Necessary and sufficient conditions for boundedness and stability of N-order difference equation

Applied Mathematics and Computation 141 (2003) 427–445 www.elsevier.com/locate/amc

Necessary and sufficient conditions for boundedness and stability of N-order difference equation M.M. El-Afifi Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt

Abstract We investigate the boundedness, persistence and asymptotic behaviour of the positive solutions of the equation xkþ1 ¼

A1 A2 An ; p1 þ p2 þ    þ pn xk xk1 xknþ1

where A1 , A2 , A3 , pi , i ¼ 1; 2; . . . ; n 2 ð0; 1Þ and Ai , i ¼ 4; 5; . . . ; n 2 ½0; 1Þ. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Stability; Boundedness; Persistence

1. Introduction The purpose of this paper is to investigate boundedness, persistence and the asymptotic behaviour of the positive solution of the equation xkþ1 ¼

A1 A2 An þ 2 þ    þ pn ; xpk1 xpk1 xknþ1

ð1:1Þ

where xi , i ¼ 0; 1; 2; . . . ; n  1, A1 , A2 , A3 , pi , i ¼ 1; 2; . . . ; n 2 ð0; 1Þ and Ai , i ¼ 4; 5; . . . ; n 2 ½0; 1Þ. The special case where Ai ¼ 0, i ¼ 4; 5; . . . ; n and pi ¼ 1, i ¼ 1; 2; 3 was investigated in [5] where it was shown that the unique positive equilibrium of Eq. (1.1) is globally asymptotically stable. 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00168-6

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

Also the special case where Ai ¼ 0, i ¼ 3; 5; . . . ; n and pi ¼ 1, i ¼ 1; 2 was investigated in [2] where it was shown that the unique positive equilibrium of Eq. (1.1) is globally asymptotically stable. First we give some results about Xkþ1 ¼

A1 1 A2 An1 þ r=2 þ r þ    þ r ; 2r xk xknþ1 xk1 xk2

ð1:2Þ

where r P 1; xi ; i ¼ 0; 1; 2; . . . ; n  1; A1 2 ð0; 1Þ Ai ; i ¼ 2; 3; . . . ; n  1 2 ½0; 1Þ:

and ð1:3Þ

For the behaviour of solutions of some related equations see [1,3,4]. Definition 1.1. We say that a solution fxk g of Eq. (1.1) is bounded and persists if there exists positive constants P and Q such that P 6 xk 6 Q

for k ¼ n þ 1; n þ 2; . . .

Definition 1.2. A positive semicycle of fxk g of Eq. (1.1) consists of ÔstringÕ of terms fxl ; xlþ1 ; . . . ; xm g, all greater than or equal to x, with l P  n þ 1 and m < 1 and such that either l ¼ n þ 1 or

l > n þ 1

and

xl1 < x

and either m ¼ 1

or

m<1

and

xmþ1 < x:

Definition 1.3. A negative semicycle of fxk g of Eq. (1.1) consists of ÔstringÕ of terms fxl ; xlþ1 ; . . . ; xm g all less than x, with l P  n þ 1 and m < 1 and such that either l ¼ n þ 1 or

l > n þ 1

and

xl1 P x

and either m ¼ 1

or

m<1

and

xmþ1 P x:

The first semicycle of a solution of Eq. (1.1) starts with the term xnþ1 and is positive if xnþ1 P x and is negative if x2 < x. A solution may have a finite or infinite number of semicycles. Consider the nonlinear difference equation xkþ1 ¼ F ðxk ; xk1 ; . . . ; xkm Þ;

k ¼ 0; 1; . . . ;

ð1:4Þ

where m is a positive integer and F is a continuous function with nonnegative values.

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

429

We will assume that x is an equilibrium point of Eq. (1.4), that is x ¼ F ðx; x; . . . ; xÞ: In general, in this paper, we are interested in scalar difference equation of the form Eq. (1.4), where the function F ðu0 ; u1 ; . . . ; um Þ has continuous partial derivatives. Then the linearized equation associated with Eq. (1.4) about an equilibrium point x is ykþ1 ¼

m X oF ðx; x; . . . ; xÞyki ; ou i i¼0

k ¼ 0; 1; . . .

ð1:5Þ

2. Boundedness and persistence Without loss of generality Eq. (1.1) may be written in the form A1 1 A2 An1 þ 2 þ p3 þ    þ pn : xpk1 xpk1 xk2 xknþ1

Xkþ1 ¼

ð2:1Þ

Theorem 2.1. Assume that xi , i ¼ 0; 1; 2; . . . ; n  1, A1 2 ð0; 1Þ, Ai ; i ¼ 2; 3; . . . ; n  1 2 ½0; 1Þ and

pi ; i ¼ 1; 2; . . . ; n 2 ð0; 1 :

ð2:2Þ

Then every solution of Eq. (2.1) is bounded and persists. Proof. Select m 6 1, such that xnþ1 ; xn ; . . . ; x0 2 ½m; M

where M ¼

1þ

Pn1 i¼1

Ai

m

:

If p ¼ maxfpi ; i ¼ 1; 2; . . . ; ng, then A1 1 A2 An1  Pn1 p1 þ  Pn1 p2 þ  Pn1 p3 þ    þ  Pn1 pn 1þ

i¼1

1þ

Ai

m

1þ

Ai

i¼1

m

i¼1

1þ

Ai

m

i¼1

Ai

m

A1 1 A2 An1 >  Pn1 p þ  Pn1 p þ  Pn1 p þ    þ  Pn1 p 1þ

i¼1

Ai

m

1þ

i¼1

Ai

m

1þ

i¼1

Ai

m

!1p Pn1 n1 X 1 þ i¼1 Ai p ¼  Pn1 p ¼ m 1 þ Ai > mp > m; 1þ

i¼1

m

Ai

i¼1

1þ

i¼1

Ai

m

ð2:3Þ

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

but x1 ¼

A1 1 An1 þ 2 þ    þ pn xp01 xp1 xnþ1

A1 1 A2 >  Pn1 p1 þ  Pn1 p2 þ  Pn1 p3 þ    1þ

i¼1

1þ

Ai

m

i¼1

Ai

1þ

m

i¼1

Ai

m

An1 þ  Pn1 pn : 1þ

i¼1

ð2:4Þ

Ai

m

Also, x1 ¼

Pn1 A1 1 An1 A1 1 An1 1 þ i¼1 Ai ¼ M: þ þ    þ 6 þ þ    þ 6 n 2 xp01 xp1 xpnþ1 m p1 m p2 mpn m ð2:5Þ

Then from (2.3)–(2.5) we have A1 1 m 6  Pn1 p1 þ  Pn1 p2 þ  1þ

i¼1

1þ

Ai

m

i¼1

Ai

m

1þ

A2 Pn1 p3 i¼1

Ai

m

An1 þ    þ  Pn1 pn 6 x1 6 M: 1þ

i¼1

Ai

m

Hence x1 2 ½m; M and the result follows by mathematical induction.



Remark 2.1. If x denotes the unique positive equilibrium of Eq. (1.2), then clearly, A1 1 A2 An1 1 x ¼ 2r þ r=2 þ r þ    þ r > r=2 ; x x x x x Pn1 which shows that 1 < x < 1 þ i¼1 Ai . So, n1 X A1 Ai : ð2:6Þ Pn1 < 1 < x < 1 þ 1 þ i¼1 Ai i¼1 Lemma 2.1. Consider Eq. (1.2) and assume that (1.3) holds. Then the following statements are true. (a) Every semicycle of a nontrivial solution of Eq. (1.2) contains at most n terms. (b) f for some N P 0, fsA1 xN 6  Pn1 2r ; 1 þ i¼1 Ai then, xN þ1 > x.

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

(c) If for some N P 0, 0

431

1

A1 B C xN 2 @  Pn1 2r ; xA; 1 þ i¼1 Ai 1 2r . then, xN þ1 >  PAn1

1þ

i¼1

Ai

Proof. (a) If for some N P 0, xN nþ1 ; . . . ; xN 1 ; xN P x with these are not equal, then xN þ1 ¼

A1 1 A2 An1 A1 1 A2 An1 þ r=2 þ r þ    þ r < 2r þ r=2 þ r þ    þ r    x x x2r x x x x xN 1 N N 2 N nþ1

¼ x: The case where xN nþ1 ; . . . ; xN 1 and xN < x is similar. (b) If for some N P 0, A1 xN 6  Pn1 2r ; 1 þ i¼1 Ai then xN þ1 ¼

A1 1 A2 An1 A1 þ r=2 þ r þ    þ r > 2r P 2r xN nþ1 xN xN xN 1 xN 2

A1 1  PAn1 2r

1þ



1þ

¼

Pn1 i¼1

Ai

4r2

A12r1

>1þ

n1 X

 ¼

!2r

i¼1

Ai

 ð2r1Þð2rþ1Þ P P 1 þ n1 A1 þ 1 þ n1 i¼1 Ai i¼2 Ai A12r1

Ai > x:

i¼1

(c) If for some N P 0, 0

1

A1 B C xN 2 @  Pn1 2r ; xA; 1 þ i¼1 Ai then xN þ1 ¼

A1 1 A2 An1 A1 A 1 þ r=2 þ r þ    þ r > 2r > 2r P  x x2r x x x xN 1 N N N 2 N nþ1

which completes the proof.



1þ

A1 Pn1 i¼1

Ai

2r

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

Remark 2.2 i(i) The lemma implies that, if for some N P 0, xN is the first term of negative semicycle such that A1 xN 6  Pn1 2r ; 1 þ i¼1 Ai then this negative semicycle has length one. (ii) If Ai ¼ 0, i ¼ 3; 4; . . . ; n  1, we have the same results as in [5]. Also, if Ai ¼ 0, i ¼ 2; 3; 4; . . . ; n  1 we have the same results as in [1]. Now consider the function 2 A1 26 f ðxÞ ¼ x2r 4    2r P n2 xr 1 þ i¼1 Ai þ An1 2

x3r =2 þ  r=2 Pn1;i6¼n2  xr 1 þ i¼1 Ai þ An2 2

A2 xr  r þ  P n1;i6¼n3 xr 1 þ i¼1 Ai þ An3 2

A3 xr  r þ  P n1;i6¼n4 xr 1 þ i¼1 Ai þ An4 2

2

An3 xr An2 x3r =2  r þ   P  r þ  þ   P n1;i6¼2 n1 xr 1 þ i¼1 Ai þ A2 xr=2 A þ 1 i i¼1 3 An1  r 7 þ  5; P n1 2r x 1 þ i¼2 Ai þ A1

ð2:7Þ

where x > 0, then clearly f is strictly decreasing in the interval ð0; 1Þ, and there exists a positive number k such that f ðkÞ ¼ 1 and ðf ðxÞ  1Þðx  kÞ < 0

for x 2 ð0; kÞ [ ðk; 1Þ:

ð2:8Þ

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

433

Lemma 2.2. Assume that (1.3) holds and set 8 > < A1 1 Ai m ¼ min k;  2r ;  2r ;  Pn1 r ; P P > n1 n1 : 1 þ i¼1 Ai 1 þ i¼1 Ai 1 þ i¼1 Ai 9 > = i ¼ 2; 3; . . . ; n  1 > ; and assume that fxk g is a solution of Eq. (1.2) such that xN < m for some k P 1: Then xN n ; xN nþ2 ; . . . ; xN 1 ; xN þ1 ; xN þ3 ; . . . ; xN þn 2 ðx; 1Þ 2 x2r N

and

< xN þnþ1 < x:

Proof. Clearly by Lemma 2.1(b) and (c), xN þ1 2 ðx; 1Þ: Also, xN þ2 ¼

A1 1 A2 An1 1 þ r=2 þ r þ    þ r > r=2 > " 2r xN þ1 xN xN 1 xN nþ2 xN

1  P1n1 2r 1þ

¼

1þ

n1 X

!r 2 Ai

>

i¼1

1þ

n1 X

#r=2

i¼1

Ai

! Ai

> x

i¼1

and xN þ3 ¼

A1 1 A2 An1 A2 þ r=2 þ r þ    þ r > r > 2r xN nþ3 xN xN þ2 xN þ1 xN

A2 2  PAn1 r

1þ



1þ

¼

>

Pn1 i¼1

Ai

r 2

A2r1 1þ

n1 X

Ai

2 3 Pn1 r2 1 ! n1 1 þ A X i i¼1 6 7 ¼4 Ai 5 1þ Ar1 2 i¼1

! Ai

i¼1

!r

> x:

i¼1

Also, by same way xi > x, i ¼ N þ 4; N þ 5; . . . ; N þ n.

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

Now we observe that xN ¼

A1 1 A2 An1 A1 þ r=2 þ r þ    þ r > 2r ; 2r xN 1 xN 2 xN 3 xN n xN 1

which gives 31=2r

2 xN 1

A1 > xN

!1=2r

6 6 >6 6 4

A1 A1

" Pn1 #2r 1þ

i¼1

7 7 !7 7 5

>1þ

n1 X

Ai > x:

i¼1

Ai

Also, by same way we have xN 2 ; xN 3 ; . . . ; xN n > x: We observe that n1 X A1 1 A2 An1 A1 þ þ þ    þ < þ 1 þ Ai r r=2 x2r xrN nþ1 x2r xN 1 xN 2 N N i¼2  Pn1  A1 þ 1 þ i¼2 Ai x2r N ¼ ; 2r xN

xN þ1 ¼

n1 X A1 1 A2 An1 1 þ þ þ    þ < þ Ai r=2 xrN 1 xrN nþ2 xr=2 x2r xN N þ1 i¼1 N P  r=2 n1 1þ A i¼1 i xN ¼ ; r=2 xN

xN þ2 ¼

n1;i6 X¼2 A1 1 A2 An1 A2 þ r=2 þ r þ    þ r < r þ1þ Ai 2r xN nþ3 xN xN þ2 xN þ1 xN i¼1  Pn1;i6¼2  r A2 þ 1 þ i¼1 Ai x N ¼ : r xN

xN þ3 ¼

By same way we have   Pn1;i6¼3  r Pn2  A3 þ 1 þ i¼1 Ai x N An1 þ 1 þ i¼1 Ai xrN xN þ4 < ; . . . ; xN þn < xrN xrN

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

435

and so xN þnþ1 ¼

A1 1 A2 An1 þ r=2 þ r þ  þ r x2r x xN þ1 xN þn1 N þn N þn2 r2 =2

2

>h

A1 x2r xN N   Pn2  r i2r þ h Pn1;i6¼n2  r ir=2 An1 þ 1 þ i¼1 Ai xN An2 þ 1 þ i¼1 Ai x N 2

A2 xrN  þh Pn1;i6¼n3  r ir An3 þ 1 þ i¼1 Ai x N r2 =2

2

þ  þ h

An3 xrN An2 xN  P  ir Pn1;i6¼2  r ir þ h n1 r=2 A2 þ 1 þ i¼1 Ai x N 1þ A xN i i¼1 2

A x2r N  n1 þh Pn1  2r ir A1 þ 1 þ i¼2 Ai xN 2

2

2r ¼ x2r N f ðxN Þ > xN :

Since, xN < m < k and f ðxÞ is decreasing function in ð0; 1Þ. Finally, from Lemma 1.1(a) and in view of the fact that xN þ1 ; xN þ2 ; xN þ3 ; . . . ; xN þn 2 ðx; 1Þ it follows that xN þnþ1 < x, which completes the proof.



Remark 2.3. If Ai ¼ 0, i ¼ 3; 4; . . . ; n  1, n ¼ 3, r ¼ 1, we have the same results as in [5]. Also, if Ai ¼ 0, i ¼ 2; 4; . . . ; n  1, n ¼ 2, r ¼ 1, the function f ðxÞ defined by Eq. (2.7) which can be written in the form 2 2

A 1 xr 26 f ðxÞ ¼ xr 4    2r P xr 1 þ n2 A þ A i n1 i¼1 2

xr =2 þ  r=2 Pn1;i6¼n2  xr 1 þ i¼1 Ai þ An2

436

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

A2 r þ  Pn1;i6¼n3  r x 1 þ i¼1 Ai þ An3 A3  r þ  P n1;i6 ¼ xr 1 þ i¼1 n4 Ai þ An4 2

An3 An2 xr =2  r þ   P  r þ  þ   P n1;i6¼2 n1 xr 1 þ i¼1 A i þ A2 xr=2 A þ 1 i i¼1 3 2

An1 xr  r 7 þ  5; P n1 2r x 1 þ i¼2 Ai þ A1 we have the following result 2

xrN < xN þðnþ1Þ < x and we have the same results as in [1]. Theorem 2.2. Assume that (1.3) holds. Then every solution of Eq. (1.2) is bounded and persists. Proof. Let m be defined as in Lemma 2.2, and set A1 1 A2 An1 þ þ þ  þ r and m2r & mr=2 mr m ' A1 1 A2 An1 E ¼ min m; 2r þ r=2 þ r þ    þ r : M M M M

M¼

Let fxk g be a solution of Eq. (1.2). Clearly if l is a lower bound of fxk g, then L¼

A1 1 A2 An1 þ r=2 þ r þ    þ r 2r l l l l

is an upper bound. Thus it suffices to show that fxk g is bounded from below. Now either E is a lower bound of fxk g or xN < E for some N P 0. We complete the proof by showing that xN is a lower bound of fxk g for k P N . Otherwise, there exists terms of the solution fxk g where k > N which are less than xN . Let xqþ1 where q > N be the first term such that xqþ1 < xN . Then it can be seen that xqnþ1 ; . . . ; xq2 ; xq1 ; xk 2 ðm; MÞ:

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

437

Thus xN > xqþ1 ¼ >

A1 1 A2 An1 þ r=2 þ þ  þ r x2r x x q2 x qnþ1 q q1

A1 1 A2 An1 þ r=2 þ r þ    þ r > E > xN 2r M M M M

and this contradiction completes the proof.



3. Existence of solutions which are not bounded and do not persist Theorem 3.1. Assume that pi pj > 1; 8i; j and & & i ¼ 1;2;.. .;ðn þ 1Þ=2; j ¼ n;n  1;... ;ðn þ 1Þ=2 if n odd; a ¼ max pi pj ; i ¼ 1;2;.. .;n=2; j ¼ n;n  1;... ;n=2 if n even: Then there exist solutions of Eq. (2.1) which are not bounded and do not persist. Proof. Let the initial condition x0 be chosen in such a way that " x0 <

Ap1n þ1

Qn2

Ap1n

Qn1

pni i¼2 Ai

i¼2

Api ni Qn1;i6¼n2

Api ni #1=ða1Þ Q pni Ap1n n1 i¼2 Ai ; þ pn Pn3 Qn1;i6¼nðtþ1Þ pni Q Q pni pni A1 Ai þ An2 Ap1n n1 þ An1 n1 t¼2 At i¼2 i¼2 Ai i¼2 Ai þ

Ap1n

i¼2

ð3:1Þ then we have x1 ¼

A1 1 A2 An1 A1 þ 2 þ p3 þ    þ pn > p1 ; xp01 xp1 x2 xnþ1 x0

x2 ¼

A1 1 A2 An1 1 þ þ 3 þ    þ pn > p2 ; xp11 xp02 xp1 xnþ2 x0

x3 ¼

A1 1 A2 An1 A2 > p3 ; p1 þ p2 þ p3 þ    þ pn x1 x2 x0 xnþ3 x0

x4 ¼

A1 1 A2 An1 A3 An1 > p4 ; . . . ; x n > pn ; p1 þ p2 þ p3 þ    þ pn x2 x3 x1 xnþ4 x0 x0

438

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

A1 1 A2 An1 þ 2 þ p3 þ    þ pn xpn1 xpn1 xn2 x1 A1 1 A A p 1 þ  p2 þ  2 p3 þ    þ  n3 pn2 <

xnþ1 ¼

An1 p x0n

þ

An2 p x0n1

An3 p x0n2

A2 p x03

An2 A pn1 þ  n1  pn

1 p x02

    1 A2 p3 pn2 þ xp02 pn1 þ x 0 3 A p2 Apn3   n2   An3 An1 pn2 p3 pn1 p2 pn p1 þ x0 ðAn2 Þ þ x0 þ    þ x0 Ap2n2 Ap1n         A1 1 A2 An3 < xa0 þ þ þ    þ 1 2 3 Apn2 Apn1 Apn3 Ap2n2   An1 þ ðAn2 Þ þ Ap1n " Q Q n2 pni ¼n2 pni pn þ1 þ Ap1n n1;i6 Ai a A1 i¼2 Ai i¼2 ¼ x0 Q n1 pni pn A1 i¼2 Ai Qn1;i6¼nðtþ1Þ pni Q Qn1 pni # pn Pn3 pni A1 Ai þ An2 Ap1n n1 þ An1 i¼2 Ai t¼2 At i¼2 i¼2 Ai : þ pn Qn1 pni A1 i¼2 Ai ¼ xp01 pn



A1 1 Apn1



A1 p x01

From (3.1), we have xnþ1 < x0 : Thus by mathematical induction we have xðnþ1Þkþðnþ1Þ < xðnþ1Þk for k P 0, and so lim xðnþ1Þk ¼ L P 0:

k!1

But it can be seen that 0 < xðnþ1Þkþðnþ1Þ < xðnþ1Þk <    < xnþ1 < x0 < 1; then lim xðnþ1Þk ¼ 0:

k!1

Thus xðnþ1Þkþ1 ¼

A1 1 A2 An1 A1 þ p2 þ 3 þ pn > 1 1 xðnþ1Þk1 xpðnþ1Þk2 xpðnþ1Þk xðnþ1Þknþ1 xpðnþ1Þk

which show that

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

439

lim xðnþ1Þkþ1 ¼ 1:

k!1

Also, by same way we have lim xðnþ1Þkþ2 ¼ lim xðnþ1Þkþ3 ¼    ¼ lim xðnþ1Þkþn ¼ 1:

k!1

k!1

k!1

Then there exist solution which is unbounded and not persists. Theorem 3.2. Assume that pffiffiffi pffiffiffi Ai P 1; pi > 2 i ¼ 1; 2; . . . ; n  1; pn > 2 and



xnþ1 ; . . . ; x1 ; x0

2 ð0; 1Þ: Then there exists solutions of Eq. (2.1) which are not bounded and do not persist. Proof. Choose p

xnþ1 ; . . . ; x1 ; x0 > max fðnMÞ i ; i ¼ 1; 2; . . . ; ng; where M ¼ maxfAi ; i ¼ 1; 2; . . . ; n  1g. Computing x1 we have A1 1 A2 An1 þ 2 þ p3 þ    þ pn xp01 xp1 x2 xnþ1 Qn;i6¼2 i Qn Qn;i6¼3 i Qn1 i i A1 i¼2 xp1i þ i¼1 xp1i þ A2 i¼1 xp1i þ    þ An1 i¼1 xp1i Qn pi ¼ : i¼1 x1i

x1 ¼

Now A1 1 A2 An1 A1 > p1 p1 þ p2 þ p3 þ    þ pn x0 x1 x1 xnþ2 x1 " Qn pi # p1 A1 i¼1 x1i ip1 : ¼h Q Qn;i6¼2 pi Qn;i6¼3 pi Q n pi pi A1 i¼2 x1i þ i¼1 x1i þ A2 i¼1 x1i þ    þ An1 n1 i¼1 x1i

x2 ¼

Computing we find " Qn x2 > M p1 Qn

hQ

n i¼2

i xp1i þ

Qn;i6¼2 i¼1

# i p1 xp1i Qn;i6¼3

i¼1

i xp1i þ

i¼1

i xp1i þ  þ

i xp1i greater than any one of the rest, then " Qn pi p1 # p2 x01 i¼1 x1i # x2 > p : p " Qn pi p1 ¼ ðnMÞ 1 ðnMÞ 1 i¼2 x1i Qn;i6¼2 i i(ii) If i¼1 xp1i greater than any one of the rest, then

ii(i) If

i¼2

Qn1 i¼1

i xp1i

i p1 :

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

" Qn pi p1 # 1 p2 x1i xp1 hi¼1 i x2 > ¼ p : Qn;i6¼2 pi p1 p ðnMÞ 1 ðnMÞ 1 i¼1 x1i (iii) If

Qn1 i¼1

x2 >

i xp1i greater than any one of the rest, then

n p1 xp1n : ðnMÞp1

In general we have p2

n p1 1 p2 1 p3 one of fx01 ; xp1 ; xp2 ; . . . xp1n g : x2 > p1 ðnMÞ

In a similar manner we can find A1 1 A2 An1 1 þ þ þ    þ pn > p2 xp21 xp12 xp03 xnþ3 x1 " Q n pi p2 # i¼1 x1i hQ > Q Qn;i6¼3 pi Qn1 pi ip2 n n;i6 ¼ 2 pi pi M p2 i¼2 x1i þ i¼1 x1i þ i¼1 x1i þ    þ i¼1 x1i

x3 ¼

i.e. p2

n p2 2 p3 2 one of fxp01 p2 ; x1 ; xp2 ; . . . ; xp1n g : x3 > p2 ðnMÞ

Also, we can write p2

xnþ1 >

n p2 n p3 n one of fxp0n p1 ; xp1 ; xp2 ; . . . ; x1n g : pn ðnMÞ

Now xnþ1 ; xn ; . . . ; x1 ; x0 > maxfðnMÞp1 ; ðnMÞp2 ; . . . ; ðnMÞpn g; then p2 1

p1 p2 1 p1 p3 1 pn p1 1 x2 > minfx01 ; x1 ; x2 ; . . . ; x1n g; .. . p2 1

pn p2 1 pn p3 1 n xnþ1 > minfxp0n p1 1 ; x1 ; x2 ; . . . ; x1n g:

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

441

Let a ¼ minfxnþ1 ; xn ; . . . ; x1 ; x0 g > 1 and b ¼ minfp12  1; p1 p2  1; . . . ; p1 pn  1; p22  1; . . . ; p2 pn  1; . . . ; pn p1  1; . . . ; pn2  1g > 1; then x2 ; x3 ; . . . ; xnþ1 > ab > a: Continuing in the same manner we find p2 1

1 p3 1 1 xnþ3 > minfxnþ1 ; xnp1 p2 1 ; xpn1 ; . . . ; x2pn p1 1 g; .. . p2 1 p1 pn 1 pn p2 1 pn p3 1 x2nþ2 > minfxnþ1 ; xn ; xn1 ; . . . ; x2n g:

Thus xnþ3 ; xnþ4 ; . . . ; x2nþ2 > minfxbnþ1 ; . . . ; xb2 g: 2

Since x2 ; x3 ; . . . ; xnþ1 > ab , then xnþ3 ; . . . ; x2nþ2 > ab . By mathematical induction we obtain k

xðnþ1Þk ; xðnþ1Þk1 ; . . . ; xðnþ1Þkðn1Þ > ab : Now, a > 1 and b > 1 imply that lim xðnþ1Þk ¼ lim xðnþ1Þk1 ¼    ¼ lim xðnþ1Þkðn1Þ ¼ 1

k!1

k!1

k!1

and so lim xðnþ1Þkþ1 ¼ 0;

k!1

which completes the proof.



4. Global stability We will now show that when (2.2) holds the unique positive equilibrium x of Eq. (2.1) is globally asymptotically stable. We first observe that the linearized equation of Eq. (2.1) is p 1 A1 p2 p3 A3 pn An1 zk þ p þ1 zk1 þ p þ1 zk2 þ    þ pn þ1 zknþ1 p þ1 1 2 3 x x x x ¼ 0; for k ¼ 0; 1; . . .

zkþ1 þ

We shall need the following theorem Theorem A (Kocic and Ladas, 1993 [4]). Assume that

ð4:1Þ

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

pi 2 R; i ¼ 1; 2; . . . ; l and

l 2 f0; 1; . . .g;

then l X

jpi j < 1

i¼1

is a sufficient condition for asymptotic stability of the difference equation xkþl þ p1 xkþl1 þ    þ pl xk ¼ 0;

k ¼ 0; 1; . . .

Theorem 4.1. Assume (2.2) holds. Then the unique positive equilibrium x of Eq. (2.1) is locally asymptotically stable. Proof. Since x ¼

A1 1 A2 An1 þ þ þ    þ pn ; xp1 xp2 xp3 x

1¼

A1 p x 1 þ1

then þ

1 xp2 þ1

þ

A2 p x 3 þ1

þ  þ

An1 : xpn þ1

Also, pi 2 ð0; 1 , i ¼ 1; 2; . . . ; n, then p1 A1 A1 < p þ1 ; p þ1 1 x x 1

p2 p x 2 þ1

<

p2 p x 2 þ1

;...;

pn An1 An1 < pn þ1 : xpn þ1 x

Thus p1 A1 p2 p3 A3 pn An1 A1 1 A2 An1 þ p þ1 þ p þ1 þ    þ pn þ1 < p þ1 þ p þ1 þ p þ1 þ    þ pn þ1 p þ1 1 2 3 1 2 3 x x x x x x x x ¼ 1; which satisfies the conditions of the asymptotically stable for the equilibrium point x in Theorem A and Theorem 1.3.5 in [4], which completes the proof.  Theorem 4.2. Assume that (2.2) holds. Then the unique positive equilibrium of Eq. (2.1) is a global attractor of all positive solutions of the equation. Proof. (a) Suppose that pi 6 1, i ¼ 1; 2; . . . ; n and solution of Eq. (2.1). Now S ¼ lim sup xk k!1

and

I ¼ lim inf xk k!1

exist since every solution is bounded and persist.

Pn

i¼1

pi 6¼ n. Let fxk g be a

M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

443

Clearly S6

A1 1 A2 An1 þ p þ p þ    þ pn p 1 2 3 I I I I

and

IP

A1 1 A2 An1 þ p þ p þ    þ pn : p 1 2 3 S S S S ð4:2Þ

From (4.2) it follows that A1 S 1p1 þ S 1p2 þ A2 S 1p3 þ    þ An1 S 1pn 6 A1 I 1p1 þ I 1p2 þ    þ An1 I 1pn :

ð4:3Þ

Now gðxÞ ¼ A1 x1p1 þ x1p2 þ A2 x1p3 þ    þ An1 x1pn is an increasing function if pi 6 1, i ¼ 1; 2; . . . ; n and S ¼ I and

Pn

i¼1

pi 6¼ n. Therefore

lim xk ¼ x:

k!1

(b) pi ¼ 1, i ¼ 1; 2; . . . ; n. For another proof of this case. Assume, for the sake of contradiction, that fxk g is a solution of Eq. (2.1) which does not converge to Ps1 x ¼ ð i¼1 Ai þ 1Þ1=2 . Let S ¼ lim sup xk k!1

and

I ¼ lim inf xk : k!1

Ps1 Then, 0 < I < S, it can be seen that, IS ¼ i¼1 Ai þ 1 ¼ x2 . In particular this implies that I < x < S. Let fxki g be a subsequence of fxk g which converge to S. Then the subsequence fxkiþ1 g contains infinitely many terms of one of the following four intervals: ð0; I ; ðI; xÞ; ðx; SÞ; ½S; 1Þ:

ð4:4Þ

Hence there exists a subsequence fxki lg of fxki g such that lim fxkil g ¼ S

ð4:5Þ

k!1

and the entire sequence is contained in one of the four sets in (4.4). It suffices to show that each of these four cases leads to a contradiction. We will give the details when fxkil þ1 g is contained in the last two sets in (4.4). The other two cases are similar and will be omitted. First assume that, xkil þ1 2 ½S; 1Þ for k P 1. Then clearly lim xkil þ1 ¼ S; lim xkil þ2 ¼ S; . . . ; lim xkil þs1 ¼ S:

k!1

k!1

k!1

This together with (4.5) and (2.1) imply that Ps1 Ai þ 1 ¼ I; lim xkil þsþ1 ¼ u1 lim xkil þs ¼ i¼1 k!1 k!1 S

and

lim xkil þsþ2 ¼ u2 ;

k!1

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M.M. El-Afifi / Appl. Math. Comput. 141 (2003) 427–445

where A1 1 A2 As1 þ þ þ  þ 2 ðI; SÞ and S S I S A1 1 A2 As1 u2 ¼ þ þ þ    þ 2 ðI; SÞ: u1 I S S

u1 ¼

Thus for some sufficiently large index l0 , xkil0 þsþ1 ; xkil0 þsþ2 2 ðI; SÞ: Let (

Ps1

Ai þ 1 m ¼ min xkil0 þsþ1 ; xkil0 þsþ2 ; maxfxkil0 þsþ1 ; xkil0 þsþ2 g

)

i¼1

and note that Ps1 I
and

i¼1

Ai þ 1 < S: m

Now by Theorem 2.1 Ps1 Ai þ 1 m 6 xk 6 i¼1 m

for k P kil0 ;

which contradicts the definitions of both I and S. It remains to consider the case where (4.5) holds and xkil þ1 2 ðx; SÞ for l P 1: If fxkil þ1 g has a subsequence which converges to S, we complete the proof as in the previous case. So we assume there exists some r 2 ð0; SÞ and the index l0 such that xkil þ1 2 ðx; S  rÞ

for l P l0 :

Now choose d 2 ð0; 1Þ,  2 ð0; rÞ and an index l1 P l such that x < xkil1 < S þ d and

A1 1 A2 An1 þ þ þ  þ > I: S Sþd Sþd Sþd

Thus xkil1 þ2 >

A1 1 A2 An1 þ þ þ  þ >I S Sþd Sþd Sþd

xkil1 þ2 <

A1 1 A2 An1 þ þ þ  þ ¼ x x x x x

and

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445

that is xkil1 þ1 2 ðx; S  rÞ and

xkil1 þ2 2 ðI; xÞ:

A contradiction may now be derived in a way similar to that in the previous case. The proof is complete.  Combining Theorems 4.2 and 4.3 we have the following result. Theorem 4.3. The equilibrium x of Eq. (2.1) is globally asymptotically stable.

References [1] R. Devault, G. Ladas, S.W. Schultz, On the recursive sequence xnþ1 ¼ ðA=xpn Þ þ ðB=xqn1 Þ, Proceedings of the Second International Conference on Difference Equations, Veszprem, Hungary, 1995, Gordon and Breach Science Publishers, in press. [2] Ch.G. Philos, I.K. Purnaras, Y.G. Sficas, Global attractivity in a nonlinear difference equations, Applied Mathematics and Computers 62 (1994) 249–258. [3] R. Devault, V.L. Kocic, G. Ladas, Global stability of a recursive sequence, Dynamic Systems and Applications 1 (1992) 13–21. [4] V.L. Kocic, G. Ladas, Global Asymptotic Behaviour of Nonlinear Difference Equations of HigheKluwer Academic Publishers, Kluwer Academic Publishers, Dordrecht, 1993. [5] H.M. El-Owaidy, A.A. Ragab, M.M. El-Afifi, On the recursive sequence xnþ1 ¼ ðA=xpn Þ þ ðB=xqn1 Þ þ ðC=xsn2 Þ, Journal of Applied Mathematics and Computation 112 (2000) 277–290.