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Oscillation and Global Asymptotic Stability
Journal of Mathematical Analysis and Applications 252, 364᎐375 Ž2000. doi:10.1006rjmaa.2000.7076, available online at http:rrwww.idealibrary.com on
Oscillation and Global Asymptotic Stability D. P. Mishev and W. T. Patula Department of Mathematics, Southern Illinois Uni¨ ersity at Carbondale, Carbondale, Illinois 62901-4408 E-mail: [email protected], [email protected] Submitted by Gerry Ladas Received April 10, 2000
We establish oscillation results and prove global asymptotic stability for the following difference equation: ynq 1 s A q
yn yny2 ⭈⭈⭈ ynyŽ2 ky2. yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1.
,
A ) 0,
k G 2,
n G 2 k.
䊚 2000 Academic Press.
Key Words: oscillation; global asymptotic stability; period 2 solution.
1. INTRODUCTION In this paper, we investigate the following difference equation: ynq 1 s A q
yn yny2 ⭈⭈⭈ ynyŽ2 ky2. yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1.
,
A ) 0,
k G 2,
n G 2 k.
Ž 1.1. The case k s 1 was studied in w2x. We prove that every solution yn4 of Ž1.1. oscillates about the equilibrium solution y s A q 1 and that yn ª A q 1 as n ª ⬁. Using this result, we give an alternative proof of a conjecture of Ladas w4, p. 312x. A brief outline of the paper is as follows. In Section 2, relevant definitions are presented. We prove that Ž1.1. is oscillatory and derive bounds for the highest possible number of elements in a positive semicycle and a negative semicycle. Section 3 proves the claim that Ž1.1. is globally asymptotically stable. That is, any solution converges to y s A q 1. In 364 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
365
GLOBAL ASYMPTOTIC STABILITY
Section 4, we use the global asymptotic stability result of Section 3 to verify a conjecture of Ladas regarding period 2 solutions.
2. OSCILLATION Consider Eq. Ž1.1.. Once 2 k positive initial conditions y 1 , . . . , y 2 k are specified, the solution of Ž1.1. is uniquely determined. The equilibrium or trivial solution, y, is the solution of Ž1.1. defined by setting yn ' c, ᭙ n. For Ž1.1., y s A q 1. Ž 2.1. We will prove that any solution of Ž1.1. oscillates around y s A q 1. That is, there will be a ‘‘string’’ of consecutive values of yn such that yn G A q 1. This set of elements will be called a positive semicycle. This will be followed by another ‘‘string’’ of consecutive solution values such that yn - A q 1. This set of elements will be designated as a negative semicycle. Oscillation means that for any N, there exist positive and negative semicycles with elements yn where n ) N. We have the following theorem. THEOREM 2.1. Consider the difference equation Ž1.1., where A ) 0 is a constant and k G 2. Then any nontri¨ ial solution yn4 oscillates about the equilibrium y s A q 1. Negati¨ e semicycles ha¨ e length at most Ž2 k q 1., and it is possible to show that such a semicycle does exist. Positi¨ e semicycles ha¨ e length at most Ž4 k y 1., and this form is unique in the sense of Ž2.13., Ž2.18., and Ž2.19. below. Moreo¨ er, positi¨ e semicycles of length Ž4 k y 1. do exist. Proof. We will prove oscillation by showing there is an upper bound for the maximum number of elements in a negative semicycle and also an upper bound for the maximum number of elements in a positive semicycle. We first relabel the subscripts in Ž1.1., which yields the following: yNq 2 k yNq2 ky2 ⭈⭈⭈ yNq2 yNq 2 kq1 s A q . Ž 2.2. yNq 2 ky1 yNq2 ky3 ⭈⭈⭈ yNq1 Subtracting one in the index leads to yNq 2 ky1 ⭈⭈⭈ yNq1 yNq 2 k s A q . yNq 2 ky2 ⭈⭈⭈ yN
Ž 2.3.
Substituting Ž2.3. into Ž2.2. gives yNq 2 kq1 s A q
AyNq 2 ky2 ⭈⭈⭈ yNq2 yNq 2 ky1 ⭈⭈⭈ yNq1
q
1 yN
.
Ž 2.4.
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MISHEV AND PATULA
We consider negative semicycles first and determine the maximum number of elements a negative semicycle can contain. Suppose yNq 1 , . . . , yNq2 k - A q 1. Suppose yNq2 kq1 - A q 1. Then Ž2.2. implies that yNq 2 k ⭈⭈⭈ yNq2 - 1. Ž 2.5. yNq 2 ky1 ⭈⭈⭈ yNq1 Increasing the index by one in Ž2.4., we have yNq 2 kq2 s A q
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq1
.
Ž 2.6.
Then Ž2.5. implies that yNq 2 ky1 ⭈⭈⭈ yNq3
)
yNq 2 k ⭈⭈⭈ yNq4 yNq2 « «
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq1 )
1 yNq1
, A
yNq1
)
,
Aq1 yNq1
) 1,
Ž 2.7.
since yNq 1 - A q 1. Now Ž2.6. and Ž2.7. imply that yNq2 kq2 ) A q 1, and so yNq 2 kq2 begins a positive semicycle. Thus, a negative semicycle can have length at most Ž2 k q 1.. One can actually construct a negative semicycle with Ž2 k q 1. elements by choosing each of yNq 1 , . . . , yNq2 k A q 1 and such that the quotient on the right of Ž2.2. - 1. Next, consider positive semicycles. Again, we determine the maximum number of elements a positive semicycle can contain. Suppose yNq 1 , . . . , yNq2 k G A q 1. Suppose also that yNq2 kq1 G A q 1. Then Ž2.2. implies that yNq 2 k ⭈⭈⭈ yNq2 G 1, yNq 2 ky1 ⭈⭈⭈ yNq1 « «
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
F
1 yNq1
A yNq1
F
,
Aq1 yNq1
F 1.
Ž 2.8.
If yNq 2 kq2 G A q 1, Ž2.6. yields that AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq2
G 1.
Ž 2.9.
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GLOBAL ASYMPTOTIC STABILITY
From Ž2.8. and Ž2.9., we conclude that AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq1
s 1.
Ž 2.10.
This together with Ž2.6. and Ž2.8. means that yNq 1 s yNq2 kq2 s A q 1.
Ž 2.11.
Next, suppose that yNq 2 kq3 G A q 1. Arguing as we did to establish Ž2.11., we can conclude that yNq 2 s yNq2 kq3 s A q 1.
Ž 2.12.
Continuing in this fashion, we have that yNq 1 s yNq2 s ⭈⭈⭈ s yNq2 ky2 s A q 1,
and
Ž 2.13.
yNq 2 kq2 s yNq2 kq3 s ⭈⭈⭈ s yNq4 ky1 s A q 1.
Ž 2.14.
By considering Ž2.2. and Ž2.13., we may write yNq 2 kq1 s A q
yNq 2 k yNq 2 ky1
.
Ž 2.15.
Then Ž2.13. and Ž2.15. imply that yNq 2 kq2 s A q
yNq 2 kq1 yNq2 ky1 ⭈⭈⭈ yNq3 yNq 2 k yNq2 ky2 ⭈⭈⭈ yNq2
sAq Aq
ž
yNq 2 kq2 s A q
yNq 2 k yNq 2 ky1
AyNq 2 ky1
Ž A q 1 . yNq 2 k
/
⭈
q
yNq2 ky1 yNq2 k Ž A q 1 . 1 Aq1
,
.
or
Ž 2.16.
Equations Ž2.14. and Ž2.16. have two expressions for yNq 2 kq2 which must be equal. Thus, Aq1sAq
AyNq 2 ky1
Ž A q 1 . yNq 2 k
q
yNq 2 ky1 s yNq2 k .
1 Aq1
,
or
Ž 2.17.
This, together with Ž2.15., implies that yNq 2 kq1 s A q 1. Also, yNq2 ky1 Žand yNq 2 k . must be ) A q 1. Otherwise, there would be 2 k consecutive values yn s A q 1. This means yn ' y s A q 1, which is not permitted.
368
MISHEV AND PATULA
Thus, the structure we have at this point is as follows: yNq 1 s ⭈⭈⭈ s yNq2 ky2 s A q 1
Ž 2.13.
yNq 2 ky1 s yNq2 k ) A q 1
Ž 2.18.
yNq 2 kq1 s ⭈⭈⭈ s yNq4 ky1 s A q 1.
Ž 2.19.
Now we can conclude that yNq 4 k - A q 1. If yNq4 k G A q 1, then by considering the elements yNq 2 ky1 , . . . , yNq4 k and arguing as we did to establish Ž2.11., we can conclude that yNq 2 ky1 s yNq4 k s A q 1. This, however, contradicts Ž2.18.. From another point of view, simply regard Ž2.13. and Ž2.18. as 2 k initial conditions. These determine a unique solution yn4 . This solution satisfies Ž2.19. and has yNq 4 k - A q 1. Thus a solution with a positive semicycle of length Ž4 k y 1. does exist. Moreover, any positive semicycle of length Ž4 k y 1. must have the form indicated in Ž2.13., Ž2.18., and Ž2.19.. This completes the proof of Theorem 2.1. We remark that when k s 1, the nature of the oscillation appears different from the case k G 2. Specifically, w2, Lemma 1x says that for k s 1, semicycles have length two or three. If k s 2, A s 1, and y 1 s y 2 s y 3 s y4 s 1, the solution defined by these initial conditions has semicycles of length one.
3. GLOBAL ASYMPTOTIC STABILITY In this section, we prove that any nontrivial solution yn4 of Ž1.1. converges to y s A q 1. From Ž1.1., we have yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1. yn s A q . Ž 3.1. yny 2 yny4 ⭈⭈⭈ yny2 k Using Ž3.1. to substitute for yn in Ž1.1. yields ynq 1 s A q A q sAq sAq ynq 1 s A q
yny 1 ⭈⭈⭈ ynyŽ2 ky1.
yny2 ⭈⭈⭈ ynyŽ2 ky2.
yny 2 ⭈⭈⭈ yny2 k
yny1 yny3 ⭈⭈⭈ ynyŽ2 ky1.
Ayny 2 ⭈⭈⭈ ynyŽ2 ky2. yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1. A yn y A 1 yny 2 k
⭈
1 yny2 k
1q
q
A yn y A
⭈
1 yny2 k .
yny2 k yny2 k ,
q
1 yny2 k
or
Ž 3.2.
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GLOBAL ASYMPTOTIC STABILITY
From Ž3.2., we have yn s A q
1 yny 2 ky1
yny 2 k s A q
A
1q
1 yny 4 ky1
,
yny1 y A 1q
and
A yny2 ky1 y A
Ž 3.3.
.
Ž 3.4.
Substituting Ž3.3. and Ž3.4. into the right hand side of Ž3.2. implies that ynq 1 s A q
1 Aq
1 yny4 ky1
1q
A
1q
A yny2 ky1 y A
1 y ny2 ky1
1q
.
A y ny1 y A
Ž 3.5. We will prove that any solution of Ž3.5. is bounded above and below by employing the same technique found in w2, Lemma 2Ži.x. LEMMA 3.1. Consider a solution of Ž3.5. determined by 2 k positi¨ e initial conditions y 1 , . . . , y 2 k . For any N G 1, let m N and MN be the min and max, respecti¨ ely, for yn , where N F n F N q 4 k q 1. Then yn F MN , ᭙ n G N, and yn G m N , ᭙ n G N. Proof. From Ž3.5., yNq 4 kq2 F A q
FAq
FAq
1 Aq
1
1q
MN
1 Aq
1
1q
A MN y A
1q
MN y A
A 1 MN
1q
A MN y A
A 1 MN y A
MN y A
1 q A Ž MN y A . s MN .
A Ž MN y A . q 1
Now use induction. A similar argument proves yn G m N , ᭙ n G N. Thus, given any solution yn4 of Ž1.1., Lemma 3.1 ensures that S and I are well defined, where S s lim sup Ž yn . nª⬁
and
I s lim inf Ž yn . . nª⬁
Ž 3.6.
Note that since yn ) A for n G 2 k q 1, Lemma 3.1 also proves that I G m N ) A,
if N G 2 k q 1.
Ž 3.7.
370
MISHEV AND PATULA
Since yn oscillates around y s A q 1, we also know that S G A q 1 G I. Our goal is to show that S s I, which would mean that any solution of Ž1.1. converges to y s A q 1. We will accomplish this by assuming there exists a solution yn4 of Ž1.1. with the following property: Assume S ) I.
Ž 3.8.
We now argue to obtain a contradiction to Ž3.8.. From Ž3.2., we have 1
S s lim sup Ž ynq 1 . F A q lim sup n
FAq
1
⭈ 1q
lim inf Ž yny 2 k . n
FAq SF
yny 2 k
n
1 I
1q
A IyA
AŽ I y A . q 1 IyA
⭈ lim sup 1 q n
A yn y A
A lim inf yn y A n
,
or
.
Ž 3.9.
Note that Ž3.7. implies I y A ) 0. In a similar fashion, from Ž3.2. we again have I s lim inf Ž ynq 1 . G A q n
IG
AŽ S y A . q 1 SyA
1 S
1q
A
,
SyA
or
.
Ž 3.10.
However, Ž3.10. holds iff IyAG IyAG SG SG
AŽ S y A . q 1 SyA 1 SyA 1 IyA
,
y
SyA
,
iff
iff
q A,
AŽ I y A . q 1 IyA
AŽ S y A .
iff .
Ž 3.11.
371
GLOBAL ASYMPTOTIC STABILITY
Thus Ž3.9. and Ž3.11. imply that Ss
AŽ I y A . q 1 IyA
,
Ž 3.12.
which is equivalent to Is
AŽ S y A . q 1
. Ž 3.13. SyA We remark that the ‘‘duality’’ between Ž3.9. and Ž3.11. which yields Ž3.12. and Ž3.13. is NOT present when k s 1. Thus the proof techniques in this paper, as compared with those in w1, 2x, are different. Note that Ž3.12. Žor Ž3.13.. implies that S)Aq1
AŽ I y A . q 1
m
IyA
)Aq1
m
A q 1 ) I.
Ž 3.14. Since we are assuming S ) I, we can change Ž3.8. to the following statement: Assume S ) A q 1 ) I. Ž 3.15. Next, let yn q14 be a subsequence such that j
yn jq1 ª S,
as j ª ⬁.
Ž 3.16.
Consider the subsequence yn j 4 . We have the following lemma. If yn jq14 is a subsequence satisfying Ž3.16., then
LEMMA 3.2.
yn j ª I,
as j ª ⬁.
Ž 3.17.
Proof. Suppose not. Then there exists an ⑀ ) 0 and a subsequence yn 4 such that yn G I q ⑀ , for m sufficiently large. Then Ž3.2., Ž3.12., jm jm and Ž3.16. imply that S s lim sup yn j mª⬁
FAq
m
q1
F A q lim sup m
1 lim inf Ž yny 2 k .
⭈ 1q
n
SFAq sAq
1 I
1q 1
A Iq⑀yA
1 yn j
m
⭈ lim sup 1 q
y2 k
A Iq⑀yA
-Aq
1 I
1q
m
,
yn j y A m
or A
IyA
s S. IyA This is a contradiction, which proves the lemma.
A
372
MISHEV AND PATULA
Continuing in this fashion, we consider the subsequence yn jy14 . LEMMA 3.3. If yn jq14 is a subsequence satisfying Ž3.16., and so necessarily yn j 4 satisfies Ž3.17., then yn jy1 ª S,
as j ª ⬁.
Ž 3.18.
Proof. Suppose not. As in Lemma 3.2, there exists an ⑀ ) 0 and a subsequence yn j y1 such that yn j y1 F S y ⑀ , for m sufficiently large. m m Then Ž3.2., Ž3.13., and Ž3.17. yield I s lim inf yn j G A q lim inf m
mª⬁
m
1
G A q lim inf
yny 2 ky1
n
IGAq sAq
1 S
1q 1
SyA
1 yn j
m
⭈ 1q
A Sy⑀yA
⭈ lim inf 1 q m
y2 ky1
A
,
Sy⑀yA 1
)Aq
S
1q
A yn j
m
y1
yA
or A
SyA
s I.
This is again a contradiction, which proves the lemma. Based on Lemmas 3.2 and 3.3, evidently we can conclude that yn jy2 ª I, yn jy3 ª S, . . . , yn jyŽ2 ky2. ª I, and yn jyŽ2 ky1. ª S, as j ª ⬁. From Ž1.1., we have yn jq1 s A q
yn j yn jy2 ⭈⭈⭈ yn jyŽ2 ky2. yn jy1 yn jy3 ⭈⭈⭈ yn jyŽ2 ky1.
.
Letting j ª ⬁ implies that SsAq
Ik Sk
,
or
Ž S y A. S k s I k . However, this is a contradiction, since Ž3.15. implies
Ž S y A. S k ) S k ) I k . Thus our assumption Ž3.8. is incorrect and S s I. This means yn ª A q 1, as n ª ⬁, for any solution yn4 of Ž1.1..
373
GLOBAL ASYMPTOTIC STABILITY
We remark that a key step in our argument was transforming Eq. Ž1.1. to the form Ž3.5.. At first glance, it appears that Ž3.5. satisfies the hypotheses of a very powerful convergence theorem in w3, Theorem 2.2x. However, it turns out that the hypotheses are not satisfied, and so a separate proof that yn ª A q 1 is necessary. 4. PERIOD 2 SOLUTIONS In this section, as a corollary to the global asymptotic stability result of Section 3, we present an alternate proof of a conjecture of Ladas. Specifically, w4, Conjecture Ž4.3.1., p. 312x proposes that every positive solution to the following difference equation converges to a period 2 solution: x nq 1 s
a xn
q
b x ny2 k
a, b ) 0,
,
k G 2,
n G 2 k q 1.
Ž 4.1.
The case k s 1 was studied and verified by DeVault et al. w2x. The case k G 2 was first proved in w3x in a much more general framework. This paper gives an alternate proof of the conjecture when k G 2. First, note that any set of Ž2 k q 1. positive initial conditions x 1 , . . . , x 2 kq1 will recursively generate a solution x n4 of Ž4.1.. Define ynq1 by setting ynq 1 s
x nq 1 x n b
.
Ž 4.2.
Then Ž4.1. can be written as x nq1 x n b
s
a b
q
xn x ny 2 k
⭈
x x x ny1 x nq 1 x n b
x ny1 x ny1 ⭈
x nyŽ2 ky1.
x ny2 x ny3
b sAq x ny 1 x ny2 b
x nyŽ2 ky1.
⭈ ⭈⭈⭈ ⭈
⭈
⭈ ⭈⭈⭈ ⭈
b x ny3 x ny4 b
⭈
bk bk
,
or
x nyŽ2 ky2. x nyŽ2 ky1.
⭈ ⭈⭈⭈ ⭈
b x nyŽ2 ky1. x ny2 k . b
In view of Ž4.2., this becomes Eq. Ž1.1., where A s arb. Consider any solution x n4 of Ž4.1.. Utilizing Ž4.2., and the asymptotic stability result of Section 3, we have lim
nª⬁
x2 n x 2 ny2
s lim
nª⬁
x 2 n x 2 ny1 x 2 ny1 x 2 ny2
s lim
nª⬁
y2 n y 2 ny1
s
Aq1 Aq1
s 1.
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MISHEV AND PATULA
Continuing in this fashion, x2 n x 2 n x 2 ny1 x 2 ny2 x 2 ny3 y 2 n y 2 ny2 lim s lim s lim s 1. nª⬁ x 2 ny4 nª⬁ x 2 ny1 x 2 ny2 x 2 ny3 x 2 ny4 nª⬁ y 2 ny1 y 2 ny3 A similar argument implies that for 0 F j F 2 k, x2 n lim s 1, nª⬁ x 2 ny2 j which means x 2 ny2 j
lim
nª⬁
s 1.
x2 n
Finally, it is clear that for 0 F j F 2 k and 0 F r F 2 k, r / j, x 2 ny2 r lim s 1. nª⬁ x 2 ny2 j
Ž 4.3.
As an analog of Lemma 3.1 and w2, Lemma 2Ži.x, we have the following result. LEMMA 4.1. Consider Ž4.1. and let N be some integer such that N ) 2 k. Define MN s max x 2 N , x 2 Ny2 , . . . , x 2 Ny2Ž2 k .4 and m N s min x 2 N , x 2 Ny2 , . . . , x 2 Ny2Ž2 k .4 . Then m N F x 2 n F MN , ᭙ n G N. Proof. From Ž4.1., x 2 Nq2 s s
a x 2 Nq1
x 2 Nq1y2 k
a a x2 N
F
b
q
q
a aq b MN
q
b x 2 Ny2 k
q
b aqb
b a x 2 Ny2 k
q
b x 2 Ny4 k
s MN .
MN
Now use induction. A similar argument with the inequalities reversed yields that x 2 n G m N , ᭙ n G N. LEMMA 4.2.
Let x n4 be a solution of Ž4.1.. Then lim nª⬁ x 2 n exists.
Proof. Lemma 4.1 implies that S and I are well defined, where S s lim sup nª⬁ x 2 n and I s lim inf nª⬁ x 2 n . We suppose S ) I and reach a contradiction. Choose N1 ) 2 k and consider the elements x 2 N1, x 2 N1y2 , . . . , x 2 N1y2Ž2 k . . At least one of these elements must be G S. If not, then Lemma 4.1
GLOBAL ASYMPTOTIC STABILITY
375
implies that ᭙ n G N1 , x 2 n F max x 2 N1, . . . , x 2 N1y4 k 4 - S, a contradiction. Similarly, at least one of x 2 N1, . . . , x 2 N1y4 k must be F I. Thus there must be at least one quotient of the form x 2 N1y2 rrx 2 N1y2 j , 0 F r F 2 k, 0 F j F 2 k, r / j, where x 2 N1y2 rrx 2 N1y2 j G SI ) 1. Next, choose N2 s N1 q 4 k q 1 and consider the elements x 2 N 2 , x 2 N 2y2 , . . . , x 2 N 2y2Ž2 k . . As above, there must be at least one quotient such that x 2 N 2y2 rrx 2 N 2y2 j G SI ) 1, where 0 F r F 2 k, 0 F j F 2 k, r / j. We can continue in this fashion to choose N3 , N4 , . . . , Nb , etc., such that x 2 N by2 r x 2 N by2 j
G
S I
) 1,
where 0 F r F 2 k, 0 F j F 2 k,
r / j. Ž 4.4.
The values of r and j may change for each Nb . For each Nb , though, there are only a finite number of possible quotients of the form Ž4.4.. Since we may let b ª ⬁, there will be at least one subsequence Nb i such that x 2 N b y2 r i
x 2 N b y2 j
G
S
i
I
) 1,
Ž 4.5.
where r and j are the same for every bi . Since we may let i ª ⬁, this contradicts Ž4.3. and proves Lemma 4.2. Thus lim nª⬁ x 2 n exists. We call it L E as in w2, Theorem 2x. Since Ž4.1. implies x 2 nq1 s
a x2 n
q
b x 2 ny2 k
,
k G 2,
Ž 4.6.
clearly lim nª⬁ x 2 nq1 exists. We call this limit LO , and Ž4.6. implies L E ⭈ LO s a q b. This yields an alternate proof of w4, Conjecture 4.3.1, p. 312x. Specifically, if we consider Eq. Ž4.1., then every positive nontrivial solution converges to a period 2 solution, where the product of the periods is Ž a q b ..
REFERENCES 1. Con Amore Problems Group, Two recurrence relations, one easy, one hard, Amer. Math. Monthly 106, No. 10 Ž1999., 967. 2. R. DeVault, G. Ladas, and S. W. Schultz, On the recursive sequence x nq 1 s Arx n q 1rx ny 2 , Proc. Amer. Math. Soc. 126, No. 11 Ž1998., 3257᎐3261. 3. H. El-Metwally, E. A. Grove, and G. Ladas, A global convergence result with applications to periodic solutions, J. Math. Anal. Appl. 245 Ž2000., 161᎐170. 4. G. Ladas, Open problems and conjectures, J. Differential Equations Appl. 4, No. 3 Ž1998., 312.
Oscillation and Global Asymptotic Stability D. P. Mishev and W. T. Patula Department of Mathematics, Southern Illinois Uni¨ ersity at Carbondale, Carbondale, Illinois 62901-4408 E-mail: [email protected], [email protected] Submitted by Gerry Ladas Received April 10, 2000
We establish oscillation results and prove global asymptotic stability for the following difference equation: ynq 1 s A q
yn yny2 ⭈⭈⭈ ynyŽ2 ky2. yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1.
,
A ) 0,
k G 2,
n G 2 k.
䊚 2000 Academic Press.
Key Words: oscillation; global asymptotic stability; period 2 solution.
1. INTRODUCTION In this paper, we investigate the following difference equation: ynq 1 s A q
yn yny2 ⭈⭈⭈ ynyŽ2 ky2. yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1.
,
A ) 0,
k G 2,
n G 2 k.
Ž 1.1. The case k s 1 was studied in w2x. We prove that every solution yn4 of Ž1.1. oscillates about the equilibrium solution y s A q 1 and that yn ª A q 1 as n ª ⬁. Using this result, we give an alternative proof of a conjecture of Ladas w4, p. 312x. A brief outline of the paper is as follows. In Section 2, relevant definitions are presented. We prove that Ž1.1. is oscillatory and derive bounds for the highest possible number of elements in a positive semicycle and a negative semicycle. Section 3 proves the claim that Ž1.1. is globally asymptotically stable. That is, any solution converges to y s A q 1. In 364 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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GLOBAL ASYMPTOTIC STABILITY
Section 4, we use the global asymptotic stability result of Section 3 to verify a conjecture of Ladas regarding period 2 solutions.
2. OSCILLATION Consider Eq. Ž1.1.. Once 2 k positive initial conditions y 1 , . . . , y 2 k are specified, the solution of Ž1.1. is uniquely determined. The equilibrium or trivial solution, y, is the solution of Ž1.1. defined by setting yn ' c, ᭙ n. For Ž1.1., y s A q 1. Ž 2.1. We will prove that any solution of Ž1.1. oscillates around y s A q 1. That is, there will be a ‘‘string’’ of consecutive values of yn such that yn G A q 1. This set of elements will be called a positive semicycle. This will be followed by another ‘‘string’’ of consecutive solution values such that yn - A q 1. This set of elements will be designated as a negative semicycle. Oscillation means that for any N, there exist positive and negative semicycles with elements yn where n ) N. We have the following theorem. THEOREM 2.1. Consider the difference equation Ž1.1., where A ) 0 is a constant and k G 2. Then any nontri¨ ial solution yn4 oscillates about the equilibrium y s A q 1. Negati¨ e semicycles ha¨ e length at most Ž2 k q 1., and it is possible to show that such a semicycle does exist. Positi¨ e semicycles ha¨ e length at most Ž4 k y 1., and this form is unique in the sense of Ž2.13., Ž2.18., and Ž2.19. below. Moreo¨ er, positi¨ e semicycles of length Ž4 k y 1. do exist. Proof. We will prove oscillation by showing there is an upper bound for the maximum number of elements in a negative semicycle and also an upper bound for the maximum number of elements in a positive semicycle. We first relabel the subscripts in Ž1.1., which yields the following: yNq 2 k yNq2 ky2 ⭈⭈⭈ yNq2 yNq 2 kq1 s A q . Ž 2.2. yNq 2 ky1 yNq2 ky3 ⭈⭈⭈ yNq1 Subtracting one in the index leads to yNq 2 ky1 ⭈⭈⭈ yNq1 yNq 2 k s A q . yNq 2 ky2 ⭈⭈⭈ yN
Ž 2.3.
Substituting Ž2.3. into Ž2.2. gives yNq 2 kq1 s A q
AyNq 2 ky2 ⭈⭈⭈ yNq2 yNq 2 ky1 ⭈⭈⭈ yNq1
q
1 yN
.
Ž 2.4.
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MISHEV AND PATULA
We consider negative semicycles first and determine the maximum number of elements a negative semicycle can contain. Suppose yNq 1 , . . . , yNq2 k - A q 1. Suppose yNq2 kq1 - A q 1. Then Ž2.2. implies that yNq 2 k ⭈⭈⭈ yNq2 - 1. Ž 2.5. yNq 2 ky1 ⭈⭈⭈ yNq1 Increasing the index by one in Ž2.4., we have yNq 2 kq2 s A q
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq1
.
Ž 2.6.
Then Ž2.5. implies that yNq 2 ky1 ⭈⭈⭈ yNq3
)
yNq 2 k ⭈⭈⭈ yNq4 yNq2 « «
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq1 )
1 yNq1
, A
yNq1
)
,
Aq1 yNq1
) 1,
Ž 2.7.
since yNq 1 - A q 1. Now Ž2.6. and Ž2.7. imply that yNq2 kq2 ) A q 1, and so yNq 2 kq2 begins a positive semicycle. Thus, a negative semicycle can have length at most Ž2 k q 1.. One can actually construct a negative semicycle with Ž2 k q 1. elements by choosing each of yNq 1 , . . . , yNq2 k A q 1 and such that the quotient on the right of Ž2.2. - 1. Next, consider positive semicycles. Again, we determine the maximum number of elements a positive semicycle can contain. Suppose yNq 1 , . . . , yNq2 k G A q 1. Suppose also that yNq2 kq1 G A q 1. Then Ž2.2. implies that yNq 2 k ⭈⭈⭈ yNq2 G 1, yNq 2 ky1 ⭈⭈⭈ yNq1 « «
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
F
1 yNq1
A yNq1
F
,
Aq1 yNq1
F 1.
Ž 2.8.
If yNq 2 kq2 G A q 1, Ž2.6. yields that AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq2
G 1.
Ž 2.9.
367
GLOBAL ASYMPTOTIC STABILITY
From Ž2.8. and Ž2.9., we conclude that AyNq 2 ky1 ⭈⭈⭈ yNq3 yNq 2 k ⭈⭈⭈ yNq2
q
1 yNq1
s 1.
Ž 2.10.
This together with Ž2.6. and Ž2.8. means that yNq 1 s yNq2 kq2 s A q 1.
Ž 2.11.
Next, suppose that yNq 2 kq3 G A q 1. Arguing as we did to establish Ž2.11., we can conclude that yNq 2 s yNq2 kq3 s A q 1.
Ž 2.12.
Continuing in this fashion, we have that yNq 1 s yNq2 s ⭈⭈⭈ s yNq2 ky2 s A q 1,
and
Ž 2.13.
yNq 2 kq2 s yNq2 kq3 s ⭈⭈⭈ s yNq4 ky1 s A q 1.
Ž 2.14.
By considering Ž2.2. and Ž2.13., we may write yNq 2 kq1 s A q
yNq 2 k yNq 2 ky1
.
Ž 2.15.
Then Ž2.13. and Ž2.15. imply that yNq 2 kq2 s A q
yNq 2 kq1 yNq2 ky1 ⭈⭈⭈ yNq3 yNq 2 k yNq2 ky2 ⭈⭈⭈ yNq2
sAq Aq
ž
yNq 2 kq2 s A q
yNq 2 k yNq 2 ky1
AyNq 2 ky1
Ž A q 1 . yNq 2 k
/
⭈
q
yNq2 ky1 yNq2 k Ž A q 1 . 1 Aq1
,
.
or
Ž 2.16.
Equations Ž2.14. and Ž2.16. have two expressions for yNq 2 kq2 which must be equal. Thus, Aq1sAq
AyNq 2 ky1
Ž A q 1 . yNq 2 k
q
yNq 2 ky1 s yNq2 k .
1 Aq1
,
or
Ž 2.17.
This, together with Ž2.15., implies that yNq 2 kq1 s A q 1. Also, yNq2 ky1 Žand yNq 2 k . must be ) A q 1. Otherwise, there would be 2 k consecutive values yn s A q 1. This means yn ' y s A q 1, which is not permitted.
368
MISHEV AND PATULA
Thus, the structure we have at this point is as follows: yNq 1 s ⭈⭈⭈ s yNq2 ky2 s A q 1
Ž 2.13.
yNq 2 ky1 s yNq2 k ) A q 1
Ž 2.18.
yNq 2 kq1 s ⭈⭈⭈ s yNq4 ky1 s A q 1.
Ž 2.19.
Now we can conclude that yNq 4 k - A q 1. If yNq4 k G A q 1, then by considering the elements yNq 2 ky1 , . . . , yNq4 k and arguing as we did to establish Ž2.11., we can conclude that yNq 2 ky1 s yNq4 k s A q 1. This, however, contradicts Ž2.18.. From another point of view, simply regard Ž2.13. and Ž2.18. as 2 k initial conditions. These determine a unique solution yn4 . This solution satisfies Ž2.19. and has yNq 4 k - A q 1. Thus a solution with a positive semicycle of length Ž4 k y 1. does exist. Moreover, any positive semicycle of length Ž4 k y 1. must have the form indicated in Ž2.13., Ž2.18., and Ž2.19.. This completes the proof of Theorem 2.1. We remark that when k s 1, the nature of the oscillation appears different from the case k G 2. Specifically, w2, Lemma 1x says that for k s 1, semicycles have length two or three. If k s 2, A s 1, and y 1 s y 2 s y 3 s y4 s 1, the solution defined by these initial conditions has semicycles of length one.
3. GLOBAL ASYMPTOTIC STABILITY In this section, we prove that any nontrivial solution yn4 of Ž1.1. converges to y s A q 1. From Ž1.1., we have yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1. yn s A q . Ž 3.1. yny 2 yny4 ⭈⭈⭈ yny2 k Using Ž3.1. to substitute for yn in Ž1.1. yields ynq 1 s A q A q sAq sAq ynq 1 s A q
yny 1 ⭈⭈⭈ ynyŽ2 ky1.
yny2 ⭈⭈⭈ ynyŽ2 ky2.
yny 2 ⭈⭈⭈ yny2 k
yny1 yny3 ⭈⭈⭈ ynyŽ2 ky1.
Ayny 2 ⭈⭈⭈ ynyŽ2 ky2. yny 1 yny3 ⭈⭈⭈ ynyŽ2 ky1. A yn y A 1 yny 2 k
⭈
1 yny2 k
1q
q
A yn y A
⭈
1 yny2 k .
yny2 k yny2 k ,
q
1 yny2 k
or
Ž 3.2.
369
GLOBAL ASYMPTOTIC STABILITY
From Ž3.2., we have yn s A q
1 yny 2 ky1
yny 2 k s A q
A
1q
1 yny 4 ky1
,
yny1 y A 1q
and
A yny2 ky1 y A
Ž 3.3.
.
Ž 3.4.
Substituting Ž3.3. and Ž3.4. into the right hand side of Ž3.2. implies that ynq 1 s A q
1 Aq
1 yny4 ky1
1q
A
1q
A yny2 ky1 y A
1 y ny2 ky1
1q
.
A y ny1 y A
Ž 3.5. We will prove that any solution of Ž3.5. is bounded above and below by employing the same technique found in w2, Lemma 2Ži.x. LEMMA 3.1. Consider a solution of Ž3.5. determined by 2 k positi¨ e initial conditions y 1 , . . . , y 2 k . For any N G 1, let m N and MN be the min and max, respecti¨ ely, for yn , where N F n F N q 4 k q 1. Then yn F MN , ᭙ n G N, and yn G m N , ᭙ n G N. Proof. From Ž3.5., yNq 4 kq2 F A q
FAq
FAq
1 Aq
1
1q
MN
1 Aq
1
1q
A MN y A
1q
MN y A
A 1 MN
1q
A MN y A
A 1 MN y A
MN y A
1 q A Ž MN y A . s MN .
A Ž MN y A . q 1
Now use induction. A similar argument proves yn G m N , ᭙ n G N. Thus, given any solution yn4 of Ž1.1., Lemma 3.1 ensures that S and I are well defined, where S s lim sup Ž yn . nª⬁
and
I s lim inf Ž yn . . nª⬁
Ž 3.6.
Note that since yn ) A for n G 2 k q 1, Lemma 3.1 also proves that I G m N ) A,
if N G 2 k q 1.
Ž 3.7.
370
MISHEV AND PATULA
Since yn oscillates around y s A q 1, we also know that S G A q 1 G I. Our goal is to show that S s I, which would mean that any solution of Ž1.1. converges to y s A q 1. We will accomplish this by assuming there exists a solution yn4 of Ž1.1. with the following property: Assume S ) I.
Ž 3.8.
We now argue to obtain a contradiction to Ž3.8.. From Ž3.2., we have 1
S s lim sup Ž ynq 1 . F A q lim sup n
FAq
1
⭈ 1q
lim inf Ž yny 2 k . n
FAq SF
yny 2 k
n
1 I
1q
A IyA
AŽ I y A . q 1 IyA
⭈ lim sup 1 q n
A yn y A
A lim inf yn y A n
,
or
.
Ž 3.9.
Note that Ž3.7. implies I y A ) 0. In a similar fashion, from Ž3.2. we again have I s lim inf Ž ynq 1 . G A q n
IG
AŽ S y A . q 1 SyA
1 S
1q
A
,
SyA
or
.
Ž 3.10.
However, Ž3.10. holds iff IyAG IyAG SG SG
AŽ S y A . q 1 SyA 1 SyA 1 IyA
,
y
SyA
,
iff
iff
q A,
AŽ I y A . q 1 IyA
AŽ S y A .
iff .
Ž 3.11.
371
GLOBAL ASYMPTOTIC STABILITY
Thus Ž3.9. and Ž3.11. imply that Ss
AŽ I y A . q 1 IyA
,
Ž 3.12.
which is equivalent to Is
AŽ S y A . q 1
. Ž 3.13. SyA We remark that the ‘‘duality’’ between Ž3.9. and Ž3.11. which yields Ž3.12. and Ž3.13. is NOT present when k s 1. Thus the proof techniques in this paper, as compared with those in w1, 2x, are different. Note that Ž3.12. Žor Ž3.13.. implies that S)Aq1
AŽ I y A . q 1
m
IyA
)Aq1
m
A q 1 ) I.
Ž 3.14. Since we are assuming S ) I, we can change Ž3.8. to the following statement: Assume S ) A q 1 ) I. Ž 3.15. Next, let yn q14 be a subsequence such that j
yn jq1 ª S,
as j ª ⬁.
Ž 3.16.
Consider the subsequence yn j 4 . We have the following lemma. If yn jq14 is a subsequence satisfying Ž3.16., then
LEMMA 3.2.
yn j ª I,
as j ª ⬁.
Ž 3.17.
Proof. Suppose not. Then there exists an ⑀ ) 0 and a subsequence yn 4 such that yn G I q ⑀ , for m sufficiently large. Then Ž3.2., Ž3.12., jm jm and Ž3.16. imply that S s lim sup yn j mª⬁
FAq
m
q1
F A q lim sup m
1 lim inf Ž yny 2 k .
⭈ 1q
n
SFAq sAq
1 I
1q 1
A Iq⑀yA
1 yn j
m
⭈ lim sup 1 q
y2 k
A Iq⑀yA
-Aq
1 I
1q
m
,
yn j y A m
or A
IyA
s S. IyA This is a contradiction, which proves the lemma.
A
372
MISHEV AND PATULA
Continuing in this fashion, we consider the subsequence yn jy14 . LEMMA 3.3. If yn jq14 is a subsequence satisfying Ž3.16., and so necessarily yn j 4 satisfies Ž3.17., then yn jy1 ª S,
as j ª ⬁.
Ž 3.18.
Proof. Suppose not. As in Lemma 3.2, there exists an ⑀ ) 0 and a subsequence yn j y1 such that yn j y1 F S y ⑀ , for m sufficiently large. m m Then Ž3.2., Ž3.13., and Ž3.17. yield I s lim inf yn j G A q lim inf m
mª⬁
m
1
G A q lim inf
yny 2 ky1
n
IGAq sAq
1 S
1q 1
SyA
1 yn j
m
⭈ 1q
A Sy⑀yA
⭈ lim inf 1 q m
y2 ky1
A
,
Sy⑀yA 1
)Aq
S
1q
A yn j
m
y1
yA
or A
SyA
s I.
This is again a contradiction, which proves the lemma. Based on Lemmas 3.2 and 3.3, evidently we can conclude that yn jy2 ª I, yn jy3 ª S, . . . , yn jyŽ2 ky2. ª I, and yn jyŽ2 ky1. ª S, as j ª ⬁. From Ž1.1., we have yn jq1 s A q
yn j yn jy2 ⭈⭈⭈ yn jyŽ2 ky2. yn jy1 yn jy3 ⭈⭈⭈ yn jyŽ2 ky1.
.
Letting j ª ⬁ implies that SsAq
Ik Sk
,
or
Ž S y A. S k s I k . However, this is a contradiction, since Ž3.15. implies
Ž S y A. S k ) S k ) I k . Thus our assumption Ž3.8. is incorrect and S s I. This means yn ª A q 1, as n ª ⬁, for any solution yn4 of Ž1.1..
373
GLOBAL ASYMPTOTIC STABILITY
We remark that a key step in our argument was transforming Eq. Ž1.1. to the form Ž3.5.. At first glance, it appears that Ž3.5. satisfies the hypotheses of a very powerful convergence theorem in w3, Theorem 2.2x. However, it turns out that the hypotheses are not satisfied, and so a separate proof that yn ª A q 1 is necessary. 4. PERIOD 2 SOLUTIONS In this section, as a corollary to the global asymptotic stability result of Section 3, we present an alternate proof of a conjecture of Ladas. Specifically, w4, Conjecture Ž4.3.1., p. 312x proposes that every positive solution to the following difference equation converges to a period 2 solution: x nq 1 s
a xn
q
b x ny2 k
a, b ) 0,
,
k G 2,
n G 2 k q 1.
Ž 4.1.
The case k s 1 was studied and verified by DeVault et al. w2x. The case k G 2 was first proved in w3x in a much more general framework. This paper gives an alternate proof of the conjecture when k G 2. First, note that any set of Ž2 k q 1. positive initial conditions x 1 , . . . , x 2 kq1 will recursively generate a solution x n4 of Ž4.1.. Define ynq1 by setting ynq 1 s
x nq 1 x n b
.
Ž 4.2.
Then Ž4.1. can be written as x nq1 x n b
s
a b
q
xn x ny 2 k
⭈
x x x ny1 x nq 1 x n b
x ny1 x ny1 ⭈
x nyŽ2 ky1.
x ny2 x ny3
b sAq x ny 1 x ny2 b
x nyŽ2 ky1.
⭈ ⭈⭈⭈ ⭈
⭈
⭈ ⭈⭈⭈ ⭈
b x ny3 x ny4 b
⭈
bk bk
,
or
x nyŽ2 ky2. x nyŽ2 ky1.
⭈ ⭈⭈⭈ ⭈
b x nyŽ2 ky1. x ny2 k . b
In view of Ž4.2., this becomes Eq. Ž1.1., where A s arb. Consider any solution x n4 of Ž4.1.. Utilizing Ž4.2., and the asymptotic stability result of Section 3, we have lim
nª⬁
x2 n x 2 ny2
s lim
nª⬁
x 2 n x 2 ny1 x 2 ny1 x 2 ny2
s lim
nª⬁
y2 n y 2 ny1
s
Aq1 Aq1
s 1.
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MISHEV AND PATULA
Continuing in this fashion, x2 n x 2 n x 2 ny1 x 2 ny2 x 2 ny3 y 2 n y 2 ny2 lim s lim s lim s 1. nª⬁ x 2 ny4 nª⬁ x 2 ny1 x 2 ny2 x 2 ny3 x 2 ny4 nª⬁ y 2 ny1 y 2 ny3 A similar argument implies that for 0 F j F 2 k, x2 n lim s 1, nª⬁ x 2 ny2 j which means x 2 ny2 j
lim
nª⬁
s 1.
x2 n
Finally, it is clear that for 0 F j F 2 k and 0 F r F 2 k, r / j, x 2 ny2 r lim s 1. nª⬁ x 2 ny2 j
Ž 4.3.
As an analog of Lemma 3.1 and w2, Lemma 2Ži.x, we have the following result. LEMMA 4.1. Consider Ž4.1. and let N be some integer such that N ) 2 k. Define MN s max x 2 N , x 2 Ny2 , . . . , x 2 Ny2Ž2 k .4 and m N s min x 2 N , x 2 Ny2 , . . . , x 2 Ny2Ž2 k .4 . Then m N F x 2 n F MN , ᭙ n G N. Proof. From Ž4.1., x 2 Nq2 s s
a x 2 Nq1
x 2 Nq1y2 k
a a x2 N
F
b
q
q
a aq b MN
q
b x 2 Ny2 k
q
b aqb
b a x 2 Ny2 k
q
b x 2 Ny4 k
s MN .
MN
Now use induction. A similar argument with the inequalities reversed yields that x 2 n G m N , ᭙ n G N. LEMMA 4.2.
Let x n4 be a solution of Ž4.1.. Then lim nª⬁ x 2 n exists.
Proof. Lemma 4.1 implies that S and I are well defined, where S s lim sup nª⬁ x 2 n and I s lim inf nª⬁ x 2 n . We suppose S ) I and reach a contradiction. Choose N1 ) 2 k and consider the elements x 2 N1, x 2 N1y2 , . . . , x 2 N1y2Ž2 k . . At least one of these elements must be G S. If not, then Lemma 4.1
GLOBAL ASYMPTOTIC STABILITY
375
implies that ᭙ n G N1 , x 2 n F max x 2 N1, . . . , x 2 N1y4 k 4 - S, a contradiction. Similarly, at least one of x 2 N1, . . . , x 2 N1y4 k must be F I. Thus there must be at least one quotient of the form x 2 N1y2 rrx 2 N1y2 j , 0 F r F 2 k, 0 F j F 2 k, r / j, where x 2 N1y2 rrx 2 N1y2 j G SI ) 1. Next, choose N2 s N1 q 4 k q 1 and consider the elements x 2 N 2 , x 2 N 2y2 , . . . , x 2 N 2y2Ž2 k . . As above, there must be at least one quotient such that x 2 N 2y2 rrx 2 N 2y2 j G SI ) 1, where 0 F r F 2 k, 0 F j F 2 k, r / j. We can continue in this fashion to choose N3 , N4 , . . . , Nb , etc., such that x 2 N by2 r x 2 N by2 j
G
S I
) 1,
where 0 F r F 2 k, 0 F j F 2 k,
r / j. Ž 4.4.
The values of r and j may change for each Nb . For each Nb , though, there are only a finite number of possible quotients of the form Ž4.4.. Since we may let b ª ⬁, there will be at least one subsequence Nb i such that x 2 N b y2 r i
x 2 N b y2 j
G
S
i
I
) 1,
Ž 4.5.
where r and j are the same for every bi . Since we may let i ª ⬁, this contradicts Ž4.3. and proves Lemma 4.2. Thus lim nª⬁ x 2 n exists. We call it L E as in w2, Theorem 2x. Since Ž4.1. implies x 2 nq1 s
a x2 n
q
b x 2 ny2 k
,
k G 2,
Ž 4.6.
clearly lim nª⬁ x 2 nq1 exists. We call this limit LO , and Ž4.6. implies L E ⭈ LO s a q b. This yields an alternate proof of w4, Conjecture 4.3.1, p. 312x. Specifically, if we consider Eq. Ž4.1., then every positive nontrivial solution converges to a period 2 solution, where the product of the periods is Ž a q b ..
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