Applied Mathematics and Computation 233 (2014) 377–382

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Global asymptotic stability of a second order rational differenceequation

http://dx.doi.org/10.1016/j.amc.2014.01.1770096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: lxhu@mail.tsnc.edu.cn (L.-X. Hu).

Lin-Xia Hu ⇑, Hong-Ming XiaDepartment of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, People’s Republic of China

a r t i c l e i n f o

Keywords:Difference equationBoundednessPeriod-two solutionStableGlobal attractorGlobally asymptotically stable

a b s t r a c t

The main goal of this paper is to investigate the global asymptotic stability of the differenceequation

ynþ1 ¼pn þ yn

pn þ yn�1; n ¼ 0;1;2; . . . ;

where

pn ¼a; if n is evenb; if n is odd

�and a > 0; b > 0; a–b

and the initial conditions y�1; y0 2 ½0;1Þ . We show that the unique equilibrium �y ¼ 1 isglobally asymptotically stable under the certain conditions.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

Our aim in this paper is to investigate the global stability of the following difference equation

ynþ1 ¼pn þ yn

pn þ yn�1; n ¼ 0;1;2; . . . ; ð1:1Þ

where

pn ¼a; if n is evenb; if n is odd

�and a > 0; b > 0; a – b

and the initial conditions y0; y�1 2 ½0;1Þ. In [10, Open Problem 4.8.12], Kulenovic and Ladas proposed the following openproblem. See also [3, Open Problem 5.26.1].

Open Problem 1.1. Let fpng1n¼0 be a periodic-two sequence of nonnegative real numbers. Investigate the global character of

all positive solutions of Eq. (1.1).Recently, systematic analysis of difference equations with periodic coefficients was considered by several authors, see, for

example Refs. [6,7,11]. Inspired by the above open problem, we investigate the global stability of Eq. (1.1) with a periodic-two coefficients, which is the non-autonomous equation that corresponds to the following autonomous equation

378 L.-X. Hu, H.-M. Xia / Applied Mathematics and Computation 233 (2014) 377–382

ynþ1 ¼pþ yn

pþ yn�1; n ¼ 0;1;2; . . . ; p > 0; y�1; y0 2 ½0;1Þ: ð1:2Þ

In [9], it is shown that the unique equilibrium �y ¼ 1 of Eq. (1.2) is globally asymptotically stable. See also [8, p.73] and [3,p.168]. Our main goal in this paper is to extend the result to the non-autonomous difference Eq. (1.1) under the certainconditions.

Before our discussion, we present some definitions and the known results which will be useful in the sequel. For the gen-eral theory of difference equations, one can refer to the monographes of Kocic and Ladas [8] and Kulenovic and Ladas [10].For other related results on nonlinear difference equations, see, for example, [1–16] and the references therein.

Consider the system:

unþ1 ¼ f ðun; vnÞ;vnþ1 ¼ gðun;vnÞ:

�n ¼ 0;1; . . . : ð1:3Þ

Let k � k be the norm of vector ðu;vÞ 2 R2. Then, we present some definitions and some useful lemmas.

Definition 1.2. The equilibrium point ð�u; �vÞ is said to be:

(i) stable if given � > 0 and N > 0 there exists d > 0 such that kðu0;v0Þ � ð�u; �vÞk < d implies that kðun;vnÞ � ð�u; �vÞk < � forall n > N, and unstable if it is not stable;

(ii) attracting if there exists g > 0 such that kðu0;v0Þ � ð�u; �vÞk < g implies that limn!1ðun;vnÞ ¼ ð�u; �vÞ;(iii) asymptotically stable if it is stable and attracting.

The main result in linearized stability analysis is following lemma which is extracted from [4], see also [10].

Lemma 1.3. Let F ¼ ðf ; gÞ be a continuously differentiable function defined on an open set D � R2.

(a) If the eigenvalues of the Jacobian matrix JFðð�u; �vÞÞ, that is, both roots of its characteristic equation

k2 � TrJFðð�u; �vÞÞkþ DetJFðð�u; �vÞÞ ¼ 0; ð1:4Þ

lie inside the unit disk, then the equilibrium ð�u; �vÞ of Eq. (1.3) is locally asymptotically stable.(b) A necessary and sufficient condition for both roots of Eq. (1.4) to lie inside the unit disk is

jTrJFðð�u; �vÞÞj < 1þ DetJFðð�u; �vÞÞ < 2:

In this case, the locally asymptotically stable equilibrium ð�u; �vÞ is also called a sink.

The following Lemma can be founded in [12].

Lemma 1.4. Let B ¼ ½a; b� � ½c; d� 2 R2. Assume that F ¼ ðf ; gÞ : B! B is a continuous function satisfying the following properties:

(i) f ðu;vÞ is non-increasing in the first variable and non-decreasing in the second variable and g is non-increasing in each of itsvariables for each ðu;vÞ 2 B;

(ii) If ðm;M; l; LÞ 2 B� B is a solution of the system

M ¼ f ðm; LÞ; m ¼ f ðM; lÞ;L ¼ gðm; lÞ; l ¼ gðM; LÞ:

�

then m ¼ M and l ¼ L.

Then the system (1.3) has a unique equilibrium ð�u; �vÞ and every solution of Eq. (1.3) with ðu0;v0Þ 2 B converges to the equilibriumð�u; �vÞ.

2. Period-two character and linearized stability

Theorem 2.1. Eq. (1.1) has no positive prime period-two solution.

Proof. Assume for the sake of contradiction that

. . . ;/;u;/;u; . . .

is a positive prime period-two solution of Eq. (1.1), then

L.-X. Hu, H.-M. Xia / Applied Mathematics and Computation 233 (2014) 377–382 379

/ ¼ aþuaþ /

; u ¼ bþ /bþu

;

from which it follows that

a/þ /2 ¼ aþu; buþu2 ¼ bþ /:

So yields

ða/þ /2Þðbþ /Þ ¼ ðaþuÞðbuþu2Þ

and

ð/�uÞ½/2 þ /uþu2 þ ðaþ bÞð/þuÞ þ ab� ¼ 0:

Thus / ¼ u, a contradiction.

The proof is complete. h

Clearly, Eq. (1.1) has the unique equilibrium �y ¼ 1. To determine the stability of the equilibrium �y, we set

un ¼ y2n�1; vn ¼ y2n for n ¼ 0;1;2; . . .

and write Eq. (1.1) in the following equivalent form

unþ1 ¼ aþvnaþun

;

vnþ1 ¼ aþabþbunþvnðaþunÞðbþvnÞ ;

(n ¼ 0;1;2; . . . : ð2:1Þ

Applying Theorem 2.1, we are sure that Eq. (2.1) has a unique equilibrium E� ¼ ð1;1Þ in the first quadrant.Consider the map T : ½0;1Þ2 ! ½0;1Þ2 associated to Eq. (2.1), i.e.

Tðu; vÞ ¼T1ðu;vÞT2ðu;vÞ

� �¼

aþvaþu

aþabþbuþvðaþuÞðbþvÞ

" #:

Calculating the partial derivatives of the function T1ðu;vÞ and T2ðu;vÞ shows that

@T1

@u¼ � aþ vðaþ uÞ2

;@T1

@v ¼1

aþ u;

@T2

@u¼ � aþ vðaþ uÞ2ðbþ vÞ

;@T2

@v ¼b� a� ab� bu

ðaþ uÞðbþ vÞ2:

Clearly, @T1@u ;

@T2@u < 0; @T1

@v > 0 for u;v > 0, and @T2@v P 0 for u 6 1� a� a=b; @T2

@v 6 0 for u P 1� a� a=b.

The Jacobian matrix of T evaluated at E� is

JTðE�Þ ¼

� 1aþ1

1aþ1

� 1ðaþ1Þðbþ1Þ � a

ðaþ1Þðbþ1Þ

" #

and its characteristic equation associated with E� is

k2 þ aþ bþ 1ðaþ 1Þðbþ 1Þ kþ

1ðaþ 1Þðbþ 1Þ ¼ 0:

Noticing that inequality j aþbþ1ðaþ1Þðbþ1Þ j < 1þ 1

ðaþ1Þðbþ1Þ < 2 holds and applying Lemma 1.3, the following statement is true.

Theorem 2.2. The equilibrium E� of Eq. (2.1) is locally asymptotically stable for all values a; b 2 ð0;1Þ.

Theorem 2.3. Eq. (2.1) has no positive prime period-two solution.

Proof. Assume for the sake of contradiction that

. . . ; ð/1;u1Þ; ð/2;u2Þ; ð/1;u1Þ; ð/2;u2Þ . . .

is a positive prime period-two solution of Eq. (2.1), then they satisfy the following systems

/1 ¼aþu2

aþ /2; /2 ¼

aþu1

aþ /1

380 L.-X. Hu, H.-M. Xia / Applied Mathematics and Computation 233 (2014) 377–382

and

u1 ¼bþ /1

bþu2; u2 ¼

bþ /2

bþu1:

Then

a/1 þ /1/2 ¼ aþu2; a/2 þ /1/2 ¼ aþu1

bu1 þu1u2 ¼ bþ /1; bu2 þu1u2 ¼ bþ /2:

So

að/1 � /2Þ ¼ u2 �u1; and bðu1 �u2Þ ¼ /1 � /2;

from which it follows that

ð/1 � /2Þðu1 �u2Þðabþ 1Þ ¼ 0:

Clearly, u1 ¼ u2 () /1 ¼ /2 and this will lead to a contradiction.The proof is complete. h

3. Global asymptotic stability

Let fðun;vnÞg be a solution of Eq. (2.1). Clearly, the second equation of system (2.1) implies that for n P 0,

vnþ1 ¼aþ abþ bun þ vn

ðaþ unÞðbþ vnÞ6

maxfaþ ab;b;1gminfab;a;bg

and

vnþ2 ¼bþ unþ2

bþ vnþ1P

bbþ vnþ1

:

Set

Sv ¼maxfaþ ab; b;1g

minfab;a;bg ; Iv ¼b

bþ Sv: ð3:1Þ

Then Iv 6 vn 6 Sv for n P 2.On the other hand, for n P 2, the first equation of system (2.1) implies that

unþ1 ¼aþ vn

aþ un6

aþ vn

a6

aþ Sv

a¼ 1þ 1

aSv

and

unþ2 ¼aþ vnþ1

aþ unþ1P

aþ b=ðbþ SvÞaþ 1þ ð1=aÞSv

¼ abþ a2bþ a2Sv

ða2 þ aþ SvÞðbþ SvÞ:

Set

Iu ¼abþ a2bþ a2Sv

ða2 þ aþ SvÞðbþ SvÞ; Su ¼ 1þ 1

aSv : ð3:2Þ

Then Iu 6 un 6 Su for n P 4.Especially, if a < b 6 aþ ab, then 1� a� a=b 6 0 and un P 1� a� a=b hold for n P 0. Using the monotonic character

that the function T2 is decreasing in u, we can get that for n P 0,

vnþ1 ¼aþ abþ bun þ vn

ðaþ unÞðbþ vnÞ6

aþ abþ bð1� a� a=bÞ þ vn

ðaþ 1� a� a=bÞðbþ vnÞ¼ b

b� a:

If b > aþ ab, then 1� a� a=b > 0 holds. Similarly, applying the monotonic character of the function T2, we get, for someN, if uN P 1� a� a=b, then vNþ1 6

bb�a, and if uN 6 1� a� a=b, then vNþ1 6

bb�a.

So in the case where b > a yields

vn 6b

b� a

and

vnþ1 ¼bþ unþ1

bþ vnP

bbþ b=ðb� aÞ ¼

b� ab� aþ 1

;

L.-X. Hu, H.-M. Xia / Applied Mathematics and Computation 233 (2014) 377–382 381

unþ1 ¼aþ vn

aþ un6

aþ vn

a6

aþ b=ðb� aÞa

¼ aðb� aÞ þ baðb� aÞ

for n P 1. Moreover, we get

unþ2 ¼aþ vnþ1

aþ unþ1P

aþ ðb� aÞ=ðb� aþ 1Þaþ ðabþ b� a2Þ=ðab� a2Þ ¼

aðab� a2 þ bÞðb� aÞða2 þ aÞðb� aÞ2 þ ða2 þ aþ bÞðb� aÞ þ b

>aðb� aÞ

ðaþ 1Þðb� aÞ þ 1:

So we have the following conclusion.

Theorem 3.1. Every solution fðun;vnÞg of Eq. (2.1) eventually enters an invariable bounded rectangle B� ¼ ½Iu; Su� � ½Iv ; Sv �, whereIv ; Sv and Iu; Su are defined by (3.1) and (3.2). Further, if b > a, then

B ¼ aðb� aÞðaþ 1Þðb� aÞ þ 1

;aðb� aÞ þ b

aðb� aÞ

� �� b� a

b� aþ 1;

bb� a

� �ð3:3Þ

is also an invariable rectangle of Eq. (2.1).Now we turn to the investigation of the global asymptotic stability of the equilibrium point E�.

Theorem 3.2. Assume that ab P 1 and b < aðbþ 1Þ þ 12 a2ðbþ 1Þ. Then the unique positive equilibrium E� of Eq. (2.1) is globally

asymptotically stable. The global stable manifold is given by WsðE�Þ ¼ fðu;vÞ : u P 0;v P 0g.

Proof. Let fðun;vnÞg be a solution of Eq. (2.1). In view of Theorems (2.2), it is sufficient to show that every nonnegative solu-tion converges to E�. In order to achieve our aim, we divide the proof of the theorem into the following two cases.

Case (i) b 6 aðbþ 1Þ.The condition that b 6 aðbþ 1Þ implies that 1� a� a=b 6 0, and thus un P 1� a� a=b for n P 0. From Theorem 3.1,

every solution fðun;vnÞgwith initial point ðu0;v0Þ 2 fðu;vÞ : u P 0;v P 0g eventually enters an invariant bounded rectangleB� ¼ ½Iu; Su� � ½Iv ; Sv �. In this rectangle B�, the function f is non-increasing in the first argument and non-decreasing in thesecond, and the function g is non-increasing in each of its arguments. Let ðm;M; l; LÞ be a solution of the following system.

M ¼ aþ Laþm

; m ¼ aþ laþM

ð3:4Þ

and

L ¼ aþ abþ bmþ lðaþmÞðbþ lÞ ; l ¼ aþ abþ bM þ L

ðaþMÞðbþ LÞ : ð3:5Þ

Then in view of Lemma 1.4, to establish attractivity, we must show that m ¼ M and l ¼ L.Eq. (3.4) implies that

L ¼ ðaþmÞM � a; and l ¼ ðaþMÞm� a ð3:6Þ

and

L� l ¼ aðM �mÞ: ð3:7Þ

Eq. (3.5) implies that

ðbþ lÞL ¼ bþ aþ laþm

; and ðbþ LÞl ¼ bþ aþ LaþM

; ð3:8Þ

from which it follows that

bðL� lÞ ¼ aþ laþm

� aþ LaþM

: ð3:9Þ

Substituting expressions (3.6) and (3.7) into Eq. (3.9) leads to the following equation

ðM �mÞ½a2 þ a3bþ a2bM þ a2bmþ ðab� 1ÞmM� ¼ 0:

Notice that ab P 1, thus m ¼ M and l ¼ L. In view of Lemma 1.4, we get that the equilibrium E� of Eq. (2.1) is a global attrac-tor. Further,

WsðE�Þ ¼ fðu; vÞ : u P 0; v P 0g:

Case (ii) aðbþ 1Þ < b < aðbþ 1Þ þ 12 a2ðbþ 1Þ.

From Theorem 3.1, we get that, under the condition that b > aðbþ 1Þ, every nonnegative solution fðun;vnÞg of Eq. (2.1)eventually enters an invariant bounded rectangle B which is defined by (3.3). Thus there exists a positive integer N such thatun P aðb�aÞ

ðaþ1Þðb�aÞþ1 for n > N.

382 L.-X. Hu, H.-M. Xia / Applied Mathematics and Computation 233 (2014) 377–382

Notice that b > aþ ab implies that b� a > ab P 1 , and b < aðbþ 1Þ þ 12 a2ðbþ 1Þ implies that aþ a

b ¼ ab ðbþ 1Þ > 2

aþ2,thus

un Paðb� aÞ

ðaþ 1Þðb� aÞ þ 1P

aaþ 2

¼ 1� 2aþ 2

> 1� a� ab; for n > N:

Therefore, in the invariant rectangle B, the function f is non-increasing in the first argument and non-decreasing in thesecond, and the function g is non-increasing in each of its arguments. Applying Lemma 1.4, then as the same argument inthe case (i), one can show that the result follows when ab P 1.

The proof is complete. h

As far as Eq. (1.1) is concerned, the statement in Theorem 3.2 can be described by the following Theorem.

Theorem 3.3. Assume that ab P 1 and b < aðbþ 1Þ þ 12 a2ðbþ 1Þ. Then the unique positive equilibrium �y ¼ 1 of Eq. (1.1) is

globally asymptotically stable.

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