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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 209309, 13 pages
doi:10.1155/2010/209309
Research Article
Oscillatory Criteria for the Two-Dimensional
Difference Systems
Jin-Fa Cheng1 and Yu-Ming Chu2
1
2
Department of Mathematics, Xiamen University, Xiamen 361005, China
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu, [email protected]
Received 5 April 2010; Revised 3 June 2010; Accepted 17 June 2010
Academic Editor: Alberto Cabada
Copyright q 2010 J.-F. Cheng and Y.-M. Chu. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We establish some necessary and sufficient conditions for oscillation of the solutions of the
following two-dimensional difference system: Δxn fn, yn , Δyn −gn, xn , where fn, u
and gn, u are strongly superlinear or sublinear functions.
1. Introduction
We consider the following two-dimensional nonlinear difference system as follows:
Δxn f n, yn ,
Δyn −gn, xn ,
1.1
where Δxn xn1 − xn , Δyn yn1 − yn , fn, u and gn, u are strongly superlinear or
sublinear functions.
Now we pose some conditions on functions f and g:
H1: ufn, u > 0 and ugn, u > 0 for u /
0;
H2: fn, u and gn, u are continuous real-valued functions, and nondecreasing with
respect to u;
2
Journal of Inequalities and Applications
H3: it is
∞
fn, ±c ±∞
1.2
nn0
for each c > 0.
Definition 1.1. Suppose that f, g : N × R → R are real-valued functions. α and β are the
quotients of positive odd numbers.
1 f and g are said to be strongly superlinear if there exist constants α > 0 and β > 0
with αβ > 1, such that fn, u/|u|α sgn u and gn, u/|u|β sgn u are nondecreasing with respect
to |u| for each fixed n ∈ N.
2 f and g are said to be strongly sublinear if there exist constants α > 0 and β > 0
with αβ < 1, such that fn, u/|u|α sgn u and gn, u/|u|β sgn u are nonincreasing with respect
to |u| for each fixed n ∈ N.
The solutions of 1.1 are said to be nonoscillatory if {xn } or {yn } is eventually positive
or negative. Otherwise the solutions are called oscillatory.
β
Some oscillation results for the difference system 1.1 in the case of gn, xn an xn
with an > 0 have been established by many authors. In particular, if fn, yn bn yn and
bn > 0, then the difference system 1.1 is reduced to the well-known second-order nonlinear
difference equation:
1
β
Δ
Δxn an xn 0.
bn
1.3
Also, if bn 1, then 1.3 becomes
β
1.4
Δ2 xn an xn 0.
Furthermore, if fn, yn bn ynα and α is a ratio of odd positive integers, then 1.1 reduces to
the well-known quasilinear difference equation:
Δ
1, 2
.
1
bn1/α
Δxn 1/α
β
an xn 0.
1.5
For 1.4, the following well-known Theorem A was established by Hooker and Patula
Journal of Inequalities and Applications
3
Theorem A. For 1.4, the following statements are true.
(1) If 0 < β < 1, then every solution of 1.4 oscillates if and only if
∞
nβ an ∞.
1.6
n1
(2) If β > 1, then every solution of 1.4 oscillates if and only if
∞
nan ∞.
1.7
n1
For 1.3, if one denotes
Bn n−1
bs
1.8
s0
and assumes that
lim Bn n→∞
∞
bs ∞,
1.9
s0
then the following theorem is proved in [3].
Theorem B. If 1.9 holds, then the following statements are true.
(1) If 0 < β < 1, then every solution of 1.3 oscillates if and only if
∞
β
Bn an ∞.
1.10
n1
(2) If β > 1, then every solution of 1.3 oscillates if and only if
∞
Bn an ∞.
1.11
n1
The problem of oscillation of second-order nonlinear difference equations has attracted
the attention of many mathematicians because of its physical applications 2, 4
. For some
results regarding the growth of solutions of some equations related to the above mentioned
see book 5
, as well as the following papers 6–8
. It is an interesting problem to extend
oscillation criteria for second-order nonlinear difference equations to the case of nonlinear
two-dimensional difference systems since such systems include, in particular, the secondorder nonlinear, half-linear, and quasilinear difference equations that are the special cases of
the nonlinear two-dimensional difference systems 5, 9, 10
.
The main purpose of this paper is to establish some necessary and sufficient conditions
for oscillation of the nonlinear two-dimensional difference systems.
4
Journal of Inequalities and Applications
2. Main Results
In order to establish our main results, we need the following lemma.
Lemma 2.1. Suppose that conditions H1–(H3 are satisfied. If {xn } and {yn } are nonoscillatory
solutions of 1.1 for n > n0 , then
sgn xn sgn yn .
2.1
Proof. Without loss of generality, we assume that xn is eventually positive; that is, xn > 0 for
n > n0 > 0. From 1.1, we clearly see that Δyn < 0, then we know that either yn > 0 or yn < 0
eventually holds.
If yn < yN1 < 0 for n > N1 > n0 , then we have
Δxn f n, yn ≤ f n, yN < 0,
2.2
summing up from N1 to n, and by 1.2 of H3, we get
xn − xN1 ≤
n−1
f s, yN −→ −∞ n −→ ∞.
2.3
sN1
This contradiction completes the proof of the lemma.
Theorem 2.2. If f and g are strongly sublinear (i.e., 0 < αβ < 1), then a necessary and sufficient
condition for 1.1 to oscillate is that
∞
nn1
g n,
n−1
fs, c
∞
2.4
sn0
for every c > 0, where n1 > n0 .
Proof. Sufficiency. If 1.1 has a nonoscillatory solution xn , then without loss of generality, we
assume that xn is eventually positive. Then, by Lemma 2.1, for n0 sufficiently large,
yn > 0,
Δxn > 0,
Δyn < 0,
for n ≥ n0 .
2.5
Since {yn } is decreasing, hence there exists c > 0 such that
yn ≤ yN ≤ c,
for N ≥ n0 .
2.6
Summing up
Δxn f n, yn
2.7
Journal of Inequalities and Applications
5
from s n0 to n − 1, we obtain
xn − xn0
n−1
n−1 f s, y
n−1
fs, c
s
α
f s, ys ≥ yn
≥ ynα
,
α
y
cα
s
sn0
sn0
sn0
n−1
c−α ynα
xn ≥
2.8
fs, c,
sn0
and so
gn, xn ≥ g
n, c−α ynα
n−1
2.9
fs, c .
sn0
Therefore
−Δyn gn, xn ≥ g
n, c−α ynα
n−1
fs, c .
2.10
sn0
Since
n−1
c−α ynα
fs, c ≤
sn0
n−1
2.11
fs, c,
sn0
we have
−Δyn ≥ g n, c−α ynα
n−1
fs, c
sn0
β
n−1
g n, c−α ynα n−1
sn0 fs, c
−α α
c yn
fs, c
β
sn0
c−α ynα n−1
fs,
c
sn0
β
n−1
g n, n−1
sn0 fs, c
−α α
c yn
≥ fs, c
β
n−1
sn0
fs,
c
sn0
2.12
n−1
αβ
yn c−αβ g n,
fs, c ,
sn0
n−1
Δyn
−αβ
− αβ ≥ c g n,
fs, c ,
yn
sn0
and let n1 > n0 , we have
n−1
in1
Δyi
− αβ
yi
≥c
−αβ
n−1
in1
g i,
i−1
sn0
fs, c .
2.13
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Journal of Inequalities and Applications
From
yn
yn1
Δyn
1
1 du αβ yn − yn1 ≥ − αβ ,
αβ
u
ξ
yn
yn1 < ξ < yn ,
2.14
∞
i−1
∞
Δyi
1
−αβ
du ≥
− αβ ≥ c
g i,
fs, c ,
αβ
yi1 u
yi
sn0
in1
in1
yn
∞
i−1
1
1
αβ
g i,
fs, c ≤ c
du < ∞,
uαβ
c
sn0
in1
2.15
we get
∞ yi
in1
which leads to a contradiction.
Necessity. If
∞
g n,
nn1
n−1
< ∞
fs, c
2.16
sn0
for some c > 0, then there exist M > n0 > 0 and c/2 > d > 0, such that
∞
g n,
n−1
fs, c
2.17
< d.
sn0
nM
Let X be the Banach space of all the real-valued sequences {xn } with the norm
|xn |
,
x sup n−1
n≥M
sn0 fs, c
2.18
let Ψ be the subset of X defined by
Ψ
{xn } ∈ X :
n−1
fs, d ≤ xn ≤
sn0
n−1
fs, 2d
2.19
sn0
and let F : Ψ → X be the operator defined by
Fxn n−1
sn0
∞
s, d gi, xi .
f
is
2.20
Journal of Inequalities and Applications
7
Then the mapping F satisfies the assumptions of Knaster’s fixed-point theorem see
11, page 8
: F maps Ψ into itself and F is increasing. The latter statement is easy to see, and
the former statement follows from
Fxn ≥
n−1
fs, d,
nn0
Fxn n−1
∞
s, d gi, xi f
sn0
≤
n−1
≤
n−1
∞
i−1
s, d g i,
fs, 2d
f
sn0
is
is
2.21
sn0
∞
i−1
n−1
≤
s, d g i,
fs, c
fs, 2d
f
sn0
sn0
is
sn0
for any {xn } ∈ Ψ. From Knaster’s fixed-point theorem, we know that there exists {xn } ∈ Ψ
such that xn Fxn . Let
yn d ∞
gs, xs ,
2.22
sn
then limn → ∞ yn d and Δyn −gn, xn . On the other hand, we have
xn Fxn n−1
f i, yi .
2.23
in0
Then by 1.2 and the continuity of function f, we have that limn → ∞ xn ∞ and Δxn fn, xn , which leads to a contradiction and the proof of Theorem 2.2 is completed.
Example 2.3. Considering the difference system,
Δxn 2n 11/3 yn1/3 ,
Δyn −
2n 3
x5/3 ,
n 1n 2 n
2.24
n ≥ n0 .
Here α 1/3, β 5/3, and f and g are strongly sublinear. It is easy to verify that the
conditions of Theorem 2.2 are satisfied and hence all solutions are oscillatory. In fact, we
clearly see that the sequence {xn , yn } {−1n , −1n1 /n 1} is such a solution for
the difference system.
Example 2.4. Considering the difference system,
Δxn n
n1
1/3
yn1/3 ,
1
Δyn − 5/3
x5/3 ,
nn
1 n
n
1
2.25
n ≥ n0 .
8
Journal of Inequalities and Applications
Here α 1/3, β 5/3, and f and g are strong sublinear. We clearly see that the conditions
of Theorem 2.2 are not satisfied and hence there exists a nonoscillatory solution. In fact, the
sequence {xn , yn } {n, n 1/n} is such a solution.
Theorem 2.5. If f and g are strongly superlinear (i.e., αβ > 1), then a necessary and sufficient
condition for 1.1 to oscillate is that
∞
f
n,
∞
gs, c
∞
2.26
sn
n1
for every c > 0.
Proof. Sufficiency. If 1.1 has a nonoscillatory solution xn , then without loss of generality, we
may assume that xn is eventually positive. Then by Lemma 2.1, we have for N1 sufficiently
large,
Δxn > 0,
xn > 0,
Δyn < 0,
for n > N1 .
2.27
Since {xn } is increasing, hence there exists c > 0 such that
xn ≥ yN ≥ c,
2.28
for N > n0 .
Summing up
Δyn −gn, xn 2.29
from s n to ∞, we have
−yn ≤ y∞ − yn −
∞
gs, xs ,
sn
yn ≥
∞
sn
gs, xs ≥
∞
gs, xs sn1
β
xs
β
xs
≥
β
xn1
∞
∞
gs, c
−β β
c
x
gs, c.
n1
cβ
sn1
sn1
2.30
Therefore
Δxn f n, yn ≥ f
β
n, c−β xn1
∞
gs, c .
sn1
2.31
Journal of Inequalities and Applications
β
From xn ≥ c and c−β xn1
∞
sn1
gs, c ≥
9
∞
sn1
gs, c, we get
β α
∞
gs,
c
f n, c−β xn1 ∞
sn1
−β β
gs, c
Δxn ≥ c xn1
α
β sn1
c−β xn1 ∞
gs,
c
sn1
α
∞
f n, ∞
sn1 gs, c
−β β
c xn1
≥ ∞
gs, c
α
sn1
sn1 gs, c
∞
αβ −αβ
xn1 c f n,
gs, c ,
Δxn
αβ
xn1
∞
Δxi
in
αβ
xi1
sn1
≥c
−αβ
f
∞
n,
2.32
gs, c ,
sn1
≥c
−αβ
∞
f
∞
i,
in
gs, c .
si1
But
xn1
1
1
Δxn
du αβ xn1 − xn ≥
, xn < ξ < xn1 ,
αβ
u
ξ
xn1 αβ
xn
∞ xs1
∞
∞
∞
1
Δxs
−αβ
du ≥
≥c
f s,
gt, c .
αβ
αβ
sn xs u
sn xs1 sn
ts1
2.33
Therefore
∞
f
sN1
s,
∞
≤c
gt, c
∞
αβ
xN1
ts1
1
du < ∞,
uαβ
2.34
which leads to a contradiction.
Necessity. If
∞
f
n,
∞
< ∞
gs, c
2.35
sn
n1
for some c > 0, then there exists M > 0 large enough, such that
∞
nM
f
n,
∞
sn
gs, c
<
c
.
2
2.36
10
Journal of Inequalities and Applications
Let X be the set of all bounded and real-valued sequences {xn } with the norm
x sup|xn |
2.37
n≥M
and Ψ be the subset of X defined by
c
Ψ {xn } ∈ X : ≤ xn ≤ c ,
2
2.38
then Ψ is a bounded, convex, and closed subset of X. Let F : Ψ → X be the operator defined
by
∞
Fxn c − f
i,
in
∞
gs, xs .
2.39
si
Then F maps Ψ into Ψ. In fact, if {xn } ∈ Ψ, then
c ≥ Fxn ≥ c −
≥c−
∞
iM
∞
i, gs, c
f
in
si
∞
c
i, gs, c ≥ .
2
si
f
∞
2.40
j
Next, we show that F is continuous. Let {xn } be a convergent sequence in Ψ such that
j
limi → ∞ xn − xn 0, then from that Ψ is closed {xn } ∈ Ψ and the definition of F, we
have
∞
∞ ∞
∞
j
j
− f i, gs, xs − Fxn f i, g s, xs
Fx
in
n
si
in
si
∞ ∞ ∞
j
≤
− f i, gs, xs .
f i, g s, xs
in
si
si
2.41
∞
∞
∞
Since ∞
in fi,
si gs, xs <
in fi,
si gs, c < ∞, now from the continuity of f and
g together with the well-known Lebesgue’s dominated convergence theorem see 11, page
263
, we know that limj → ∞ Fxj n − Fxn 0 for xj − x → 0.
Journal of Inequalities and Applications
11
Finally, we show that FΨ is precompact. Let {xm } ∈ Ψ, {xn } ∈ Ψ, then for large enough
m > n we have
∞
∞
∞
∞
|Fxm − Fxn | f i, gs, xs − f i, gs, xs im
si
in
si
m
∞
f i, gs, xs in
si
≤
m
2.42
∞
i, gs, c < ε
f
in
si
for any ε > 0. From Schauder’s fixed-point theorem see 11
, we know that there exists
{xn } ∈ Ψ such that xn Fxn .
Let
yn ∞
gs, xs ,
2.43
sn
then limn → ∞ yn 0 and Δyn −gn, xn . On the other hand, we have
xn Fxn c −
∞
f s, ys .
2.44
sn
Therefore, limn → ∞ xn c and Δxn fn, yn , which leads to a contradiction. The proof of
Theorem 2.5 is completed.
Example 2.6. Considering the difference system,
Δxn 23n1 yn3 ,
Δyn −
3 3
x ,
2n n
n ≥ n0 .
2.45
Here α 3, β 3, and f and g are strongly suplinear. We clearly see that the conditions
of Theorem 2.5 are satisfied and hence all solutions are oscillatory. In fact, the sequence
{xn , yn } {−1n , −1n1 /2n } is such a solution.
Example 2.7. Considering the difference system,
Δxn n
n1
2/3
yn2/3 ,
1
Δyn − 5/3
x5/3 ,
nn
1 n
n
1
2.46
n ≥ n0 .
12
Journal of Inequalities and Applications
Here α 2/3, β 5/3, and f and g are strong sublinear. However, it is easy to see that the
conditions of Theorem 2.5 are not satisfied and hence there exists a nonoscillatory solution.
In fact, the sequence {xn , yn } {n, n 1/n} is such a solution.
β
If we set fn, yn bn ynα , gn, xn an xn , then the difference system 1.1 is reduced
to 1.5. From Theorems 2.2 and 2.5, we get the following results for 1.5.
Corollary 2.8. If 0 < αβ < 1, then every solution of 1.5 oscillation if and only if
∞
nn1
an
n−1
β
bs
∞,
2.47
sn0
where n1 > n0 .
Corollary 2.9. If αβ > 1, then every solution of 1.5 oscillation if and only if
α
∞
∞
bn
as
∞.
n1
2.48
sn
Remark 2.10. It is easy to see that Theorems A and B are the special cases of our Corollaries
2.8 and 2.9, respectively.
Acknowledgment
The authors wish to thank the anonymous referees for their very careful reading of the paper
and fruitful comments and suggestions.
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