IJMMS 28:5 (2001) 301–306
PII. S0161171201007323
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
OSCILLATION OF NONLINEAR DELAY
DIFFERENCE EQUATIONS
JIANCHU JIANG
(Received 16 March 2000)
Abstract. We obtain some oscillation criteria for solutions of the nonlinear delay differ
αj
ence equation of the form xn+1 − xn + pn m
j=1 xn−k = 0.
j
2000 Mathematics Subject Classification. 39A10.
1. Introduction. Consider the nonlinear delay difference equation
xn+1 − xn + pn
m
αj
xn−kj = 0,
n = 0, 1, 2, . . . ,
(1.1)
j=1
where pn ≥ 0, n = 0, 1, 2, . . . , 0 ≤ k1 ≤ k2 ≤ · · · ≤ km are integers, αj > 0 are rational numbers with denominator of positive odd integers for each j = 1, 2, . . . , m and
m
j=1 αj = 1.
Equation (1.1) is a discrete analogue of the following first-order nonlinear delay
differential equation
x (t) + p(t)
m
αj
= 0,
x t − τj
(1.2)
j=1
where p(t) ∈ C([t0 , ∞), [0, ∞)), 0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τm , and αj are the same as in (1.1).
For (1.2), the oscillation of its solutions has been extensively studied in the literature,
see, for example [2, 4, 11, 12].
When k1 = k2 = · · · = km = k, (1.1) reduces to the linear delay difference equation
xn+1 − xn + pn xn−k = 0,
n = 0, 1, 2, . . . .
(1.3)
Recently, there has been a lot of activity concerning the oscillatory behavior of (1.3).
See, for example, [1, 3, 5, 6, 7, 8, 9, 10]. In particular, [7] proved that every solution of
(1.3) is oscillatory provided
∞
n=0


n+k
i=n

pi ln 
n+k
i=n
pi + 1 − sign
n+k
i=n

pi  −
n+k
i=n+1

pi ln 
n+k
i=n+1
pi + 1 − sign
n+k
i=n+1

pi  = ∞.
(1.4)
Condition (1.4) improves many previous well-known results. Furthermore, (1.4) fits
n−1
n−1
the case when i=n−k pi − (k/(k + 1))k+1 oscillate or i=n−k pi ≤ (k/(k + 1))k+1 .
302
JIANCHU JIANG
Our main aim in this note is to generalize condition (1.4) to (1.1).
2. Main results
Theorem 2.1. Assume that




n+kj
n+kj
n+kj
∞ 
m
m
m
αj
pi ln 
αj
pi + 1 − sign 
αj
pi 

n=0
j=1
−
i=n
m
j=1

n+kj
αj
j=1
pi ln 
i=n+1
m
i=n
j=1

n+kj
αj
j=1
pi + 1 − sign 
i=n+1
i=n
n+kj
m
αj
j=1
i=n+1


pi  = ∞.

(2.1)
Then every solution of (1.1) oscillates.
Proof. Assume, by way of contradiction, that (1.1) has an eventually positive
solutions {xn }. Then there exists an integer n1 > 0 such that xn−km > 0, xn+1 ≤ xn ,
n ≥ n1 . We define the functions p(t) and x(t) as follows:
p(t) = pn ,
x(t) = xn + (t − n) xn+1 − xn ,
n ≤ t < n + 1, n = 0, 1, 2, . . . .
(2.2)
Let x (t) denote derivation on the right. Then
x (t) ≤ 0,
x(t) > 0,
x (t) = xn+1 − xn
t ≥ n1 ,
for n ≤ t < n + 1, n = 0, 1, 2, . . . .
(2.3)
Hence, (1.1) can be rewritten as
x (t) + p(t)
m
αj
= 0,
x t − kj
t ≥ 0,
(2.4)
j=1
where here and in the sequel, [·] denotes the greatest integer function.
Set λ(t) = −x (t)/x(t) for t ≥ n1 . Then λ(t) ≥ 0 for t ≥ n1 , and from (2.4) we have

λ(t) = p(t) exp 

t
m
αj
j=1
λ(s)ds ,
t ≥ n 1 + km ,
[t−kj ]
(2.5)
or
λ(t)
m
j=1
[t+kj +1]
αj
p(s)ds
t

≥ p(t)
m
j=1
[t+kj +1]
αj
t


λ(s)ds  exp 
m
j=1
t
αj

λ(s)ds ,
[t−kj ]
(2.6)
t ≥ n 1 + km .
One can easily show that
φ(r )ex ≥ φ(r )x + φ(r ) ln(er + 1 − sign r ),
r ≥ 0, x ≥ R,
(2.7)
where φ(0) = 0 and ϕ(r ) ≥ 0 for r > 0. By the definition of p(t), we see that p(t) is
OSCILLATION OF NONLINEAR DELAY DIFFERENCE EQUATIONS
303
nonnegative and right-continuous. Therefore, it follows from
m
[t+kj +1]
αj
t
j=1
p(s)ds = 0
(2.8)
that p(t) = 0. Applying inequalities (2.7) to the right side of (2.6), we obtain
λ(t)
m
[t+kj +1]
αj
j=1
p(s)ds ≥ p(t)
t
t
m
αj
j=1
[t−kj ]
λ(s)ds + p(t) ln A(t),
n ≥ n 1 + km ,
(2.9)
where
A(t) = e
[t+kj +1]
m
αj
t
j=1

p(s)ds + 1 − sign 
[t+kj +1]
m
αj

p(s)ds .
(2.10)
t
j=1
Set n2 = n1 + km . Integrating both sides of (2.9) from n2 to N > n2 + 2km , we have
[t+kj +1]
N
m
αj
λ(t)
p(s)ds dt
n2
j=1
t
≥
m
t
N
αj
j=1
[t−kj ]
n2
(2.11)
N
p(t)
λ(s)ds dt +
p(t) ln A(t)dt.
n2
Interchanging the order of integration, we get
t
N
p(t)
[t−kj ]
n2
λ(s)ds dt ≥
=
N−kj
[s+kj +1]
λ(s)
n2
p(t)dt ds
s
(2.12)
[t+kj +1]
N−kj
λ(t)
n2
p(s)ds dt.
t
Substituting this into (2.11), we have
m
[t+kj +1]
N
αj
λ(t)
N−kj
j=1
t
N
p(s)ds dt ≥
p(t) ln A(t)dt.
(2.13)
n ≥ n1 .
(2.14)
n2
From (1.1) we have
α
m
1−αm
xn−k
≤ 0,
xn+1 − xn + pn xn
m
α
Set yn = xn m for n ≥ n1 . Then
yn+1 − yn + αm pn yn−km ≤ 0,
It follows that αm
n
i=n−km
t
(2.15)
pi ≤ 1, and so
[t+km +1]
αm
n ≥ n 1 + km .
p(s)ds ≤ αm
[t]+k
m
i=[t]
pi ≤ 1,
t ≥ n2 .
(2.16)
304
JIANCHU JIANG
Recall that k1 ≤ k2 ≤ · · · ≤ km , therefore
[t+kj +1]
αj
t
p(s)ds ≤
αj
,
αm
t ≥ n2 , j = 1, 2, . . . , m.
(2.17)
Substituting this into (2.13), we obtain
N
m
αj N
λ(t)dt ≥
p(t) ln A(t)dt,
α
N−kj
n2
j=1 m
(2.18)
αj 
N
x N − kj
 ≥ αm
p(t) ln A(t)dt.
x(N)
n2
j=1
(2.19)
or

ln 
m
It follows that
αj
∞
x N − kj
≥ exp αm
p(t) ln A(t)dt .
lim
N→∞
x(N)
n2
i=1
m
(2.20)
Let E = {n ≥ n2 | pn > 0}. Then
∞
p(t) ln A(t)dt
n2
=
=
∞ n+1

n=n2 n
∞
m
p(t) lne
[t+kj +1]
αj
pn

ln e
n
n=n2
m

αj 
j=1

− sign 
=
n+1
pn
n=n2

ln e
n
m
j=1
n+kj +1
n
m
m
j=1


αj 
t
p(s)ds −
n+kj +1
n
n
i=n

p(s)dsdt
t

p(s)ds  + 1
t
p(s)ds −

p(s)ds dt
n

 n+k
j
αj 
pi − pn (t − n)dt
i=n

 n+k
n+1
m
j
= pn
ln e
αj 
pi − pn (t − n)dt
j=1
αj
i=n
− sign 
n
[t+kj +1]

 n+k
j
αj 
pi − pn (t − n) + 1

E
m
j=1
j=1
∞
p(s)ds+1−sign
t
j=1
n+1

OSCILLATION OF NONLINEAR DELAY DIFFERENCE EQUATIONS
=

E
n+kj
m

αj
j=1
−
m
∞
n=n2


αj
pi ln 
i=n+1
n+kj
αj
j=1
j=1
pi ln 
n+kj
m
αj
pi ln 
i=n+1

pi + 1 − sign 
i=n+1
n+kj
m

pi 
i=n
αj
j=1
n+kj
αj
αj
j=1

i=n
m
j=1


n+kj
m
pi ln 
n+kj
m
−

i=n
j=1
=
305
pi + 1 − sign 
αj
αj
m
n+kj
αj

pi + 1 − sign 
i=n+1

pi 
i=n+1
j=1
n+kj
j=1
n+kj
j=1

i=n
m
m

pi 
i=n
m
j=1
n+kj
αj

pi .
i=n+1
(2.21)
From (2.1) and (2.20), we have
αj
x N − kj
= ∞.
N→∞
x(N)
j=1
lim
m
(2.22)
On the other hand, it follows from (2.1) that
lim sup pn > 0.
(2.23)
n→∞
By (2.15), we have
lim sup pn ≤ lim sup
n→∞
n→∞
1
yn
1
,
=
αm yn−km
αm lim inf n→∞ yn−km /yn
(2.24)
which, together with (2.23) yields
lim inf
n→∞
yn−km
< ∞,
yn
(2.25)
that is,
αm
x N − km
< ∞,
lim inf
N→∞
x(N)
(2.26)
and so
lim inf
N→∞
αj
x N − kj
x N − km
< ∞,
≤ lim inf
N→∞
x(N)
x(N)
j=1
m
which contradicts (2.22) and so the proof is completed.
(2.27)
306
JIANCHU JIANG
From Theorem 2.1 we have immediately.
Corollary 2.2. Assume that there exists an integer N ≥ 0 such that
m
n+kj
αj
j=1
pi > 0
for n ≥ N,
(2.28)
i=n
and that
∞
n=N


m
j=1
n+kj
αj
j=n

pi ln 
m
j=1
n+kj
αj

pi  −
i=n
m
j=1
n+kj
αj
i=n+1

pi ln 
m
j=1
n+ki
αj

pi  = ∞. (2.29)
i=n+1
Then every solution of (1.1) oscillates.
Clearly, when k1 = k2 = · · · = km , condition (2.1) reduces to (1.4).
Acknowledgement. This work was supported by the Science Foundation of
Hunan Educational Committee of China.
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Jianchu Jiang: Department of Mathematics, Loudi Teachers’ College, Loudi, Hunan
417000, China
E-mail address: ldjjc@mail.ld.hn.cn