On the dynamics of the nonlinear rational difference ...

Journal of the Egyptian Mathematical Society (2015) xxx, xxx–xxx

Egyptian Mathematical Society

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ORIGINAL ARTICLE

On the dynamics of the nonlinear rational difference bxnk equation xnþ1 ¼ Axn þ Bxnk þ Cxnl þ dxnk exnl M.A. El-Moneam a b

a,*

, E.M.E. Zayed

b

Mathematics Department, Faculty of Science and Arts in Farasan, Jazan University, Saudi Arabia Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt

Received 11 June 2014; accepted 5 November 2014

KEYWORDS Difference equations; Prime period two solution; Boundedness character; Locally asymptotically stable; Global attractor; Global stability

Abstract In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation

xnþ1 ¼ Axn þ Bxnk þ Cxnl þ

bxnk ; dxnk exnl

n ¼ 0; 1; 2; . . .

where the coefficients A; B; C; b; d; e 2 ð0; 1Þ, while k and l are positive integers. The initial conditions xl ; . . . ; xk ; . . . ; x1 ; x0 are arbitrary positive real numbers such that k < l. Some numerical examples will be given to illustrate our results. 2010 MATHEMATICS SUBJECT CLASSIFICATION:

39A10; 39A11; 39A99; 34C99

ª 2015 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction The qualitative study of difference equations is a fertile research area and increasingly attracts many mathematicians. This topic draws its importance from the fact that many real life phenomena are modeled using difference equations. Examples from economy, biology, etc. can be found in [1–6]. It is known that nonlinear difference equations are capable * Corresponding author. E-mail addresses: [email protected] (M.A. El-Moneam), [email protected] (E.M.E. Zayed). Peer review under responsibility of Egyptian Mathematical Society.

of producing a complicated behavior regardless its order. This can be easily seen from the family xnþ1 ¼ gl ðxn Þ, l > 0, n P 0. This behavior is ranging according to the value of l, from the existence of a bounded number of periodic solutions to chaos. There has been a great interest in studying the global attractivity, the boundedness character and the periodicity nature of nonlinear difference equations. For example, in the articles [7–13] closely related global convergence results were obtained which can be applied to nonlinear difference equations in proving that every solution of these equations converges to a period two solution. For other closely related results (see [14–23]) and the references cited therein. The study of these equations is challenging and rewarding and is still in its infancy. We believe that the nonlinear rational difference equations are of paramount importance in their own right. Furthermore the results about such equations offer prototypes for the development of

Production and hosting by Elsevier http://dx.doi.org/10.1016/j.joems.2014.11.002 1110-256X ª 2015 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Please cite this article in press as: M.A. El-Moneam, E.M.E. Zayed, On the dynamics of the nonlinear rational nk xnþ1 ¼ Axn þ Bxnk þ Cxnl þ dxnkbxex , Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.11.002 nl

difference

equation

2

M.A. El-Moneam, E.M.E. Zayed

the basic theory of the global behavior of nonlinear difference equations. The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation xnþ1 ¼ Axn þ Bxnk þ Cxnl þ

bxnk ; dxnk exnl

n

¼ 0; 1; 2; . . .

ð1Þ

where the coefficients A; B; C; b; d; e 2 ð0; 1Þ, while k and l are positive integers. The initial conditions xl ; . . . ; xk ; . . . ; x1 ; x0 are arbitrary positive real numbers such that k < l. Note that the special cases of Eq. (1) have been studied discussed in [24] when B ¼ C ¼ 0, and k ¼ 0; l ¼ 1; b is replaced by b and in [25] when B ¼ C ¼ 0, and k ¼ 0; b is replaced by b and in [26] when A ¼ C ¼ 0; l ¼ 0; b is replaced by b and in [27] when B ¼ C ¼ 0; l ¼ 0. Our interest now is to study behavior of the solutions of Eq. (1) in its general form. For the related work (see [28–30]). Let us now recall some well known results [2] which will be useful in the sequel. Definition 1. Consider a difference equation in the form xnþ1 ¼ Fðxn ; xnk ; xnl Þ;

n ¼ 0; 1; 2; . . .

ð2Þ

where F is a continuous function, while k and l are positive integers such that k < l. An equilibrium point xe of this equation is a point that satisfies the condition xe ¼ Fð xe; xe; xeÞ. That is, the constant sequence fxn g with xn ¼ xe for all n P k P l is a solution of that equation. Definition 2. Let xe 2 ð0; 1Þ be an equilibrium point of Eq. (2). Then we have (i) An equilibrium point ex of Eq. (2) is called locally stable if for every e > 0 there exists d > 0 such that, if xl ; . . . ; xk ; . . . ; x1 ; x0 2 ð0; 1Þ with jxl e xj þ þjxk e x j þ þ jx1 ex j þ jx0 e x j < d, then jxn ex j < e for all n P k P l. (ii) An equilibrium point e x of Eq. (2) is called locally asymptotically stable if it is locally stable and there exists c > 0 such that, if xl ; . . . ; xk ; . . . ; x1 ; x0 2 ð0; 1Þ with xj x j þ jx0 e x j þ . . . þ jx1 e x j þ . . . þ jxk e jxl e < c, then lim xn ¼ xe:

n!1

x of Eq. (2) is called a global (iii) An equilibrium point e attractor if for every xl ; . . . ; xk ; . . . ; x1 ; x0 2 ð0; 1Þ we have lim xn ¼ xe:

to be periodic with prime period r if r is the smallest positive integer having this property. Definition 4. Eq. (2) is called permanent and bounded if there exists numbers m and M with 0 < m < M < 1 such that for any initial conditions xl ; . . . ; xk ; . . . ; x1 ; x0 2 ð0; 1Þ there exists a positive integer N which depends on these initial conditions such that m 6 xn 6 M for all n P N: Definition 5. The linearized equation of Eq. (2) about the equilibrium point xe is defined by the equation

znþ1 ¼ q0 zn þ q1 znk þ q2 znl ¼ 0;

ð3Þ

where q0 ¼

@Fð xe; xe; xeÞ ; @xn

q1 ¼

@Fð xe; xe; xeÞ ; @xnk

q2 ¼

@Fð xe; xe; xeÞ : @xnl

The characteristic equation associated with Eq. (3) is qðkÞ ¼ klþ1 q0 kl q1 klk q2 ¼ 0:

ð4Þ

Theorem 1 [2]. Assume that F is a C1 – function and let xe be an equilibrium point of Eq. (2). Then the following statements are true. (i) If all roots of Eq. (4) lie in the open unit disk jkj < 1, then the equilibrium point ex is locally asymptotically stable. (ii) If at least one root of Eq. (4) has absolute value greater than one, then the equilibrium point e x is unstable. (iii) If all roots of Eq. (4) have absolute value greater than one, then the equilibrium point e x is a source. Theorem 2 [3]. Assume that q0 ; q1 and q2 2 R. Then ð5Þ

jq0 j þ jq1 j þ jq2 j < 1;

is a sufficient condition for the asymptotic stability of Eq. (2). Theorem 3 [2]. Consider the difference Eq. (2). Let xe 2 I be an equilibrium point of Eq. (2). Suppose also that (i) F is a nondecreasing function in each of its arguments. (ii) The function F satisfies the negative feedback property ½Fðx; x; xÞ xðx xeÞ < 0 for all x 2 I f xeg;

where I is an open interval of real numbers. Then xe is global attractor for all solutions of Eq. (2).

2. The local stability of the solutions

n!1

(iv) An equilibrium point e x of Eq. (2) is called globally asymptotically stable if it is locally stable and a global attractor. x of Eq. (2) is called unstable if it is (v) An equilibrium point e not locally stable. Definition 3. A sequence fxn g1 n¼l is said to be periodic with period r if xnþr ¼ xn for all n P p. A sequence fxn g1 n¼l is said

The equilibrium point xe of Eq. (1) is the positive solution of the equation xe ¼ ðA þ B þ CÞ xe þ

b xe ; ðd eÞ xe

ð6Þ

where d – e. If ½ðA þ B þ CÞ 1ðe dÞ > 0, then the only positive equilibrium point xe of Eq. (1) is given by xe ¼

b : ½ðA þ B þ CÞ 1ðe dÞ

Please cite this article in press as: M.A. El-Moneam, E.M.E. Zayed, On the dynamics of the nonlinear rational nk xnþ1 ¼ Axn þ Bxnk þ Cxnl þ dxnkbxex , Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.11.002 nl

ð7Þ

difference

equation

Dynamics of the nonlinear rational difference equation Let us now introduce a continuous F : ð0; 1Þ3 !ð0; 1Þ which is defined by Fðu0 ; u1 ; u2 Þ ¼ Au0 þ Bu1 þ Cu2 þ

function

bu1 ; ðdu1 eu2 Þ

provided du1 – eu2 . Consequently, we get 8 @Fðe x ;e x ;e xÞ > ¼ A ¼ q0 ; > @u0 > > > > > < @Fðe x ;e x ;e xÞ Þ1 ¼ B e½ðAþBþC ¼ q1 ; @u1 ðedÞ > > > > > > > : @Fðex ;ex ;ex Þ ¼ C þ e½ðAþBþCÞ1 ¼ q ; @u2

ðedÞ

3 Proof. Following the proof of Theorem 5, we deduce that if k and l are both odd positive integers, then xnþ1 ¼ xnk ¼ xnl . It follows from Eq. (1) that

ð8Þ P ¼ AQ þ ðB þ CÞP

b ; ðe dÞ

ð16Þ

b : ðe dÞ

ð17Þ

and ð9Þ

Q ¼ AP þ ðB þ CÞQ

By subtracting (17) from (16), we get

2

ðP QÞ½A ðB þ CÞ þ 1 ¼ 0:

ð18Þ

where e – d. Thus, the linearized equation of Eq. (1) about xe takes the form znþ1 q0 zn q1 znk q1 znl ¼ 0; ð10Þ

Since A ðB þ CÞ þ 1 – 0, then P ¼ Q. This is a contradiction. Thus, the proof is now completed. h

where q0 ; q1 and q2 are given by (9).

Theorem 7. If k is even and l is odd positive integers, then Eq. (1) has prime period two solution if the condition

Theorem 4. Assume that e – d; A þ B þ C – 1 and

ð1 CÞð3e dÞ < ðe þ dÞðA þ BÞ;

jAðe dÞj þ jBðe dÞ e½ðA þ B þ CÞ 1j

is valid, provided C < 1 and eð1 CÞ dðA þ BÞ > 0.

þ jCðe dÞ þ e½ðA þ B þ CÞ 1j < je dj;

ð19Þ

ð11Þ

then the equilibrium point (7) of Eq. (1) is locally asymptotically stable.

Proof. If k is even and l is odd positive integers, then xn ¼ xnk and xnþ1 ¼ xnl . It follows from Eq. (1) that P ¼ ðA þ BÞQ þ CP

Proof. From (9) and (11) we deduce that jq0 j þ jq1 j þ jq2 j < 1, and hence the proof follows with the aid of Theorem 2. h

ð20Þ

bP : ðeQ dPÞ

ð21Þ

and Q ¼ ðA þ BÞP þ CQ

3. Periodic solutions

bQ ; ðeP dQÞ

Consequently, we get In this section, we study the existence of periodic solutions of Eq. (1). The following theorem states the necessary and sufficient conditions that the Eq. (1) has periodic solutions of prime period two.

eP2 dPQ ¼ eðA þ BÞPQ dðA þ BÞQ2 þ eCP2 CdPQ bQ;

ð22Þ

and Theorem 5. If k and l are both even positive integers, then Eq. (1) has no prime period two solution.

eQ2 dPQ ¼ eðA þ BÞPQ dðA þ BÞP2 þ eCQ2

Proof. Assume that there exist distinct positive solutions

By subtracting (23) from (22), we get ð12Þ

. . . ; P; Q; P; Q; . . .

of prime period two of Eq. (1). If k and l are both even positive integers, then xn ¼ xnk ¼ xnl . It follows from Eq. (1) that b ; ð13Þ P ¼ ðA þ B þ CÞQ ðe dÞ and Q ¼ ðA þ B þ CÞP

b : ðe dÞ

PþQ¼

ð23Þ

b ; ½eð1 CÞ dðA þ BÞ

ð24Þ

where eð1 CÞ dðA þ BÞ > 0. By adding (22) and (23), we obtain PQ ¼

eb2 ð1 CÞ ðe þ dÞ½ð1 CÞ þ ðA þ BÞ½eð1 CÞ dðA þ BÞ2

; ð25Þ

ð14Þ

where C < 1. Assume that P and Q are two positive distinct real roots of the quadratic equation

By subtracting (14) from (13), we get ðP QÞ½ðA þ B þ CÞ þ 1 ¼ 0:

CdPQ bP:

ð15Þ

Since A þ B þ C þ 1 – 0, then P ¼ Q. This is a contradiction. Thus, the proof is now completed. h Theorem 6. If k and l are both odd positive integers and A þ 1 – B þ C, then Eq. (1) has no prime period two solution.

t2 ðP þ QÞt þ PQ ¼ 0:

ð26Þ

Thus, we deduce that ðP þ QÞ2 > 4PQ:

ð27Þ

Substituting (24) and (25) into (17), we get the condition (19). Thus, the proof is now completed. h


difference

equation

4


Theorem 8. If k is odd and l is even positive integers, then Eq. (1) has prime period two solution if the condition ðA þ CÞð3e dÞ < ðe þ dÞð1 BÞ;

ð28Þ xNþ1 ¼ AxN þ BxNk þ CxNl þ

is valid, provided B < 1 and dð1 BÞ eðA þ CÞ > 0. Proof. If k is odd and l is even positive integers, then xnþ1 ¼ xnk and xn ¼ xnl . It follows from Eq. (1) that P ¼ ðA þ CÞQ þ BP

bP ; ðeQ dPÞ

ð29Þ

bQ : ðeP dQÞ

ð30Þ

b ; ½dð1 BÞ eðA þ CÞ

ð31Þ

bxNk : dxNk exNl

eb2 ðA þ CÞ ðe þ dÞ½ð1 BÞ þ ðA þ CÞ½dð1 BÞ eðA þ CÞ2

; ð32Þ

where B < 1. Substituting (31) and (32) into (27), we get the condition (28). Thus, the proof is now completed. h 4. Boundedness of the solutions

But, it is easy to see that dxNk exNl P b e, then for b – e, we get b : be

Theorem 9. Let fxn g be a solution of Eq. (1). Then the following statements are true: (i) Suppose b < d and for some N P 0; the initial conditions b xNlþ1 ; . . . ; xNkþ1 ; . . . ; xN1 ; xN 2 ; 1 ; d

b bxNk xNþ1 P ðA þ B þ CÞ þ : d dxNk exNl But, one can see that dxNk exNl 6 d we get

are valid, then for b – e and d – be, we have the inequality

6 ðA þ B þ C Þ þ

b ; ðb e Þ

ð33Þ

for all n P N. (ii) Suppose b > d and for some N P 0; the initial conditions b xNlþ1 ; . . . ; xNkþ1 ; . . . ; xN1 ; xN 2 1; ; d are valid, then for b – e and d2 – be, we have the inequality b b b2 ; 6 xn 6 ðA þ B þ CÞ þ 2 ðb eÞ d d be

for all n P N.

ð37Þ 2

be , d

then for d2 – be,

ð38Þ

From (36) and (38) we deduce for all n P N that the inequality (33) is valid. Hence, the proof of part (i) is completed. Similarly, if 1 6 xN 6 bd, then we can prove part (ii) which is omitted here for convenience. Thus, the proof is now completed. h

In this section we study the global asymptotic stability of the positive solutions of Eq. (1). Theorem 10. If 0 < A þ B þ C < 1 and e – d, then the equilibrium point xe given by (7) of Eq. (1) is global attractor. Proof. We consider the following function

Fðx; y; zÞ ¼ Ax þ By þ Cz þ

2

ðA þ B þ C Þ þ

ð36Þ

5. Global stability

In this section, we investigate the boundedness of the positive solutions of Eq. (1).

b b2 6 xn ðA þ B þ CÞ þ 2 d d be

ð35Þ

b b2 : xNþ1 P ðA þ B þ CÞ þ 2 d d be

where dð1 BÞ eðA þ CÞ > 0, PQ ¼

bxNk dxNk exNl

Similarly, we can show that

Consequently, we get PþQ¼

6AþBþCþ

xNþ1 6 A þ B þ C þ

and Q ¼ ðA þ CÞP þ BQ

Proof. First of all, if for some N P 0, bd 6 xN 6 1 and b – e, we have

ð34Þ

by ; ðdy ezÞ

ð39Þ

where dy – ez, provided that Bðdy ezÞ2 > ez and Cðdy ezÞ2 þ bey > 0. It is easy to verify the condition (i) of Theorem 3. Let us now verify the condition (ii) of Theorem 3 as follows: b x ½Fðx; x; xÞ xðx xeÞ ¼ ðA þ B þ CÞx ed b x ½ðA þ B þ CÞ 1ðe dÞ 2 xðe dÞ½ðA þ B þ CÞ 1 b ¼ ed 1 : ½ðA þ B þ CÞ 1 ð40Þ Since 0 < A þ B þ C < 1 and e – d, then we deduce from (40) that ½Fðx; x; xÞ xðx xeÞ < 0:


ð41Þ

difference

equation

Dynamics of the nonlinear rational difference equation

5

plot of y(n+1)=(A*y(n)+B*y(n−2)+C*y(n−4))+((b*y(n−2))/(d*y(n−2)−e*y(n−4)))

plot of y(n+1)=(A*y(n)+B*y(n−2)+C*y(n−4))+((b*y(n−2))/(d*y(n−2)−e*y(n−4))) 20

307

x 10

10.7105

18

solution of y(n+1)

solution of y(n+1)

16

5.3553

14 12 10 8 6 4

0

2 0

50

100

150

200

n−iteration

0

0

50

100

150

200

n−iteration n2 xnþ1 ¼ 300xn þ 200xn þ 200xn2 þ 100xn4 þ 30xn250xþ20x . n4

Fig. 1

Fig. 3

n2 xnþ1 ¼ 0:4xn þ 0:3xn2 þ 0:2xn4 þ xn25xþ2x . n4

plot of y(n+1)=(A*y(n)+B*y(n−1)+C*y(n−3))+((b*y(n−1))/(d*y(n−1)−e*y(n−3))) 306

14

Example 2. Fig. 2, shows that Eq. (1) has no prime period two solutions if k ¼ 1, l ¼ 3, x3 ¼ 1, x2 ¼ 2, x1 ¼ 3, x0 ¼ 4, A ¼ 100, B ¼ 50, C ¼ 25, b ¼ 5, d ¼ 3, e ¼ 2.

x 10

solution of y(n+1)

12

Example 3. Fig. 3, shows that Eq. (1) is globally asymptotically stable if x4 ¼ 1, x3 ¼ 2, x2 ¼ 3, x1 ¼ 4, x0 ¼ 5, A ¼ 0:4, B ¼ 0:3, C ¼ 0:2, b ¼ 5, d ¼ 1, e ¼ 2.

10 8 6

7. Conclusions

4 2 0

0

50

100

150

200

n−iteration

Fig. 2

5xn1 xnþ1 ¼ 100xn þ 50xn1 þ 25xn3 þ 3xn1 . þ2xn3

According to Theorem 3, xe is global attractor. Thus, the proof is now completed. h On combining the two Theorems 4 and 10, we have the result. Theorem 11. The equilibrium point xe given by (7) of Eq. (1) is globally asymptotically stable.

We have discussed some properties of the nonlinear rational difference Eq. (1), namely the periodicity, the boundedness and the global stability of the positive solutions of this equation. We gave some figures to illustrate the behavior of these solutions. Our results in this article can be considered as a more generalization than the results obtained in Refs. [24– 27]. Note that Example 1 verifies Theorem 5 which shows that if k and l are both even positive integers, then Eq. (1) has no prime period two solution. But Example 2 verifies Theorem 6 which shows that if k and l are both odd positive integers, then Eq. (1) has no prime period two solution, while Example 3 verifies Theorem 11 which shows that Eq. (1) is globally asymptotically stable. Acknowledgments The authors wish to thank the referees for their suggestions and comments.

6. Numerical examples References In order to illustrate the results of the previous section and to support our theoretical discussions, we consider some numerical examples in this section. These examples represent different types of qualitative behavior of solutions of Eq. (1). Example 1. Fig. 1, shows that Eq. (1) has no prime period two solutions if k ¼ 2, l ¼ 4, x4 ¼ 1, x3 ¼ 2, x2 ¼ 3; x1 ¼ 4, x0 ¼ 5, A ¼ 300, B ¼ 200, C ¼ 100, b ¼ 50, d ¼ 30, e ¼ 20.

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equation

6


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difference

equation