We use o.ns/, for a given nonpositive reals, as a measure of approximation

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Vol. 19 (2018), No. 2, pp. 1047–1061 DOI: 10.18514/MMN.2018.2735

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENCE EQUATIONS WITH DEVIATING ARGUMENT

JANUSZ MIGDA AND MAGDALENA NOCKOWSKA-ROSIAK Received 04 November, 2018

Abstract. We consider nonlinear second order difference equations with deviating argument of the form

.rnxn/Danf .xnC 1; xnC 2; : : : ; xnC m/Cbn:

We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use o.n^s/, for a given nonpositive reals, as a measure of approximation.

2010Mathematics Subject Classification: 39A10; 39A22

Keywords: difference equation, prescribed asymptotic behavior, bounded solution, convergent solution, quasidifference, deviating argument

1. INTRODUCTION

In this paper we consider the nonlinear second order difference equation with deviating argument of the form

.rnxn/Danf .xnC 1; xnC 2; : : : ; xnC m/Cbn (E) rn; an; bn2R; rn> 0; 2Z; m2N; f WR^m!R:

HereN,Z,Rdenote the set of positive integers, the set of all integers, and the set of real numbers, respectively. By a solution of (E) we mean a sequencexWN!R satisfying (E) for largen.

An important issue in the asymptotic theory of ordinary and delay differential equations is constructing sufficient conditions which ensure the existence of solutions with prescribed asymptotic behavior. From this point of view many authors studied second order differential equations with deviating argument of the form

rx⁰0

.t /Da.t /f .x.0.t /// (1.1) or

x⁰⁰.t /Df .t; x.1.t //; : : : ; x.m.t /// (1.2)

c 2018 Miskolc University Press

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whererWR!.0;C1/,i.t /! 1ast! 1foriD0; : : : ; m, see e.g. [1], [2], [3], [11], and the references therein. Since difference equations can be treated as discret- ization of differential equations, the existence of solutions with prescribed asymptotic behavior of difference equations was studied in literature too, see for example, in [4], [12], [13] and the references therein.

In papers [4–9], the first author presented a new theory of the study of asymptotic properties of solutions of difference equations of the form

²xnDanf .x_.n//Cbn

in which o.n^s/, fors0, is used a measure of approximation. In the paper [10], we extend some of results on difference equations of Sturm-Liouville type of the form

.rnxn/Danf .x_.n//Cbn:

This paper is a continuation of these investigations. Our main goal is to present sufficient conditions for the existence of a solutionxof equation (E) such that

xnDc.r₁¹C Cr_{n 1}¹ /CdCo.n^s/; (1.3) wherec; d 2Rands2. 1; 0. We give also sufficient conditions for a given solutionxof equation (E) to have an asymptotic property (1.3).

The paper is organized as follows. In Section 2, we introduce notation and present some preliminary lemmas. Next, Section 3 is devoted to our first main result Theorem 1, some consequences of it and the example which proves that one of assumptions in main theorem is essential. In Section 4 we prove our second main result and some corollaries from it. Moreover, this section includes the example which proves that one of assumptions of main theorem is not “too” strong.

2. PRELIMINARIES

The space of all sequencesxWN!Rwe denote byR^N. Ifx; y inR^N, then xy andjxjdenotes the sequences defined by.xy/nDxnynandjxjⁿD jxnj, respectively.

Moreover,

kxk Dsupfjxnj Wn2Ng:

For any sequencex2R^Nwe define a sequencexWN!R^mby x_nD

(.0; 0; : : : ; 0/ forn < m C1 .xnC 1; xnC 2; : : : ; xnC m/ fornm C1:

We use the symboldmto denote the max-metric onR^mdefined by dm.u; v/Dmaxfju1 v1j; : : : ;jum vmjg:

Moreover, B.u; ˛/denotes the closed ball of radius˛centered at a pointu2R^m. We say that a functiongWR^m!Ris bounded at infinity if there exists a real number such thatgis bounded on the set

Œ;1/ Œ;1/DŒ;1/^m:

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In the same way, the boundedness at minus infinity can be defined.

Lemma 1. Ify2R^Nand.rnyn/D0, then there exist real constantsc; dsuch that

ynDc

n 1

X

jD1

1

r_j Cd (2.1)

for anyn. Ifc; d 2Randy2R^Nis defined by(2.1), then.rnyn/D0.

Proof. We leave an easy proof of this lemma to the reader.

Lemma 2([10, Lemma 3]). Assumes2. 1; 0, t2Œs;1/,r_n¹DO.n^t/,a2 R^N, and

1

X

nD1

n¹^C^{t s}janj<1; then

1

X

nD1

1 n^srn

1

X

jDn

jajj<1:

Lemma 3. Assume s2. 1; 0,t 2Œs;1/, r_n¹DO.n^t/,a2R^N, and at least one of the following conditions is satisfied

.a/ lim inf

n!1n

janj janC1j 1

> 2Ct s; .b/ lim inf

n!1 nlog janj

janC1j> 2Ct s:

Then

1

X

nD1

1 n^srn

1

X

jDn

jajj<1:

Proof. Using [7, Lemma 6.3] in case (a), or [7, Lemma 6.4] in case (b) we obtain

1

X

nD1

n¹^C^{t s}janj<1:

By Lemma2we get the result.

Lemma 4. Assume2R,r_n¹DO.n/,a2R^N, and at least one of the following conditions is satisfied

.a/ lim sup

n!1

janC1j

janj < 1; .b/ lim sup

n!1

pn

janj< 1:

Then

1

X

nD1

1 n^srn

1

X

jDn

jajj<1

for anys2. 1; 0.

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Proof. Lets2. 1; 0. Choose a real numbert such thatt >max.s; /. Then r_n¹DO.n^t/and, using the ratio test in case (a), or the root test in case (b) we get

1

X

nD1

n¹^C^{t s}janj<1:

Hence the assertion is a consequence of Lemma2.

Lemma 5([10, Lemma 5]). If

1

X

kD1

1 r_k

1

X

iDk

juij<1;

then

1

X

kD1

ju_kj

k

X

iD1

1 ri

<1 and

1

X

kDn

1 rk

1

X

iDk

juij

1

X

kDn

ju_kj

k

X

iD1

1 ri

for anyn2N.

Lemma 6([4, Lemma 4.7]). Assumey; WN!R, and lim

n!1_nD0. In the set X D fx2R^NW jx yj jjgwe define a metric by the formula

d.x; ´/D kx ´k: (2.2)

Then any continuous mapHWX !X has a fixed point.

3. SOLUTIONS WITH PRESCRIBED ASYMPTOTIC BEHAVIOR

In this section we establish various conditions under which for a given solutiony of the equation.rnyn/D0and a given nonpositive realsthere exists a solution xof (E) such thatxnDynCo.n^s/.

Theorem 1. Assumes2. 1; 0,yis a solution of the equation.rnyn/D0,

1

X

kD1

1 k^srk

1

X

jDk

.jajj C jbjj/ <1; q2N; ˛2.0;1/; U D

1

[

nDq

B.y_n; ˛/;

andf is continuous and bounded onU. Then there exists a solutionx of (E)such thatxnDynCo.n^s/.

Proof. Forn2Nandx2R^Nlet

F .x/.n/Danf .x_n/Cbn: (3.1) There existsL > 0, such that

jf .u/j L (3.2)

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for anyu2U. Sinces0we have

1

X

kD1

1 rk

1

X

jDk

.Ljajj C jbjj/ <1: Let

Y D fx2R^NW jx yj ˛g; 2R^N; nD

1

X

kDn

1 r_k

1

X

jDk

.Ljajj C jbjj/:

Ifx2Y, thenx_n2U for largen. Hence the sequence.f .x_n//is bounded for any x2Y. Define sequencesw; gby

wj DLjajj C jbjj; gnD

1

X

kDn

1 k^sr_k

1

X

jDk

wj:

Then

n ^snDn ^s

1

X

kDn

1 r_k

1

X

jDk

wj D

1

X

kDn

1 n^sr_k

1

X

jDk

wj

1

X

kDn

1 k^sr_k

1

X

jDk

wj Dgn: (3.3) Using the assumption

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/ <1

we getgnDo.1/. Hence, by (3.3),

nDn^sgnDn^so.1/Do.n^s/:

Therefore there exists an indexpsuch that

n˛ and nC mq for np:

Let

X D fx2R^NW jx yj andxnDynforn < pg; H WY !R^N; H.x/.n/D

(yn forn < p

ynCP₁

kDn 1 r_k

P₁

jDkF .x/.j / fornp:

Note thatXY. Ifx2X, then fornpwe have jH.x/.n/ ynj D

ˇ ˇ ˇ ˇ ˇ ˇ

1

X

kDn

1 r_k

1

X

jDk

F .x/.j / ˇ ˇ ˇ ˇ ˇ ˇ

1

X

kDn

1 r_k

1

X

jDk

jF .x/.j /j n:

ThereforeHXX. Letx2X, and" > 0. By Lemma5, we have

1

X

kD1

jakj

k

X

iD1

1 ri

<1:

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Choose an indexqpand a positive constant such that L

1

X

kDq

ja_kj

k

X

iD1

1 ri

< " and

q

X

kDp

ja_kj

k

X

iD1

1 ri

< ":

Let

C D

q

[

nD1

B.y_n; ˛/:

Since C is a compact subset of R^m, f is uniformly continuous on C. Choose a positiveısuch that ifu1; u22C anddm.u1; u2/ < ı, then

jf .u1/ f .u2/j< : Choose´2X such thatkx ´k< ı. Then

kH x H ´k D sup

np

ˇ ˇ ˇ ˇ ˇ ˇ

1

X

kDn

1 r_k

1

X

jDk

.F .x/.j / F .´/.j //

ˇ ˇ ˇ ˇ ˇ ˇ

1

X

kDp

1 rk

1

X

jDk

jF .x/.j / F .´/.j /j

1

X

kDp

1 rk

1

X

jDk

jajjjf .x_j/ f .´_j/j:

By Lemma5

1

X

kDp

1 r_k

1

X

jDk

jajjjf .x_j/ f .´_j/j

1

X

kDp

ja_kjjf .x_k/ f .´_k/j

k

X

iD1

1 ri

:

Hence

kH x H ´k

q

X

kDp

ja_kj

k

X

iD1

1 ri C2L

1

X

kDq

ja_kj

k

X

iD1

1 ri

< 3":

Therefore the mapH WX !X is continuous with respect to the metric defined by (2.2). By Lemma6there exists a pointx2X such thatxDH x. Then fornpwe have

xnDynC

1

X

kDn

1 rk

1

X

jDk

F .x/.j /:

Hence

.rnxn/D.rnyn/C 0

@rn 0

@

1

X

kDn

1 r_k

1

X

jDk

F .x/.j / 1 A 1 A

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fornp. Define a sequenceG by GnD

1

X

kDn

1 rk

1

X

jDk

F .x/.j /:

Then

rnGnDrn

0

@ 1 rn

1

X

jDn

F .x/.j / 1 A;

.rnGn/DF .x/.n/Danf .x_n/CbnDanf .xnC 1; xnC 2; : : : ; xnC m/Cbn

for largen. Thereforexis a solution of (E). Sincex2X andnDo.n^s/, we have

xnDynCo.n^s/.

Corollary 1. Assumef is continuous,s2. 1; 0, and

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/ <1:

Then for any bounded solutionyof the equation.rnyn/D0there exists a solution xof (E)such thatxnDynCo.n^s/.

Proof. Assumeyis a bounded solution of the equation.rnyn/D0. Then the set

U D

1

[

nD1

B.y_n; 1/

is bounded. Hencef is continuous and bounded onU. By Theorem1there exists a

solutionxof (E) such thatxnDynCo.n^s/.

Corollary 2. Assumef is continuous,s2. 1; 0, and

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/ <1:

Then for any real constantdthere exists a solutionxof(E)such thatxnDdCo.n^s/.

Proof. Any constant sequence is a bounded solution of the equation.rnyn/D 0. Hence the assertion is a consequence of Corollary1.

Corollary 3. Assumef is continuous,s2. 1; 0,

1

X

kD1

1

r_k <1; and

1

X

kD1

1 k^sr_k

1

X

jDk

.ja_jj C jb_jj/ <1:

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Then for anyc; d2Rthere exists a solutionxof (E)such that xnDc

n 1

X

jD1

1

rj CdCo.n^s/:

Proof. Define a sequenceyby ynDc

n 1

X

jD1

1 rj Cd:

By Lemma1, y is a solution of the equation .rnyn/D0. By assumption, the sequenceyis bounded. Using Corollary1we get the result.

Corollary 4. Assumef is continuous and bounded,s2. 1; 0, and

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/ <1:

Then for any solutiony of the equation.rnyn/D0 there exists a solutionx of (E)such thatxnDynCo.n^s/.

Proof. This corollary is an immediate consequence of Theorem1.

Corollary 5. Assumef is continuous,s2. 1; 0,t2Œs;1/,r_n¹DO.n^t/, and

1

X

nD1

n¹^C^{t s}.janj C jbnj/ <1:

Proof. This corollary is a consequence of Lemma2and Corollary1.

Corollary 6. Assumef is continuous,s2. 1; 0,t2Œs;1/,r_n¹DO.n^t/, and lim inf

n!1n

janj C jbnj janC1j C jbnC1j 1

> 2Ct s:

Proof. This corollary is a consequence of Lemma3and Corollary5.

Corollary 7. Assumef is continuous and bounded at infinity,2R, 1

rn DO.n/ and lim sup

n!1

janC1j C jbnC1j janj C jbnj < 1:

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Then for any positivec, any reald and anys2. 1; 0there exists a solutionxof (E)such that

xnDc

n 1

X

jD1

1

rj CdCo.n^s/:

Proof. Letc2.0;1/,d 2R,s2. 1; 0. Define a sequenceyby ynDc

n 1

X

jD1

1 rj Cd:

By Lemma1,yis a solution of the equation.rnyn/D0. By Lemma4,

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/ <1:

If the series

1

X

nD1

1 rn

(3.4) is convergent then the sequenceyis bounded and, using Corollary1we get the result.

Assume the series (3.4) is divergent. Sincec > 0, we haveyn! 1. There exists a real number˛such thatf is bounded onŒ˛;1/^m. There exists an indexqsuch that ynC m> ˛C1for anynq. Let

U D

1

[

nDq

B.y_n; 1/:

ThenU Œ˛;1/^m. Hencef is continuous and bounded onU. Now, using Theorem

1we obtain the result.

Now we present an example that proves the assumption that the function f is bounded on some “neighborhood” of .y_n/ such that .yn/ solves .rnyn/ =0 in Theorem1, is essential.

Example1. AssumemD2,

rnDn ¹; anD2 ⁿ; bnD0; D0; sD0; f .x; y/Dx³Cexp.2y/:

Then equation (E) takes the form

.n ¹xn/D2 ⁿ x_{n 1}³ Cexp.2xn 2/

: (3.5)

Let

ynD

n 1

X

kD1

1 r_k D

n 1

X

kD1

kD n.n 1/

2 :

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Thenf is continuous and not bounded on S1 nDq

B.y_n; ˛/for any q2Nand˛ > 0.

Moreover,

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/D

1

X

kD1

k

1

X

jDk

1 2^j D2

1

X

kD1

k 2^k <1: Assumexis a solution of (3.5) such that

xnDynC´n;

for largenand´nDo.n^s/Do.1/. Since._n¹yn/D0for largen, we have .¹_n´n/D.¹_nxn/D2 ⁿ x_{n 1}³ Cexp.2xn 2/

> 0

for largen. Therefore, the sequence ¹_n´nis eventually increasing, and there exists the limit

D lim

n!1 1

n´n> 1:

If <1, then the sequence_n¹´nis convergent inR. Hence the series

1

X

nD1

._n¹xn/D

1

X

nD1

.¹_n´n/

is convergent. On the other hand for largen

.¹_nxn/D2 ⁿ x³_{n 1}Cexp.2xn 2/

2 ⁿexp.2xn 2/:

Since

xn yn

2 for largen, we get that

._n¹´n/ > 2 ⁿexp..n 2/.n 3//:

HenceD 1. Therefore ¹_n´n> 1for largenand we get

1

X

nD1

´n

1

X

nD1

nD 1:

On the other hand, since´n!0, the seriesP₁

nD1´nis convergent.

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4. ASYMPTOTIC BEHAVIOR OF SOLUTIONS

In this section we present sufficient conditions for a given solutionxof equation (E) to have an asymptotic propertyxnDynCo.n^s/, where y is a solution of the equation.rnyn/D0ands2. 1; 0.

Theorem 2. Assume x is a solution of (E) such that the sequence .f .x_n// is bounded,

s2. 1; 0; and

1

X

kD1

1 k^srk

1

X

jDk

.jajj C jbjj/ <1: Then there exists a solutionyof the equation.rnyn/D0such that

x_nDy_nCo.n^s/:

Proof. Define a sequenceuby

unD.rnxn/Danf .x_n/Cbn: Since the sequence.f .x_n//is bounded, we have

1

X

kD1

1 k^sr_k

1

X

jDk

jujj<1:

Define sequencesw; y; ´by wnD

1

X

kDn

1 r_k

1

X

jDk

uj; ynDxn wn; ´nD

1

X

kDn

1 k^sr_k

1

X

jDk

uj:

Then

n ^sjwnj n ^s

1

X

kDn

1 rk

1

X

jDk

jujj D

1

X

kDn

1 n^srk

1

X

jDk

jujj

1

X

kDn

1 k^srk

1

X

jDk

jujj Do.1/:

HencewnDo.n^s/. Moreover .rnwn/D

0

@rn 0

@

1

X

kDn

1 rk

1

X

jDk

uj

1 A 1 AD

0

@rn

0

@ 1 rn

1

X

jDn

uj

1 A

1 ADun

and we obtain

.rnyn/D.rnxn/ .rnwn/Dun unD0; xnDynCwnDynCo.n^s/:

Corollary 8. Assumef is locally bounded,s2. 1; 0, and

1

X

kD1

1 k^sr_k

1

X

jDk

.ja_jj C jb_jj/ <1:

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Then for any bounded solutionxof (E)there exists a solutiony of the equation .rnyn/D0

such thatxnDynCo.n^s/.

Proof. Assume x is a bounded solution of (E). Then the sequence x is also bounded. Sincef is locally bounded, the sequence.f .x_n//is bounded. Hence the

result follows from Theorem2.

Corollary 9. Assumef is bounded,s2. 1; 0, and

1

X

kD1

1 k^sr_k

1

X

jDk

.jajj C jbjj/ <1:

Then for any solutionxof (E)there exists a solutionyof the equation .rnyn/D0

such thatxnDynCo.n^s/.

Proof. The assertion is an immediate consequence of Theorem2.

Corollary 10. Assumef is bounded and

1

X

kD1

1

r_k C jakj C jbkj

<1: Then any solutionxof (E)is convergent.

Proof. LetsD0and letxbe a solution of (E). By assumption the seriesP₁

kD11=r_k is convergent and the sequenceudefined byu_kDP₁

jDk.jajj C jbjj/is convergent to zero. Hence

1

X

kD1

1 r_k

1

X

jDk

.jajj C jbjj/ <1: (4.1) Therefore, by Corollary9and Lemma1, there exist real constantsc; dsuch that

xnDc

n 1

X

jD1

1

rj CdCo.n^s/:

SincesD0andP₁

kD11=r_k<1, we get limn!1xnDcP₁

kD11=r_kCd. Corollary 11. Assumef is locally bounded and

1

X

kD1

1

r_k C ja_kj C jb_kj

<1: Then any bounded solutionxof (E)is convergent.

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Proof. LetsD0and letxbe a bounded solution of (E). As in the proof of Corol- lary10we obtain (4.1). Using Corollary8and Lemma1we get the result.

Corollary 12. Assumef is locally bounded,s2. 1; 0,t2Œs;1/,r_n¹DO.n^t/,

and 1

X

nD1

Then for any bounded solution x of (E) there exists a solution y of the equation .rnyn/D0such thatxnDynCo.n^s/.

Proof. The assertion is a consequence of Lemma2and Corollary8.

Corollary 13. Assumef is locally bounded,s2. 1; 0,t2Œs;1/,r_n¹DO.n^t/,

1

X

nD1

1

rn D 1; and

1

X

nD1

Then for any bounded solutionxof (E)there exists a real constantd such that xnDdCo.n^s/:

Proof. Assumex be a bounded solution of (E). By Lemma 2 and Corollary 8, there existc; d2Rsuch that

xnDc

n 1

X

jD1

1

rj CdCo.n^s/:

Sincexis bounded andP₁

nD11=rnD 1, we havecD0.

Corollary 14. Assumef is locally bounded, 1

rn DO.1/; and

1

X

nD1

n.janj C jbnj/ <1: Then any bounded solution of (E)is convergent.

Proof. Lett DsD0 and letx be a bounded solution of (E). By Corollary 12 there exists a solutiony of the equation.r_ny_n/D0 such thatx_nDy_nCo.1/.

Thenyis a bounded sequence. By Lemma1any bounded solutionyof the equation

.rnyn/D0is convergent. Hencexis convergent.

Corollary 15. Assumef is bounded at infinity,s2. 1; 0, t2Œs;1/,r_n¹D O.n^t/,

lim inf

n!1nlog janj C jbnj

janC1j C jbnC1j > 2Ct s;

andxis a solution of (E)such thatlimn!1xnD 1. Then there exists a solutiony of the equation.rnyn/D0such thatxnDynCo.n^s/.

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Proof. Sincef is bounded at infinity and limn!1xnD 1, the sequence.f .x_n//

is bounded. Using Lemma3and Theorem2we get the result.

Corollary 16. Assumef is locally bounded,2R,r_n¹DO.n/, lim sup

n!1

janC1j C jbnC1j janj C jbnj < 1;

andxis a bounded solution of (E). Then for anys2. 1; 0there exists a solution yof the equation.rnyn/D0such thatxnDynCo.n^s/.

Proof. This assertion is a consequence of Lemma4and Corollary8.

Now we present an example that proves the assumption

1

X

kD1

1 k^srk

1

X

jDk

jajj<1:

is not enough in Theorem2.

Example2. AssumemD2,sD0,D0, rnDn²; anD 1

n²; bnD2nC1 2

n²; f .x; y/D x

jxj C1C yC3 jyj C2: Then equation (E) takes the form

.n²xn/D 1 n²

xn 1

jxn 1j C1C xn 2C3 jxn 2j C2

C2nC1 2

n²: (4.2) Notice thatf is bounded and

1

X

kD1

1 k^sr_k

1

X

jDk

ja_jj D

1

X

kD1

1 k²

1

X

jDk

1 j² <1:

Moreover, the sequencexnDn, is a solution of (4.2). On the other hand, any solution of the equation.n²yn/D0is of the form

ynDc

n 1

X

kD1

1

r_kCd Dc

n 1

X

kD1

1 k²Cd

for somec; d2R. Hence any solution of.n²yn/D0is convergent, which means thatxcannot be approximated by any solution of the equation.n²yn/D0.

(15)

REFERENCES

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10.1016/j.cam.2005.11.038.

[2] M. Bohner and S. Stevi´c, “Asymptotic behavior of second-order dynamic equations,”Appl. Math.

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10.1017/S0017089502001143.

[4] J. Migda, “Approximative solutions of difference equations,”Electron. J. Qual. Theory Differ.

Equ., no. 13, pp. 1–26, 2014, doi:10.14232/ejqtde.2014.1.13.

[5] J. Migda, “Iterated remainder operator, tests for multiple convergence of series and solutions of difference equations,”Adv. Difference Equ., no. 189, pp. 1–18, 2014, doi: 10.1186/1687-1847- 2014-189.

[6] J. Migda, “Approximative solutions to difference equations of neutral type,”Appl. Math. Comput., vol. 268, pp. 763–774, 2015, doi:10.1016/j.amc.2015.06.097.

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Theory Differ. Equ., no. 32, pp. 1–26, 2015, doi:10.14232/ejqtde.2015.1.32.

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[9] J. Migda and M. Migda, “Qualitative approximation of solutions to discrete Volterra equations,”

Electron. J. Qual. Theory Differ. Equ., no. 3, pp. 1–27, 2018, doi:10.14232/ejqtde.2018.1.3.

[10] J. Migda and M. Nockowska-Rosiak, “Asymptotic properties of solutions to difference equations of Sturm-Liouville type,” Appl. Math. Comput., vol. 340, pp. 126–137, 2019, doi:

10.1016/j.amc.2018.08.001.

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10.1017/S0017089507003667.

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Difference Equ. Appl., vol. 22, no. 1, pp. 107–139, 2016, doi:10.1080/10236198.2015.1077815.

[13] S. Stevi´c, “Asymptotic behaviour of second-order difference equation,”ANZIAM J., vol. 46, no. 1, pp. 157–170, 2004, doi:10.1017/S1446181100013742.

Authors’ addresses

Janusz Migda

Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Pozna´n, Poland

E-mail address:migda@amu.edu.pl

Magdalena Nockowska-Rosiak

Institute of Mathematics, Lodz University of Technology, Wólczańska 215, 90-924 Łód´z, Poland E-mail address:magdalena.nockowska@p.lodz.pl