PERGAMON
Applied
Mathematics
Letters
Applied Mathematics Letters 13 (2000) 1-6
www.elsevier.nl/locate/aml
The Asymptotic Stability
of Difference Equations
I. KOV .CSV(JLGYI
D e p a r t m e n t of M a t h e m a t i c s and C o m p u t i n g , University of Veszpr@m
P.O. Box 158, 8201 Veszpr6m, H u n g a r y
kovacsv©almos, vein. hu
( R e c e i v e d M a y 1999; a c c e p t e d J u l y 1 9 9 9 )
C o m m u n i c a t e d by R. P. Agarwal
A b s t r a c t - - I n this paper, the linear delay difference equation x n + l - x n = - - a n X n _ k , where an > 0
(n > 0) any nonnegative coefficient sequence, is considered and its stability properties are investigated.
The obtained result is extended to the nonlinear difference equation Xn+l - x n = f n ( x n - k ) . @ 1999
Elsevier Science Ltd. All rights reserved.
Keywords--Difference
equation, Stability, Asymptotic behavior, Rational recursive sequences.
1. I N T R O D U C T I O N
Ladas et al. [1] considered the following k + 1St-order linear delay difference equation:
Xn_t_ 1 -- X n ~ - - a n X n _ k ,
where k > 1 and an _ 0 (n > 0) any nonnegative coefficient sequence. They gave a sufficient
condition for its asymptotic stability proving that if
(C1)
~---~ a n ~ O0
n=l
and
lim sup ~
aj < 1
n---*oo 3 = r t _ k
hold, then for every solution (xn)n~__l of (El)
Xn ~ 0,
Supported in part by Hungarian National Foundation for Scientific Research Grant No. T 019846.
0893-9659/1999/$ - see front matter. @ 1999 Elsevier Science Ltd. All rights reserved.
PII: S0893-9659(99)00136-6
Typeset by ~43/tS-TEX
2
I. KOV~,CSVOLGYI
as n --~ oo. Later on, several authors extended and generalized this result by replacing (C2) with
a weaker condition or a condition of a slightly different type. Instead of (C2) Erbe, Xia and
Yu [2] assumed t h a t
3
1
(C2')
lim sup
aj < ~ + 2 (k + 1----~)'
n--*aa j = n - k
while Gy6ri and Pituk [3] supposed t h a t
n-1
aj < 1.
lim sup E
n--~oo j = n - k
(C2")
Yu and Cheng [4] showed t h a t if
lim sup
aj < z,
n - ~ j = n - 2k
(c2'")
then the s a m e convergence holds.
2. L I N E A R
EQUATION
In this paper, we state and prove the following.
THEOREM 1. A s s u m e that
n=0
and
n-1
E
limsup
n---*aa
j=n-
aj <
7
(c2'"')
2k
oo of ( E I )
hold. Then for every solution ( X n)n=l
Xn --~ O,
as
n
---~ O 0 .
oo
PROOF. Let x - k , . . . , x0 be any given initial sequence and (X~)n=
1 the corresponding solution
of (El). If (xn) nonoscillatory, its convergence is an obvious consequence of (C1) as shown, e.g.,
in [1]. If (x~) oscillates, we proceed in the following way. By (C2""), there is a real number
c < 7/4 and an index no E N, such that
n--1
j=n-2k
for n > no. Choose c such t h a t 1 _< e and, for simplicity, assume no = k. Set
M :=
max
-k<n<2k
IZn].
We will show t h a t there is an increasing sequence of indices nl < nz < r t 3 . . . , for which
sup I x ~ [ < M
c-
,
(j=0,1,2,...)
n>nj
holds. So, first, we demonstrate that the solution is bounded by M, i.e.,
sup Ix~l <_ M.
n>_-k
(1)
Asymptotic Stability
For contradiction, suppose t h a t there exists an index N _> 2k such t h a t
max
Ixal < M < I x N + l l .
-k<j<N
W i t h o u t any restriction of the generality we m a y assume t h a t
(2)
M < XN+~.
Since
0 ~ X N + 1 -- X N ~
--aNXN-k
(3)
and aN >_ O, we get
XN-k
(41)
< O,
and so
ZN+I --32N-k > ~.1,
(5)
which will t u r n out to be a contradiction. By (El),
N-1
XN+ 1 :
2 g N _ k Jrj=N-k
(s)
N-l
:
Z N _ k --
Z
ajXj-k
-- a N X N - k .
j=N-k
Choose N - 2k <_/V _< N - k - 1 such t h a t Z N = minN-2k<_j <_N - k - l Z j , and for short, denote p =
N-1
~ j = N - k aj. Obviously, - M _< aN. We will also show t h a t z~;T < 0. Suppose, for contradiction,
t h a t xj >_ 0 for every N - 2 k
<_ j _< N - k 1. This, together with the nonnegativity o f a j
( N - 2k < j < N - k - 1), implies t h a t
X x - k >_ X N - k + l >_ " " >_ a:s,.
On tile other hand, since X N - k < 0 and aN <_ C,
XN+l
-- X N
<_ - - e x ; V - k
holds, and so
XN+I ~.~ X N
-- C 3 ; N _ k
XN-k
-- C 2 ; N - k
_< (1 - c) Z N - k ,
which, considering - M
_< X N - k < O, implies
x x + , <_ M ( c - 1)
_<M,
and so, xN < 0.
A p p l y now condition (C2'"') to a N - k , . . . ,
so, t o g e t h e r with X g - k < O,
XN+ 1 ~
:
aN in (6). This yields t h a t a N - k + ' '
X N _ k -(1
-
(c -
x19 p - (c - p) X N - k
p))
-
+ a~\' < c and
(7)
4
I. KOV~,CSVOLGYI
We have two cases.
CASE 1. 1 -- (c -- p) _< 0. Using - M < Xg, XN-k < 0, and 1 - (c - p) < 0,
XN+I <_ M ( ( c - p) - I) + Mp
= M ( c - 1)
which contradicts (2).
CASE 2. 1 -- ( c - - p ) > 0. Now,
N-k-1
X N - k = X~ +
~
j=9
(Zj+I - x j )
N-k-1
X fiI --
E
ajXj-k
j=9
N-k-1
<x~+M
E
aj
j=9
and, applying condition (C2'"') to a g , . . . , aN-k-l,
X N - k ~__ XI~ ~- M (c -
p).
So, t a k i n g (7) into account
XN+l _< (1 -- (C -- p)) (X~ + M (c - ; ) ) - z ~ ;
: x ~ (1 - c) + M (c - p) (1 - (c - ; ) ) .
Sincec>l
and-M<x
9 <0,
< - M (1 - c) + M ( c - p )
= M (_p2 + ( 2 c - 1 ) p -
(1 - ( c - p))
c2 + 2 c - 1).
(8)
For fixed c the right-hand side term attains its m a x i m u m value at p = c - 1/2 and so
XN+I
<_ M ( c - ~ ) ,
which contradicts (2).
In the very same way, multiplying the above equations by - 1 we get
- M < XN+ 1
and so SUPn>__klXn] < M has been proven.
Now we show t h a t after the first oscillation of ( X ~)n=2k has been made, i.e., for the smallest
index n l _> 2k such t h a t x,~ 1 • x~l+l < 0,
sup
n_>nl+l
Asymptotic Stability
5
holds. We can use the above arguments. Suppose that there exists an index 2k < nl + 1 ~: N
such that
max
xj < M (c-
nlTl<_j<N
~ ) < XN+I.
--
Then x N - k < 0, which implies that XN+l -- X N - k > M ( c - 3/4). We choose fil as above and
show that - M <_ x 9 < 0 and this implies, in both of the above cases, that x N + l <_ M ( c - 3/4).
It can be shown in the very same way that M ( c - 3/4) <_ XN+I.
So far, we have proved that SUPn>__klXnl <_ M and suPn>_nl+l[Xnl <_ M ( c - 3/4). Using again
that (x~) oscillatory and the above argument, we get that there exists an index n2 such that
supn>_n2+llXn I < M ( c - 3/4) 2, and so on. Since (c - 3/4)J -~ 0 (j -~ oc), our theorem has been
proven.
|
REMARK 2. Note that if the delay in (El) is k = 1, then condition (C2')
3
1
7
limsup(an_ln_~o + a n ) < ~ + 2(1 + 1~ - 4'
coincides with our condition (C2"")
7
limsup (a~-2 + a n - l ) < - .
n--* OG
4
REMARK 3. The obtained result is sharp, as shown in the following example. Let
Xn+l
-- Xn
=
--anXn-1
with periodically changing coefficient-sequence
1
a4m = ~,
5
a4m+l ---- ~,
1
a4ra+2 = 2'
a4m+3 = O,
(m E N)
-I
for which limn--.oc E ~ -j=n-2k
aj = 7/4, and initial condition
x-1 = x 0 = 1 .
This is the sharp case, obtained by putting p = c - 1/2 in (8). Its solution
1
3
2'
4'-1'-1'-~'1'1'
1
1
3
2'-5'""
is bounded, but does not tend to 0. If we set, however, a4m+l = 5/4 - ~ (~ > 0), then (x,,) ~ O.
On the other hand, if setting a4m+l = 5/4 -t- ~ (C > 0), then (xn)~°°__1 is unbounded.
REMARK 4. Condition (C2'"') is independent of (C2'"). In the above example, set a4m+l = 9/8.
Then (C2"') cannot be applied, while (C2'"') implies convergence.
3. N O N L I N E A R
EQUATION
We are now about to generalize the result obtained in the previous section to the nonlinear
equation in [5]. Consider
xn+l - xn = fn ( x n - k )
(E2)
with sign-condition
f n ( X ) <_0,
X
(x~O)
and
fn(O)=O.
(SC)
6
I. KOV~.CSV6LGYI
THEOREM 5. Assume that (SC) fulfills and for some nonnegative sequence fin >_0 (n > O)
oo
#n Ixl _< [A (x) l,
(x e R)
and
& =
(C3)
n=0
and for another nonnegative sequence an >_0 (n > O)
n-1
IA(x)] < a n l x [ ,
(xER)
and
limsup
E
aj <
7
(c4)
j=n-2k
hold. Then for every solution (Xn)n°°=l Of (E2)
Xn --4 0~
~s n ---+ oK).
The proof is very similar to that of Theorem 1. The terms a n X n - k should be replaced with
fn(Xn-k), when referring to an > 0 earlier, sign-condition (SC) should now be referred to.
Condition (C3) implies that the nonoscillatory solutions converge. When (xn) oscillates, equation (3) can be replaced with
0 < XN+ 1 -- X N -~ f N ( X N - k ) .
Sign-condition (SC) implies that
X N - k < O.
Equation (6) may be rewritten as
N-1
XNq-1 ~ X N - k --
f# ( X j - k ) -- f N ( X N - a )
j=N-k
and then, applying (C4), can be dominated by
N-1
XN+I ~ X N - k +
a j x j _ k q- a N X N _ k.
Z
j=N-k
The remaining steps of the proof are the same ones as in the proof of Theorem 1, after replacing
aj with a j, so they are omitted.
REFERENCES
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2. L.H. Erbe, H. Xia and J.S. Yu, Global stability of a linear nonautonomous delay difference equation, J. Difference Eqns. and Appl. 1 (2), 151-161, (1995).
3. I. Gy6ri and M. Pituk, Asymptotic stability in a linear delay difference equation, In Proceedings of SICDEA,
Veszprdm, Hungary, August 6-11, 1995, Gordon and Breach Science, Langhorne, PA, (1997).
4. J.S. Yu and S.S. Cheng, A stability criterion for a neutral difference equation with delay, Appl. Math. Lett.
7' (6), 75-80, (1994).
5. M.-P. Chen and B. Liu, Asymptotic behavior of solutions of first order nonlinear delay difference equations,
J. Comp. Math. Appl. 32 (4), 9-13, (1996).
6. R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker,
New York, (1992).
7. V. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,
Kluwer Academic, Dordrecht, (1993).