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Asymptotic Stability Condition for a Class of Linear Delay Difference Equations of Higher Order
Journal of Mathematical Analysis and Applications 248, 83᎐96 Ž2000. doi:10.1006rjmaa.2000.6868, available online at http:rrwww.idealibrary.com on
Asymptotic Stability Condition for a Class of Linear Delay Difference Equations of Higher Order Ryuzou Ogita Osaka Prefectural Seijyou Technical Senior High School, Osaka 536-0021, Japan
and Hideaki Matsunaga and Tadayuki Hara1 Department of Mathematical Sciences, Osaka Prefecture Uni¨ ersity, Sakai 599-8531, Japan Submitted by William F. Ames Received April 20, 1999
In this paper we obtain a necessary and sufficient condition for the asymptotic stability of the zero solution of the linear delay difference equation N
x nq 1 y x n q p
Ý x nykqŽ jy1.l s 0,
n s 0, 1, 2, . . . ,
js1
by using root-analysis for the characteristic equation. Here, p is a real number and k, l, and N are positive integers with k ) Ž N y 1. l. 䊚 2000 Academic Press
1. INTRODUCTION In this paper we study the asymptotic stability of the linear delay difference equation of higher order, N
x nq 1 y x n q p
Ý x nykqŽ jy1.l s 0,
n s 0, 1, 2, . . . ,
Ž 1.1.
js1
where p is a real number and k, l, and N are positive integers with k ) Ž N y 1. l. 1 Supported in part by Grant-in-Aid for Scientific Research 11640210. E-mail: [email protected].
83 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
84
OGITA, MATSUNAGA, AND HARA
In the case N s 1, it is known w5᎐7x that the zero solution of the difference equation x nq 1 y x n q px nyk s 0,
n s 0, 1, 2, . . . ,
Ž 1.2.
is asymptotically stable if and only if k
0 - p - 2 cos
2k q 1
.
Ž 1.3.
This nice result is proved by using the fact that the zero solution of the linear difference equation is asymptotically stable if and only if all the roots of its characteristic equation are inside the unit disk. By the application of Rouche’s ´ theorem for the characteristic equation, the following is easily shown w1; 3, pp. 12᎐13x: The zero solution of the difference equation N
x nq 1 y x n q
Ý pj x nyk
j
s 0,
n s 0, 1, 2, . . . ,
Ž 1.4.
js1
where p1 , p 2 , . . . , pN are positi¨ e numbers and k 1 , k 2 , . . . , k N are positi¨ e integers, is asymptotically stable pro¨ ided that N
Ý k j pj - 1.
Ž 1.5.
js1
The condition Ž1.5., however, is not necessary. Indeed, Rouche’s ´ theorem cannot give the necessary condition. Also, several kinds of the necessary and sufficient conditions given by using the Routh᎐Hurwitz criterion are much too complicated to even verify the condition Ž1.3.. The purpose of this paper is to give a necessary and sufficient condition for the asymptotic stability of the zero solution of Ž1.1. which is more restrictive than Ž1.4.. Our main result is stated as follows: THEOREM 1.1. Let k, l, and N be positi¨ e integers with k ) Ž N y 1. l. Then the zero solution of Ž1.1. is asymptotically stable if and only if 2 sin 0-p-
2M
sin where M s 2 k q 1 y Ž N y 1. l.
l
ž / ž / sin
Nl
ž / 2M
2M
Ž 1.6.
85
LINEAR DELAY DIFFERENCE EQUATION
Note that the condition Ž1.3. is a special case of the condition Ž1.6.. Our research is motivated by a result on delay differential equations in w2x: For the linear differential equation with N delays, xX Ž t . q a
N
Ý x Ž t y q Ž j y 1. ␦ . s 0,
t G 0,
Ž 1.7.
js1
where and ␦ are positi¨ e numbers with ) Ž N y 1. ␦ , the zero solution of Ž1.7. is asymptotically stable if and only if
0-a-
L
sin
␦
ž / ž /
sin
2L N␦
,
2L
where L s 2 y Ž N y 1. ␦ . We see that there is some resemblance between the above inequality and Ž1.6.. For the proof of our main theorem, we first examine the value of p and the location of the roots z of the characteristic equation when the roots are on the unit circle. Next we regard the root z as the algebraic function of p and then compute the derivative of the absolute value of the root z by p to observe its behavior. Finally, we determine the range of p when all the roots are inside the unit disk to get the inequality Ž1.6..
2. PROOF OF THEOREM The characteristic equation of Ž1.1. is given by the form F Ž z . ' z kq 1 y z k q p Ž z Ž Ny1.l q ⭈⭈⭈ qz l q 1 . s 0.
Ž 2.1.
Clearly, for p s 0, the roots of Ž2.1. are 0 Žmultiplicity k . and 1 Žsimple.. We will discuss the location of the roots of Ž2.1. as p varies in order to prove Theorem 1.1. The first proposition deals with the value of p and the roots of Ž2.1. on the unit circle. Hereafter, we put M s 2 k q 1 y Ž N y 1. l. PROPOSITION 2.1. Let z be a root of Ž2.1. on the unit circle except for z s 1. Then the root z and the real number p are expressed as z s emi,
where m s
Ž 2 m q 1. M
Ž 2.2.
86
OGITA, MATSUNAGA, AND HARA
and sin pm s 2 Ž y1 .
m
m
l m
ž / ž / 2
sin
ž
sin
Nl m 2
2
Ž 2.3.
/
for some m s 0, 1, . . . , M y 1, respecti¨ ely. Con¨ ersely, if p is gi¨ en by Ž2.3., z s e m i is a root of Ž2.1.. Proof. First, we note that the value z satisfying z N l s 1 and z l / 1, namely z Ž Ny1.l q ⭈⭈⭈ qz l q 1 s 0,
Ž 2.4.
is not the root of Ž2.1. because Ž2.4. implies that we cannot find the real number p. Thus Ž2.1. may be written as psy
z k Ž z y 1. z Ž Ny1.l q ⭈⭈⭈ qz l q 1
.
Ž 2.5.
Since p is real, we have psy
z k Ž z y 1. z Ž Ny1.l q ⭈⭈⭈ qz l q 1
sy
zyky1qŽ Ny1.l Ž 1 y z . 1 q z l q ⭈⭈⭈ qz Ž Ny1.l
Ž 2.6.
because of z s 1rz. From Ž2.5. and Ž2.6., we obtain z 2 kq1yŽ Ny1.l s y1, which implies that Ž2.2. is valid for m s 0, 1, . . . , M y 1 except for the integers m satisfying e N l m i s 1 and e l m i / 1 where m s Ž2 m q 1.rM. Next we will show that Ž2.3. holds. In the case z N l / 1, by Ž2.5. we get psy sy
z k Ž z y 1. Ž z l y 1. z Nl y 1 e k m i Ž e m i y 1. Ž e l m i y 1. e N l m i y 1
s ye Ž kyN l r2q1r2ql r2. m i s ye Ž2 mq1. i r2 s 2 Ž y1 .
m
Ž e m i r2 y ey m i r2 . Ž e l m i r2 y eyl m i r2 . e N l m i r2 y eyN l m i r2
2 i sin Ž m r2 . 2 i sin Ž l m r2 . 2 i sin Ž Nl m r2 .
sin Ž m r2 . sin Ž l m r2 . sin Ž Nl m r2 .
' pm .
87
LINEAR DELAY DIFFERENCE EQUATION
In the case z l s 1, put l m s 2 q where q is some positive integer, then we can obtain p s 2 Ž y1 .
mqq Ž Ny1 .
sin Ž m r2 .
Ž 2.7.
N
in a similar way. Note that Ž2.7. is the limit of pm as l m tends to 2 q . Hence, we conclude that Ž2.3. is valid for m s 0, 1, . . . , M y 1 except for the integers m satisfying e N l m i s 1 and e l m i / 1. Conversely, if p is given by Ž2.3., then it is clear that z s e m i is a root of Ž2.1.. The proof is complete. Now we regard p as a holomorphic function of z, that is, pŽ z . s y
z k Ž z y 1. z Ž Ny1.l q ⭈⭈⭈ qz l q 1
.
Differentiating pŽ z ., we have dp dz
z ky 1 Ž k Ž z y 1 . q z .
sy
z Ž Ny1.l q ⭈⭈⭈ qz l q 1
q
z ky 1 Ž z y 1 . Ž N y 1 . lz Ž Ny1.l q ⭈⭈⭈ qlz l 4
Ž z Ž Ny1.l q ⭈⭈⭈ qz l q 1 .
2
.
Ž 2.8.
Then we have the following lemma. LEMMA 2.1. The deri¨ ati¨ e of pŽ z . at z s e m i is nonzero. Proof. Suppose that dprdz s 0. Dividing Ž2.8. by pŽ z .rz, we get kq
z zy1
y
l Ž N y 1 . z Ž Ny1.l q ⭈⭈⭈ qz l 4 z Ž Ny1.l q ⭈⭈⭈ qz l q 1
s 0.
Ž 2.9.
In view of z s 1rz, Ž2.9. yields kq
1 1yz
y
l Ž N y 1 . q Ž N y 2 . z l q ⭈⭈⭈ qz Ž Ny2.l 4 1 q z l q ⭈⭈⭈ qz Ž Ny1.l
Thus, adding Ž2.9. and Ž2.10., we obtain 2 k q 1 y Ž N y 1 . l s 0, which contradicts k ) Ž N y 1. l. This completes the proof.
s 0. Ž 2.10.
88
OGITA, MATSUNAGA, AND HARA
Remark 2.1. Lemma 2.1 shows that the roots of Ž2.1. on the unit circle are simple. From the property of holomorphic functions, there exists a neighborhood of z s e m i where the mapping pŽ z . is one-to-one. Therefore the inverse function of pŽ z . exists locally. Let z be expressed in polar form, that is, z s re i . Then we have dz
s
dp
z
ž
r
dr dp
q ir
d dp
/
,
which implies that the value of drrdp is the real part of Ž rrz .Ž dzrdp. if p moves, keeping real value. Next, we will observe how the roots of Ž2.1. cross the unit circle when the real number p varies. PROPOSITION 2.2. The absolute ¨ alues of the roots of Ž2.1. at z s e m i increase as < p < increases. Proof. Let r be the absolute value of z. It suffices to show that dr dp
⭈p)0
Ž 2.11.
at z s e m i. In the case z N l / 1, we express pŽ z . as pŽ z . s y
z k Ž z y 1. Ž z l y 1. z Nl y 1
sy
zkf Ž z. z Nl y 1
,
where f Ž z . s Ž z y 1.Ž z l y 1.. Then we have dp dz
sy
z ky 1 g Ž z .
Ž z N l y 1.
2
,
where g Ž z . s Ž kf Ž z . q zf X Ž z ..Ž z N l y 1. y Nlz N l f Ž z .. For the sake of convenience, putting w s yŽ z N l y 1. 2rŽ z k g Ž z .., we obtain dr dp
s Re
r dz
ž / z dp
s r Re w.
Hence, we shall compute the value of Re w. Note that f Ž z. s
f Ž z. z
lq1
,
fXŽ z. s
hŽ z . zl
,
89
LINEAR DELAY DIFFERENCE EQUATION
where hŽ z . s l q 1 y lz y z l. By using the relation above and z M s y1, we get zkgŽ z. s s
1 z
k
½ž
kf Ž z . q
1 z
fXŽ z.
1
/ž
z
Nl
y1 y
/
Nl z Nl
f Ž z.
5
Ž kf Ž z . q h Ž z . . Ž 1 y z N l . y Nlf Ž z . z kq N lqlq1
sy
Ž kf Ž z . q h Ž z . . Ž 1 y z N l . y Nlf Ž z . z 2 N lyk
.
Then it follows that Re w s
wqw 2 2
Ž z N l y 1. Ž z N l y 1. sy q 2 zkgŽ z. zkgŽ z. 1
½
2
5
2
sy
z k g Ž z . Ž z N l y 1. q z k g Ž z . Ž z N l y 1. 2 gŽ z.
2
2
2
sy
Ž z N l y 1. z k 2z
2 Nl
gŽ z.
2
y Ž kf Ž z . q h Ž z . . Ž 1 y z N l . q Nlf Ž z . q Ž kf Ž z . q zf X Ž z . . Ž z N l y 1 . y Nlz N l f Ž z . 4
3
sy
Ž z N l y 1. z k 2z
2Nl
gŽ z.
2
Ž 2 kf Ž z . q h Ž z . q zf X Ž z . y Nlf Ž z . . .
Taking note that 2 kf Ž z . q h Ž z . q zf X Ž z . y Nlf Ž z . s Mf Ž z . , we have 4
Re w s
Ž z N l y 1 . M yz k f Ž z . 2 z2Nl g Ž z.
2
⭈
z Nl y 1
4
s
Ž z N l y 1 . Mp 2 z2 Nl g Ž z.
2
.
90
OGITA, MATSUNAGA, AND HARA
Now, by z s e m i, it is easily seen that
Ž z N l y 1. z
2Nl
4 2
s Ž 2 cos Nl m y 2 . ) 0,
and therefore we arrive at dr dp
2
s
2 r Ž cos Nl m y 1 . Mp gŽ z.
2
.
This implies that Ž2.11. is valid at z s e m i. In the case z l s 1, by an argument similar to that above, we can obtain dr dp
s
2 rN 2 Mp
Ž M q 1. z y M q 1
2
,
which implies that Ž2.11. is also valid at z s e m i. This completes the proof. Furthermore, we will determine the positive minimum value of pm given by Ž2.3.. PROPOSITION 2.3. The ¨ alue of p 0 is the positi¨ e minimum ¨ alue of pm for m s 0, 1, . . . , M y 1. To prove Proposition 2.3, we need the following two lemmas which will be proved in the Appendix. LEMMA 2.2.
Let 0 - - r2, then the inequality sin x sin y G sin sin xy
holds for all x, y g Ž1, rŽ2 ... LEMMA 2.3.
Let N be a positi¨ e integer, then the inequality sin Nt sin t
FN
holds for all t. Proof of Proposition 2.3. It is clear that p 0 is positive by the definition. Now, we have only to consider the values of pm which correspond to the roots z s e m i for m s 0, 1, . . . , wŽ M y 1.r2x lying in the upper half plane, because the conjugate root e m i also corresponds to the same value of pm . Our argument is divided into three cases.
91
LINEAR DELAY DIFFERENCE EQUATION
N s 1. It follows from Ž2.3. that
Case I:
m
pm s 2 Ž y1 . sin
Ž 2 m q 1. 2 Ž 2 k q 1.
for m s 0, 1, . . . , k.
Thus we can immediately conclude that p 0 s 2 sin
ž
2 Ž 2 k q 1.
s 2 cos
k 2k q 1
/
is the positive minimum value of pm . Case II: N s 2. It suffices to show that the value of 1rpm attains its maximum when m s 0, that is, 1 p0
1
G
Ž 2.12.
pm
for m s 0, 1, . . . , wŽ M y 1.r2x. Since z l s yz 2 kq1 and z s e m i, the value of pm can be written as pm s
z k Ž z y 1. z
2 kq1
y1
s
z 1r2 y zy1r2 z
kq1r2
yky1r2
yz
s
sin Ž m r2 . sin Ž Ž 2 k q 1 . m r2 .
.
This implies 1 p0
s
sin Ž 2 k q 1 . sin
1
,
pm
s
sin Ž 2 k q 1 . Ž 2 m q 1 . sin Ž 2 m q 1 .
,
where s rŽ2 M .. Note that 0 - - r2 and 1 F 2 M y Ž 2 k q 1. F
2
,
1 F 2m q 1 F
2
,
because of k ) l. Hence it follows from Lemma 2.2 that sin Ž 2 M y Ž 2 k q 1 . . sin Ž 2 m q 1 . G sin sin Ž 2 M y Ž 2 k q 1 . . Ž 2 m q 1 . . Taking account of Ž2 M y Ž2 k q 1.. s y Ž2 k q 1. , we obtain Ž2.12. for m s 0, 1, . . . , wŽ M y 1.r2x. Case III:
N G 3. To end the proof, we will show that < pm < G p 0
Ž 2.13.
92
OGITA, MATSUNAGA, AND HARA
for m s 0, 1, . . . , wŽ M y 1.r2x. For simplicity, we set s rŽ2 M . again, then 0 - Ž2 m q 1. F r2 and therefore Ž2.3. becomes sin Ž 2 m q 1 . l
< pm < s 2 sin Ž 2 m q 1 .
.
sin Ž 2 m q 1 . Nl
By Lemma 2.3 and Jordan’s inequality, 2
sin
F
for 0 F F
F1
,
2
we have 2
< pm < G 2 ⭈
Ž 2 m q 1. ⭈
1
4 Ž 2 m q 1.
s
N
.
N
Ž 2.14.
Now we estimate p 0 from the above. Subcase ŽIIIa.:
Nl F r2. We have p0 s
2 sin sin l sin Nl
F
2 ⭈ l
Ž 2r . Nl
s
N
,
which, together with Ž2.14., implies that Ž2.13. holds for m s 0, 1, . . . , wŽ M y 1.r2x. Subcase ŽIIIb.:
Nl ) r2. We note that
Nl s
Nl
-
2M
Nl
-
2 Ž N y 1. l q 1
N
Ž 2.15.
2 Ny1
because of k ) Ž N y 1. l. By using sin Nl s sinŽ y Nl ., we get p0 F
2 ⭈ l
s
Ž 2r . Ž y Nl .
l 2 y Nl
.
Ž 2.16.
From Ž2.15., the inequalities Ž2.14. and Ž2.16. yield < pm < p0
G )
4 Ž 2 m q 1.
N 4 Ž 2 m q 1.
2
ž
⭈
y Nl l
2
2 Ž N y 1. N
s
4 Ž 2 m q 1.
y1 s
/
2
ž
Nl
4 Ž 2 m q 1.
2
y1
ž
/
1y
2 N
/
.
93
LINEAR DELAY DIFFERENCE EQUATION
Here we find the following: Ži. If N G 12 and m G 1, then Ž2.13. holds. Žii. If N G 4 and m G 2, then Ž2.13. holds. Žiii. If N s 3 and m G 4, then Ž2.13. holds. Consequently, we have only to consider the remainder cases. In the case N G 4 and m s 1, Ž2.15. implies l - r6 and hence we have 2 sin 3 sin 3l
< p1 < s
2
G2⭈
sin 3 Nl
3 ⭈
2
3l s
72 l 2
2
.
From Ž2.15., the above inequality and Ž2.16. yield < p1 < p0
G
72 l 2
2
⭈
y Nl l
2
)
72
ž
3
y
N
2 Ny1
/
G
24
2
) 1.
In the case N s 3 and 1 F m F 3, Ž2.15. and the assumption of Subcase ŽIIIb. imply r6 - l - r4 and we have < pm < p0
s
sin Ž 2 m q 1 . l sin 3l sin Ž 2 m q 1 . sin 3 Ž 2 m q 1 . l sin l
sin
.
Ž 2.17.
By Lemma 2.3, we estimate the first factor of Ž2.17. into sin Ž 2 m q 1 . l sin 3l sin 3 Ž 2 m q 1 . l sin l
G
1 sin 3l 3 sin l
s
1 3
<3 y 4 sin 2 l < )
The second factor of Ž2.17. can be estimated by sin Ž 2 m q 1 . sin
)
Ž 2r . Ž 2 m q 1 .
s
2 Ž 2 m q 1.
Therefore we obtain < pm < p0
2 Ž 2 m q 1.
)
3
)1
for m s 2, 3.
Finally, if m s 1 and p1 ) 0, using r6 - l - r4, we get p1 p0
s
3 y 4 sin 2 l sin Ž 2 m q 1 . 4 sin 3l y 3 2
sin
This completes the proof of Proposition 2.3.
)1⭈
6
) 1.
.
1 3
.
94
OGITA, MATSUNAGA, AND HARA
Now, we are in a position to prove Theorem 1.1. Proof of Theorem 1.1. First of all, we note that if p - 0, then there exists a positive root z# of Ž2.1. such that z# ) 1 because lim z ªq⬁ F Ž z . s q⬁ and F Ž1. s pN - 0. Recall that the roots of Ž2.1. with p s 0 are 0 Žmultiplicity k . and 1 Žsimple.. We claim here that if p ) 0 is sufficiently small, then all the roots of Ž2.1. are inside the unit disk. In fact, let z1Ž p . be the branch of the root of Ž2.1. with z1Ž0. s 1, then it follows from Ž2.8. that dz1 dp
s yN - 0. ps 0
By the continuity of the roots with respect to p, this implies that our claim is valid. In addition, Proposition 2.3 shows that p 0 is a positive minimum value of p such that a root of Ž2.1. intersects the unit circle as p increases from 0. Also, by virtue of Proposition 2.2, we obtain that if p G p 0 then there exists a root zU of Ž2.1. such that < zU < G 1. From the preceding argument, we conclude that all the roots of Ž2.1. are inside the unit disk if and only if 0 - p - p 0 . That is, the zero solution of Ž1.1. is asymptotically stable if and only if the condition Ž1.6. holds. The proof of Theorem 1.1 is complete. Finally, we consider the linear difference equation x nq 1 y x n q p Ž x nyk 1 q x nyk 2 . s 0,
n s 0, 1, 2, . . . ,
Ž 2.18.
where p is a real number and k 1 , k 2 are positive integers. By setting N s 2 and l s < k 1 y k 2 <, we have the following result as a corollary of Theorem 1.1, which corresponds to the result in w8, p. 65x and w4, p. 87x for a linear delay differential equation with two delays. COROLLARY 2.1. Let k 1 and k 2 be positi¨ e integers. Then the zero solution of Ž2.18. is asymptotically stable if and only if
sin 0-pcos
2 Ž k1 q k 2 q 1. . Ž k1 y k 2 . 2 Ž k1 q k 2 q 1.
LINEAR DELAY DIFFERENCE EQUATION
95
APPENDIX In this appendix we give proofs of Lemmas 2.2 and 2.3. Proof of Lemma 2.2. Set the domain D s Ž x, y . : 1 F x F rŽ2 ., 1 F y F rŽ2 .4 and put the function G Ž x, y . s sin x sin y y sin sin xy . Then GŽ x, y . is continuously differentiable in D and its minimum is attained at the boundary ⭸ D or is relatively minimum. Obviously, GŽ x, y . G 0 in ⭸ D. An easy calculation yields that
⭸G ⭸x ⭸G ⭸y
s cos x sin y y y cos xy sin , s sin x cos y y x cos xy sin .
If GŽ x, y . is relatively minimum at Ž x, y ., then ⭸ Gr⭸ x s ⭸ Gr⭸ y s 0. This implies that x, y satisfy x tan x
y
s
tan y
.
Ž A.1.
It is easy to see that the function Ž x . s xrŽtan x . is strictly decreasing on Ž1, rŽ2 .., and thus ŽA.1. implies x s y. Hence, if we show that G Ž x, x . s sin 2 x y sin x 2 sin G 0 then the proof will be complete. To this end, it is sufficient to verify the inequality sin 2 x
Ž x .
2
G
sin x 2 sin
x 2
.
Ž A.2.
In the case x 2 G 2 , for F x F r2, we have sin 2 x
Ž x .
2
y
sin x 2 sin x 2
G
2
ž /
2
y
1 2
) 0.
In the case F x 2 F 2 , it is clear that ŽA.2. holds because of sin x 2 F 0. In the case 0 - x 2 - , consider the function Ž . s logŽŽsin .r ..
96
OGITA, MATSUNAGA, AND HARA
Then
X Ž . s
Ž y tan . cos sin
Y Ž . s
- 0,
sin 2 y 2 2 Ž sin .
- 0,
and thus Ž . is a decreasing and upward convex function. Therefore, using Ž x 2 q .r2 G x and the properties of Ž ., we obtain
Ž x . G
ž
x 2 q 2
/
G
Ž x 2 . q Ž . 2
.
This becomes 2 Ž x . G Ž x 2 . q Ž ., namely 2 log
sin x x
G log
sin x 2 x 2
q log
sin
,
and we arrive at ŽA.2.. The proof of Lemma 2.2 is now complete. Proof of Lemma 2.3. Let N and k be positive integers. Define Ž t . s Žsin Nt .rsin t and Ž k . s lim t ª k Ž t ., then the function Ž t . is continuously differentiable and its period is 2 . We also note that Žyt . s Ž t . and < Ž y t .< s < Ž t .<. It is easily seen that
Ž t. s
FN
for t g 0,
2
.
Hence, these facts above imply the assertion of Lemma 2.3.
REFERENCES 1. K. L. Cooke and I. Gyori, ¨ Numerical approximation of the solution of delay differential equations on an infinite interval using piecewise constant arguments, Comput. Math. Appl. 28 Ž1994., 81᎐92. 2. T. Hara, R. Miyazaki, and T. Morii, Asymptotic stability condition for linear differentialdifference equations with N delays, Dynam. Systems Appl. 6 Ž1997., 493᎐506. 3. V. L. Kocic and G. Ladas, ‘‘Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,’’ Kluwer Academic, Dordrecht, 1993. 4. Y. Kuang, ‘‘Delay Differential Equations with Applications in Population Dynamics,’’ Academic Press, San Diego, 1993. 5. S. A. Kuruklis, The asymptotic stability of x nq 1 y ax n q bx nyk s 0, J. Math. Anal. Appl. 188 Ž1994., 719᎐731. 6. S. A. Levin and R. M. May, A note on difference-delay equations, Theoret. Population Biol. 9 Ž1976., 178᎐187. 7. V. G. Papanicolaou, On the asymptotic stability of a class of linear difference equations, Math. Magazine 69 Ž1996., 34᎐43. 8. G. Stepan, ´ ´ ‘‘Retarded Dynamical Systems: Stability and Characteristic Functions,’’ Longman, Harlow, UK, 1989.
Asymptotic Stability Condition for a Class of Linear Delay Difference Equations of Higher Order Ryuzou Ogita Osaka Prefectural Seijyou Technical Senior High School, Osaka 536-0021, Japan
and Hideaki Matsunaga and Tadayuki Hara1 Department of Mathematical Sciences, Osaka Prefecture Uni¨ ersity, Sakai 599-8531, Japan Submitted by William F. Ames Received April 20, 1999
In this paper we obtain a necessary and sufficient condition for the asymptotic stability of the zero solution of the linear delay difference equation N
x nq 1 y x n q p
Ý x nykqŽ jy1.l s 0,
n s 0, 1, 2, . . . ,
js1
by using root-analysis for the characteristic equation. Here, p is a real number and k, l, and N are positive integers with k ) Ž N y 1. l. 䊚 2000 Academic Press
1. INTRODUCTION In this paper we study the asymptotic stability of the linear delay difference equation of higher order, N
x nq 1 y x n q p
Ý x nykqŽ jy1.l s 0,
n s 0, 1, 2, . . . ,
Ž 1.1.
js1
where p is a real number and k, l, and N are positive integers with k ) Ž N y 1. l. 1 Supported in part by Grant-in-Aid for Scientific Research 11640210. E-mail: [email protected].
83 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
84
OGITA, MATSUNAGA, AND HARA
In the case N s 1, it is known w5᎐7x that the zero solution of the difference equation x nq 1 y x n q px nyk s 0,
n s 0, 1, 2, . . . ,
Ž 1.2.
is asymptotically stable if and only if k
0 - p - 2 cos
2k q 1
.
Ž 1.3.
This nice result is proved by using the fact that the zero solution of the linear difference equation is asymptotically stable if and only if all the roots of its characteristic equation are inside the unit disk. By the application of Rouche’s ´ theorem for the characteristic equation, the following is easily shown w1; 3, pp. 12᎐13x: The zero solution of the difference equation N
x nq 1 y x n q
Ý pj x nyk
j
s 0,
n s 0, 1, 2, . . . ,
Ž 1.4.
js1
where p1 , p 2 , . . . , pN are positi¨ e numbers and k 1 , k 2 , . . . , k N are positi¨ e integers, is asymptotically stable pro¨ ided that N
Ý k j pj - 1.
Ž 1.5.
js1
The condition Ž1.5., however, is not necessary. Indeed, Rouche’s ´ theorem cannot give the necessary condition. Also, several kinds of the necessary and sufficient conditions given by using the Routh᎐Hurwitz criterion are much too complicated to even verify the condition Ž1.3.. The purpose of this paper is to give a necessary and sufficient condition for the asymptotic stability of the zero solution of Ž1.1. which is more restrictive than Ž1.4.. Our main result is stated as follows: THEOREM 1.1. Let k, l, and N be positi¨ e integers with k ) Ž N y 1. l. Then the zero solution of Ž1.1. is asymptotically stable if and only if 2 sin 0-p-
2M
sin where M s 2 k q 1 y Ž N y 1. l.
l
ž / ž / sin
Nl
ž / 2M
2M
Ž 1.6.
85
LINEAR DELAY DIFFERENCE EQUATION
Note that the condition Ž1.3. is a special case of the condition Ž1.6.. Our research is motivated by a result on delay differential equations in w2x: For the linear differential equation with N delays, xX Ž t . q a
N
Ý x Ž t y q Ž j y 1. ␦ . s 0,
t G 0,
Ž 1.7.
js1
where and ␦ are positi¨ e numbers with ) Ž N y 1. ␦ , the zero solution of Ž1.7. is asymptotically stable if and only if
0-a-
L
sin
␦
ž / ž /
sin
2L N␦
,
2L
where L s 2 y Ž N y 1. ␦ . We see that there is some resemblance between the above inequality and Ž1.6.. For the proof of our main theorem, we first examine the value of p and the location of the roots z of the characteristic equation when the roots are on the unit circle. Next we regard the root z as the algebraic function of p and then compute the derivative of the absolute value of the root z by p to observe its behavior. Finally, we determine the range of p when all the roots are inside the unit disk to get the inequality Ž1.6..
2. PROOF OF THEOREM The characteristic equation of Ž1.1. is given by the form F Ž z . ' z kq 1 y z k q p Ž z Ž Ny1.l q ⭈⭈⭈ qz l q 1 . s 0.
Ž 2.1.
Clearly, for p s 0, the roots of Ž2.1. are 0 Žmultiplicity k . and 1 Žsimple.. We will discuss the location of the roots of Ž2.1. as p varies in order to prove Theorem 1.1. The first proposition deals with the value of p and the roots of Ž2.1. on the unit circle. Hereafter, we put M s 2 k q 1 y Ž N y 1. l. PROPOSITION 2.1. Let z be a root of Ž2.1. on the unit circle except for z s 1. Then the root z and the real number p are expressed as z s emi,
where m s
Ž 2 m q 1. M
Ž 2.2.
86
OGITA, MATSUNAGA, AND HARA
and sin pm s 2 Ž y1 .
m
m
l m
ž / ž / 2
sin
ž
sin
Nl m 2
2
Ž 2.3.
/
for some m s 0, 1, . . . , M y 1, respecti¨ ely. Con¨ ersely, if p is gi¨ en by Ž2.3., z s e m i is a root of Ž2.1.. Proof. First, we note that the value z satisfying z N l s 1 and z l / 1, namely z Ž Ny1.l q ⭈⭈⭈ qz l q 1 s 0,
Ž 2.4.
is not the root of Ž2.1. because Ž2.4. implies that we cannot find the real number p. Thus Ž2.1. may be written as psy
z k Ž z y 1. z Ž Ny1.l q ⭈⭈⭈ qz l q 1
.
Ž 2.5.
Since p is real, we have psy
z k Ž z y 1. z Ž Ny1.l q ⭈⭈⭈ qz l q 1
sy
zyky1qŽ Ny1.l Ž 1 y z . 1 q z l q ⭈⭈⭈ qz Ž Ny1.l
Ž 2.6.
because of z s 1rz. From Ž2.5. and Ž2.6., we obtain z 2 kq1yŽ Ny1.l s y1, which implies that Ž2.2. is valid for m s 0, 1, . . . , M y 1 except for the integers m satisfying e N l m i s 1 and e l m i / 1 where m s Ž2 m q 1.rM. Next we will show that Ž2.3. holds. In the case z N l / 1, by Ž2.5. we get psy sy
z k Ž z y 1. Ž z l y 1. z Nl y 1 e k m i Ž e m i y 1. Ž e l m i y 1. e N l m i y 1
s ye Ž kyN l r2q1r2ql r2. m i s ye Ž2 mq1. i r2 s 2 Ž y1 .
m
Ž e m i r2 y ey m i r2 . Ž e l m i r2 y eyl m i r2 . e N l m i r2 y eyN l m i r2
2 i sin Ž m r2 . 2 i sin Ž l m r2 . 2 i sin Ž Nl m r2 .
sin Ž m r2 . sin Ž l m r2 . sin Ž Nl m r2 .
' pm .
87
LINEAR DELAY DIFFERENCE EQUATION
In the case z l s 1, put l m s 2 q where q is some positive integer, then we can obtain p s 2 Ž y1 .
mqq Ž Ny1 .
sin Ž m r2 .
Ž 2.7.
N
in a similar way. Note that Ž2.7. is the limit of pm as l m tends to 2 q . Hence, we conclude that Ž2.3. is valid for m s 0, 1, . . . , M y 1 except for the integers m satisfying e N l m i s 1 and e l m i / 1. Conversely, if p is given by Ž2.3., then it is clear that z s e m i is a root of Ž2.1.. The proof is complete. Now we regard p as a holomorphic function of z, that is, pŽ z . s y
z k Ž z y 1. z Ž Ny1.l q ⭈⭈⭈ qz l q 1
.
Differentiating pŽ z ., we have dp dz
z ky 1 Ž k Ž z y 1 . q z .
sy
z Ž Ny1.l q ⭈⭈⭈ qz l q 1
q
z ky 1 Ž z y 1 . Ž N y 1 . lz Ž Ny1.l q ⭈⭈⭈ qlz l 4
Ž z Ž Ny1.l q ⭈⭈⭈ qz l q 1 .
2
.
Ž 2.8.
Then we have the following lemma. LEMMA 2.1. The deri¨ ati¨ e of pŽ z . at z s e m i is nonzero. Proof. Suppose that dprdz s 0. Dividing Ž2.8. by pŽ z .rz, we get kq
z zy1
y
l Ž N y 1 . z Ž Ny1.l q ⭈⭈⭈ qz l 4 z Ž Ny1.l q ⭈⭈⭈ qz l q 1
s 0.
Ž 2.9.
In view of z s 1rz, Ž2.9. yields kq
1 1yz
y
l Ž N y 1 . q Ž N y 2 . z l q ⭈⭈⭈ qz Ž Ny2.l 4 1 q z l q ⭈⭈⭈ qz Ž Ny1.l
Thus, adding Ž2.9. and Ž2.10., we obtain 2 k q 1 y Ž N y 1 . l s 0, which contradicts k ) Ž N y 1. l. This completes the proof.
s 0. Ž 2.10.
88
OGITA, MATSUNAGA, AND HARA
Remark 2.1. Lemma 2.1 shows that the roots of Ž2.1. on the unit circle are simple. From the property of holomorphic functions, there exists a neighborhood of z s e m i where the mapping pŽ z . is one-to-one. Therefore the inverse function of pŽ z . exists locally. Let z be expressed in polar form, that is, z s re i . Then we have dz
s
dp
z
ž
r
dr dp
q ir
d dp
/
,
which implies that the value of drrdp is the real part of Ž rrz .Ž dzrdp. if p moves, keeping real value. Next, we will observe how the roots of Ž2.1. cross the unit circle when the real number p varies. PROPOSITION 2.2. The absolute ¨ alues of the roots of Ž2.1. at z s e m i increase as < p < increases. Proof. Let r be the absolute value of z. It suffices to show that dr dp
⭈p)0
Ž 2.11.
at z s e m i. In the case z N l / 1, we express pŽ z . as pŽ z . s y
z k Ž z y 1. Ž z l y 1. z Nl y 1
sy
zkf Ž z. z Nl y 1
,
where f Ž z . s Ž z y 1.Ž z l y 1.. Then we have dp dz
sy
z ky 1 g Ž z .
Ž z N l y 1.
2
,
where g Ž z . s Ž kf Ž z . q zf X Ž z ..Ž z N l y 1. y Nlz N l f Ž z .. For the sake of convenience, putting w s yŽ z N l y 1. 2rŽ z k g Ž z .., we obtain dr dp
s Re
r dz
ž / z dp
s r Re w.
Hence, we shall compute the value of Re w. Note that f Ž z. s
f Ž z. z
lq1
,
fXŽ z. s
hŽ z . zl
,
89
LINEAR DELAY DIFFERENCE EQUATION
where hŽ z . s l q 1 y lz y z l. By using the relation above and z M s y1, we get zkgŽ z. s s
1 z
k
½ž
kf Ž z . q
1 z
fXŽ z.
1
/ž
z
Nl
y1 y
/
Nl z Nl
f Ž z.
5
Ž kf Ž z . q h Ž z . . Ž 1 y z N l . y Nlf Ž z . z kq N lqlq1
sy
Ž kf Ž z . q h Ž z . . Ž 1 y z N l . y Nlf Ž z . z 2 N lyk
.
Then it follows that Re w s
wqw 2 2
Ž z N l y 1. Ž z N l y 1. sy q 2 zkgŽ z. zkgŽ z. 1
½
2
5
2
sy
z k g Ž z . Ž z N l y 1. q z k g Ž z . Ž z N l y 1. 2 gŽ z.
2
2
2
sy
Ž z N l y 1. z k 2z
2 Nl
gŽ z.
2
y Ž kf Ž z . q h Ž z . . Ž 1 y z N l . q Nlf Ž z . q Ž kf Ž z . q zf X Ž z . . Ž z N l y 1 . y Nlz N l f Ž z . 4
3
sy
Ž z N l y 1. z k 2z
2Nl
gŽ z.
2
Ž 2 kf Ž z . q h Ž z . q zf X Ž z . y Nlf Ž z . . .
Taking note that 2 kf Ž z . q h Ž z . q zf X Ž z . y Nlf Ž z . s Mf Ž z . , we have 4
Re w s
Ž z N l y 1 . M yz k f Ž z . 2 z2Nl g Ž z.
2
⭈
z Nl y 1
4
s
Ž z N l y 1 . Mp 2 z2 Nl g Ž z.
2
.
90
OGITA, MATSUNAGA, AND HARA
Now, by z s e m i, it is easily seen that
Ž z N l y 1. z
2Nl
4 2
s Ž 2 cos Nl m y 2 . ) 0,
and therefore we arrive at dr dp
2
s
2 r Ž cos Nl m y 1 . Mp gŽ z.
2
.
This implies that Ž2.11. is valid at z s e m i. In the case z l s 1, by an argument similar to that above, we can obtain dr dp
s
2 rN 2 Mp
Ž M q 1. z y M q 1
2
,
which implies that Ž2.11. is also valid at z s e m i. This completes the proof. Furthermore, we will determine the positive minimum value of pm given by Ž2.3.. PROPOSITION 2.3. The ¨ alue of p 0 is the positi¨ e minimum ¨ alue of pm for m s 0, 1, . . . , M y 1. To prove Proposition 2.3, we need the following two lemmas which will be proved in the Appendix. LEMMA 2.2.
Let 0 - - r2, then the inequality sin x sin y G sin sin xy
holds for all x, y g Ž1, rŽ2 ... LEMMA 2.3.
Let N be a positi¨ e integer, then the inequality sin Nt sin t
FN
holds for all t. Proof of Proposition 2.3. It is clear that p 0 is positive by the definition. Now, we have only to consider the values of pm which correspond to the roots z s e m i for m s 0, 1, . . . , wŽ M y 1.r2x lying in the upper half plane, because the conjugate root e m i also corresponds to the same value of pm . Our argument is divided into three cases.
91
LINEAR DELAY DIFFERENCE EQUATION
N s 1. It follows from Ž2.3. that
Case I:
m
pm s 2 Ž y1 . sin
Ž 2 m q 1. 2 Ž 2 k q 1.
for m s 0, 1, . . . , k.
Thus we can immediately conclude that p 0 s 2 sin
ž
2 Ž 2 k q 1.
s 2 cos
k 2k q 1
/
is the positive minimum value of pm . Case II: N s 2. It suffices to show that the value of 1rpm attains its maximum when m s 0, that is, 1 p0
1
G
Ž 2.12.
pm
for m s 0, 1, . . . , wŽ M y 1.r2x. Since z l s yz 2 kq1 and z s e m i, the value of pm can be written as pm s
z k Ž z y 1. z
2 kq1
y1
s
z 1r2 y zy1r2 z
kq1r2
yky1r2
yz
s
sin Ž m r2 . sin Ž Ž 2 k q 1 . m r2 .
.
This implies 1 p0
s
sin Ž 2 k q 1 . sin
1
,
pm
s
sin Ž 2 k q 1 . Ž 2 m q 1 . sin Ž 2 m q 1 .
,
where s rŽ2 M .. Note that 0 - - r2 and 1 F 2 M y Ž 2 k q 1. F
2
,
1 F 2m q 1 F
2
,
because of k ) l. Hence it follows from Lemma 2.2 that sin Ž 2 M y Ž 2 k q 1 . . sin Ž 2 m q 1 . G sin sin Ž 2 M y Ž 2 k q 1 . . Ž 2 m q 1 . . Taking account of Ž2 M y Ž2 k q 1.. s y Ž2 k q 1. , we obtain Ž2.12. for m s 0, 1, . . . , wŽ M y 1.r2x. Case III:
N G 3. To end the proof, we will show that < pm < G p 0
Ž 2.13.
92
OGITA, MATSUNAGA, AND HARA
for m s 0, 1, . . . , wŽ M y 1.r2x. For simplicity, we set s rŽ2 M . again, then 0 - Ž2 m q 1. F r2 and therefore Ž2.3. becomes sin Ž 2 m q 1 . l
< pm < s 2 sin Ž 2 m q 1 .
.
sin Ž 2 m q 1 . Nl
By Lemma 2.3 and Jordan’s inequality, 2
sin
F
for 0 F F
F1
,
2
we have 2
< pm < G 2 ⭈
Ž 2 m q 1. ⭈
1
4 Ž 2 m q 1.
s
N
.
N
Ž 2.14.
Now we estimate p 0 from the above. Subcase ŽIIIa.:
Nl F r2. We have p0 s
2 sin sin l sin Nl
F
2 ⭈ l
Ž 2r . Nl
s
N
,
which, together with Ž2.14., implies that Ž2.13. holds for m s 0, 1, . . . , wŽ M y 1.r2x. Subcase ŽIIIb.:
Nl ) r2. We note that
Nl s
Nl
-
2M
Nl
-
2 Ž N y 1. l q 1
N
Ž 2.15.
2 Ny1
because of k ) Ž N y 1. l. By using sin Nl s sinŽ y Nl ., we get p0 F
2 ⭈ l
s
Ž 2r . Ž y Nl .
l 2 y Nl
.
Ž 2.16.
From Ž2.15., the inequalities Ž2.14. and Ž2.16. yield < pm < p0
G )
4 Ž 2 m q 1.
N 4 Ž 2 m q 1.
2
ž
⭈
y Nl l
2
2 Ž N y 1. N
s
4 Ž 2 m q 1.
y1 s
/
2
ž
Nl
4 Ž 2 m q 1.
2
y1
ž
/
1y
2 N
/
.
93
LINEAR DELAY DIFFERENCE EQUATION
Here we find the following: Ži. If N G 12 and m G 1, then Ž2.13. holds. Žii. If N G 4 and m G 2, then Ž2.13. holds. Žiii. If N s 3 and m G 4, then Ž2.13. holds. Consequently, we have only to consider the remainder cases. In the case N G 4 and m s 1, Ž2.15. implies l - r6 and hence we have 2 sin 3 sin 3l
< p1 < s
2
G2⭈
sin 3 Nl
3 ⭈
2
3l s
72 l 2
2
.
From Ž2.15., the above inequality and Ž2.16. yield < p1 < p0
G
72 l 2
2
⭈
y Nl l
2
)
72
ž
3
y
N
2 Ny1
/
G
24
2
) 1.
In the case N s 3 and 1 F m F 3, Ž2.15. and the assumption of Subcase ŽIIIb. imply r6 - l - r4 and we have < pm < p0
s
sin Ž 2 m q 1 . l sin 3l sin Ž 2 m q 1 . sin 3 Ž 2 m q 1 . l sin l
sin
.
Ž 2.17.
By Lemma 2.3, we estimate the first factor of Ž2.17. into sin Ž 2 m q 1 . l sin 3l sin 3 Ž 2 m q 1 . l sin l
G
1 sin 3l 3 sin l
s
1 3
<3 y 4 sin 2 l < )
The second factor of Ž2.17. can be estimated by sin Ž 2 m q 1 . sin
)
Ž 2r . Ž 2 m q 1 .
s
2 Ž 2 m q 1.
Therefore we obtain < pm < p0
2 Ž 2 m q 1.
)
3
)1
for m s 2, 3.
Finally, if m s 1 and p1 ) 0, using r6 - l - r4, we get p1 p0
s
3 y 4 sin 2 l sin Ž 2 m q 1 . 4 sin 3l y 3 2
sin
This completes the proof of Proposition 2.3.
)1⭈
6
) 1.
.
1 3
.
94
OGITA, MATSUNAGA, AND HARA
Now, we are in a position to prove Theorem 1.1. Proof of Theorem 1.1. First of all, we note that if p - 0, then there exists a positive root z# of Ž2.1. such that z# ) 1 because lim z ªq⬁ F Ž z . s q⬁ and F Ž1. s pN - 0. Recall that the roots of Ž2.1. with p s 0 are 0 Žmultiplicity k . and 1 Žsimple.. We claim here that if p ) 0 is sufficiently small, then all the roots of Ž2.1. are inside the unit disk. In fact, let z1Ž p . be the branch of the root of Ž2.1. with z1Ž0. s 1, then it follows from Ž2.8. that dz1 dp
s yN - 0. ps 0
By the continuity of the roots with respect to p, this implies that our claim is valid. In addition, Proposition 2.3 shows that p 0 is a positive minimum value of p such that a root of Ž2.1. intersects the unit circle as p increases from 0. Also, by virtue of Proposition 2.2, we obtain that if p G p 0 then there exists a root zU of Ž2.1. such that < zU < G 1. From the preceding argument, we conclude that all the roots of Ž2.1. are inside the unit disk if and only if 0 - p - p 0 . That is, the zero solution of Ž1.1. is asymptotically stable if and only if the condition Ž1.6. holds. The proof of Theorem 1.1 is complete. Finally, we consider the linear difference equation x nq 1 y x n q p Ž x nyk 1 q x nyk 2 . s 0,
n s 0, 1, 2, . . . ,
Ž 2.18.
where p is a real number and k 1 , k 2 are positive integers. By setting N s 2 and l s < k 1 y k 2 <, we have the following result as a corollary of Theorem 1.1, which corresponds to the result in w8, p. 65x and w4, p. 87x for a linear delay differential equation with two delays. COROLLARY 2.1. Let k 1 and k 2 be positi¨ e integers. Then the zero solution of Ž2.18. is asymptotically stable if and only if
sin 0-pcos
2 Ž k1 q k 2 q 1. . Ž k1 y k 2 . 2 Ž k1 q k 2 q 1.
LINEAR DELAY DIFFERENCE EQUATION
95
APPENDIX In this appendix we give proofs of Lemmas 2.2 and 2.3. Proof of Lemma 2.2. Set the domain D s Ž x, y . : 1 F x F rŽ2 ., 1 F y F rŽ2 .4 and put the function G Ž x, y . s sin x sin y y sin sin xy . Then GŽ x, y . is continuously differentiable in D and its minimum is attained at the boundary ⭸ D or is relatively minimum. Obviously, GŽ x, y . G 0 in ⭸ D. An easy calculation yields that
⭸G ⭸x ⭸G ⭸y
s cos x sin y y y cos xy sin , s sin x cos y y x cos xy sin .
If GŽ x, y . is relatively minimum at Ž x, y ., then ⭸ Gr⭸ x s ⭸ Gr⭸ y s 0. This implies that x, y satisfy x tan x
y
s
tan y
.
Ž A.1.
It is easy to see that the function Ž x . s xrŽtan x . is strictly decreasing on Ž1, rŽ2 .., and thus ŽA.1. implies x s y. Hence, if we show that G Ž x, x . s sin 2 x y sin x 2 sin G 0 then the proof will be complete. To this end, it is sufficient to verify the inequality sin 2 x
Ž x .
2
G
sin x 2 sin
x 2
.
Ž A.2.
In the case x 2 G 2 , for F x F r2, we have sin 2 x
Ž x .
2
y
sin x 2 sin x 2
G
2
ž /
2
y
1 2
) 0.
In the case F x 2 F 2 , it is clear that ŽA.2. holds because of sin x 2 F 0. In the case 0 - x 2 - , consider the function Ž . s logŽŽsin .r ..
96
OGITA, MATSUNAGA, AND HARA
Then
X Ž . s
Ž y tan . cos sin
Y Ž . s
- 0,
sin 2 y 2 2 Ž sin .
- 0,
and thus Ž . is a decreasing and upward convex function. Therefore, using Ž x 2 q .r2 G x and the properties of Ž ., we obtain
Ž x . G
ž
x 2 q 2
/
G
Ž x 2 . q Ž . 2
.
This becomes 2 Ž x . G Ž x 2 . q Ž ., namely 2 log
sin x x
G log
sin x 2 x 2
q log
sin
,
and we arrive at ŽA.2.. The proof of Lemma 2.2 is now complete. Proof of Lemma 2.3. Let N and k be positive integers. Define Ž t . s Žsin Nt .rsin t and Ž k . s lim t ª k Ž t ., then the function Ž t . is continuously differentiable and its period is 2 . We also note that Žyt . s Ž t . and < Ž y t .< s < Ž t .<. It is easily seen that
Ž t. s
FN
for t g 0,
2
.
Hence, these facts above imply the assertion of Lemma 2.3.
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