128 (2003)
MATHEMATICA BOHEMICA
No. 3, 309–317
OSCILLATION OF A NONLINEAR DIFFERENCE EQUATION
WITH SEVERAL DELAYS
,
,
, Xiangtan
(Received June 10, 2002)
Abstract. In this paper we consider the nonlinear difference equation with several delays
(axn+1 + bxn )k − (cxn )k +
m
pi (n)xkn−σi = 0
i=1
where a, b, c ∈ (0, ∞), k = q/r, q, r are positive odd integers, m, σi are positive integers,
{pi (n)}, i = 1, 2, . . . , m, is a real sequence with pi (n) > 0 for all large n, and lim inf pi (n) =
n→∞
pi < ∞, i = 1, 2, . . . , m. Some sufficient conditions for the oscillation of all solutions of the
above equation are obtained.
Keywords: nonlinear difference equtions, oscillation, eventually positive solutions, characteristic equation
MSC 2000 : 39A10
1. Introduction
Consider the nonlinear difference equation
(1)
(axn+1 + bxn )k − (cxn )k +
m
X
pi (n)xkn−σi = 0
i=1
where a, b, c ∈ (0, ∞), c > b, k = q/r, q, r are positive odd integers, m, σi are positive
integers, {pi (n)} are real sequences with pi (n) > 0 for all large n, and lim inf pi (n) =
n→∞
Project supported by Hunan Provincial Natural Science Foundation of China (02JJY2003)
and Research Fund of Hunan Provincial Education Department (01B003).
309
pi < ∞, i = 1, 2, . . . , m. It is easy to see that if c < b then (1) cannot have an
eventually positive solution. The corresponding “limiting” equation of (1) is
k
(2)
k
(axn+1 + bxn ) − (cxn ) +
m
X
pi xkn−σi = 0
i=1
with the characteristic equation
k
(3)
k
(aλ + b) − c +
m
X
pi λ−kσi = 0.
i=1
For the special case where k = a = 1, c = b + 1, equation (1) reduces to the linear
difference equation
m
X
xn+1 − xn +
pi (n)xn−σi = 0.
i=1
There have been a lot of activities concerning the oscillation of solutions of linear
difference equations. But there have been few results for the oscillation of solutions
of the nonlinear equation (1). Under the condition that 0 < c−b
a 6 1, a sufficient
condition of nonexistence of eventually positive solutions for (1) was obtained in
[5], [6]. In this paper we obtain several new sufficient conditions for oscillation of
all solutions of (1) by removing the condition c−b
a 6 1. A sufficient and necessary
condition for oscillation of all solutions of (2) is obtained as well.
A solution {xn } of equation (1) is said to oscillate about zero or simply to oscillate
if the terms xn of the sequence {xn } are neither eventually all positive nor eventually
all negative. Otherwise, the solution is called nonoscillatory.
2. Main results
Lemma 1 [3]. If x, y are positive numbers and x 6= y, then
rxr−1 (x − y) > xr − y r > ry r−1 (x − y) for r > 1.
Theorem 1. If (3) has no positive roots, then every solution of (1) oscillates.
. By way of contradiction, assume that {xn } is an eventually positive
solution of (1). By (1) we have
(4)
310
xn+1
a
+b
xn
k
m
X
xn−σi
−c +
pi (n)
xn
i=1
k
k
= 0.
For sufficiently large n, set
xn+1
xn
= βn . Then eventually
0 < βn 6
c−b
,
a
and (4) yields
k
(5)
k
(aβn + b) − c +
m
X
pi (n)
Y
σi
−1
βn−j
j=1
i=1
k
= 0.
Set
lim sup βn = β.
n→∞
It follows from (5) that 0 < β 6
c−b
a .
We also claim that
(aβ + b)k − ck +
m
X
pi β −kσi 6 0.
i=1
Indeed, by virtue of lim inf pi (n) = pi < ∞, for every ε ∈ (0, 1) there exists an nε > 0
n→∞
such that
pi (n) > (1 − ε)pi for i = 1, 2, . . . , m and n > nε .
Therefore,
(aβn + b)k − ck + (1 − ε)
m
X
pi
i=1
Y
σi
j=1
−1
βn−j
k
6 0 for n > nε .
Let Nε > nε be such that
βn < (1 + ε)β
for n > Nε .
Then
(aβn + b)k − ck + (1 − ε)
m
X
pi (1 + ε)−kσi β −kσi 6 0 for n > Nε + σi ,
i=1
i.e.,
(aβn + b)k 6 ck − (1 − ε)
m
X
pi (1 + ε)−kσi β −kσi ,
i=1
which implies
(aβ + b)k 6 ck − (1 − ε)
m
X
pi (1 + ε)−kσi β −kσi .
i=1
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As this is true for every ε ∈ (0, 1), it follows that
(aβ + b)k 6 ck −
(6)
m
X
pi β −kσi ,
i=1
which proves our claim. Set
F (λ) = (aλ + b)k − ck +
(7)
m
X
pi λ−kσi .
i=1
Then F (0+) = +∞ and F (β) 6 0. It follows that (3) has a positive root. This
contradiction completes the proof.
Corollary 1. Every solution of (2) oscillates if and only if (3) has no positive
roots.
. Sufficiency can be directly derived from Theorem 1. So it suffices to
prove Necessity. Suppose that (3) has a positive root λ.
Let
xn = λn , n = 1, 2, . . . .
Then we have
(axn+1 + bxn )k − (cxn )k +
m
X
i=1
m
X
pi xkn−σi = λkn (aλ + b)k − ck +
pi λ−kσi = 0.
i=1
Hence, {xn } is an eventually positive solution of (2). This contradiction completes
the proof.
Corollary 2. Assume that lim inf pi (n) = pi < ∞, i = 1, 2, . . . , m, k > 1, and
n→∞
that
m
X pi akσi (kσi + 1)kσi +1
> 1.
kck−1 (c − b)kσi +1 (kσi )kσi
i=1
Then every solution of (1) oscillates.
. Let
F (λ) = (aλ + b)k − ck +
m
X
pi λ−kσi .
i=1
Then
F (λ) > 0 for λ >
312
c−b
.
a
For λ <
c−b
a ,
by Lemma 1, we have
k
k
F (λ) > {c − (aλ + b) } − 1 +
Since
min
0<λ< c−b
a
m
X
i=1
λ−kσi
k−1
kc
(c − aλ − b)
=
pi λ−kσi
.
kck−1 (c − aλ − b)
akσi (kσi + 1)kσi +1
,
− b)kσi +1 (kσ i )kσi
kck−1 (c
by the condition of Corollary 2 we get
k
k
F (λ) > {c −(aλ+b) } −1+
m
X
i=1
pi akσi (kσi + 1)kσi +1
kck−1 (c − b)kσi +1 (kσ i )kσi
> 0 for λ <
c−b
.
a
By Theorem 1, every solution of (1) oscillates. The proof is completed.
Define a sequence {λl } by
(8)
1
c−b
, λl+1 =
a
a
λ1 =
ck −
m
X
i
pi λ−kσ
l
i=1
k1
− b , l = 1, 2, . . .
Lemma 2. Suppose that {λl } is defined by (8). Then 0 < λ∗ 6 λl 6
lim λl = λ∗ , where λ∗ is the largest root of equation (3) on (0, c−b
a ].
c−b
a
and
l→∞
. First, we prove the sequence {λl } is nonincreasing. Since
λ2 =
1
a
ck −
m
X
i
pi λ−kσ
1
i=1
k1
c−b
−b <
= λ1 ,
a
hence, by induction, supposing that λl 6 λl−1 , we have
λl+1
1
=
a
k
c −
m
X
i
pi λ−kσ
l
i=1
k1
1
−b 6
a
k
c −
m
X
i
pi λ−kσ
l−1
i=1
k1
− b = λl .
Hence, the sequence {λl } is nonincreasing. From (3) it is obvious that λ1 > λ∗ , and
by induction
λl+1
1
=
a
k
c −
m
X
i=1
i
pi λ−kσ
l
k1
1
−b >
a
k
c −
m
X
i
pi λ−kσ
∗
i=1
k1
− b = λ∗ .
Hence, {λl } is nonincreasing and bounded. Therefore, lim λl exists. Letting l → ∞
l→∞
in (8) and noting that λ∗ is the largest root of (3), we conclude lim λl = λ∗ . The
proof is completed.
l→∞
313
Theorem 2. Assume that (3) has positive roots, λ∗ is the largest root of (3) and
(9)
lim sup
n→∞
a
1
k
k X
m
m
X
1
k(1−σi )
−kσi
> ck .
p
(n
+
1)λ
p
(n)λ
+
b
+
i
∗
i
∗
ck − bk i=1
i=1
Then every solution of (1) oscillates.
. Suppose that {xn } is an eventually positive solution of (1). Then there
exists n1 > 0 such that xn > 0 and xn−σi > 0, i = 1, 2, . . . , m, for n > n1 . From (1)
we have
m
X
(axn+1 + bxn )k − (cxn )k = −
pi (n)xkn−σi 6 0,
i=1
i.e.,
(axn+1 + bxn )k 6 (cxn )k .
Since k = q/r, q, r are positive odd integers, we have
(10)
where θ =
xn+1 6 θxn
c−b
a .
for n > n1 ,
By virtue of lim inf pi (n) = pi < ∞, for every ε ∈ (0, 1) there exists
n→∞
an nε > n1 such that
(11)
pi (n) > (1 − ε)pi
for i = 1, 2, . . . , m and n > nε .
Define a sequence {µl (ε)} by
1
µ1 (ε) = θ, µl+1 (ε) =
a
k
c − (1 − ε)
m
X
pi (µl (ε))
−kσi
i=1
k1
− b , l = 1, 2, . . . .
From (10) we get
xkn−σi > (µ1 (ε))−kσi xkn
(12)
for n > nε + σi .
From (1), (11), and (12) we have
(axn+1 + bxn )k 6 (cxn )k − (1 − ε)
m
X
pi (µ1 (ε))−kσi xkn ,
i=1
i.e.,
xn+1
314
1
6
a
k
c − (1 − ε)
m
X
i=1
pi (µ1 (ε))
−kσi
k1
− b xn .
That is
xn+1 6 µ2 (ε)xn , n > nε + σi ,
which gives
xn 6 µ2 (ε)xn−1 6 (µ2 (ε))2 xn−2 6 . . . 6 (µ2 (ε))σi xn−σi , n > nε + 2σi ,
i.e.,
xn−σi > (µ2 (ε))−σi xn .
Repeating the above process, we obtain
xn+1 6
1
a
ck −
m
X
pi (µl−1 (ε))−kσi
i=1
k1
− b xn ,
i.e.,
(13)
xn+1 6 µl (ε)xn , n > nε + (l − 1)σi .
Since lim µl (ε) = λl and lim λl = λ∗ , for a sequence {εl } with εl > 0 and εl → 0 as
ε→0
l→∞
l → ∞, by (13) there exists a sequence {nl } such that nl → ∞ as l → ∞ and
(14)
xn+1 6 (λ∗ + εl )xn , n > nl ,
and
(15)
xn−σi > (λ∗ + εl )−σi xn , n > nl + σi .
On the other hand, from (1) we have
(bk − ck )xkn +
m
X
pi (n)xkn−σi 6 0,
i=1
i.e.,
(16)
(ck − bk )xkn >
m
X
pi (n)xkn−σi .
i=1
From (15) and (16) we obtain
(ck − bk )xkn >
m
X
pi (n)(λ∗ + εl )k(1−σi ) xkn−1 ,
i=1
315
i.e.,
k1
m
X
xn
1
k(1−σi )
p
(n)(λ
+
ε
)
, n > n l + σi .
> k
i
∗
l
xn−1
c − bk i=1
(17)
From (1) we have
ck =
(18)
axn+1 + bxn
xn
k
+
m
X
i=1
pi (n)
xkn−σi
.
xkn
From (15), (17) and (18) we obtain
c >
a
k
(19)
1
k
k
m
X
1
k(1−σi )
p
(n
+
1)(λ
+
ε
)
+
b
i
∗
l
ck − bk i=1
m
X
+
pi (n)(λ∗ + εl )−kσi .
i=1
Let l → ∞, then (19) implies
lim sup a
n→∞
1
k
k X
m
m
X
1
k(1−σi )
−kσi
pi (n)λ∗
6 ck ,
+b +
pi (n + 1)λ∗
ck − bk i=1
i=1
which contradicts (9) and completes the proof.
"!
1. Theorems 1 and 2 extend the results on linear difference equations
in [2], [9].
# $%'&)(*
1. Consider equation (1). Let
4
3
, p1 (1) = , p1 (n + 2) = p1 (n) for n = 0, 1, 2, . . . ,
3
4
3
1
p2 (0) = , p2 (1) = , p2 (n + 2) = p2 (n) for n = 0, 1, 2, . . . ,
4
2
k = 3, a = 1, b = 1, c = 2, σ1 = 1, σ2 = 2, m = 2.
p1 (0) =
Then
lim inf p1 (n) =
n→∞
and
2
X
i=1
3
1
, lim inf p2 (n) = ,
n→∞
4
2
pi akσi (kσi + 1)kσi +1
= 1.32 . . . > 1.
kck−1 (c − b)kσi +1 (kσi )kσi
Thus according to Corollary 2, every solution of (1) oscillates.
316
# $%'&)(*
2. Consider equation (1). Let
p1 (0) = 1, p1 (1) = 7, p1 (n + 2) = p1 (n)
for n = 0, 1, 2, . . . ,
k = 3, a = 1, b = 1, c = 2, σ1 = 1, m = 1.
Then
lim inf p1 (n) = 1.
n→∞
Hence, the characteristic equation (3) has the largest root λ∗ = 0.8586. Therefore,
lim sup a
n→∞
1
k X
k
m
m
X
1
k(1−σi )
−kσi
+
b
+
p
(n)λ
= 9.5799 > 8.
p
(n
+
1)λ
i
i
∗
∗
ck − bk i=1
i=1
Thus according to Theorem 2, every solution of (1) oscillates.
+-, .)0/1(*02"3'.4
. The authors thank the referee for useful comments and
suggestions.
References
[1] R. P. Agarwal, P. J. Y. Wong: Advanced Topics in Difference Equations. Kluwer Academic Publishers, 1997.
[2] I. Györi, G. Ladas: Oscillation Theory of Delay Differential Equations with Applications.
Oxford Science Publications, 1991.
[3] G. H. Hardy, J. E. Littlewood, G. Polya: Inequalities. Second edition, Cambridge University Press, 1952.
[4] B. Li: Discrete oscillations. J. Differ. Equations Appl. 2 (1996), 389–399.
[5] S. T. Liu, S. S. Cheng: Oscillation criteria for a nonlinear difference equation. Far East
J. Math. Sci. 5 (1997), 387–392.
[6] J. Yang, X. P. Guan et al.: Nonexistence of positive solution of certain nonlinear difference equations. Acta Math. Appl. Sinica 23 (2000), 562–567. (In Chinese.)
[7] J. S. Yu, B. G. Zhang, X. Z. Qian: Oscillations of delay differential equations with oscillatory coefficients. J. Math. Anal. Appl. 177 (1993), 432–444.
[8] Y. Zhou, B. G. Zhang: Oscillations of delay differential equations in a critical state. Comput. Math. Appl. 39 (2000), 71–89.
[9] B. G. Zhang, Yong Zhou: Oscillation of delay differential equations with several delays.
To appear in Comput. Math. Appl.
[10] B. G. Zhang, Y. J. Sun: Existence of nonoscillatory solutions of a class of nonlinear difference equations with a forced term. Math. Bohem. 126 (2001), 639–647.
Authors’ addresses: X. N. Luo, Yong Zhou, C. F. Li, Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, P. R. China, e-mail: yzhou@xtu.edu.cn.
317