Journal of Difference Equations and Applications, Vol. 11, No. 2, February 2005, 133–150
Dynamics of a higher order nonlinear rational difference equation YOU-HUI SU†‡, WAN-TONG LI‡*§ and STEVO STEVIC´{ †Department of Mathematics, Hexi University, Zhangye, Gansu 734000, P.R. China ‡Department of Mathematics, Lanzhou University Lanzhou, Gansu, 730000, P.R. China {Mathematical Institute of Serbian Academy of Science, Knez Mihailova 35/I, 11000 Beograd, Serbia (Received 30 July 2004; in final form 27 September 2004) In this paper, we study the global attractivity, the invariant intervals, the periodic and oscillatory character of the difference equation a þ bxn ; n ¼ 0; 1; . . .; Axn þ Bxn2k where a, b, A, B are positive real numbers, k $ 1 is a positive integer, and the initial conditions x2k,. . .,x21,x0 are nonnegative real numbers such that x2k or x0 or both are positive real numbers. We show that the positive equilibrium of the difference equation is a global attractor. As a corollary, our main result confirms a conjecture proposed by Kulenovic´ et al. (2003) [The dynamics of xnþ1 ¼ ða þ bxn Þ=ðA þ Bxn þ Cxn21 Þ facts and conjectures, Computational Mathematics Applications, 45, 1087–1099]. xnþ1 ¼
Keywords: Difference equation; Invariant interval; Global attractor; Globally asymptotically stable; Oscillatory AMS 2000 Subject Classification: 39A10
1. Introduction and preliminaries In this paper, we consider the following rational difference equation xnþ1 ¼
a þ bxn ; Axn þ Bxn2k
n ¼ 0; 1; . . .;
ð1Þ
where a, b, A, B are positive real numbers, k $ 1 is a positive integer, and the initial conditions x2k ; . . .; x21 ; x0 are nonnegative real numbers such that x2k or x0 or both are positive real numbers. Clearly xn . 0
for
n $ 1:
In the sequel, we will only consider positive solutions of equation (1).
*Corresponding author. Email:
[email protected] §Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online q 2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/10236190512331319352
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Equation (1) has a unique positive equilibrium pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b þ b 2 þ 4aðA þ BÞ : x¼ 2ðA þ BÞ When k ¼ 1; equation (1) reduces to xnþ1 ¼
a þ bxn ; Axn þ Bxn21
n ¼ 0; 1; . . . :
ð2Þ
Clearly, equations (1) and (2) have the same positive equilibrium x: In [2], the authors investigate the local asymptotic stability and the global asymptotic stability of the positive equilibrium x under certain conditions. We summarize their results as follows: Theorem A [2] (i) The unique positive equilibrium x of equation (2) is locally asymptotically stable for all positive values of the parameters. (ii) The unique positive equilibrium x of equation (2) is globally asymptotically stable for all positive values of the parameters provided the condition is satisfied: Bb 2 # 4aA 2 þ Ab 2 :
ð3Þ
In 2003, Kulenovic´ et al. [7] considered equation (2) and among other things, offered the following conjecture. Conjecture A [7, Conjecture 2, p.1092] (2) is globally asymptotically stable.
The unique positive equilibrium x of equation
Furthermore, Kulenovic´ and Ladas [6] suggested the following open problem: Open Problem Assume that p; q [ ½0; 1Þ and k [ {2; 3; . . .}: Investigate the global behavior of all positive solutions of the difference equation ynþ1 ¼
yn þ p ; yn þ qyn2k
n ¼ 0; 1; . . . :
Motivated by the conjecture and the open problem, we investigate the global attractivity, the invariant intervals, the periodic and oscillatory character of all positive solutions of equation (1). Our results show that the positive equilibrium of equation (1) is a global attractor for all positive values of the parameters. As a corollary, we prove that the above conjecture is true. Other related results on rational difference equations can be found in refs. [1,3 – 6,8 –26]. For the sake of convenience, we now present some definitions and known lemmas which will be useful in the sequel. Consider the difference equation xnþ1 ¼ Fðxn ; xn21 ; . . .; xn2k Þ;
n ¼ 0; 1; . . .;
ð4Þ
where k $ 1 is a positive integer, and the function Fðu0 ; u1 ; . . .; uk Þ has continuous partial derivatives.
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A point x is called an equilibrium of equation (4) if x ¼ Fðx; x; . . .; xÞ: That is xn ¼ x for n $ 2 k is a solution of equation (4), or equivalently, x is a fixed point of F. A real interval J is called an invariant interval of equation (4), if x2k ; . . .; x0 [ J ) xn [ J for all n . 0; that is, every solution of equation (4) with initial conditions in J remains in J. The linearized equation associated with equation (4) about the equilibrium point x is xnþ1 ¼
k X ›F ðx; x; . . .; xÞ xn2i ; › ui i¼0
n ¼ 0; 1; . . . :
Its characteristic equation is
l kþ1 ¼
Theorem B
k X ›F ðx; x; . . .; xÞl k2i : › ui i¼0
[8] Assume that a; b [ R, and k [ N. Then jaj þ jbj , 1
ð5Þ
is a sufficient condition for the asymptotic stability of the difference equation xnþ1 2 axn þ bxn2k ¼ 0;
n ¼ 0; 1; . . . :
ð6Þ
The following three results will be useful in establishing the global attractivity of the equilibrium point x of equation (1). Theorem C
[3] Consider the difference equation ynþ1 ¼ f ð yn ; yn2k Þ;
n ¼ 0; 1; . . .;
ð7Þ
where k [ {1; 2; . . .}: Assume that f : ½a; b £ ½a; b ! ½a; b is a continuous function satisfying the following properties: (a) f (x,y) is nonincreasing in each of its arguments. (b) If (m,M) [ [a,b] £ [a,b] is a solution of the system f ðm; mÞ ¼ M and f ðM; MÞ ¼ m; then m ¼ M: Then equation (7) has a unique positive equilibrium y and every solution of equation (7) converges to y: Theorem D [8] Let F [ C½ð0; 1Þ; ð0; 1Þ be a nonincreasing function, and let x denote the unique fixed point of F. Then the following statements are equivalent: (i) x is the only fixed point of F 2 in (0,1); (ii) x is a global attractor of all positive solutions of the equation xnþ1 ¼ Fðxn Þ; with x0 [ ð0; 1Þ;
n ¼ 0; 1; 2; . . .;
ð8Þ
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(iii) If l and m are positive numbers such that FðmÞ # l # x # m # FðlÞ; then
l ¼ m ¼ x: Theorem E [8] Let F; H [ C½ð0; 1Þ; ð0; 1Þ be nonincreasing functions in (0,1), and let x [ ð0; 1Þ be such that FðxÞ ¼ HðxÞ ¼ x; and ½HðxÞ 2 FðxÞ ðx 2 xÞ # 0 for x $ 0: Assume that x is the only fixed point of H 2 in (0,1). Then x is also the only fixed point of F 2 in (0,1). Theorem F [8,9] Consider equation (8), where F is a decreasing function which maps some interval I into itself. Assume that F has negative Schwarzian derivative " # 00 0 F ðxÞ 1 F 00 ðxÞ2 SFðxÞ ¼ 2 F 0 ðxÞ 2 F 0 ðxÞ everywhere on I, except for points x*, where F 0 ðx*Þ ¼ 0: Then the positive equilibrium x of equation (8), is global attractor of all positive solutions of the equation.
2. Main results 2.1 Linearized stability and periodic character In this subsection, we consider the linearized stability and periodic character of the positive solutions of equation (1). The change of variables xn ¼ ðb=AÞyn reduces equation (1) to the difference equation ynþ1 ¼
yn þ p ; yn þ qyn2k
n ¼ 0; 1; . . .;
ð9Þ
where p ¼ aA=b 2 ; q ¼ B=A; and the initial conditions y2k ; . . .; y21 ; y0 are nonnegative real numbers such that y2k or y0 or both are positive real numbers. Clearly, the positive equilibrium of equation (9) is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 4pð1 þ qÞ : ð10Þ y¼ 2ð1 þ qÞ The linearized equation associated with equation (9) about y is Z nþ1 2
qy 2 p q Z n2k ¼ 0; Zn þ ð y þ pÞð1 þ qÞ 1þq
n ¼ 0; 1; . . .;
and its characteristic equation is z kþ1 2
qy 2 p q ¼ 0: zk þ ð y þ pÞð1 þ qÞ 1þq
From this, we have the following result.
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Theorem 2.1 The positive equilibrium y of equation (9) is locally asymptotically stable provided that one of the following three conditions is satisfied: (i) k . 1 and p $ q; (ii) k . 1 and p , q # 1; (iii) k . 1, p , q and 1 , q , 1 þ 4p. If p $ q, then in view of equation (10) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð1 þ 2qÞ2 1 þ 1 þ 4pð1 þ qÞ 1 þ ð1 þ 2pÞ2 1 þ p p # # : 1¼ #y¼ ¼ 2ð1 þ qÞ 1þq q 2ð1 þ qÞ 2ð1 þ qÞ
Proof
A From this and by Theorem B it follows (i). If p , q, then from equation (10) it follows that p # y # 1; q Since qy 2 p q yðq 2 1Þ 2 2p 21¼ þ , 0; ð y þ pÞð1 þ qÞ 1 þ q ð1 þ qÞð y þ pÞ if q # 1, then by Theorem B we see that (ii) holds. By some calculations it is easy to see that the condition q , 4p þ 1 is equivalent to ðq 2 1Þy , 2p:
ð11Þ
which implies (iii). In the following, we will consider the periodic character of equation (9). Theorem 2.2 Proof
Equation (9) has no positive solutions with prime period two.
Assume, for the sake of contradiction, that :; F; C; F; C; . . .
is a prime period two solution of equation (9). If k is odd, then ynþ1 ¼ yn2k and F and C satisfy the system qF2 þ FC ¼ C þ p
and qC2 þ FC ¼ F þ p;
Subtracting these two equations, we obtain ðF 2 C Þ ðqF þ qC þ 1Þ ¼ 0: Since qF þ qC þ 1 . 0, we have F ¼ C: This is a contradiction. If k is even, then yn ¼ yn2k and F and C satisfy the system qFC þ FC ¼ C þ p and which implies F ¼ C; a contradiction.
qCF þ FC ¼ F þ p: A
2.2 Oscillatory character In this subsection, we investigate the oscillatory character of equation (9). Before stating our results, we give the definitions of positive and negative semicycles of a solution of equation (9) relative to an equilibrium y:
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A positive semicycle of a solution {yn} of equation (9) consists of a ‘string’ of terms {yl, ylþ1,. . .,ym}, all greater than or equal to the equilibrium y; with l $ 2 k and m # 1 and such that either l ¼ 2k;
or l . 2k
either m ¼ 1;
or
and yl21 , y;
and m,1
and
ymþ1 , y:
A negative semicycle of a solution {yn} of equation (9) consists of a ‘string’ of terms {yl, ylþ1,. . .,ym}, all less than the equilibrium y; with 1 $ 2 k and m # 1 and such that either l ¼ 2k;
or l . 2k
and
yl21 $ y
either m ¼ 1;
or
m,1
and
ymþ1 $ y:
and
The first semicycle of a solution starts with the terms y2k and is positive if y2k $ y and is negative if y2k , y: A solution {yn }1 n¼2k of equation (9) is called oscillatory with respect to the point y; if for every n0 [ N there exists a n1 $ n0 such that ð yn1 2 yÞð yn1 þ1 2 yÞ # 0: The solution {yn }1 n¼2k is called nonoscillatory if it is not oscillatory. Theorem 2.3
The following statements are true.
(i) Let {yn }1 n¼2k be a solution of equation (9) such that for some n0 $ 0; either yn $ y for
n $ n0
ð12Þ
yn # y for
n $ n0 :
ð13Þ
or
Then for n $ n0 þ k, the sequence {yn} is monotonic and limn!1 yn ¼ y: (ii) Every nontrivial positive solution of equation (9) is oscillatory with respect to the positive equilibrium y: Furthermore, any semicycle of such solutions has at most 2k þ 1 terms. (iii) Let {yn }1 n¼2k be a solution of equation (9) which is oscillatory with respect to the positive equilibrium y; then the extreme point in any semicycle occurs in one of the first k þ 1 terms of the semicycle. Proof (i) Assume that equation (12) holds. The case when equation (13) holds is similar and will be omitted. Then for n $ n0 þ k, ynþ1 ¼
1 þ ypn 1 þ py yn þ p ¼ yn ; ¼ yn # yn yn þ qyn2k yn þ qyn2k y þ qy
ð14Þ
that is, {yn} is nonincreasing for n $ n0 þ k. Hence {yn} converges to the positive equilibrium y: (ii) Assume for the sake of contradiction that equation (9) has a nontrivial positive solution {yn} which is not oscillatory with respect to y: Then for some n0 $ 0, either equation (12) or equation (13) holds. Suppose that equation (12) holds (the case where equation (13) holds
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is similar and will be omitted ). In view of part (i), {yn} is nonincreasing for n $ n0 þ k and so yn0 þk $ yn0 þ2k $ y; where at least one of these inequalities is strict. If, for example, yn0 þ2k . y; then using the fact that the function gðxÞ ¼ ð p þ xÞ=ðx þ qxÞ is strictly decreasing on the interval (0,1), we have p þ yn0 þ2k p þ yn0 þ2k y ¼ gð yÞ . gðyn0 þ2k Þ ¼ $ ¼ yn0 þ2kþ1 ; yn0 þ2k þ qyn0 þ2k yn0 þ2k þ qyn0 þk which is a contradiction. Similarly, we can obtain a contradiction if yn0 þk . yn0 þ2k : Therefore every nontrivial positive solution of equation (9) is oscillatory about the positive equilibrium y: The above analysis also shows that no semicycle contains more than 2k þ 1 terms. (iii) We give the proof for a positive semicycle. The proof for a negative semicycle is similar and will be omitted. Let ys, ysþ1,. . .,ysþk be the first k þ 1 terms in a positive semicycle and s is a positive integer number. Then ys21 , y # ys ; ysþ1 ; . . .; ysþk ; ysþkþ1
1 þ ys pþk 1 þ py ysþk þ p ¼ ysþk : ¼ ¼ ysþk # ysþk ysþk þ qys ysþk þ qys y þ qy
If ysþkþ1 $ y; then ysþkþ2 ¼
ysþkþ1 þ p # ysþkþ1 : ysþkþ1 þ qysþ1
If ysþkþ2 $ y; then similarly ysþkþ3 # ysþkþ2 : By induction, we see that part (iii) is true. The proof is complete. A Theorem 2.4 Assume that p ¼ q and {yn} is a nontrivial solution of equation (9), then the following results are true. (i) If ð y2k ; . . .; y21 Þ [ ½0; 1Þk and y0 [ (0,1), or ð y2kþ1 ; . . .; y0 Þ [ ½0; 1Þk and y2k [ ð0; 1Þ; then every semicycle of a solution {yn} of equation (9) has k þ 1 terms. (ii) If ð y2k ; . . .; y0 Þ [ ð1; 1Þkþ1 ; then every semicycle of a solution {yn} of equation (9) has k þ 1 terms. Proof
(i) Since p ¼ q; then y ¼ 1: Hence, if ð y2k ; . . .; y21 Þ [ ½0; 1Þk and y0 [ (0,1), then y0 þ p y0 þ p ¼ 1; . y0 þ py2k y0 þ p y1 þ p y1 þ p y2 ¼ ¼ 1; . y1 þ py2kþ1 y1 þ p ··· yk þ p yk þ p ykþ1 ¼ ¼ 1; . yk þ py0 yk þ p ykþ1 þ p ykþ1 þ p ykþ2 ¼ ¼ 1; , ykþ1 þ py1 ykþ1 þ p ··· y2kþ1 þ p y2kþ1 þ p y2kþ2 ¼ ¼ 1: , y2kþ1 þ pykþ1 y2kþ1 þ p y1 ¼
By induction we complete the proof of part (i). (ii) The proof is similar and will be omitted. The proof is complete.
A
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2.3 Invariant intervals and global attractivity This subsection is concerned with invariant intervals and global attractivity of equation (9). First, we study the invariant intervals of equation (9) when p – q: Theorem 2.5 Let {yn }1 n¼2k be a positive solution of equation (9), then the following statements are true. (i) Assume that p . q, and the initial conditions ð y2k ; . . .; y0 Þ [ ½1; p=q kþ1 : Then yn [ ½1; p=q for all n $ 0, that is, the interval ½1; p=q is an invariant interval of equation (9). (ii) Assume that p , q, and the initial conditions ð y2k ; . . .; y0 Þ [ ½ p=q; 1 kþ1 : Then yn [ ½ p=q; 1 for all n $ 0, that is, the interval ½ p=q; 1 is an invariant interval of equation (9). Proof
(i) Let f ðu; vÞ ¼ ðu þ pÞ=ðu þ qvÞ: If u; v [ ½1; p=q ; then since p . q we have 1¼
uþp uþp p # f ðu; vÞ # # : uþp uþq q
Using this we obtain the result in this case. (ii) The proof is similar to the proof of (i) and will be omitted. The proof is complete.
A
In the following, we study the global attractivity of the positive solutions of equation (9). From our results it follows that Conjecture A is true. Let {yn }1 n¼2k be a solution of equation (9), then ynþ1 2 1 ¼ q
ynþ1 2
p q
2 yn2k
yn þ qyn2k
;
for
p ðq 2 pÞyn þ pqð1 2 yn2k Þ ¼ ; q qð yn þ qyn2k Þ
ð15Þ
n $ 0;
for
ð16Þ
n $ 0;
and yn 2 ynþ2ðkþ1Þ ¼
ð yn 2 1Þ½qyn ynþ2kþ1 þ ynþk ynþ2kþ1 þ qynþk yn 2 pq qð ynþk þ pÞ þ ynþ2kþ1 ð ynþk þ qyn Þ
:
ð17Þ
Thus, the following two results are the direct consequences of equations (15) – (17). Lemma 2.1 Assume that p . q and let {yn }1 n¼2k be a solution of equation (9). Then the following statements are true: (i) If for some N $ 0, yN , 1; then yN , yNþ2ðkþ1Þ , p=q: (ii) If for some N $ 0, yN . p=q; then 1 , yNþ2ðkþ1Þ , yN : (iii) If for some N $ 0, 1 # yN # p=q; then 1 # yNþ2ðkþ1Þ # p=q: Lemma 2.2 Assume that p , q and let {yn }1 n¼2k be a solution of equation (9). Then the following statements are true: (i) If for some N $ 0, yN , p=q; then yN , yNþ2ðkþ1Þ , 1: (ii) If for some N $ 0, yN . 1, then yN . yNþ2ðkþ1Þ . p=q: (iii) If for some N $ 0, p=q # yN # 1; then p=q # yNþ2ðkþ1Þ # 1:
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Now we consider the case p ¼ q: If p ¼ q; then the positive equilibrium of equation (9) is y ¼ 1 and so ynþ1 2 1 ¼ p
1 2 yn2k for n $ 0; yn þ pyn2k
ð18Þ
and yn 2 ynþ2ðkþ1Þ ¼ ð yn 2 1Þ
½pyn ynþ2kþ1 þ ynþk ynþ2kþ1 þ pynþk : pð ynþk þ pÞ þ ynþ2kþ1 ð ynþk þ pyn Þ
ð19Þ
In view of equations (18) and (19), it is easy to see that the following result holds. Lemma 2.3 Assume that p ¼ q and let {yn }1 n¼2k be a solution of equation (9). Then the following statements are true: (i) If for some N $ 0, yN , 1, then yN , yNþ2ðkþ1Þ , 1: (ii) If for some N $ 0, yN . 1, then yN . yNþ2ðkþ1Þ . 1: (iii) If for some N $ 0, yN ¼ 1; then yNþkþ1 ¼ 1: Let ( Fð yÞ ¼ where
Lemma 2.4
maxy#z#y Gð y; zÞ;
for 0 , y # y
min y#z#y Gð y; zÞ;
for y . y
;
" # p p k21 1 þ pz 1 þ y 1 þ y Gð y; zÞ ¼ z : z þ qy y þ qz y þ qy
ð20Þ
ð21Þ
F [ C½ð0; 1Þ; ð0; 1Þ and is strictly decreasing in (0,1).
Proof Obviously, the function G( y,z) is continuous for ð y; zÞ [ ð0; 1Þ; strictly decreasing in y for y . 0 and satisfies Fð yÞ ¼ Gð y; yÞ ¼ y: First we prove that F is strictly decreasing in (0,1). If y1 , y2 , y; then we choose z2 [ ½ y2 ; y such that Fð y2 Þ ¼ max Gð y2 ; zÞ ¼ Gð y2 ; z2 Þ; y2 #z#y
and so Fð y2 Þ ¼ Gð y2 ; z2 Þ , Gð y1 ; z2 Þ # max Gð y1 ; zÞ ¼ Fð y1 Þ: y1 #z#y
If y , y1 , y2 ; then, by using a similar method, we see that Fð y1 Þ . Fð y2 Þ: If y1 , y , y2 ; then Fð y2 Þ , Fð yÞ , Fð y1 Þ: Therefore, F is strictly decreasing.
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Next, we prove that F is continuous. Otherwise there exists a point y0 . 0 such that F is discontinuous at y0. We assume that y0 # y: The case where y0 $ y is similar and will be omitted. Now if y0 ¼ y; let { yn} be any sequence of points in ð0; y with limit y and {zn} be any sequence of points with yn # zn # y
and
Fð yn Þ ¼ Gð yn ; zn Þ;
then lim zn ¼ y:
n!1
Since G is continuous at y; then lim Fð yn Þ ¼ lim Gð yn ; zn Þ ¼ Gð y; yÞ ¼ y ¼ Fð yÞ;
n!1
n!1
which shows that F is continuous at y: So it remains to consider the case: 0 , y0 , y: If F is discontinuous at y0, then there must exist 11 . 0 such that for every d1 . 0, there exists y1 . 0 such that j y 0 2 y 1 j , d1
and jFð y0 Þ 2 Fð y1 Þj $ 11 :
ð22Þ
Since G( y,z) is uniformly continuous on ½1; y £ ½1; y for arbitrary small 1 . 0, then for every 12 . 0 there exists d2 . 0 such that 0 y 2 y 00 , d2 and z 0 2 z 00 , d2 ) Gð y 0 ; z 0 Þ 2 Gð y 00 ; z 00 Þ , 12 : We now assume that y1 , y0 . The case where y1 . y0 is similar and will be omitted. Taking 12 ¼ 11 =2; d1 ¼ d2 and let z0 [ ½y0 ; y be such that Fð y0 Þ ¼ max Gð y0 ; zÞ ¼ Gð y0 ; z0 Þ: y0 #z#y
Choose z1 [ ½ y1 ; y such that jz0 2 z1 j , d1 : Then 11 11 Gð y0 ; z0 Þ 2 , Gð y1 ; z1 Þ , Gð y0 ; z0 Þ þ : 2 2 But Fð y1 Þ $ Gð y1 ; z1 Þ . Gð y0 ; z0 Þ 2
11 11 ¼ Fð y0 Þ 2 ; 2 2
and so 0 , Fð y0 Þ 2 Fð y1 Þ ,
11 ; 2
which contradicts equation (22) and completes the proof. Now we state our main result. Theorem 2.6
A
The equilibrium y of equation (9) is a global attractor.
Proof First, we consider the case p ¼ q: It follows from Lemma 2.3 that each of the 2(k þ 1) subsequences {y2ðkþ1Þnþi }1 n¼0
for
i ¼ 1; 2; . . .; 2ðk þ 1Þ
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is either identically equal to 1 or else it is strictly monotonically convergent and its limit is greater than zero. Set Li ¼ lim y2ðkþ1Þnþi n!1
for
i ¼ 1; 2; . . .; 2ðk þ 1Þ:
Then :; L1 ; L2 ; . . .; L2ðkþ1Þ ; . . .
ð23Þ
is a periodic solution of equation (9) with period 2(k þ 1). Letting n ! 1 in equation (19) and using the fact Li . 0,i ¼ 1; . . .; 2ðk þ 1Þ; we obtain Li ¼ 1
for
i ¼ 1; 2; . . .; 2ðk þ 1Þ;
and so lim yn ¼ 1:
n!1
Next, we consider the case p – q: When p . q, recall that from Theorem 2.3, ½1; p=q is an invariant interval. In this interval, the function f ðu; vÞ ¼ ðu þ pÞ=ðu þ qvÞ is decreasing in both variables and so it follows from Theorem C that every solution of equation (9) with k þ 1 consecutive values in ½1; p=q ; converges to y: If the solution is not eventually in ½1; p=q ; there are three cases to be considered. Case (i). If for some N $ 0, yN . p=q; then there are two cases to be considered. If yNþ2ðkþ1Þn $ p=q; for every n [ N, then by Lemma 2.1 we have yNþ2ðkþ1Þ ðn21Þ . yNþ2ðkþ1Þn . 1; hence the subsequence {yNþ2ðkþ1Þn } is strictly monotonically decreasingly convergent. If for some n0, yNþ2ðkþ1Þn0 , p=q; then by Lemma 2.1 we obtain that {yNþ2ðkþ1Þn } is eventually in the interval ½1; p=q : Case (ii). If for some N $ 0, yN , 1, then there are two cases to be considered. If yNþ2ðkþ1Þn , 1; for every n [ N; then by Lemma 2.1 we obtain p yNþ2ðkþ1Þ ðn21Þ , yNþ2ðkþ1Þn , ; q which implies that the subsequence {yNþ2ðkþ1Þn } is convergent. If for some n0 , yNþ2ðkþ1Þn0 . 1; then by Lemma 2.1 we obtain that {yNþ2ðkþ1Þn } is eventually in the interval ½1; p=q : Case (iii). If for some N $ 0, 1 # yN # p=q; then by Lemma 2.1 it follows that 1 # yNþ2ðkþ1Þn # p=q: Assume that there is a subsequence {yNþ2ðkþ1Þn } such that yNþ2ðkþ1Þn $ p=q; or yNþ2ðkþ1Þn # 1; for every n 2 N: Then its limit S satisfies S $ p=q; or S # 1. Taking limit on both sides of equation (17), we obtain a contradiction. Hence, for all N [ {1; 2; . . .; 2ðk þ 1Þ} the subsequences {yNþ2ðkþ1Þn } are eventually in the interval ½1; p=q : By Theorem C the sequence {yn} converges to y: Now, we consider the case p , q. Similar to the case p . q, we can see that every solution of equation (9) must eventually lie in the interval ½p=q; 1 : Let {yn} be a positive solution of equation (9). By Theorem 2.3, if {yn} is not oscillatory then lim yn ¼ y:
n!1
ð24Þ
So it remains to establish that equation (24) holds when {yn} is oscillatory. To this end, let {ypi þ1 ; ypi þ2 ; . . .; yqi } be the i th positive semicycle of {yn} followed by the i th negative
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semicycle {yqi þ1 ; yqi þ2 ; . . .; yqi þs }: Let yMi and ymi , be the extreme values in these two semicycles, respectively, with smallest possible indices Mi and mi . Then by Theorem 2.3. M i 2 pi # k þ 1 yMi ¼ ypi
MY i 21
1 þ ypj
j ¼ pi
yj þ qyj2k
and
mi 2 qi # k þ 1;
and
ymi ¼ yqi
mY i 21
1 þ ypj
j¼qi
yj þ qyj2k
:
From above we know when p – q, there exist positive numbers K1 and K2 such that every solution of equation (9) must eventually lie in the interval [K1, K2]. Let
l ¼ lim inf yn ¼ lim inf ymi ; n!1
i!1
m ¼ lim sup yn ¼ lim sup yMi : n!1
ð25Þ
i!1
Then K1 # l # y # m # K2: To complete the proof, it suffices to show that l ¼ y ¼ m: From equation (25) it follows that for arbitrary small 1 . 0, then there exists positive integer n0 such that
l 2 1 # yn # m þ 1 for n $ n0 2 k:
ð26Þ
yMi # Gðl 2 1; ypi Þ;
ð27Þ
We now claim that
where G is given by equation (21). Indeed, if M i 2 pi , k þ 1;
ð28Þ
then yM i ¼ ypi
# ypi
p
MY i 21
1 þ yj
j¼pi
yj þ qyj2k 1 þ ypp
1 þ ypj
MY i 21
i
ypi þ qðl 2 1Þ j¼p þ1 yj þ qðl 2 1Þ i
# ypi
# ypi
"
1 þ ypp
i
1 þ py
#M i 2pi 21
ypi þ qðl 2 1Þ y þ qðl 2 1Þ "
1 þ ypp
i
1 þ py
ypi þ qðl 2 1Þ y þ qðl 2 1Þ
But 1 þ py y þ qypi
$
1 þ py y þ qy
¼ 1;
#k21 :
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and so yM i # ypi
"
1 þ ypp
i
#k21
1 þ py
ypi þ qðl 2 1Þ y þ qðl 2 1Þ
1 þ py y þ qypi
¼ Gðl 2 1; ypi Þ;
which means that equation (27) is established if equation (28) holds. On the other hand, if equation (28) does not hold, then M i 2 pi ¼ k þ 1 and in this case " # MY i 22 1 þ yMp 1 þ ypp 1 þ ypj i i21 yM i ¼ ypi ypi þ qypi 2k j¼p þ1 yj þ qyj2k yMi21 þ qyMi212k i
# ypi
"
1 þ ypp
i
1 þ py
#k21
ypi þ qðl 2 1Þ y þ qðl 2 1Þ
1 þ py y þ qypi
¼ Gðl 2 1; ypi Þ; which completes the proof that equation (27) holds. Since
l 2 1 , ypi , y; then it follows from equation (27) that yMi # Gðl 2 1; ypi Þ #
max
l21#ypi #y
Gðl 2 1; ypi Þ ¼ Fðl 2 1Þ:
Therefore, as 1 is arbitrary small, yM i # FðlÞ and so
m # FðlÞ: In a similar way, we show that
l $ FðmÞ: Hence, we have FðmÞ # l # y # m # FðlÞ: Now we consider properties of the function #k21
" 1 þ py zþp p Gð y; zÞ ¼ 1þ : ðz þ qyÞð y þ qzÞ y y þ qy Since › zþp 2qz 2 þ qyy 2 2pqz 2 py 2 pq 2 y ; ¼ ›z ðz þ qyÞð y þ qzÞ ðqz 2 þ zy þ qyy þ q 2 yzÞ2 then there are two cases to be considered. Case (i) If qyy , 2pqz þ py þ pq 2 y þ qz 2 ;
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then the function zþp ðz þ qyÞð y þ qzÞ is decreasing in z and
Fð yÞ ¼
8 < maxy#z#y Gð y; zÞ; for 0 , y # y : miny#z#y Gð y; zÞ;
for y . y
" #k 1 þ py yþp y þ p y þ qy k ¼ : ¼ yð1 þ qÞ y þ qy yð1 þ qÞ y þ qy In order to apply Theorem E, we need to show that y is the only fixed point of F 2 in (0,1). Let y þ p y þ qy N Hð yÞ ¼ yð1 þ qÞ y þ qy
where N ¼ nk; n is a sufficiently large positive integer:
By Lemma 2.4, we know that the function Hð yÞ [ C½ð0; 1Þ; ð0; 1Þ is strictly decreasing in (0,1) and Hð yÞ ¼ Fð yÞ ¼ y: Let L ¼ HðMÞ and M . 0 is the fixed point of H 2( y), that is H 2 ðMÞ ¼ M: Then L þ p y þ qy N ¼M HðLÞ ¼ Lð1 þ qÞ y þ qL
and
M þ p y þ qy N HðMÞ ¼ ¼ L; Mð1 þ qÞ y þ qM
hence Mþp Lþp : N ¼ ðy þ qMÞ ð y þ qLÞN Let RðxÞ ¼
xþp ; ð y þ qxÞN
then R0 ðxÞ ¼
y þ ðq 2 NqÞx 2 Npq ,0 ð y þ qxÞNþ1
since n is a sufficiently large positive integer. Thus the function RðxÞ is strictly decreasing and M ¼ L; which implies that every fixed point H 2 is fixed point of H. Since H is decreasing, H(x) . 0 for x [ ð0; 1Þ; Hðþ1Þ ¼ 0; it follows that H has a unique fixed point y: Thus H 2 has a unique fixed point.
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For y . 0 we know that #
" y þ p y þ qy k y þ qy ðn21Þk 21 ð y 2 yÞ Hð yÞ 2 Fð yÞ ð y 2 yÞ ¼ yð1 þ qÞ y þ qy y þ qy q ð y 2 yÞ2 ð y þ pÞð y þ qyk Þ yð1 þ qÞð y þ qyÞnk £ ð y þ qyÞðn21Þk21 þ ð y þ qyÞðn21Þk22 ð y þ qyÞ
¼2
þ· · · þ ð y þ qyÞðn21Þk21 # 0: By Theorem E, y is also the only fixed point of F 2 in (0,1). By Theorem D, we have
l ¼ y ¼ m: Case (ii) If qyy $ 2pqz þ py þ pq 2 y þ qz 2 ; then the function zþp ðz þ qyÞð y þ qzÞ is nondecreasing in z and Fð yÞ ¼
8 < maxy#z#y Gð y; zÞ; for 0 , y # y : miny#z#y Gð y; zÞ; "
¼y
1 þ py
for y . y
#k
y þ qy
:
If k ¼ 1; let L ¼ FðMÞ and M . 0 is the fixed point of F 2, that is to say, F 2 ðMÞ ¼ M; then " # 1 þ py FðMÞ ¼ y ¼L y þ qM and " FðLÞ ¼ y
1 þ py y þ qL
# ¼ M;
and hence y þ qM y þ qL ¼ : M L Let IðxÞ ¼
y þ qx ; x
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then 0
I ðxÞ ¼
y þ qx y ¼ 2 2 , 0: x x
Thus, the function IðxÞ is strictly decreasing and M ¼ L; which implies that every fixed point of F 2 is fixed point of F. Since F is decreasing, F(x) . 0 for x [ ð0; 1Þ; Fðþ1Þ ¼ 0; it follows that F has a unique fixed point y: Thus, F 2 has a unique fixed point. Hence y is the only fixed point of F 2 in (0,1). By Theorem D, we have
l ¼ y ¼ m: For the case k . 1, to complete the proof it remains to show that y is the global attractor of all positive solutions of the equation ynþ1 ¼ Fð yn Þ;
n ¼ 0; 1; 2; . . . :
In view of Theorem F, since Fð yÞ ¼ y; it is sufficient to show that there exists an interval I , R such that Fð yÞ [ I for y [ I and that Schwarizan derivative of F is negative. Let k0 ¼ yð1 þ qÞk and I ¼ ð0; k0 Þ: Since F( y) is decreasing, then for every y [ I, we have " #k 1 þ py 0 , Fð yÞ ¼ y y þ qy " # Fð0þ Þ ¼ y
#k 1 þ py y
¼ yð1 þ qÞk ¼ k0 ;
which means that FðIÞ , I: Now we prove that Schwarizan derivative of F is negative. We have 0 " # 10 p k 1 þ y A ¼ 2Ckqð y þ qyÞ2k21 ; F 0 ð yÞ ¼ @y y þ qy F 00 ð yÞ ¼ Cðk þ 1Þkq 2 ð y þ qyÞ2k22 ; where
p k C ¼y 1þ : y From this it follows that F 00 ð yÞ 2ðk þ 1Þq ¼ F 0 ð yÞ y þ qy
SFð yÞ ¼
¼
and
0 F 00 ð yÞ ðk þ 1Þq 2 ¼ : 0 F ð yÞ ð y þ qyÞ2
00 0 F ð yÞ 1 F 00 ð yÞ 2 2 F 0 ð yÞ 2 F 0 ð yÞ ðk þ 1Þq 2 1 ðk þ 1Þq 2 2ðk 2 2 1Þq 2 2 ¼ , 0: ð y þ qyÞ2 2 y þ qy 2ð y þ qyÞ2
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By Theorem D, we have
l ¼ y ¼ m: The proof is complete.
A
In view of Theorem 2.1, we have the following result. Theorem 2.7 Assume that one of the following three conditions is satisfied: (i) k . 1 and p $ q; (ii) k . 1 and p , q # 1; (iii) k . 1, p , q and 1 , q , 1 þ 4p. Then the equilibrium y of equation (9) is globally asymptotically stable. If k ¼ 1, then Theorem 2.6 and Theorem A implies the following result. Corollary 2.1
The equilibrium y of equation (2) is globally asymptotically stable.
Notice that the corollary confirms Conjecture A.
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