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Global asymptotic stability for two recursive difference equations
Applied Mathematics and Computation 150 (2004) 481–492 www.elsevier.com/locate/amc
Global asymptotic stability for two recursive difference equations q Xianyi Li
a,b,* ,
Deming Zhu
b
a b
School of Mathematics and Physics, Nanhua University, Hengyang 421001, PR China Department of Mathematics, East China Normal University, Shanghai 200062, PR China
Abstract Two sufficient conditions are obtained for the global asymptotic stability of the following two recursive difference equations xn xn1 þ xn2 þ a ; n ¼ 0; 1; 2; . . . ; xnþ1 ¼ xn þ xn1 xn2 þ a and xnþ1 ¼
xn1 þ xn xn2 þ a ; xn xn1 þ xn2 þ a
n ¼ 0; 1; 2; . . . ;
where a 2 ½0; 1Þ and the initial values x2 ; x1 ; x0 2 ð0; 1Þ. Some known results are included. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Recursive difference equation; Global asymptotic stability; Semicycle
1. Introduction The qualitative analysis of recursive difference equations has been the object of the recent study. For example, see [1–12] and the references cited therein.
q Project by NNSF of China (grant: 10071022), Mathematical Tianyuan Foundation of China (grant: TY10026002-01-05-03), and Shanghai Priority Academic Discipline. * Corresponding author. Address: Department of Mathematics, East China Normal University, Shanghai 200062, PR China. E-mail addresses: [email protected] (X. Li), [email protected] (D. Zhu).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00286-8
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In [3], Ladas put forward to investigate the global asymptotic stability of the following rational difference equation xnþ1 ¼
xn þ xn1 xn2 ; xn xn1 þ xn2
n ¼ 0; 1; . . . ;
ðE1 Þ
where the initial values x2 ; x1 ; x0 2 ð0; 1Þ. The authors of this paper obtained in [4] a general result implying the global asymptotic stability of (E1 ). Recently, Tim Nesemann [5] studied the following difference equation xnþ1 ¼
xn1 þ xn xn2 ; xn xn1 þ xn2
n ¼ 0; 1; . . . ;
ðE2 Þ
where the initial values x2 ; x1 ; x0 2 ð0; 1Þ, and mainly utilized the strong negative feedback property of [6] to derive the following result. Theorem A. The positive equilibrium point of Eq. (E2 ) is globally asymptotically stable. To be motivated by the above studies, in this paper we consider the following two equations xnþ1 ¼
xn xn1 þ xn2 þ a ; xn þ xn1 xn2 þ a
n ¼ 0; 1; 2; . . .
ð1Þ
xnþ1 ¼
xn1 þ xn xn2 þ a ; xn xn1 þ xn2 þ a
n ¼ 0; 1; 2; . . . ;
ð2Þ
and
where a 2 ½0; 1Þ and the initial values x2 ; x1 ; x0 2 ð0; 1Þ. Obviously, Eq. (1) has a reverse form of Eq. (E1 ) as a ¼ 0, and Eq. (2) is an extension of Eq. (E2 ). Eqs. (1) and (2) are interesting in their own right. To the best of our knowledge, however, Eqs. (1) and (2) with a > 0 have not been investigated so far. Therefore, theoretically, it is meaningful to study their qualitative properties. Because of the similarity of the structure between Eqs. (1) and (E1 ), whether are their properties similar or not? Accordingly, a conjecture naturally arises. Conjecture B. Assume that a 2 ½0; 1Þ. The positive equilibrium of Eq. (1) is globally asymptotically stable. Analogously, Whether or not is Eq. (2) globally asymptotically stable like Eq. (E2 ) either? Along a new road in this note, we give a positive answer to the above two problems. Whereas, it is extremely difficult to use the method in the known
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literature, such as [1,2,5,6], to obtain the global asymptotic stability of Eq. (1) or (2). Just as what we watch, the structure between Eqs. (1) and (2) is also very similar. But, as we will see in the following that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of the two equations to successively occur is very different. Thus, it is necessary to consider these two equations separately. Our main results are the following. Theorem 1.1. Assume that a 2 ½0; 1Þ. Then the positive equilibrium of Eq. (1) is globally asymptotically stable. Theorem 1.2. Assume that a 2 ½0; 1Þ. Then the positive equilibrium of Eq. (2) is globally asymptotically stable. It is easy to see that Eq. (1) (or (2)) has a unique positive equilibrium x ¼ 1. In the following we always denote by x the unique positive equilibrium point of Eq. (1) (or (2)). Definition 1.3. A positive semicycle of a solution fxn g1 n¼2 of Eq. (1) (or (2)) consists of a ‘‘string’’ of terms fxl ; xlþ1 ; . . . ; xm g, all greater than or equal to the equilibrium x, with l P 2 and m 6 1 such that either l ¼ 2 or l > 2 and
xl1 < x
either m ¼ 1 or m < 1
xmþ1 < x:
and and
1
A negative semicycle of a solution fxn gn¼2 of Eq. (1) (or (2)) consists of a ‘‘string’’ of terms fxl ; xlþ1 ; . . . ; xm g, all less than x, with l P 2 and m 6 1 such that either l ¼ 2 or l > 2 and
xl1 P x
either m ¼ 1 or m < 1
xmþ1 P x:
and and 1
Definition 1.4. A solution fxn gn¼2 of Eq. (1) (or (2)) is said to be eventually positive (negative) if xn x > 0 (<0) eventually holds. For the other concepts in this paper, see [1,2].
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2. Several lemmas In order to simplify the proofs of our main results, we need the following lemmas. We first consider Eq. (1). Lemma 2.1. A positive solution fxn g1 n¼1 of Eq. (1) is eventually equal to 1 if and only if ðx1 1Þðx0 1Þðx2 x0 Þ ¼ 0:
ð3Þ
Proof. Assume that (3) holds. Then according to Eq. (1), it is easy to see that the following conclusions hold: (a) if x0 ¼ 1, then xn ¼ 1 for n P 2; (b) if x1 ¼ 1, then xn ¼ 1 for n P 1; (c) if x2 ¼ x0 , then xn ¼ 1 for n P 3. Conversely, assume that ðx1 1Þðx0 1Þðx2 x0 Þ 6¼ 0:
ð4Þ
Then we show xn 6¼ 1
for any n P 1:
Assume the contrary that for some N P 1, xN ¼ 1 and that xn 6¼ 1 for 1 6 n 6 N 1:
ð5Þ
Clearly, 1 ¼ xN ¼
xN 1 xN 2 þ xN 3 þ a ; xN 1 þ xN 2 xN 3 þ a
which implies xN 1 ¼ xN 3 and by (4), N P 2. Thus, from xN 3 ¼ xN 1 ¼
xN 2 xN 3 þ xN 4 þ a ; xN 2 þ xN 3 xN 4 þ a
one can obtain ðxN 3 1ÞðxN 4 ðxN 3 þ 1Þ þ aÞ ¼ 0, which contradicts (5).
Remark 2.1. If the initial conditions do not satisfy equality (3), then, for any solution fxn g of Eq. (1), xn 6¼ 1 for n P 1 and xn 6¼ xn2 for n P 0. Lemma 2.2. Let fxn g1 n¼2 be a positive solution of Eq. (1) which is not eventually equal to 1. Then the following conclusions are true:
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(a1) ðxnþ1 1Þðxn1 1Þðxn xn2 Þ > 0, for n P 0; (b1) ðxnþ1 xn1 Þðxn1 1Þ < 0, for n P 0; (c1) ðxnþ1 1Þðxn1 1Þðxn2 1Þ < 0, for n P 1. Proof. In view of Eq. (1), we obtain xnþ1 xn1 ¼
ð1 xn1 Þðxn2 ð1 þ xn1 Þ þ aÞ ; xn þ xn1 xn2 þ a
n ¼ 0; 1; 2; . . .
and xnþ1 1 ¼
ðxn xn2 Þðxn1 1Þ ; xn þ xn1 xn2 þ a
n ¼ 0; 1; 2; . . .
From this, inequalities (a1) and (b1) follow. Inequality (c1) is an immediate consequence of inequalities (a1) and (b1). The proof is complete. Now we analyze the situation of the semicycles of nontrivial solutions of Eq. (1). Lemma 2.3. Assume that x2 ; x1 ; x0 2 ð0; 1Þ. Let fxn g be an arbitrary solution of Eq. (1). Then xn < 1 for all n P 1 if and only if x0 > x2 . Proof. If x0 > x2 , then, according to Lemma 2.2(a1) and (c1), xn < 1 for all n P 1. Conversely, if xn < 1 for all n P 1, then, still by Lemma 2.2(a1), one can see x0 > x2 . From Lemma 2.3, it is easily seen that, except perhaps for the first semicycle, a negative semicycle of a solution contains at most two terms. Considering positive semicycles, it is clear again by Lemma 2.2(c1) that, except perhaps for the first semicycle, a positive semicycle contains at most three terms. More precisely, we have the following result. Lemma 2.4. Consider a solution of Eq. (1) which is not eventually less than or equal to 1. Then, with the possible exception of the first semicycle, the following affirmations hold: (d1) every negative semicycle consists of one or two terms; every positive semicycle consists of one or three terms; (e1) every negative semicycle of length one is followed by a positive semicycle of length three; every negative semicycle of length two is followed by a positive semicycle of length one; (f1) every positive semicycle of length one is followed by a negative semicycle of length one; every positive semicycle of length three is followed by a negative semicycle of length two;
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Proof. By Lemma 2.2(c1) and Lemma 2.3, one can see that a negative semicycle of a solution which is not eventually less than or equal to 1, has one or two terms and, by Lemma 2.2(c1), a positiive semicycle has at most three terms. Obviously, for some p P 0, one of the following cases must occur: Case 1: xp2 < 1, xp1 > 1 and xp < 1. It follows from Lemma 2.2 (c1) that xpþ1 > 1, xpþ2 > 1, xpþ3 > 1 and xpþ4 < 1. Then we have xpþ5 < 1. It means that a positive semicycle of length 1 is followed by a negative semicycle of length 1, which in turn is followed by a positive semicycle of length 3. Case 2: xp2 < 1, xp1 > 1 and xp > 1. We have, again in view of Lemma 2.2(c1), that xpþ1 > 1, xpþ2 < 1, xpþ3 < 1 and xpþ4 > 1. It implies xpþ5 < 1, and hence, a positive semicycle of length 3 is followed by a negative semicycle of length 2, which in turn is followed by a positive semicycle of length 1. The proof is complete. Remark 2.2. It follows from Lemma 2.4 that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of Eq. (1) to successively occur is . . . , 3, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, . . . Similarly, we can derive the following results for Eq. (2). 1
Lemma 2.5. A positive solution fxn gn¼1 of Eq. (2) is eventually equal to 1 if and only if ðx0 1Þðx1 x2 Þ ¼ 0:
ð6Þ
Remark 2.3. If the initial conditions do not satisfy equality (6), then xn 6¼ 1 for n P 0 and xn 6¼ xn1 for n P 1 for any solution fxn g of Eq. (2). Lemma 2.6. Let fxn g1 n¼2 be a positive solution of Eq. (1) which is not eventually equal to 1. Then the following conclusions are true: (a2) ðxnþ1 1Þðxn 1Þðxn1 xn2 Þ < 0, for n P 0; (b2) ðxnþ1 xn Þðxn 1Þ < 0, for n P 0; (c2) ðxnþ1 1Þðxn 1Þðxn2 1Þ > 0, for n P 2. Lemma 2.7. Assume that x2 ; x1 ; x0 2 ð1; 1Þ. Let fxn g be a solution of Eq. (2). Then xn > 1 for all n P 1 if and only if x1 < x2 . The proofs of Lemmas 2.5–2.7 are completely similar to those of Lemmas 2.1–2.3 and so omitted here.
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We are now in a position to analyze in detail the situation of the semicycles of nontrivial solutions of Eq. (2). Based on Lemma 2.6, we have the following result.
Lemma 2.8. With the possible exception of the first semicycle, the following affirmations are true for any solution of Eq. (2) which is not eventually greater than or equal to 1. (d2) Every positive semicycle has one or two terms; every negative semicycle has one or three terms; (e2) every positive semicycle with length one is followed by a negative semicycle with length one; every positive semicycle with length two is followed by a negative semicycle with length three; (f2) every negative semicycle with length one is followed by a positive semicycle with length two; every negative semicycle with length three is followed by a positive semicycle with length one.
Proof. Due to Lemma 2.6(c2) and Lemma 2.7, one can see that a positive semicycle of a solution which is not eventually greater than or equal to 1, has one or two terms and, again due to Lemma 2.6(c2), a negative semicycle has at most three terms. Obviously, for some p P 0, one of the following cases has to appear: Case 1: xp2 < 1, xp1 > 1 and xp < 1. Applying Lemma 2.6(c2) to the case we have that xpþ1 > 1, xpþ2 > 1, xpþ3 < 1 and xpþ4 < 1, which leads to xpþ5 < 1. Hence, we know a positive semicycle of length 1 is followed by a negative semicycle of length 1, which in turn is followed by a positive semicycle of length 2.Case 2: xp2 < 1, xp1 > 1 and xp > 1. Using Lemma 2.6(c2) again, we get xpþ1 < 1, xpþ2 < 1, xpþ3 < 1, xpþ4 > 1 and xpþ5 < 1. Therefore, a positive semicycle of length 2 is followed by a negative semicycle of length 3, and the later in turn is followed by a positive semicycle of length 1. The proof is complete.
Remark 2.4. It is clear from Lemma 2.8 that the rule for the lengths of positive and negative semicycles of nontrivial solutions of Eq. (2) to alternately occur is . . . , 2, 3, 1, 1, 2, 3,1,1, 2, 3, 1, 1, . . ., which, compared with Remark 2.2, manifests that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of the two equations to occur is very different.
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3. Proofs of theorems We now begin with the proofs of these two theorems. First, show that the positive equilibrium point of Eq. (1) is globally asymptotically stable. Proof of Theorem 1.1. We must prove that the positive equilibrium point x of Eq. (1) is both locally asymptotically stable and globally attractive. The linearized equation of Eq. (1) about the positive equilibrium x ¼ 1 is ynþ1 ¼ 0 yn þ 0 yn1 þ 0 yn2 ;
n ¼ 0; 1; . . .
By virtue of [2, Remark 1.3.7], x is locally asymptotically stable. It remains to 1 verify that every positive solution fxn gn¼2 of Eq. (1) converges to 1 as n ! 1. Namely, we want to prove lim xn ¼ x ¼ 1:
ð7Þ
n!1
If the solution fxn g1 n¼2 of Eq. (1) is trivial, then (7) obviously holds. If the solution is nonoscillatory about the positive equilibrium point x of Eq. (1), then according to Lemma 2.3, we know that the solution is actually an eventually negative one. Furthermore, we can easily derive from Lemma 1 2.2(b1) that the subsequences fx2n g and fx2nþ1 g of the solution fxn gn¼2 of Eq. (1) are eventually monotonically increasing and bounded from the above. Therefore, the limits limn!1 x2n ¼ L and limn!1 x2nþ1 ¼ M exist and are finite. Note x2n ¼
x2n1 x2n2 þ x2n3 þ a x2n1 þ x2n2 x2n3 þ a
and x2nþ1 ¼
x2n x2n1 þ x2n2 þ a ; x2n þ x2n1 x2n2 þ a
take the limits on both sides of the above equalities and obtain L¼
LM þ M þ a ¼1 M þ LM þ a
and
M¼
LM þ L þ a ¼ 1: L þ ML þ a
That is to say, fxn g tends to the positive equilibrium point x of Eq. (1), and so (7) holds. Therefore, it suffices to prove that (7) holds for the solution to be strictly oscillatory. Consider now fxn g to be strictly oscillatory about the positive equilibrium point x of Eq. (1). By virtue of Lemma 2.4, one understands that every positive semicycle of this solution has one or three terms and every negative semicycle, except perhaps for the first, has one or two terms. Every positive semicycle of length one is followed by a negative semicycle of length one, which is followed by a positive semicycle of length three, in turn followed by the negative semicycle length two. For the convenience of statement, without loss of generality, we use the following denotation. We denote by xp the term of a positive semicycle of length one, followed by xpþ1 , which is the term of a negative semicycle of length
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one. Afterwards there is the positive semicycle: xpþ2 , xpþ3 , ,xpþ4 , in turn followed by the negative semicycle: xpþ5 , xpþ6 , and so on. Following this way, we get the following positive and negative semicycles: fxpþ7n g; fxpþ7nþ1 g; fxpþ7nþ2 ; xpþ7nþ3 ; xpþ7nþ4 g; fxpþ7nþ5 ; xpþ7nþ6 g; n ¼ 0; 1; . . . We have the following observations: (O1) xpþ7nþ4 < xpþ7nþ3 < xpþ7nþ2 ; xpþ7nþ6 > xpþ7nþ5 ; (O2) xpþ7nþ1 xpþ7nþ3 < 1, xpþ7nþ5 xpþ7nþ3 > 1, xpþ7nþ6 xpþ7nþ4 > 1; (O3) xpþ7nþ8 > xpþ7nþ6 . In fact, the inequality (O1) is followed straightforward from Lemma 2.2(b1), while (O2) is a direct consequence of the observations of xpþ7nþ2 xpþ7nþ1 þ xpþ7n þ a xpþ7nþ2 þ xpþ7nþ1 xpþ7n þ a xpþ7nþ2 xpþ7nþ1 þ xpþ7n þ a 1 < ¼ ; xpþ7nþ2 x2pþ7nþ1 þ xpþ7nþ1 xpþ7n þ axpþ7nþ1 xpþ7nþ1
xpþ7nþ3 ¼
xpþ7nþ4 xpþ7nþ3 þ xpþ7nþ2 þ a xpþ7nþ4 þ xpþ7nþ3 xpþ7nþ2 þ a xpþ7nþ4 xpþ7nþ3 þ xpþ7nþ2 þ a 1 > ¼ ; 2 xpþ7nþ4 xpþ7nþ3 þ xpþ7nþ3 xpþ7nþ2 þ axpþ7nþ3 xpþ7nþ3
xpþ7nþ5 ¼
and xpþ7nþ5 xpþ7nþ4 þ xpþ7nþ3 þ a xpþ7nþ5 þ xpþ7nþ4 xpþ7nþ3 þ a xpþ7nþ54 xpþ7nþ4 þ xpþ7nþ3 þ a 1 > ¼ : xpþ7nþ5 x2pþ7nþ4 þ xpþ7nþ4 xpþ7nþ3 þ axpþ7nþ4 xpþ7nþ4
xpþ7nþ6 ¼
Immediately, we also obtain xpþ7nþ8 ¼
xpþ7nþ7 xpþ7nþ6 þ xpþ7nþ5 þ a > xpþ7nþ6 ; xpþ7nþ7 þ xpþ7nþ6 xpþ7nþ5 þ a
n ¼ 0; 1; 2; . . .
This verifies (O3). Combining the above observations, we derive 1 (R1) xpþ7nþ8 > xpþ7nþ6 > xpþ7nþ5 > xpþ7nþ3 > xpþ7nþ1 ; n ¼ 0; 1; 2; . . . and 1 1 (R2) xpþ7nþ6 > xpþ7nþ4 > xpþ7nþ3 > xpþ7nþ1 ; n ¼ 0; 1; 2; . . .
From (R1), one can see that fxpþ7nþ1 g1 n¼0 is increasing with upper bound 1. So, the limit limn!1 xpþ7nþ1 ¼ L exists and is finite. Accordingly, by view of (R1), we obtain
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lim xpþ7nþ6 ¼ lim xpþ7nþ5 ¼ L
n!1
n!1
and
1 lim xpþ7nþ3 ¼ : L
n!1
Thus, in light of (R2), it follows that 1 lim xpþ7nþ4 ¼ lim xpþ7nþ3 ¼ : n!1 L
n!1
Now, we verify that L ¼ 1. To this end note xpþ7nþ6 ¼
xpþ7nþ5 xpþ7nþ4 þ xpþ7nþ3 þ a ; xpþ7nþ5 þ xpþ7nþ4 xpþ7nþ3 þ a
take the limits on both sides of the above equality and obtain L¼
L L1 þ L1 þ a ; L þ L1 L1 þ a
which yields ðL 1ÞðL þ 1 þ aÞ ¼ 0 and so L ¼ 1. Analogously, by taking limits on both sides of the equality xpþ7nþ7 ¼
xpþ7nþ6 xpþ7nþ5 þ xpþ7nþ4 þ a ; xpþ7nþ6 þ xpþ7nþ5 xpþ7nþ4 þ a
we derive lim xpþ7nþ7 ¼
n!1
L L þ L1 þ a ¼ 1: L þ L L1 þ a
Again, by taking limits on both sides of the equality xpþ7nþ9 ¼
xpþ7nþ8 xpþ7nþ7 þ xpþ7nþ6 þ a ; xpþ7nþ8 þ xpþ7nþ7 xpþ7nþ6 þ a
one derives lim xpþ7nþ9 ¼
n!1
LLþ1þa ¼ 1: LþLþa
Up to now, we have shown limn!1 xpþ7nþk ¼ 1, k ¼ 1; 2; . . . ; 7. So, the proof for Theorem 1.1, is complete. Next, we show that the positive equilibrium point x of Eq. (2) is globally asymptotically stable. Proof of Theorem 1.2. Similar to the previous proof of Theorem 1.1, we only need to prove that lim xn ¼ x ¼ 1
n!1
holds when the solution fxn g of Eq. (2) is strictly oscillatory.
ð8Þ
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It is clear from Remark 2.4 that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of Eq. (2) to occur is . . . , 2, 3, 1, 1, 2, 3,1,1, 2, 3, 1, 1,. . . We denote by xp the term of a positive semicycle of length one, followed by xpþ1 , which is the term of a negative semicycle of length one. Afterwards there is the positive semicycle of length two: xpþ2 , xpþ3 , in turn followed by the negative semicycle of length three: xpþ4 , xpþ5 , xpþ6 , and so on. So, we can turn and turn about getting the following positive and negative semicycles: fxpþ7n g; fxpþ7nþ1 g; fxpþ7nþ2 ; xpþ7nþ3 g; fxpþ7nþ4 ; xpþ7nþ5 ; xpþ7nþ6 g; n ¼ 0; 1; . . . We claim that the following inequalties are valid. (O1) xpþ7nþ3 < xpþ7nþ2 ; xpþ7nþ6 > xpþ7nþ5 > xpþ7nþ4 ; (O2) xpþ7nþ4 xpþ7nþ3 > 1, xpþ7nþ6 xpþ7nþ7 < 1, xpþ7nþ8 xpþ7nþ7 > 1; (O3) xpþ7nþ9 < xpþ7nþ7 . These can be obtained by the definition of positive and negative semicycles and Eq. (2) and Lemma 2.6. Combining the above observations, we derive (R1) xpþ7nþ9 < xpþ7nþ7 <
1 1 1 < < < xpþ7nþ3 < xpþ7nþ2 ; xpþ7nþ6 xpþ7nþ5 xpþ7nþ4
n ¼ 0; 1; 2; . . . 1
From (R1), one can see that fxpþ7nþ2 gn¼0 is decreasing with lower bound 1. So, the limit limn!1 xpþ7nþ2 ¼ L exists and is finite. Accordingly, by view of (R1), we obtain lim xpþ7nþ7 ¼ lim xpþ7nþ3 ¼ L
n!1
n!1
and
lim xpþ7nþ6 ¼ lim xpþ7nþ5
n!1
n!1
1 ¼ lim xpþ7nþ4 ¼ : n!1 L Then, we check that L ¼ 1. Eq. (2) implies that xpþ7nþ7 ¼
xpþ7nþ5 þ xpþ7nþ6 xpþ7nþ4 þ a : xpþ7nþ5 xpþ7nþ6 þ xpþ7nþ4 þ a
Thus, one can take the limits on both sides of the above equality and obtain 1
þ L1 L1 þ a ¼ 1: L1 þ L1 þ a L
L ¼ L1
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Again, by taking limits on both sides of the equality xpþ7nþ8 ¼
xpþ7nþ6 þ xpþ7nþ7 xpþ7nþ5 þ a ; xpþ7nþ6 xpþ7nþ7 þ xpþ7nþ5 þ a
one derives lim xpþ7nþ8 ¼
n!1
1þ1þa ¼ 1: 1þ1þa
Up to now, we have shown limn!1 xpþ7nþk ¼ 1, k ¼ 2; . . . ; 8, which completes the proof of Theorem 1.2.
References [1] R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992. [2] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [3] G. Ladas, Open problems and conjectures, J. Differen. Equat. Appl. (1998), 4.1. [4] Li Xianyi, Zhu Deming, Global asymptotic stability of a kind of nonlinear delay difference equations, Appl. Math.––JCU, Ser. B 17 (2) (2002) 178–183. [5] Tim Nesemann, Positive solution difference equations: some results and applications, Nonlinear Anal. 47 (2001) 4707–4717. [6] A.M. Amleh, N. Kruse, G. Ladas, On a class of difference equations with strong negative feedback, J. Differen. Equat. Appl. 5 (1999) 497–515. [7] G. Ladas, Progress report on xn¼1 ¼ ða þ bxn þ cxn1 Þ=ðA þ Bxn þ Cxn1 Þ, J. Differen. Equat. Appl. 5 (1995) 211–215. [8] A.M. Amleh, E.A. Grove, D.A. Georgiou, G. Ladas, On the recursive sequence xnþ1 ¼ a þ xn1 =xn , J. Math. Anal. Appl. 233 (1999) 790–798. [9] C. Gibbons, M.R.S. Kulenovic, G. Ladas, On the recursive sequence xnþ1 ¼ ða þ bxn1 Þ= ðc þ xn Þ, Math. Sci. Res. Hot-line 4 (2) (2000). [10] Li Xianyi, Xiao Gongfu, Periodicity and strict oscillation for generalized Lyness equations, Appl. Math. Mech. 21 (4) (2000) 455–460. [11] Li Xianyi, Several properties of a kind of nonlinear delay difference equations, Appl. Math. 15 (1) (2000) 27–30. [12] Li Xianyi et al., A conjecture by G. Ladas, Appl. Math.––JCU, Ser. B 13 (1) (1998) 39–44.
Global asymptotic stability for two recursive difference equations q Xianyi Li
a,b,* ,
Deming Zhu
b
a b
School of Mathematics and Physics, Nanhua University, Hengyang 421001, PR China Department of Mathematics, East China Normal University, Shanghai 200062, PR China
Abstract Two sufficient conditions are obtained for the global asymptotic stability of the following two recursive difference equations xn xn1 þ xn2 þ a ; n ¼ 0; 1; 2; . . . ; xnþ1 ¼ xn þ xn1 xn2 þ a and xnþ1 ¼
xn1 þ xn xn2 þ a ; xn xn1 þ xn2 þ a
n ¼ 0; 1; 2; . . . ;
where a 2 ½0; 1Þ and the initial values x2 ; x1 ; x0 2 ð0; 1Þ. Some known results are included. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Recursive difference equation; Global asymptotic stability; Semicycle
1. Introduction The qualitative analysis of recursive difference equations has been the object of the recent study. For example, see [1–12] and the references cited therein.
q Project by NNSF of China (grant: 10071022), Mathematical Tianyuan Foundation of China (grant: TY10026002-01-05-03), and Shanghai Priority Academic Discipline. * Corresponding author. Address: Department of Mathematics, East China Normal University, Shanghai 200062, PR China. E-mail addresses: [email protected] (X. Li), [email protected] (D. Zhu).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00286-8
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In [3], Ladas put forward to investigate the global asymptotic stability of the following rational difference equation xnþ1 ¼
xn þ xn1 xn2 ; xn xn1 þ xn2
n ¼ 0; 1; . . . ;
ðE1 Þ
where the initial values x2 ; x1 ; x0 2 ð0; 1Þ. The authors of this paper obtained in [4] a general result implying the global asymptotic stability of (E1 ). Recently, Tim Nesemann [5] studied the following difference equation xnþ1 ¼
xn1 þ xn xn2 ; xn xn1 þ xn2
n ¼ 0; 1; . . . ;
ðE2 Þ
where the initial values x2 ; x1 ; x0 2 ð0; 1Þ, and mainly utilized the strong negative feedback property of [6] to derive the following result. Theorem A. The positive equilibrium point of Eq. (E2 ) is globally asymptotically stable. To be motivated by the above studies, in this paper we consider the following two equations xnþ1 ¼
xn xn1 þ xn2 þ a ; xn þ xn1 xn2 þ a
n ¼ 0; 1; 2; . . .
ð1Þ
xnþ1 ¼
xn1 þ xn xn2 þ a ; xn xn1 þ xn2 þ a
n ¼ 0; 1; 2; . . . ;
ð2Þ
and
where a 2 ½0; 1Þ and the initial values x2 ; x1 ; x0 2 ð0; 1Þ. Obviously, Eq. (1) has a reverse form of Eq. (E1 ) as a ¼ 0, and Eq. (2) is an extension of Eq. (E2 ). Eqs. (1) and (2) are interesting in their own right. To the best of our knowledge, however, Eqs. (1) and (2) with a > 0 have not been investigated so far. Therefore, theoretically, it is meaningful to study their qualitative properties. Because of the similarity of the structure between Eqs. (1) and (E1 ), whether are their properties similar or not? Accordingly, a conjecture naturally arises. Conjecture B. Assume that a 2 ½0; 1Þ. The positive equilibrium of Eq. (1) is globally asymptotically stable. Analogously, Whether or not is Eq. (2) globally asymptotically stable like Eq. (E2 ) either? Along a new road in this note, we give a positive answer to the above two problems. Whereas, it is extremely difficult to use the method in the known
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literature, such as [1,2,5,6], to obtain the global asymptotic stability of Eq. (1) or (2). Just as what we watch, the structure between Eqs. (1) and (2) is also very similar. But, as we will see in the following that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of the two equations to successively occur is very different. Thus, it is necessary to consider these two equations separately. Our main results are the following. Theorem 1.1. Assume that a 2 ½0; 1Þ. Then the positive equilibrium of Eq. (1) is globally asymptotically stable. Theorem 1.2. Assume that a 2 ½0; 1Þ. Then the positive equilibrium of Eq. (2) is globally asymptotically stable. It is easy to see that Eq. (1) (or (2)) has a unique positive equilibrium x ¼ 1. In the following we always denote by x the unique positive equilibrium point of Eq. (1) (or (2)). Definition 1.3. A positive semicycle of a solution fxn g1 n¼2 of Eq. (1) (or (2)) consists of a ‘‘string’’ of terms fxl ; xlþ1 ; . . . ; xm g, all greater than or equal to the equilibrium x, with l P 2 and m 6 1 such that either l ¼ 2 or l > 2 and
xl1 < x
either m ¼ 1 or m < 1
xmþ1 < x:
and and
1
A negative semicycle of a solution fxn gn¼2 of Eq. (1) (or (2)) consists of a ‘‘string’’ of terms fxl ; xlþ1 ; . . . ; xm g, all less than x, with l P 2 and m 6 1 such that either l ¼ 2 or l > 2 and
xl1 P x
either m ¼ 1 or m < 1
xmþ1 P x:
and and 1
Definition 1.4. A solution fxn gn¼2 of Eq. (1) (or (2)) is said to be eventually positive (negative) if xn x > 0 (<0) eventually holds. For the other concepts in this paper, see [1,2].
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2. Several lemmas In order to simplify the proofs of our main results, we need the following lemmas. We first consider Eq. (1). Lemma 2.1. A positive solution fxn g1 n¼1 of Eq. (1) is eventually equal to 1 if and only if ðx1 1Þðx0 1Þðx2 x0 Þ ¼ 0:
ð3Þ
Proof. Assume that (3) holds. Then according to Eq. (1), it is easy to see that the following conclusions hold: (a) if x0 ¼ 1, then xn ¼ 1 for n P 2; (b) if x1 ¼ 1, then xn ¼ 1 for n P 1; (c) if x2 ¼ x0 , then xn ¼ 1 for n P 3. Conversely, assume that ðx1 1Þðx0 1Þðx2 x0 Þ 6¼ 0:
ð4Þ
Then we show xn 6¼ 1
for any n P 1:
Assume the contrary that for some N P 1, xN ¼ 1 and that xn 6¼ 1 for 1 6 n 6 N 1:
ð5Þ
Clearly, 1 ¼ xN ¼
xN 1 xN 2 þ xN 3 þ a ; xN 1 þ xN 2 xN 3 þ a
which implies xN 1 ¼ xN 3 and by (4), N P 2. Thus, from xN 3 ¼ xN 1 ¼
xN 2 xN 3 þ xN 4 þ a ; xN 2 þ xN 3 xN 4 þ a
one can obtain ðxN 3 1ÞðxN 4 ðxN 3 þ 1Þ þ aÞ ¼ 0, which contradicts (5).
Remark 2.1. If the initial conditions do not satisfy equality (3), then, for any solution fxn g of Eq. (1), xn 6¼ 1 for n P 1 and xn 6¼ xn2 for n P 0. Lemma 2.2. Let fxn g1 n¼2 be a positive solution of Eq. (1) which is not eventually equal to 1. Then the following conclusions are true:
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(a1) ðxnþ1 1Þðxn1 1Þðxn xn2 Þ > 0, for n P 0; (b1) ðxnþ1 xn1 Þðxn1 1Þ < 0, for n P 0; (c1) ðxnþ1 1Þðxn1 1Þðxn2 1Þ < 0, for n P 1. Proof. In view of Eq. (1), we obtain xnþ1 xn1 ¼
ð1 xn1 Þðxn2 ð1 þ xn1 Þ þ aÞ ; xn þ xn1 xn2 þ a
n ¼ 0; 1; 2; . . .
and xnþ1 1 ¼
ðxn xn2 Þðxn1 1Þ ; xn þ xn1 xn2 þ a
n ¼ 0; 1; 2; . . .
From this, inequalities (a1) and (b1) follow. Inequality (c1) is an immediate consequence of inequalities (a1) and (b1). The proof is complete. Now we analyze the situation of the semicycles of nontrivial solutions of Eq. (1). Lemma 2.3. Assume that x2 ; x1 ; x0 2 ð0; 1Þ. Let fxn g be an arbitrary solution of Eq. (1). Then xn < 1 for all n P 1 if and only if x0 > x2 . Proof. If x0 > x2 , then, according to Lemma 2.2(a1) and (c1), xn < 1 for all n P 1. Conversely, if xn < 1 for all n P 1, then, still by Lemma 2.2(a1), one can see x0 > x2 . From Lemma 2.3, it is easily seen that, except perhaps for the first semicycle, a negative semicycle of a solution contains at most two terms. Considering positive semicycles, it is clear again by Lemma 2.2(c1) that, except perhaps for the first semicycle, a positive semicycle contains at most three terms. More precisely, we have the following result. Lemma 2.4. Consider a solution of Eq. (1) which is not eventually less than or equal to 1. Then, with the possible exception of the first semicycle, the following affirmations hold: (d1) every negative semicycle consists of one or two terms; every positive semicycle consists of one or three terms; (e1) every negative semicycle of length one is followed by a positive semicycle of length three; every negative semicycle of length two is followed by a positive semicycle of length one; (f1) every positive semicycle of length one is followed by a negative semicycle of length one; every positive semicycle of length three is followed by a negative semicycle of length two;
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Proof. By Lemma 2.2(c1) and Lemma 2.3, one can see that a negative semicycle of a solution which is not eventually less than or equal to 1, has one or two terms and, by Lemma 2.2(c1), a positiive semicycle has at most three terms. Obviously, for some p P 0, one of the following cases must occur: Case 1: xp2 < 1, xp1 > 1 and xp < 1. It follows from Lemma 2.2 (c1) that xpþ1 > 1, xpþ2 > 1, xpþ3 > 1 and xpþ4 < 1. Then we have xpþ5 < 1. It means that a positive semicycle of length 1 is followed by a negative semicycle of length 1, which in turn is followed by a positive semicycle of length 3. Case 2: xp2 < 1, xp1 > 1 and xp > 1. We have, again in view of Lemma 2.2(c1), that xpþ1 > 1, xpþ2 < 1, xpþ3 < 1 and xpþ4 > 1. It implies xpþ5 < 1, and hence, a positive semicycle of length 3 is followed by a negative semicycle of length 2, which in turn is followed by a positive semicycle of length 1. The proof is complete. Remark 2.2. It follows from Lemma 2.4 that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of Eq. (1) to successively occur is . . . , 3, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, . . . Similarly, we can derive the following results for Eq. (2). 1
Lemma 2.5. A positive solution fxn gn¼1 of Eq. (2) is eventually equal to 1 if and only if ðx0 1Þðx1 x2 Þ ¼ 0:
ð6Þ
Remark 2.3. If the initial conditions do not satisfy equality (6), then xn 6¼ 1 for n P 0 and xn 6¼ xn1 for n P 1 for any solution fxn g of Eq. (2). Lemma 2.6. Let fxn g1 n¼2 be a positive solution of Eq. (1) which is not eventually equal to 1. Then the following conclusions are true: (a2) ðxnþ1 1Þðxn 1Þðxn1 xn2 Þ < 0, for n P 0; (b2) ðxnþ1 xn Þðxn 1Þ < 0, for n P 0; (c2) ðxnþ1 1Þðxn 1Þðxn2 1Þ > 0, for n P 2. Lemma 2.7. Assume that x2 ; x1 ; x0 2 ð1; 1Þ. Let fxn g be a solution of Eq. (2). Then xn > 1 for all n P 1 if and only if x1 < x2 . The proofs of Lemmas 2.5–2.7 are completely similar to those of Lemmas 2.1–2.3 and so omitted here.
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We are now in a position to analyze in detail the situation of the semicycles of nontrivial solutions of Eq. (2). Based on Lemma 2.6, we have the following result.
Lemma 2.8. With the possible exception of the first semicycle, the following affirmations are true for any solution of Eq. (2) which is not eventually greater than or equal to 1. (d2) Every positive semicycle has one or two terms; every negative semicycle has one or three terms; (e2) every positive semicycle with length one is followed by a negative semicycle with length one; every positive semicycle with length two is followed by a negative semicycle with length three; (f2) every negative semicycle with length one is followed by a positive semicycle with length two; every negative semicycle with length three is followed by a positive semicycle with length one.
Proof. Due to Lemma 2.6(c2) and Lemma 2.7, one can see that a positive semicycle of a solution which is not eventually greater than or equal to 1, has one or two terms and, again due to Lemma 2.6(c2), a negative semicycle has at most three terms. Obviously, for some p P 0, one of the following cases has to appear: Case 1: xp2 < 1, xp1 > 1 and xp < 1. Applying Lemma 2.6(c2) to the case we have that xpþ1 > 1, xpþ2 > 1, xpþ3 < 1 and xpþ4 < 1, which leads to xpþ5 < 1. Hence, we know a positive semicycle of length 1 is followed by a negative semicycle of length 1, which in turn is followed by a positive semicycle of length 2.Case 2: xp2 < 1, xp1 > 1 and xp > 1. Using Lemma 2.6(c2) again, we get xpþ1 < 1, xpþ2 < 1, xpþ3 < 1, xpþ4 > 1 and xpþ5 < 1. Therefore, a positive semicycle of length 2 is followed by a negative semicycle of length 3, and the later in turn is followed by a positive semicycle of length 1. The proof is complete.
Remark 2.4. It is clear from Lemma 2.8 that the rule for the lengths of positive and negative semicycles of nontrivial solutions of Eq. (2) to alternately occur is . . . , 2, 3, 1, 1, 2, 3,1,1, 2, 3, 1, 1, . . ., which, compared with Remark 2.2, manifests that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of the two equations to occur is very different.
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3. Proofs of theorems We now begin with the proofs of these two theorems. First, show that the positive equilibrium point of Eq. (1) is globally asymptotically stable. Proof of Theorem 1.1. We must prove that the positive equilibrium point x of Eq. (1) is both locally asymptotically stable and globally attractive. The linearized equation of Eq. (1) about the positive equilibrium x ¼ 1 is ynþ1 ¼ 0 yn þ 0 yn1 þ 0 yn2 ;
n ¼ 0; 1; . . .
By virtue of [2, Remark 1.3.7], x is locally asymptotically stable. It remains to 1 verify that every positive solution fxn gn¼2 of Eq. (1) converges to 1 as n ! 1. Namely, we want to prove lim xn ¼ x ¼ 1:
ð7Þ
n!1
If the solution fxn g1 n¼2 of Eq. (1) is trivial, then (7) obviously holds. If the solution is nonoscillatory about the positive equilibrium point x of Eq. (1), then according to Lemma 2.3, we know that the solution is actually an eventually negative one. Furthermore, we can easily derive from Lemma 1 2.2(b1) that the subsequences fx2n g and fx2nþ1 g of the solution fxn gn¼2 of Eq. (1) are eventually monotonically increasing and bounded from the above. Therefore, the limits limn!1 x2n ¼ L and limn!1 x2nþ1 ¼ M exist and are finite. Note x2n ¼
x2n1 x2n2 þ x2n3 þ a x2n1 þ x2n2 x2n3 þ a
and x2nþ1 ¼
x2n x2n1 þ x2n2 þ a ; x2n þ x2n1 x2n2 þ a
take the limits on both sides of the above equalities and obtain L¼
LM þ M þ a ¼1 M þ LM þ a
and
M¼
LM þ L þ a ¼ 1: L þ ML þ a
That is to say, fxn g tends to the positive equilibrium point x of Eq. (1), and so (7) holds. Therefore, it suffices to prove that (7) holds for the solution to be strictly oscillatory. Consider now fxn g to be strictly oscillatory about the positive equilibrium point x of Eq. (1). By virtue of Lemma 2.4, one understands that every positive semicycle of this solution has one or three terms and every negative semicycle, except perhaps for the first, has one or two terms. Every positive semicycle of length one is followed by a negative semicycle of length one, which is followed by a positive semicycle of length three, in turn followed by the negative semicycle length two. For the convenience of statement, without loss of generality, we use the following denotation. We denote by xp the term of a positive semicycle of length one, followed by xpþ1 , which is the term of a negative semicycle of length
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one. Afterwards there is the positive semicycle: xpþ2 , xpþ3 , ,xpþ4 , in turn followed by the negative semicycle: xpþ5 , xpþ6 , and so on. Following this way, we get the following positive and negative semicycles: fxpþ7n g; fxpþ7nþ1 g; fxpþ7nþ2 ; xpþ7nþ3 ; xpþ7nþ4 g; fxpþ7nþ5 ; xpþ7nþ6 g; n ¼ 0; 1; . . . We have the following observations: (O1) xpþ7nþ4 < xpþ7nþ3 < xpþ7nþ2 ; xpþ7nþ6 > xpþ7nþ5 ; (O2) xpþ7nþ1 xpþ7nþ3 < 1, xpþ7nþ5 xpþ7nþ3 > 1, xpþ7nþ6 xpþ7nþ4 > 1; (O3) xpþ7nþ8 > xpþ7nþ6 . In fact, the inequality (O1) is followed straightforward from Lemma 2.2(b1), while (O2) is a direct consequence of the observations of xpþ7nþ2 xpþ7nþ1 þ xpþ7n þ a xpþ7nþ2 þ xpþ7nþ1 xpþ7n þ a xpþ7nþ2 xpþ7nþ1 þ xpþ7n þ a 1 < ¼ ; xpþ7nþ2 x2pþ7nþ1 þ xpþ7nþ1 xpþ7n þ axpþ7nþ1 xpþ7nþ1
xpþ7nþ3 ¼
xpþ7nþ4 xpþ7nþ3 þ xpþ7nþ2 þ a xpþ7nþ4 þ xpþ7nþ3 xpþ7nþ2 þ a xpþ7nþ4 xpþ7nþ3 þ xpþ7nþ2 þ a 1 > ¼ ; 2 xpþ7nþ4 xpþ7nþ3 þ xpþ7nþ3 xpþ7nþ2 þ axpþ7nþ3 xpþ7nþ3
xpþ7nþ5 ¼
and xpþ7nþ5 xpþ7nþ4 þ xpþ7nþ3 þ a xpþ7nþ5 þ xpþ7nþ4 xpþ7nþ3 þ a xpþ7nþ54 xpþ7nþ4 þ xpþ7nþ3 þ a 1 > ¼ : xpþ7nþ5 x2pþ7nþ4 þ xpþ7nþ4 xpþ7nþ3 þ axpþ7nþ4 xpþ7nþ4
xpþ7nþ6 ¼
Immediately, we also obtain xpþ7nþ8 ¼
xpþ7nþ7 xpþ7nþ6 þ xpþ7nþ5 þ a > xpþ7nþ6 ; xpþ7nþ7 þ xpþ7nþ6 xpþ7nþ5 þ a
n ¼ 0; 1; 2; . . .
This verifies (O3). Combining the above observations, we derive 1 (R1) xpþ7nþ8 > xpþ7nþ6 > xpþ7nþ5 > xpþ7nþ3 > xpþ7nþ1 ; n ¼ 0; 1; 2; . . . and 1 1 (R2) xpþ7nþ6 > xpþ7nþ4 > xpþ7nþ3 > xpþ7nþ1 ; n ¼ 0; 1; 2; . . .
From (R1), one can see that fxpþ7nþ1 g1 n¼0 is increasing with upper bound 1. So, the limit limn!1 xpþ7nþ1 ¼ L exists and is finite. Accordingly, by view of (R1), we obtain
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lim xpþ7nþ6 ¼ lim xpþ7nþ5 ¼ L
n!1
n!1
and
1 lim xpþ7nþ3 ¼ : L
n!1
Thus, in light of (R2), it follows that 1 lim xpþ7nþ4 ¼ lim xpþ7nþ3 ¼ : n!1 L
n!1
Now, we verify that L ¼ 1. To this end note xpþ7nþ6 ¼
xpþ7nþ5 xpþ7nþ4 þ xpþ7nþ3 þ a ; xpþ7nþ5 þ xpþ7nþ4 xpþ7nþ3 þ a
take the limits on both sides of the above equality and obtain L¼
L L1 þ L1 þ a ; L þ L1 L1 þ a
which yields ðL 1ÞðL þ 1 þ aÞ ¼ 0 and so L ¼ 1. Analogously, by taking limits on both sides of the equality xpþ7nþ7 ¼
xpþ7nþ6 xpþ7nþ5 þ xpþ7nþ4 þ a ; xpþ7nþ6 þ xpþ7nþ5 xpþ7nþ4 þ a
we derive lim xpþ7nþ7 ¼
n!1
L L þ L1 þ a ¼ 1: L þ L L1 þ a
Again, by taking limits on both sides of the equality xpþ7nþ9 ¼
xpþ7nþ8 xpþ7nþ7 þ xpþ7nþ6 þ a ; xpþ7nþ8 þ xpþ7nþ7 xpþ7nþ6 þ a
one derives lim xpþ7nþ9 ¼
n!1
LLþ1þa ¼ 1: LþLþa
Up to now, we have shown limn!1 xpþ7nþk ¼ 1, k ¼ 1; 2; . . . ; 7. So, the proof for Theorem 1.1, is complete. Next, we show that the positive equilibrium point x of Eq. (2) is globally asymptotically stable. Proof of Theorem 1.2. Similar to the previous proof of Theorem 1.1, we only need to prove that lim xn ¼ x ¼ 1
n!1
holds when the solution fxn g of Eq. (2) is strictly oscillatory.
ð8Þ
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491
It is clear from Remark 2.4 that the rule for the numbers of terms of positive and negative semicycles of nontrivial solutions of Eq. (2) to occur is . . . , 2, 3, 1, 1, 2, 3,1,1, 2, 3, 1, 1,. . . We denote by xp the term of a positive semicycle of length one, followed by xpþ1 , which is the term of a negative semicycle of length one. Afterwards there is the positive semicycle of length two: xpþ2 , xpþ3 , in turn followed by the negative semicycle of length three: xpþ4 , xpþ5 , xpþ6 , and so on. So, we can turn and turn about getting the following positive and negative semicycles: fxpþ7n g; fxpþ7nþ1 g; fxpþ7nþ2 ; xpþ7nþ3 g; fxpþ7nþ4 ; xpþ7nþ5 ; xpþ7nþ6 g; n ¼ 0; 1; . . . We claim that the following inequalties are valid. (O1) xpþ7nþ3 < xpþ7nþ2 ; xpþ7nþ6 > xpþ7nþ5 > xpþ7nþ4 ; (O2) xpþ7nþ4 xpþ7nþ3 > 1, xpþ7nþ6 xpþ7nþ7 < 1, xpþ7nþ8 xpþ7nþ7 > 1; (O3) xpþ7nþ9 < xpþ7nþ7 . These can be obtained by the definition of positive and negative semicycles and Eq. (2) and Lemma 2.6. Combining the above observations, we derive (R1) xpþ7nþ9 < xpþ7nþ7 <
1 1 1 < < < xpþ7nþ3 < xpþ7nþ2 ; xpþ7nþ6 xpþ7nþ5 xpþ7nþ4
n ¼ 0; 1; 2; . . . 1
From (R1), one can see that fxpþ7nþ2 gn¼0 is decreasing with lower bound 1. So, the limit limn!1 xpþ7nþ2 ¼ L exists and is finite. Accordingly, by view of (R1), we obtain lim xpþ7nþ7 ¼ lim xpþ7nþ3 ¼ L
n!1
n!1
and
lim xpþ7nþ6 ¼ lim xpþ7nþ5
n!1
n!1
1 ¼ lim xpþ7nþ4 ¼ : n!1 L Then, we check that L ¼ 1. Eq. (2) implies that xpþ7nþ7 ¼
xpþ7nþ5 þ xpþ7nþ6 xpþ7nþ4 þ a : xpþ7nþ5 xpþ7nþ6 þ xpþ7nþ4 þ a
Thus, one can take the limits on both sides of the above equality and obtain 1
þ L1 L1 þ a ¼ 1: L1 þ L1 þ a L
L ¼ L1
492
X. Li, D. Zhu / Appl. Math. Comput. 150 (2004) 481–492
Again, by taking limits on both sides of the equality xpþ7nþ8 ¼
xpþ7nþ6 þ xpþ7nþ7 xpþ7nþ5 þ a ; xpþ7nþ6 xpþ7nþ7 þ xpþ7nþ5 þ a
one derives lim xpþ7nþ8 ¼
n!1
1þ1þa ¼ 1: 1þ1þa
Up to now, we have shown limn!1 xpþ7nþk ¼ 1, k ¼ 2; . . . ; 8, which completes the proof of Theorem 1.2.
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