- Email: [email protected]
On the higher order rational recursive sequence xn=Axn-k+Bxn-3k
Applied Mathematics and Computation 173 (2006) 710–723 www.elsevier.com/locate/amc
On the higher order rational A B þ xn3k recursive sequence xn ¼ xnk Majid Jaberi Douraki a,1, Mehdi Dehghan Mohsen Razzaghi a,b a
a,*
,
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iran b Department of Mathematics and Statistics, Mississippi State University, Mississippi state, MS 39762, U.S.A
Abstract The main purpose of the current paper is to prove that every positive solution of the delay difference equation xn ¼
A B þ ; xnk xn3k
where A, B 2 (0, 1), the initial conditions x3k+1, x3k+2, . . ., x0 2 (0, 1) and k 2 {1, 2, 3, . . .}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion. 2005 Elsevier Inc. All rights reserved. Keywords: Semi-cycle; Difference equations; Stability; Global attractivity
*
Corresponding author. E-mail addresses: [email protected] (M. Jaberi Douraki), [email protected] (M. Dehghan), [email protected] (M. Razzaghi). 1 Department of Mathematics and Statistics, University of Laval, Quebec, Canada. 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.007
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1. Preliminaries Consider the recursive sequence A B xn ¼ þ ; n ¼ 1; 2; . . . ; ð1Þ xnk xn3k where A, B 2 (0, 1), k 2 {1, 2, 3, . . .}, and the initial conditions x3k+1, x3k+2, . . ., x0, are arbitrary positive real numbers. We investigate the periodic character of the positive solution of Eq. (1) and we shall show that the sequence {xn} converges eventually to a period-k solution. This confirms Conjecture 4.8.1 in [10,11] given by Ladas and his co-workers for the above mentioned particular case. Basically, we generalize some of the results due to DeVault et al. in [7,10] related to the difference equation A 1 xn ¼ þ ; n ¼ 1; 2; . . . ; ð2Þ xn1 xn3 where x2, x1, x0, A 2 (0, 1). DeVault showed that the solution of Eq. (2) is periodic and eventually converges to a period-2 solution. Observe that Eq. (1) reduces to Eq. (2) when k = 1 and B = 1. Periodicity, and some other important properties of specific classes of difference equations, was investigated in [4–6,8,14]. In particular, the boundedness and global stability received more attention recently [3–5,7,13,14]. More information on nonlinear higher order recursive sequences can be found in [1,2,9–13]. Let {xn} be a positive solution of Eq. (1), and xn xnk yn ¼ . B Then Eq. (1) turns into the following: y y n ¼ C þ nk ; n ¼ 1; 2; . . . ; ð3Þ y n2k where C = A/B 2 (0, 1), k 2 {1, 2, 3, . . .}, the initial conditions y2k+1, y2k+2, . . ., y0 are arbitrary positive real numbers, and fy n g1 n¼2kþ1 satisfies Eq. (3). In the rest of this paper we need the following definitions: Let I be an interval of real numbers, and f : I · I ! I be a continuous differentiable function. Consider the difference equation [15–22] y n ¼ f ðy nk ; y n2k Þ;
n ¼ 1; 2; . . . ;
ð4Þ
where y2k+1, . . ., y1, y0 2 I. Definition 1. The equilibrium point y ¼ C þ 1 of Eq. (4) is the point that satisfies the condition y ¼ f ðy ; y Þ, or equivalently, y is a fixed point of f. So the sequence fy n g1 y for all n P 2k + 1 is a solution of Eq. (3). n¼2kþ1 with y n ¼
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Definition 2. Let I be an interval of real numbers, the equilibrium y of Eq. (3) is said to be (i) locally stable if for every positive e > 0, there exists d > 0, such that for all y2k+1, . . ., y1, y0 2 I, with jy 2kþ1 y j þ þ jy 1 y j þ jy 0 y j < d, we have jy n y j < e, for all n P 2k + 1. (ii) locally asymptotically stable if it is locally stable, and there exists c > 0, such that for all y2k+1, . . ., y1, y0 2 I with jy 2kþ1 y j þ þ jy 1 y j þ jy 0 y j < c, we have Limn!1 y n ¼ y . (iii) a global attractor if for all y2k+1, . . ., y1, y0 2 I, we have Limn!1 y n ¼ y . (iv) globally asymptotically stable if it is locally stable and a global attractor. (v) unstable if it is not locally stable. (vi) a source, or a repeller, if there exists r > 0 such that for all y2k+1, . . ., y1, y0 2 I with 0 < jy 2kþ1 y j þ þ jy 1 y j þ jy 0 y j < r, there exists N P 1 such that jy N y j P r. Clearly a source is an unstable equilibrium.
2. Local stability of Eq. (3) In this section, we discuss locally asymptotically stable of Eq. (3). Let x f ðx; yÞ ¼ C þ ; y assume that p :¼ pðy ; y Þ ¼
of ðy ; y Þ 1 1 ¼ ¼ y C þ 1 ox
q :¼ qðy ; y Þ ¼
of ðy ; y Þ 1 1 ¼ ¼ ; y oy Cþ1
and
denote the partial derivates of f(x, y) evaluated at the equilibrium point y of Eq. (3). Then the equation zn ¼pznk þ qzn2k ¼
1 1 znk zn2k Cþ1 Cþ1
ð5Þ
is called the Linearized equation associated with Eq. (3) about the equilibrium point y . Its characteristic equation is
k2k
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1 1 kk þ ¼ 0. Cþ1 Cþ1
ð6Þ
By substituting h = kk, it turns to the following equation: h2
1 1 hþ ¼ 0. Cþ1 Cþ1
ð7Þ
We observe that if all roots of Eq. (6) lie inside the open unit disk jkj < 1, then the equilibrium y is locally asymptotically stable; equivalently all roots of Eq. (7) lie inside the open unit disk jhj < 1. The following facts are well known (see [9]). Theorem A (Linearized stability) (a) If both roots of the quadratic Eq. (7) lie in the open unit disk jhj < 1, then the equilibrium y of Eq. (3) is locally asymptotically stable. (b) If at least one root of Eq. (7) has absolute value greater than one, then the equilibrium y of Eq. (3) is unstable. (c) A necessary and sufficient condition for both roots of Eq. (7) to lie in the open unit disk jhj < 1, is jpj < 1 q 6 2. In this case the locally asymptotically stable equilibrium y is also called a sink. (d) A necessary and sufficient condition for both roots of Eq. (7) to have absolute value greater than one is: jqj > 1
and
jpj < j1 qj.
In this case the equilibrium point y is called a repeller. (e) A necessary and sufficient condition for one root of Eq. (7) to have absolute value less than one and the other root of Eq. (3) to have absolute value greater than one is p2 þ 4q > 0 and
jpj > j1 qj.
In this case the unstable equilibrium y is called a saddle point. It follows from part (c) of Theorem A, that the equilibrium y is locally asymptotically stable for all C > 0. 3. Analysis of semi-cycle and global stability of Eq. (3) In this section, first we investigate the semi-cycle of Eq. (3). Next we consider the global behavior of solutions of it and we prove that the equilibrium y is globally asymptotically stable for all C > 0.
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Definition 3. Let fy n g1 n¼2kþ1 be a positive solution of Eq. (3). A positive semicycle of fy n g1 consists of a ‘‘string’’ of terms {yl, yl+1 . . ., ym}, all greater n¼2kþ1 than or equal to the equilibrium y , with l P 2k + 1 and m 6 1 and such that either l ¼ 2k þ 1;
or l > 2k þ 1;
and
y l1 < y ;
and either m ¼ 1;
or m < 1;
and
y mþ1 < y .
Definition 4. Let fy n g1 n¼2kþ1 be a positive solution of Eq. (3). A negative semicycle of fy n g1 consists of a ‘‘string’’ of terms {yl, yl+1 . . ., ym}, all less than n¼2kþ1 or equal to the equilibrium y , with l P 2k + 1 and m 6 1 and such that either l ¼ 2k þ 1;
or l > 2k þ 1;
and
y l1 P y ;
and either m ¼ 1;
or m < 1;
and
y mþ1 P y .
1
Definition 5. A solution fy n gn¼2kþ1 of Eq. (3) is called nonoscillatory if there exists N P 1 such that either y n > y for all n P N ; or y n < y for all n P N . 1 fy n gn¼2kþ1
is called oscillatory if it is not nonoscillatory.
Lemma 1. Let {yn} be a nontrivial positive solution of Eq. (3). Then the following statements hold: (i) {yn} oscillates about the equilibrium y with a semi-cycle of length at most 3k. (ii) {ynk+i} oscillates about the equilibrium y with a semi-cycle of length at most 2 or 3, for i = 0, 1, . . ., k 1, and the extreme point in a semi-cycle occurs in the first or second term. (iii) For n > 3k, we have C < yn < C þ
1þC . C
ð8Þ
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Proof (i) Let {yn} be a solution of Eq. (3). We show that every positive semi-cycle, except possibly the first, has 3k terms. The case of negative semi-cycle is similar. Let y N P y be the first term in a positive semi-cycle. If yN+k < yN, then y N þ2k ¼ A þ
y N þk < A þ 1 ¼ y . yN
So in this case the lemma is true. If yN+k P yN, then y y N þ2k ¼ A þ N þk P A þ 1 ¼ y yN and y N þ2k ¼ A þ
y N þk y y 6 A þ N þk ¼ A þ N þk < y N þk . y yN Aþ1
Therefore y 6 y N þ2k < y N þk , and we finally have y y N þ3k ¼ A þ N þ2k < A þ 1 ¼ y . y N þk 1
This shows that the positive semi-cycle of fy n gn¼2kþ1 has length at most 3k. From this, we can deduce that every solution {yn} of Eq. (3) oscillates about the equilibrium y . 1 (ii) Let fy nkþi gn¼1 be a solution of Eq. (3), for i = 0, 1, . . ., k 1. We show that every positive semi-cycle, except possibly the first, has 2 or 3 terms as part (i). Assume i 2 {0, 1, . . ., k 1}, and let y Nkþi P y be the first term in a positive semi-cycle. Then y ðN 1Þkþi < y , and y y ðN þ1Þkþi ¼ A þ Nkþi > A þ 1 ¼ y P y ðN 1Þkþi . y ðN 1Þkþi If y(N+1)k+i > yNk+i, then y ðN þ2Þkþi > y , and y ðN þ1Þkþi y ðN þ1Þkþi y ðN þ2Þkþi ¼ A þ < y ðN þ1Þkþi . 6Aþ y y Nkþi As a result y < y ðN þ2Þkþi < y ðN þ1Þkþi , and y ðN þ3Þkþi < y . In this case, the positive semi-cycle of {ynk+i} has length 3. If y(N+1)k+i < yNk+i then y ðN þ1Þkþi < y , the positive semi-cycle of {ynk+i} has length 2. From the above discussion, it is clear that the extreme point in a semi-cycle of {ynk+i} occurs in the first or second term. (iii) Clearly yn > C, for n > 0. Assume N > 2k, so we have yN C 1 1 ¼ þ <1þ . C y N k y N k y N 2k
ð9Þ
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By Eqs. (3) and (9), we gain y N þk ¼ C þ
yN 1þC .
This completes the proof.
Theorem 1. Assume that k 2 {1, 2, 3, . . .} and the initial conditions y2k+1, y2k+2, . . ., y0, C are arbitrary positive real numbers. Then the positive equilibrium y ¼ C þ 1 of Eq. (3) is globally asymptotically stable. Proof. We saw that y ¼ C þ 1 is locally asymptotically stable. So it remains to show that y ¼ C þ 1 is a global attractor which means that if {yn} is a nontrivial positive solution of Eq. (3), then Lim y n ¼ C þ 1. n!1
To prove its global attractivity, we define the sequences {Lnk+i} and {Unk+i} as follows: Li ¼ C
and
Ui ¼ C þ
1þC C
for i ¼ 0; 1; . . . ; k 1;
and for n = 0, 1, . . ., k 1, Lðnþ1Þkþi ¼ C þ
1þC U nkþi
and
U ðnþ1Þkþi ¼ C þ
1þC . Lðnþ1Þkþi
We observe that {Lnk+i} and {Unk+i} are sequences of upper and lower bounds for the solution {ynk+i} of Eq. (3), for i = 0, 1, . . ., k 1. We also have Lðnþ1Þkþi ¼ C þ
1þC C þ L1þC nkþi
for i ¼ 0; 1; . . . ; k 1;
ð10Þ
and U ðnþ1Þkþi ¼ C þ
1þC C þ U1þC nkþi
for i ¼ 0; 1; . . . ; k 1.
ð11Þ
One can see that {Lnk+i} is increasing and {Unk+i} is decreasing, for i = 0, 1, . . ., k 1, which means that: Li < Lkþi < < Lnkþi < < y < < U nkþi < < U kþi < U i . Thus Lim Lnþi ¼ Li 6 y and n!1
Lim U nþi ¼ U i 6 y . n!1
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By replacing Li and U i in Eqs. (10) and (11), we obtain: 1þC 1þC Li ¼ C þ and U i ¼ C þ for i ¼ 0; 1; . . . ; k 1. 1þC Cþ L C þ 1þC U i
i
Simple computations show that we have a unique solution U i ¼ Li ¼ y , for i = 0, 1, . . ., k 1. This completes the proof. h 4. The periodic behavior of the solution of Eq. (1) Here we show that the equilibrium of the solution of Eq. (1) is periodic with period k. From Eq. (1), we obtain x2n ¼
A B þ ; x2nk x2n3k
ð12Þ
and x2nk ¼
A B þ . x2n2k x2n4k
ð13Þ
So it follows that x2n ¼
A B þ A . B B þ x2n4k x2n4k þ x2n6k x2n2k A
ð14Þ
The following statements outline properties of the Eq. (1). Their proofs are based on Eq. (14). Lemma 2. Let {xn} be a positive solution of Eq. (1). Then the following statements hold. (i) For N > 0, let mN ¼ Minfx2N 4k ; x2N 2k ; x2N g; and M N ¼ Maxfx2N 4k ; x2N 2k ; x2N g. Then mN 6 x2N þ2lk 6 M N
for l P 1.
(ii) There exist positive numbers m and M such that m < xn < M, for n = 0, 1, . . . (iii) x2nkþi ¼ 1 for i ¼ 0; 1; . . . ; k 1. Lim n!1 xð2n2Þkþi
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Proof (i) First we define the function f by A B þ A B. B þ þ x y y z
f ðx; y; zÞ ¼ A
Note that f is increasing in x, y, and z. So x2N þ2k ¼
A x2N
A B A B þ A 6 A þ A ¼ MN . B B B þ x2N 2k x2N 2k þ x2N 4k M N þ M N M N þ MBN
It can be simply deduced by induction that x2N+2lk 6 MN for all l P 1. Similarly, x2N+2lk P mN for all l P 1. (ii) The proof is similar to (i). (iii) Recall that y 2n ¼ x2n xB2nk converges to the equilibrium, and therefore the sequence x2nkþi xð2n1Þkþi y 2nkþi x2nþi ¼ ¼ ; y 2nkkþi xð2n1Þkþi xð2n2Þkþi xð2n2Þkþi converges to 1. So Lim n!1
x2nkþi ¼ 1. xð2n2Þkþi
The following theorem is the main result of this paper. Theorem 2. Let {xn} be a positive solution of Eq. (1). Then there exist positive constants l0, l1, l2, . . ., l2k1 such that lilk+i = A + B and Lim x2nkþi ¼ li n!1
for i ¼ 0; 1; . . . ; k 1.
Proof. We show that the sequences x2nk+i for i = 0, 1, . . ., k 1 are Cauchy. By Lemma 2 and Eq. (13), we have x2nkþi P x2nkþi x2nk2kþi . 1 x M 2nk2kþi
Hence Lim jx2nkþi x2nk2kþi j ¼ 0 n!1
for i ¼ 0; 1; . . . ; k 1.
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Since {xn} is bounded by Lemma 2, Limn!1x2nk+i exists and is a positive number li. Finally, from Eq. (13) we observe that li lki1 ¼ A þ B. This completes the proof.
h
5. Numerical discussion In this section, we present two test examples to support our theoretical discussion. Example 1. Assume that Eq. (1) hold, k = 2 and A = B = 1. So the equation reduces to the following: xn ¼
1 1 þ . xn2 xn6
Using the change of variable y n ¼ xn xn2 ; the corresponding equation will be as follows: y y n ¼ 1 þ n2 . y n4 In this part, to illustrate the result of this paper a numerical example is given, which was carried out using Maple 7. We assume A = B = 1 and the initial points x5, x4, . . ., x0 in the following table. n
xn
n
xn
n
xn
n
xn
5 4 3 2 1 0 1 2 3 4 5
3.05 1.7 0.03 10 1 4 1.327868852 0.8382352941 34.08641975 1.292982456 1.029337197
20 21 22 23 24 25 26 27 28 29 30
1.874541882 0.7964310198 1.092014157 2.578755699 1.810166430 0.8111423400 1.085899066 2.4884307681 0.836634946 0.7896436056 1.096909405
50 51 52 53 54 55 56 57 58 59 60
1.092287726 2.504462888 1.831370239 0.7984982925 1.092110372 2.505197171 1.831168060 0.7984573881 1.092138753 2.504765819 1.831292940
90 91 92 93 94 95 96 97 98 99 100
1.092146771 2.504889089 1.831255943 0.7984382900 1.092146760 2.504889883 1.831255710 0.7984384186 1.092146696 2.504889688 1.831255772
720
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By Theorem 1, the equilibrium point y ¼ A þ B ¼ 2 is globally asymptotically stable and by Theorem 2, we have l 0 l2 ¼ 2
l1 l3 ¼ 2;
and
ð15Þ
where Lim x4n ¼ l0
Lim x4nþ2 ¼ l2 ;
and
n!1
Lim x4nþ1 ¼ l1
n!1
and
n!1
Lim x4nþ3 ¼ l3 . n!1
As we see, for large n, Eq. (15) converges to the exact solution. We compute the error for Eq. (15) E ¼ jli l2þi x4nþi x4nþ2i j for i ¼ 0; 1. The following table shows the corresponding errors. n
xn · xn+2
E
n
xn · xn+2
E
52 53 54 55 56
2.00005843 2.00029318 1.99988960 2.00039566 1.99983763
0.00005843 0.00029318 0.0001104 0.00039566 0.00016237
94 95 96 97 98
1.99999999 2.00000031 1.99999994 2.00000016 1.99999987
0.00000001 0.00000031 0.00000006 0.00000016 0.00000013
Example 2. Let k = 3, A = 2, and B = 1, so Eq. (1) will be in the following form: xn ¼
2 xn3
þ
1 xn9
.
Using the change of variable y n ¼ xn xn3 ; we will have yn ¼ 2 þ
y n3 . y n9
Let x8, x7, . . ., x0 be the initial points in the following table:
xn
xn · xn+3
E
n
xn
xn · xn+3
E
8 7 6 5 4 3 2 1 0 1 2 3 4 5 6
2 4.2 1 1.78 11 0.1 3 6 0.003 1.166666667 0.5714285714 667.6666666 2.282467532 3.590909091 10.00299551
3.52 46.2 0.1 5.28 66 0.0003 3.50000001 3.428571428 2.002999999 2.662878788 2.051948051 6678.666668 2.7608225101 2.5984848489 3336.3318362
0.52 43.2 2.9 2.28 63 2.997 0.50000001 0.42857142 0.9970000 0.3371212 0.9480519 6648.6666 0.2391774 0.4015151 3333.3318
40 41 42 43 44 45 50 51 52 92 93 94 95 96 97
2.455977836 4.223516072 0.00988088907 1221671980 0.7105214509 303.0713419 0.7104728048 303.2734699 2.455737009 0.7104989319 303.1987610 2.455734427 4.222384829 0.00989450112 1.221630466
3.000399305 3.000898767 2.994614311 3.000036823 2.999858851 2.999696525 2.999931535 3.000666931 3.000022755 2.999999911 3.000000482 2.999999992 2.999999998 3.000000098 3.000000009
0.00039930 0.00089876 0.0053856 0.00003682 0.0001411 0.0003034 0.0000684 0.00066693 0.00002275 0.00000008 0.00000048 0.000000008 0.000000002 0.000000098 0.000000009
M. Jaberi Douraki et al. / Appl. Math. Comput. 173 (2006) 710–723
n
721
722
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where E ¼ jli l3þi x6nþi x6nþ3i j for i ¼ 0; 1; 2 and Lim x6n ¼ l0 ; n!1
Lim x6nþ3 ¼ l3 ; n!1
Lim x6nþ1 ¼ l1 ; n!1
Lim x6nþ4 ¼ l4 ; n!1
Lim x6nþ2 ¼ l2 ; n!1
Lim x6nþ5 ¼ l5 . n!1
We can see, for large n, E converges to zero.
6. Conclusion The results of this paper are generalizations of those of DeVault [2]. This solves the conjecture of Ladas and his co-workers [5,6] for some special cases. However, the general case is still open. By a change of variable, we deduce a new difference equation which involves some interesting results in its own right. For instance, it has a unique positive equilibrium which is oscillatory and globally attractive. This was used to show that every positive solution of Eq. (1) converges to a periodic-k solution.
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[12] V. Lakshmikantham, D. Trigiante, Theory of Difference Equations, Numerical Methods and Applications, Academic Press, New York, 1988. [13] H. Sedaghat, Geometric stability conditions for higher order difference equations, Journal of Mathematical Analysis and Applications 224 (1998) 225–272. [14] W.-S. He, W.-T. Li, X.-X. Yan, Global attractivity of the difference equation xn+1 = a + xnk/ xn, Applied Mathematics and Computation 151 (3) (2004) 879–885. [15] H. Sedaghat, Nonlinear Difference Equations, Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003. nkþ1 þcxn2kþ1 [16] M. Dehghan, M. Jaberi Douraki, On the recursive sequence xnþ1 ¼ aþbx Bxnkþ1 þCxn2kþ1 , Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2005.01.004. [17] M. Dehghan, M. Jaberi Douraki, Dynamics of a rational difference equation using both theoretical and computational approaches, Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2004.09.009. [18] M. Dehghan, M. Nasri, M.R. Razvan, Global stability of a deterministic model for HIV infection in vivo, submitted for publication. [19] M. Nasri, M. Dehghan, M. Jaberi Douraki, Study of a system of non-linear difference equations arising in a deterministic model for HIV infection, Applied Mathematics and Computation. [20] M. Jaberi Douraki, M. Dehghan, M. Razzaghi, The qualitative behavior of solutions of a nonlinear difference equation, Applied Mathematics and Computation, in press, doi:10.1016/ j.amc.2004.12.017. [21] M. Jaberi Douraki, M. Dehghan, A. Razavi, On the global behavior of higher order recursive sequences, Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2004.09.065. [22] M. Jaberi Douraki, The study of some classes of nonlinear difference equations, M.Sc. Thesis, Department of Applied Mathematics, Amirkabir University of Technology, Tehran, Iran, July 2004.
On the higher order rational A B þ xn3k recursive sequence xn ¼ xnk Majid Jaberi Douraki a,1, Mehdi Dehghan Mohsen Razzaghi a,b a
a,*
,
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iran b Department of Mathematics and Statistics, Mississippi State University, Mississippi state, MS 39762, U.S.A
Abstract The main purpose of the current paper is to prove that every positive solution of the delay difference equation xn ¼
A B þ ; xnk xn3k
where A, B 2 (0, 1), the initial conditions x3k+1, x3k+2, . . ., x0 2 (0, 1) and k 2 {1, 2, 3, . . .}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion. 2005 Elsevier Inc. All rights reserved. Keywords: Semi-cycle; Difference equations; Stability; Global attractivity
*
Corresponding author. E-mail addresses: [email protected] (M. Jaberi Douraki), [email protected] (M. Dehghan), [email protected] (M. Razzaghi). 1 Department of Mathematics and Statistics, University of Laval, Quebec, Canada. 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.007
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1. Preliminaries Consider the recursive sequence A B xn ¼ þ ; n ¼ 1; 2; . . . ; ð1Þ xnk xn3k where A, B 2 (0, 1), k 2 {1, 2, 3, . . .}, and the initial conditions x3k+1, x3k+2, . . ., x0, are arbitrary positive real numbers. We investigate the periodic character of the positive solution of Eq. (1) and we shall show that the sequence {xn} converges eventually to a period-k solution. This confirms Conjecture 4.8.1 in [10,11] given by Ladas and his co-workers for the above mentioned particular case. Basically, we generalize some of the results due to DeVault et al. in [7,10] related to the difference equation A 1 xn ¼ þ ; n ¼ 1; 2; . . . ; ð2Þ xn1 xn3 where x2, x1, x0, A 2 (0, 1). DeVault showed that the solution of Eq. (2) is periodic and eventually converges to a period-2 solution. Observe that Eq. (1) reduces to Eq. (2) when k = 1 and B = 1. Periodicity, and some other important properties of specific classes of difference equations, was investigated in [4–6,8,14]. In particular, the boundedness and global stability received more attention recently [3–5,7,13,14]. More information on nonlinear higher order recursive sequences can be found in [1,2,9–13]. Let {xn} be a positive solution of Eq. (1), and xn xnk yn ¼ . B Then Eq. (1) turns into the following: y y n ¼ C þ nk ; n ¼ 1; 2; . . . ; ð3Þ y n2k where C = A/B 2 (0, 1), k 2 {1, 2, 3, . . .}, the initial conditions y2k+1, y2k+2, . . ., y0 are arbitrary positive real numbers, and fy n g1 n¼2kþ1 satisfies Eq. (3). In the rest of this paper we need the following definitions: Let I be an interval of real numbers, and f : I · I ! I be a continuous differentiable function. Consider the difference equation [15–22] y n ¼ f ðy nk ; y n2k Þ;
n ¼ 1; 2; . . . ;
ð4Þ
where y2k+1, . . ., y1, y0 2 I. Definition 1. The equilibrium point y ¼ C þ 1 of Eq. (4) is the point that satisfies the condition y ¼ f ðy ; y Þ, or equivalently, y is a fixed point of f. So the sequence fy n g1 y for all n P 2k + 1 is a solution of Eq. (3). n¼2kþ1 with y n ¼
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Definition 2. Let I be an interval of real numbers, the equilibrium y of Eq. (3) is said to be (i) locally stable if for every positive e > 0, there exists d > 0, such that for all y2k+1, . . ., y1, y0 2 I, with jy 2kþ1 y j þ þ jy 1 y j þ jy 0 y j < d, we have jy n y j < e, for all n P 2k + 1. (ii) locally asymptotically stable if it is locally stable, and there exists c > 0, such that for all y2k+1, . . ., y1, y0 2 I with jy 2kþ1 y j þ þ jy 1 y j þ jy 0 y j < c, we have Limn!1 y n ¼ y . (iii) a global attractor if for all y2k+1, . . ., y1, y0 2 I, we have Limn!1 y n ¼ y . (iv) globally asymptotically stable if it is locally stable and a global attractor. (v) unstable if it is not locally stable. (vi) a source, or a repeller, if there exists r > 0 such that for all y2k+1, . . ., y1, y0 2 I with 0 < jy 2kþ1 y j þ þ jy 1 y j þ jy 0 y j < r, there exists N P 1 such that jy N y j P r. Clearly a source is an unstable equilibrium.
2. Local stability of Eq. (3) In this section, we discuss locally asymptotically stable of Eq. (3). Let x f ðx; yÞ ¼ C þ ; y assume that p :¼ pðy ; y Þ ¼
of ðy ; y Þ 1 1 ¼ ¼ y C þ 1 ox
q :¼ qðy ; y Þ ¼
of ðy ; y Þ 1 1 ¼ ¼ ; y oy Cþ1
and
denote the partial derivates of f(x, y) evaluated at the equilibrium point y of Eq. (3). Then the equation zn ¼pznk þ qzn2k ¼
1 1 znk zn2k Cþ1 Cþ1
ð5Þ
is called the Linearized equation associated with Eq. (3) about the equilibrium point y . Its characteristic equation is
k2k
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713
1 1 kk þ ¼ 0. Cþ1 Cþ1
ð6Þ
By substituting h = kk, it turns to the following equation: h2
1 1 hþ ¼ 0. Cþ1 Cþ1
ð7Þ
We observe that if all roots of Eq. (6) lie inside the open unit disk jkj < 1, then the equilibrium y is locally asymptotically stable; equivalently all roots of Eq. (7) lie inside the open unit disk jhj < 1. The following facts are well known (see [9]). Theorem A (Linearized stability) (a) If both roots of the quadratic Eq. (7) lie in the open unit disk jhj < 1, then the equilibrium y of Eq. (3) is locally asymptotically stable. (b) If at least one root of Eq. (7) has absolute value greater than one, then the equilibrium y of Eq. (3) is unstable. (c) A necessary and sufficient condition for both roots of Eq. (7) to lie in the open unit disk jhj < 1, is jpj < 1 q 6 2. In this case the locally asymptotically stable equilibrium y is also called a sink. (d) A necessary and sufficient condition for both roots of Eq. (7) to have absolute value greater than one is: jqj > 1
and
jpj < j1 qj.
In this case the equilibrium point y is called a repeller. (e) A necessary and sufficient condition for one root of Eq. (7) to have absolute value less than one and the other root of Eq. (3) to have absolute value greater than one is p2 þ 4q > 0 and
jpj > j1 qj.
In this case the unstable equilibrium y is called a saddle point. It follows from part (c) of Theorem A, that the equilibrium y is locally asymptotically stable for all C > 0. 3. Analysis of semi-cycle and global stability of Eq. (3) In this section, first we investigate the semi-cycle of Eq. (3). Next we consider the global behavior of solutions of it and we prove that the equilibrium y is globally asymptotically stable for all C > 0.
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Definition 3. Let fy n g1 n¼2kþ1 be a positive solution of Eq. (3). A positive semicycle of fy n g1 consists of a ‘‘string’’ of terms {yl, yl+1 . . ., ym}, all greater n¼2kþ1 than or equal to the equilibrium y , with l P 2k + 1 and m 6 1 and such that either l ¼ 2k þ 1;
or l > 2k þ 1;
and
y l1 < y ;
and either m ¼ 1;
or m < 1;
and
y mþ1 < y .
Definition 4. Let fy n g1 n¼2kþ1 be a positive solution of Eq. (3). A negative semicycle of fy n g1 consists of a ‘‘string’’ of terms {yl, yl+1 . . ., ym}, all less than n¼2kþ1 or equal to the equilibrium y , with l P 2k + 1 and m 6 1 and such that either l ¼ 2k þ 1;
or l > 2k þ 1;
and
y l1 P y ;
and either m ¼ 1;
or m < 1;
and
y mþ1 P y .
1
Definition 5. A solution fy n gn¼2kþ1 of Eq. (3) is called nonoscillatory if there exists N P 1 such that either y n > y for all n P N ; or y n < y for all n P N . 1 fy n gn¼2kþ1
is called oscillatory if it is not nonoscillatory.
Lemma 1. Let {yn} be a nontrivial positive solution of Eq. (3). Then the following statements hold: (i) {yn} oscillates about the equilibrium y with a semi-cycle of length at most 3k. (ii) {ynk+i} oscillates about the equilibrium y with a semi-cycle of length at most 2 or 3, for i = 0, 1, . . ., k 1, and the extreme point in a semi-cycle occurs in the first or second term. (iii) For n > 3k, we have C < yn < C þ
1þC . C
ð8Þ
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Proof (i) Let {yn} be a solution of Eq. (3). We show that every positive semi-cycle, except possibly the first, has 3k terms. The case of negative semi-cycle is similar. Let y N P y be the first term in a positive semi-cycle. If yN+k < yN, then y N þ2k ¼ A þ
y N þk < A þ 1 ¼ y . yN
So in this case the lemma is true. If yN+k P yN, then y y N þ2k ¼ A þ N þk P A þ 1 ¼ y yN and y N þ2k ¼ A þ
y N þk y y 6 A þ N þk ¼ A þ N þk < y N þk . y yN Aþ1
Therefore y 6 y N þ2k < y N þk , and we finally have y y N þ3k ¼ A þ N þ2k < A þ 1 ¼ y . y N þk 1
This shows that the positive semi-cycle of fy n gn¼2kþ1 has length at most 3k. From this, we can deduce that every solution {yn} of Eq. (3) oscillates about the equilibrium y . 1 (ii) Let fy nkþi gn¼1 be a solution of Eq. (3), for i = 0, 1, . . ., k 1. We show that every positive semi-cycle, except possibly the first, has 2 or 3 terms as part (i). Assume i 2 {0, 1, . . ., k 1}, and let y Nkþi P y be the first term in a positive semi-cycle. Then y ðN 1Þkþi < y , and y y ðN þ1Þkþi ¼ A þ Nkþi > A þ 1 ¼ y P y ðN 1Þkþi . y ðN 1Þkþi If y(N+1)k+i > yNk+i, then y ðN þ2Þkþi > y , and y ðN þ1Þkþi y ðN þ1Þkþi y ðN þ2Þkþi ¼ A þ < y ðN þ1Þkþi . 6Aþ y y Nkþi As a result y < y ðN þ2Þkþi < y ðN þ1Þkþi , and y ðN þ3Þkþi < y . In this case, the positive semi-cycle of {ynk+i} has length 3. If y(N+1)k+i < yNk+i then y ðN þ1Þkþi < y , the positive semi-cycle of {ynk+i} has length 2. From the above discussion, it is clear that the extreme point in a semi-cycle of {ynk+i} occurs in the first or second term. (iii) Clearly yn > C, for n > 0. Assume N > 2k, so we have yN C 1 1 ¼ þ <1þ . C y N k y N k y N 2k
ð9Þ
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By Eqs. (3) and (9), we gain y N þk ¼ C þ
yN 1þC .
This completes the proof.
Theorem 1. Assume that k 2 {1, 2, 3, . . .} and the initial conditions y2k+1, y2k+2, . . ., y0, C are arbitrary positive real numbers. Then the positive equilibrium y ¼ C þ 1 of Eq. (3) is globally asymptotically stable. Proof. We saw that y ¼ C þ 1 is locally asymptotically stable. So it remains to show that y ¼ C þ 1 is a global attractor which means that if {yn} is a nontrivial positive solution of Eq. (3), then Lim y n ¼ C þ 1. n!1
To prove its global attractivity, we define the sequences {Lnk+i} and {Unk+i} as follows: Li ¼ C
and
Ui ¼ C þ
1þC C
for i ¼ 0; 1; . . . ; k 1;
and for n = 0, 1, . . ., k 1, Lðnþ1Þkþi ¼ C þ
1þC U nkþi
and
U ðnþ1Þkþi ¼ C þ
1þC . Lðnþ1Þkþi
We observe that {Lnk+i} and {Unk+i} are sequences of upper and lower bounds for the solution {ynk+i} of Eq. (3), for i = 0, 1, . . ., k 1. We also have Lðnþ1Þkþi ¼ C þ
1þC C þ L1þC nkþi
for i ¼ 0; 1; . . . ; k 1;
ð10Þ
and U ðnþ1Þkþi ¼ C þ
1þC C þ U1þC nkþi
for i ¼ 0; 1; . . . ; k 1.
ð11Þ
One can see that {Lnk+i} is increasing and {Unk+i} is decreasing, for i = 0, 1, . . ., k 1, which means that: Li < Lkþi < < Lnkþi < < y < < U nkþi < < U kþi < U i . Thus Lim Lnþi ¼ Li 6 y and n!1
Lim U nþi ¼ U i 6 y . n!1
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717
By replacing Li and U i in Eqs. (10) and (11), we obtain: 1þC 1þC Li ¼ C þ and U i ¼ C þ for i ¼ 0; 1; . . . ; k 1. 1þC Cþ L C þ 1þC U i
i
Simple computations show that we have a unique solution U i ¼ Li ¼ y , for i = 0, 1, . . ., k 1. This completes the proof. h 4. The periodic behavior of the solution of Eq. (1) Here we show that the equilibrium of the solution of Eq. (1) is periodic with period k. From Eq. (1), we obtain x2n ¼
A B þ ; x2nk x2n3k
ð12Þ
and x2nk ¼
A B þ . x2n2k x2n4k
ð13Þ
So it follows that x2n ¼
A B þ A . B B þ x2n4k x2n4k þ x2n6k x2n2k A
ð14Þ
The following statements outline properties of the Eq. (1). Their proofs are based on Eq. (14). Lemma 2. Let {xn} be a positive solution of Eq. (1). Then the following statements hold. (i) For N > 0, let mN ¼ Minfx2N 4k ; x2N 2k ; x2N g; and M N ¼ Maxfx2N 4k ; x2N 2k ; x2N g. Then mN 6 x2N þ2lk 6 M N
for l P 1.
(ii) There exist positive numbers m and M such that m < xn < M, for n = 0, 1, . . . (iii) x2nkþi ¼ 1 for i ¼ 0; 1; . . . ; k 1. Lim n!1 xð2n2Þkþi
718
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Proof (i) First we define the function f by A B þ A B. B þ þ x y y z
f ðx; y; zÞ ¼ A
Note that f is increasing in x, y, and z. So x2N þ2k ¼
A x2N
A B A B þ A 6 A þ A ¼ MN . B B B þ x2N 2k x2N 2k þ x2N 4k M N þ M N M N þ MBN
It can be simply deduced by induction that x2N+2lk 6 MN for all l P 1. Similarly, x2N+2lk P mN for all l P 1. (ii) The proof is similar to (i). (iii) Recall that y 2n ¼ x2n xB2nk converges to the equilibrium, and therefore the sequence x2nkþi xð2n1Þkþi y 2nkþi x2nþi ¼ ¼ ; y 2nkkþi xð2n1Þkþi xð2n2Þkþi xð2n2Þkþi converges to 1. So Lim n!1
x2nkþi ¼ 1. xð2n2Þkþi
The following theorem is the main result of this paper. Theorem 2. Let {xn} be a positive solution of Eq. (1). Then there exist positive constants l0, l1, l2, . . ., l2k1 such that lilk+i = A + B and Lim x2nkþi ¼ li n!1
for i ¼ 0; 1; . . . ; k 1.
Proof. We show that the sequences x2nk+i for i = 0, 1, . . ., k 1 are Cauchy. By Lemma 2 and Eq. (13), we have x2nkþi P x2nkþi x2nk2kþi . 1 x M 2nk2kþi
Hence Lim jx2nkþi x2nk2kþi j ¼ 0 n!1
for i ¼ 0; 1; . . . ; k 1.
M. Jaberi Douraki et al. / Appl. Math. Comput. 173 (2006) 710–723
719
Since {xn} is bounded by Lemma 2, Limn!1x2nk+i exists and is a positive number li. Finally, from Eq. (13) we observe that li lki1 ¼ A þ B. This completes the proof.
h
5. Numerical discussion In this section, we present two test examples to support our theoretical discussion. Example 1. Assume that Eq. (1) hold, k = 2 and A = B = 1. So the equation reduces to the following: xn ¼
1 1 þ . xn2 xn6
Using the change of variable y n ¼ xn xn2 ; the corresponding equation will be as follows: y y n ¼ 1 þ n2 . y n4 In this part, to illustrate the result of this paper a numerical example is given, which was carried out using Maple 7. We assume A = B = 1 and the initial points x5, x4, . . ., x0 in the following table. n
xn
n
xn
n
xn
n
xn
5 4 3 2 1 0 1 2 3 4 5
3.05 1.7 0.03 10 1 4 1.327868852 0.8382352941 34.08641975 1.292982456 1.029337197
20 21 22 23 24 25 26 27 28 29 30
1.874541882 0.7964310198 1.092014157 2.578755699 1.810166430 0.8111423400 1.085899066 2.4884307681 0.836634946 0.7896436056 1.096909405
50 51 52 53 54 55 56 57 58 59 60
1.092287726 2.504462888 1.831370239 0.7984982925 1.092110372 2.505197171 1.831168060 0.7984573881 1.092138753 2.504765819 1.831292940
90 91 92 93 94 95 96 97 98 99 100
1.092146771 2.504889089 1.831255943 0.7984382900 1.092146760 2.504889883 1.831255710 0.7984384186 1.092146696 2.504889688 1.831255772
720
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By Theorem 1, the equilibrium point y ¼ A þ B ¼ 2 is globally asymptotically stable and by Theorem 2, we have l 0 l2 ¼ 2
l1 l3 ¼ 2;
and
ð15Þ
where Lim x4n ¼ l0
Lim x4nþ2 ¼ l2 ;
and
n!1
Lim x4nþ1 ¼ l1
n!1
and
n!1
Lim x4nþ3 ¼ l3 . n!1
As we see, for large n, Eq. (15) converges to the exact solution. We compute the error for Eq. (15) E ¼ jli l2þi x4nþi x4nþ2i j for i ¼ 0; 1. The following table shows the corresponding errors. n
xn · xn+2
E
n
xn · xn+2
E
52 53 54 55 56
2.00005843 2.00029318 1.99988960 2.00039566 1.99983763
0.00005843 0.00029318 0.0001104 0.00039566 0.00016237
94 95 96 97 98
1.99999999 2.00000031 1.99999994 2.00000016 1.99999987
0.00000001 0.00000031 0.00000006 0.00000016 0.00000013
Example 2. Let k = 3, A = 2, and B = 1, so Eq. (1) will be in the following form: xn ¼
2 xn3
þ
1 xn9
.
Using the change of variable y n ¼ xn xn3 ; we will have yn ¼ 2 þ
y n3 . y n9
Let x8, x7, . . ., x0 be the initial points in the following table:
xn
xn · xn+3
E
n
xn
xn · xn+3
E
8 7 6 5 4 3 2 1 0 1 2 3 4 5 6
2 4.2 1 1.78 11 0.1 3 6 0.003 1.166666667 0.5714285714 667.6666666 2.282467532 3.590909091 10.00299551
3.52 46.2 0.1 5.28 66 0.0003 3.50000001 3.428571428 2.002999999 2.662878788 2.051948051 6678.666668 2.7608225101 2.5984848489 3336.3318362
0.52 43.2 2.9 2.28 63 2.997 0.50000001 0.42857142 0.9970000 0.3371212 0.9480519 6648.6666 0.2391774 0.4015151 3333.3318
40 41 42 43 44 45 50 51 52 92 93 94 95 96 97
2.455977836 4.223516072 0.00988088907 1221671980 0.7105214509 303.0713419 0.7104728048 303.2734699 2.455737009 0.7104989319 303.1987610 2.455734427 4.222384829 0.00989450112 1.221630466
3.000399305 3.000898767 2.994614311 3.000036823 2.999858851 2.999696525 2.999931535 3.000666931 3.000022755 2.999999911 3.000000482 2.999999992 2.999999998 3.000000098 3.000000009
0.00039930 0.00089876 0.0053856 0.00003682 0.0001411 0.0003034 0.0000684 0.00066693 0.00002275 0.00000008 0.00000048 0.000000008 0.000000002 0.000000098 0.000000009
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n
721
722
M. Jaberi Douraki et al. / Appl. Math. Comput. 173 (2006) 710–723
where E ¼ jli l3þi x6nþi x6nþ3i j for i ¼ 0; 1; 2 and Lim x6n ¼ l0 ; n!1
Lim x6nþ3 ¼ l3 ; n!1
Lim x6nþ1 ¼ l1 ; n!1
Lim x6nþ4 ¼ l4 ; n!1
Lim x6nþ2 ¼ l2 ; n!1
Lim x6nþ5 ¼ l5 . n!1
We can see, for large n, E converges to zero.
6. Conclusion The results of this paper are generalizations of those of DeVault [2]. This solves the conjecture of Ladas and his co-workers [5,6] for some special cases. However, the general case is still open. By a change of variable, we deduce a new difference equation which involves some interesting results in its own right. For instance, it has a unique positive equilibrium which is oscillatory and globally attractive. This was used to show that every positive solution of Eq. (1) converges to a periodic-k solution.
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