.g., an equation

Δ 2 u ( k ) ± k α u m 1 ( k ) ( Δ u ( k ) ) m 2 = 0 ,

with real numbers m 1 , m 2 , can be treated similarly. A challenge for future investigations is to consider a delayed discrete equation of Emden-Fowler type with a single delay as follows:

Δ 2 u ( k ) ± k α u m ( k − l ) = 0 ,

where l is a positive integer. An approach similar to that used in the article can be useful if Lemma 1 is replaced by one reflecting the presence of a delay (we refer to [11,16]). Another challenge is to extend the method developed to include discrete equations of higher order, e.g., to equation

Δ ℓ u ( k ) ± k α u m ( k ) = 0 ,

where ℓ > 2 .

4.2 Comparisons with studies [4,14,15]

Let us compare the above results with those of our previous investigations related to the discrete Emden-Fowler equation (1) and published in the studies [4,14,15]. The basic scheme of all investigations is the following. The transformations (9)–(11), where a ± and b ± are computed by formulas (4) and (12), are used for transforming equation (1) into an auxiliary system of equations (15) and (16). Then, some particular results of those published in the studies of Diblík [10,12] are applied to investigate the system of equations (15) and (16). A crucial role in applying Lemma 1 is played by a proper choice of functions b i ( k ) , c i ( k ) , i = 1 , 2 . In the article, these functions (equation (22)) are defined as follows:

(89) b 1 ( k ) ≔ − ε 1 k γ , c 1 ( k ) ≔ ε 2 k γ , b 2 ( k ) ≔ − ε 3 k γ , c 2 ( k ) ≔ ε 4 k γ ,

where γ ∈ ( 0 , 1 ) is fixed and positive constants ε j , j = 1 , … , 4 , are specified in Theorems 1 and 2. Below, the specification of functions b i and c i , i = 1 , 2 by equation (89) is referred to as a nonconstant case.

If Lemma 1 is applicable and lim k → ∞ b i ( k ) = lim k → ∞ c i ( k ) = 0 , i = 1 , 2 , then, lim k → ∞ Y i ( k ) = 0 , i = 1 , 2 , and formulas (9)–(11), as indicated in Theorems 1 and 2, give inequalities (38)–(40) and (70)–(72). Note that a necessary condition for the applicability of Lemma 1 is existence of a solution of inequalities (37) or (69).

The choice of functions b i ( k ) and c i ( k ) , i = 1 , 2 , in our previous studies [14,15] is the same as (89), while in the study by Astashova et al. [4], constant functions

(90) b 1 ( k ) ≔ − ε 1 , c 1 ( k ) ≔ ε 2 , b 2 ( k ) ≔ − ε 3 , c 2 ( k ) ≔ ε 4 ,

ε j and j = 1 , … , 4 , are applied. Below, the specification of functions b i and c i , i = 1 , 2 , by equation (90) is referred to as a constant case.

Both sets of functions (as in equations (89) and (90)) lead to the same asymptotic formula (7) (although inequalities (38)–(40) and (70)–(72) are more exact than those derived in the study by Astashova et al. [4]). Therefore, it has a sense to compare the results derived with those in the studies [4,14,15].

Analyzing the hypotheses in the articles considered, we conclude that the main assumptions deal with the cases s + 1 > 0 , s + 1 < 0 , m s > 0 , and m s < 0 . Table 1 shows which of the main assumptions are used in the articles mentioned, referring to the relevant theorems.

Table 1

The main assumptions used and related references

The case	m s < 0	m s > 0
s + 1 > 0	[15, Theorem 2] (nonconstant case)	[14, Theorem 1] (nonconstant case)
	[4, Theorem 5.1] (constant case)	[4, Theorem 5.1] (constant case)
s + 1 < 0	Theorem 1 (nonconstant case)	Theorem 2 (nonconstant case)

In each of the above articles, specific criteria guaranteeing the existence of the desired solutions are derived if, in addition to the above main assumptions, some specific ones are considered. Below is a list of these.

Article [15]. This article deals with the nonconstant case; we refer to Theorem 2. The considerations show that if, in addition to s + 1 > 0 , m s < 0 , an inequality

α 2 ( 1 + m ) + α ( m 2 + 8 m − 1 ) + 8 m 2 > 0

is assumed, the desired solutions exist.

Article [14]. In this article, Theorem 1 and Corollary 1 dealing with the nonconstant case are crucial. The considerations show that if inequalities

( s > 0 ) ∧ ( m > 0 ) or ( − 1 < s < 0 ) ∧ ( m < 0 ) ,

being specifications of inequalities ( s + 1 > 0 ) ∧ ( m s > 0 ) , are assumed, then the desired solutions exist if

(91) m s < ( s + 2 ) ( s + 3 ) s + 1 .

It is proved in Corollary 2 that, for inequality (91) to hold, it is sufficient that at least one of the following restrictions (92)–(95) holds:

(92) m ∈ ( − 7 − 4 3 , − 7 + 4 3 ) , − 2 < α < − m − 1 ,

(93) 0 < m < 1 , α < − 2 ,

(94) m > 1 , − 2 < α < ( − ( m − 1 ) + ( m − 1 ) 2 + 16 m ) ⁄ 2 ,

(95) − 2 < α < − m − 1 , m < 0 , ( m − 1 ) 2 + 16 m > 0 ,

where either

α < ( − ( m − 1 ) − ( m − 1 ) 2 + 16 m ) ⁄ 2

α > ( − ( m − 1 ) + ( m − 1 ) 2 + 16 m ) ⁄ 2 .

Article [4]. This article deals with the constant case. From Theorem 5.1 (and from its analysis in Section 6), provided m s > 0 , the desired solutions will exist if

m > 1 , − 2 < α < 0 ,

m < 0 , − 2 < α < min { 0 , − m − 1 } ,

0 < m < 1 , α < − 2 .

Provided that m s < 0 , the desired solutions will exist if one of the following sets of inequalities (96)–(99) holds:

(96) m > 1 , α > − m − 1 , α < − 2 , α > − 4 m m + 1 ,

(97) − 1 < m < 0 , α < − 2 , α < − 4 m 1 + m ,

(98) m < − 1 , α < − 2 , α > − 4 m 1 + m ,

(99) 0 < m < 1 , α > − 2 , α < − 4 m 1 + m , α < − m − 1 .

Remark 3

Table 1 does not refer to results dealing with the constant case when s + 1 < 0 . We show that the technique used is not applicable in this situation. Assuming the choice of b i and c i , i = 1 , 2 , by formulas (90), for i = 1 , an attempt to satisfy inequality (19) in Lemma 1, leads to the inequalities

F 1 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω b 1 = F 1 ( k , b 1 ( k ) , Y 1 ) ∣ b 2 ( k ) ≤ Y 1 ≤ c 2 ( k ) = F 1 ( k , − ε 1 , Y 1 ) ∣ − ε 3 ≤ Y 1 ≤ ε 4 = − s + 1 k + O 1 k 2 ( ε 1 + Y 1 ) ∣ − ε 3 ≤ Y 1 ≤ ε 4 < − s + 1 k + O 1 k 2 ( ε 1 + ε 4 ) < b 1 ( k + 1 ) − b 1 ( k ) = − ε 1 + ε 1 = 0 ,

implying ε 1 + ε 4 < 0 . This contradicts the positivity of both ε 1 and ε 4 . Similarly, the inequality (20), where i = 1 , will hold if

F 1 ( k , Y 0 , Y 1 ) ∣ ( k , Y 0 , Y 1 ) ∈ Ω c 1 = F 1 ( k , c 1 ( k ) , Y 1 ) ∣ b 2 ( k ) ≤ Y 1 ≤ c 2 ( k ) = F 1 ( k , ε 2 , Y 1 ) ∣ − ε 3 ≤ Y 1 ≤ ε 4 = − s + 1 k + O 1 k 2 ( − ε 2 + Y 1 ) ∣ − ε 3 ≤ Y 1 ≤ ε 4 > − s + 1 k + O 1 k 2 ( − ε 2 − ε 3 ) > c 1 ( k + 1 ) − c 1 ( k ) = ε 2 − ε 2 = 0 ,

and therefore, it implies ε 2 + ε 3 < 0 . This contradicts the positivity of ε 2 and ε 3 .

The above overview compares the results obtained with those derived in the studies [4,14,15]. From equations (89) and (90), two sets of functions of the type b i and c i , i = 1 , 2 , were used to obtain the desired asymptotic behavior of a solution to equation (1) with all the admissible values of parameters m and α covering a part of the ( m , α ) -plane. Concerning the nonconstant case, we refer to Figures 3 and 4 (where the central part of the previous figure is enlarged), summarizing the results of the article and those presented in the our previous studies [14] (the domains shown in yellow) and [15] (the domains shown in blue). Figure 5 graphically compares the results related to the nonconstant case derived in the article and [14,15] (the domains shown in yellow) with those related to the constant case derived in the study by Astashova et al. [4] (the domains dashed in green). It can be seen (and exactly proved as well) that all results the study by Astashova et al. [4] are covered by the results of the present article and by those in our previous studies [14,15].

$Figure 3 Admissible values of m m and α \alpha in the ( m , α ) \left(m,\alpha ) -plane.$

Figure 3

Admissible values of m and α in the ( m , α ) -plane.

$Figure 4 Admissible values of m m and α \alpha in the ( m , α ) \left(m,\alpha ) -plane – zoom.$

Figure 4

Admissible values of m and α in the ( m , α ) -plane – zoom.

Figure 5

Results for the constant and nonconstant cases – a comparison.

It is an open question if, for other values of parameters m and α , equation (1) has a solution with a similar asymptotic behavior. A set of functions b i and c i , i = 1 , 2 , different from the sets of equations (89) and (90) might perhaps be used to solve this problem.

4.3 Other comparisons

Let us compare the results derived with another one, not yet discussed. In the very interesting article by Erbe et al. [18], the Emden-Fowler equation is considered on time scales. Note that discrete equations are equations defined on discrete time scales. Then, considering only discrete time scales, given some conditions for nonlinearity, it is proved that there exists a solution that, for k → ∞ , is asymptotically equivalent to a straight line. In the paper [4], coauthored by the authors a detailed analysis of [18] is performed. The independence of the results is established on those derived in [18]. For the given results, a comparison can be made in much the same way.

In the study by Kharkov [20], asymptotic representations are considered of the so-called P ( λ ) -solutions of the equation

Δ 2 y n = α p n ∣ y n ∣ σ sign y n ,

where α ∈ { ± 1 } , σ ∈ R ⧹ { 0 , 1 } , and { p n } is a positive sequence. The results are applied to the equation

Δ 2 y n = α n k ∣ y n ∣ σ sign y n ,

and, among others, the condition

(100) α ( k + 2 ) ( k + σ + 1 ) > 0

must be fulfilled. Adopting notation from [20], we state that (despite the equations considered being not equivalent), e.g., for equation (61), where the upper sign variant + is considered, we have α = − 1 , k = − 27 ⁄ 20 , and σ = 1 ⁄ 2 so that inequality (100) does not hold. Therefore, the results are independent.

Different equations or asymptotic problems with similar topics are studied in the articles [8,21,24]. Migda [24] studied difference equations of Emden-Fowler type

Δ m x n = a n f ( x σ ( n ) ) + b n ,

and assuming that f is a power-type function with y being a solution to the equation Δ m y n = b n , conditions are found guaranteeing the existence of a solution x such that x n = y n + o ( n s ) , where s < 0 .

Cecchi et al. [8] in their study considered a class of equations of Emden-Fowler type

Δ ( a n ∣ Δ x n ∣ α sign Δ x n ) + b n ∣ Δ x n + 1 ∣ β sgn Δ x n + 1 = 0 ,

where α > 0 , β > 0 , and { a n } and { b n } are positive sequences. Among others, the existence of nonoscillatory solutions is studied.

A full classification of positive solutions of the equation

(101) Δ 2 y n = α p n ∣ y n + 1 ∣ σ sign y n + 1 ,

where α ∈ { ± 1 } , σ ∈ R ⧹ { 0 , 1 } , and lim n → ∞ ( n Δ p n ) ⁄ p n = k ∈ R ⧹ { − 2 , − 1 − σ } , is given in study by Kharkov [21]. Here, rather than the “direct” discretization (6), a different one is used. Therefore, the classes of the equations (1) and (101) are different.

Note that, by discretizing continuous equations, we obtain discrete equations. These can be regarded as algorithms for numerically solving the initial continuous equations. In conclusion of the comparisons, we refer to a few research and surveys [1,2,7,17,25,26] where a variety of results on the asymptotic behavior of solutions to some classes of difference equations can be found.

jakovi300195@yandex.ru

Acknowledgments

The authors would like to express their sincere gratitude to the editor and referees for their comments that have improved the present article in many aspects.

Funding information: The first author has been supported by the projects of specific university research FAST-S-22-7867 (Faculty of Civil Engineering, Brno University of Technology) and FEKT-S-23-8179 (Faculty of Electrical Engineering and Communication, Brno University of Technology). The second author has been supported by the project of specific university research FEKT-S-23-8179 (Faculty of Electrical Engineering and Communication, Brno University of Technology).
Conflict of interest: The authors state that there is no conflict of interest.

References

[1] R. P. Agarwal, Difference equations and inequalities. Theory, methods and applications, 2nd edition, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 2000. 10.1201/9781420027020Search in Google Scholar

[2] R. P. Agarwal, M. Bohner, S. R. Grace, and D. O’Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, 2005. 10.1155/9789775945198Search in Google Scholar

[3] E. Akin-Bohner and J. Hoffacker, Oscillation properties of an Emden-Fowler type equation on discrete time scales, J. Difference Equ. Appl. 9 (2003), no. 6, 603–612. 10.1080/1023619021000053575Search in Google Scholar

[4] I. V. Astashova, J. Diblík, and E. Korobko, Existence of a solution of discrete Emden-Fowler equation caused by continuous equation, Discrete Contin. Dyn. Syst. Ser. S 14 (2021), no. 12, 4159–4178. 10.3934/dcdss.2021133Search in Google Scholar

[5] I. Astashova, M. Bartůšek, Z. Došlá, and M. Marini, Asymptotic proximity to higher order nonlinear differential equations, Adv. Nonlinear Anal. 11 (2022), no. 1, 1598–1613. 10.1515/anona-2022-0254Search in Google Scholar

[6] R. Bellman, Stability Theory of Differential Equations, Dover Publications, Inc., New York, 2008. Search in Google Scholar

[7] S. Bodine and D. A. Lutz, Asymptotic Integration of Differential and Difference Equations, Lecture Notes in Mathematics 2129, Springer, Cham, 2015. 10.1007/978-3-319-18248-3Search in Google Scholar

[8] M. Cecchi, Z. Došlá, and M. Marini, On oscillation and nonoscillation properties of Emden-Fowler difference equations, Cent. Eur. J. Math. 7 (2009), no. 2, 322–334. 10.2478/s11533-009-0014-7Search in Google Scholar

[9] M. H. M. Christianen, A. J. E. M. Janssen, M. Vlasiou, and B. Zwart, Asymptotic analysis of Emden-Fowler type equation with an application to power flow models, Indagationes Mathematicae 34 (2022), no. 5, 1146–1180. 10.1016/j.indag.2022.12.001Search in Google Scholar

[10] J. Diblík, Asymptotic behavior of solutions of discrete equations, Funct. Differ. Equ. 11 (2004), no. 1–2, 37–48. Search in Google Scholar

[11] J. Diblík, Bounded solutions to systems of fractional discrete equations, Adv. Nonlinear Anal. 11 (2022), no. 1, 1614–1630. 10.1515/anona-2022-0260Search in Google Scholar

[12] J. Diblík, Discrete retract principle for systems of discrete equations, Comput. Math. Appl. 42 (2001), 515–528. 10.1016/S0898-1221(01)00174-2Search in Google Scholar

[13] J. Diblík and I. Hlavičková, Asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation, Adv. Stud. Pure Math. 53 (2009), 23–32. 10.2969/aspm/05310023Search in Google Scholar

[14] J. Diblík and E. Korobko, On a discrete variant of the Emden-Fowler equation, Mathematics, Information Technologies and Applied Sciences 2021 post-conference proceedings of extended versions of selected papers (2021). Search in Google Scholar

[15] J. Diblík and E. Korobko, New conditions for existence of the solution to the discrete Emden-Fowler type equation with power asymptotics, Mathematics, Information Technologies and Applied Sciences 2022, post-conference proceedings (2022). Search in Google Scholar

[16] J. Diblík and J. Vítovec, Asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points, Appl. Math. Comput. 238 (2014), 289–299. 10.1016/j.amc.2014.04.021Search in Google Scholar

[17] S. N. Elaydi, An introduction to difference equations, 3rd edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. Search in Google Scholar

[18] L. Erbe, J. Baoguo, and A. Peterson, On the asymptotic behavior of solutions of Emden-Fowler equations on time scales, Ann. Mat. Pura Appl. 191 (2012), no. 4, 205–217. 10.1007/s10231-010-0179-5Search in Google Scholar

[19] M. Galewski, Dependence on parameters for a discrete Emden-Fowler equation, Appl. Math. Comput. 218 (2011), no. 4, 1247–1253. 10.1016/j.amc.2011.06.005Search in Google Scholar

[20] V. Kharkov, Asymptotic representations of a class of solutions of a second-order difference equation with a power nonlinearity, Ukraiiin. Mat. Zh. 61 (2009), no. 6, 839–854444; translation in Ukrainian Math. J.61 (2009), no. 6, 994–1012. Search in Google Scholar

[21] V. Kharkov, Positive solutions of the Emden-Fowler difference equation, J. Difference Equ. Appl. 19 (2013), no. 2, 234–260. 10.1080/10236198.2011.634805Search in Google Scholar

[22] V. Kharkov and A. Berdnikov, Asymptotic representations of solutions of k-th order Emden-Fowler difference equation, J. Difference Equ. Appl. 21 (2015), no. 9, 840–853. 10.1080/10236198.2015.1051479Search in Google Scholar

[23] W. T. Li, X. L. Fan, and C. K. Zhong, Positive solutions of discrete Emden-Fowler equation with singular nonlinear term, Dynam. Systems Appl. 9 (2000), no. 2, 247–254. Search in Google Scholar

[24] J. Migda, Asymptotic properties of solutions to difference equations of Emden-Fowler type, Electron. J. Qual. Theory Differ. Equ. 77 (2019), 1–17. 10.14232/ejqtde.2019.1.77Search in Google Scholar

[25] M. A. Radin, Difference Equations for Scientists and Engineering: Interdisciplinary Difference Equations, World Scientific Publishing, Singapore, 2019. 10.1142/11349Search in Google Scholar

[26] S. Saker, Oscillation Theory of Delay Differential and Difference Equations Second and Third Orders, VDM Verlag Dr. Mü ller, 2010. Search in Google Scholar

Received: 2023-04-25

Revised: 2023-05-25

Accepted: 2023-06-13

Published Online: 2023-10-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2023-0105

Keywords for this article

discrete equation; Emden-Fowler equation; equation of the second-order; asymptotic behavior; nonlinearity

Creative Commons

BY 4.0