Volume 2010, Article ID 209309,13pages doi:10.1155/2010/209309
Research Article
Oscillatory Criteria for the Two-Dimensional
Difference Systems
Jin-Fa Cheng
1and Yu-Ming Chu
21Department of Mathematics, Xiamen University, Xiamen 361005, China 2Department 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
Copyrightq2010 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, wherefn, u
andgn, uare strongly superlinear or sublinear functions.
1. Introduction
We consider the following two-dimensional nonlinear difference system as follows:
Δxnf
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 functionsfandg:
H1:ufn, u>0 andugn, u>0 foru /0;
H3: it is
∞
nn0
fn,±c ±∞ 1.2
for eachc >0.
Definition 1.1. Suppose thatf, g : N×R → R are real-valued functions. αand β are the quotients of positive odd numbers.
1f andgare said to be strongly superlinear if there exist constantsα >0 andβ >0 withαβ >1, such thatfn, u/|u|αsgnuandgn, u/|u|βsgnuare nondecreasing with respect
to|u|for each fixedn∈N.
2f andg are said to be strongly sublinear if there exist constantsα > 0 andβ > 0 withαβ <1, such thatfn, u/|u|αsgnuandgn, u/|u|βsgnuare nonincreasing with respect
to|u|for each fixedn∈N.
The solutions of1.1are said to be nonoscillatory if{xn}or{yn}is eventually positive
or negative. Otherwise the solutions are called oscillatory.
Some oscillation results for the difference system1.1in the case ofgn, xn anxβn
with an > 0 have been established by many authors. In particular, iffn, yn bnyn and
bn>0, then the difference system1.1is reduced to the well-known second-order nonlinear
difference equation:
Δ
1
bnΔxn
anxβn0. 1.3
Also, ifbn1, then1.3becomes
Δ2x
nanxβn0. 1.4
Furthermore, iffn, yn bnynαandαis a ratio of odd positive integers, then1.1reduces to
the well-known quasilinear difference equation:
Δ
1
b1n/α
Δxn1/α
anxβn0. 1.5
For1.4, the following well-known Theorem A was established by Hooker and Patula
Theorem A. For1.4, the following statements are true.
(1) If0< β <1, then every solution of 1.4oscillates if and only if
∞
n1
nβan∞. 1.6
(2) Ifβ >1, then every solution of 1.4oscillates if and only if
∞
n1
nan∞. 1.7
For1.3, if one denotes
Bn n−1
s0
bs 1.8
and assumes that
lim
n→ ∞Bn ∞
s0
bs∞, 1.9
then the following theorem is proved in [3].
Theorem B. If 1.9holds, then the following statements are true. (1) If0< β <1, then every solution of 1.3oscillates if and only if
∞
n1
Bnβan∞. 1.10
(2) Ifβ >1, then every solution of 1.3oscillates if and only if
∞
n1
Bnan∞. 1.11
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
second-order 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
2. Main Results
In order to establish our main results, we need the following lemma.
Lemma 2.1. Suppose that conditionsH1–(H3are satisfied. If{xn}and{yn}are nonoscillatory
solutions of 1.1forn > n0, then
sgnxnsgnyn. 2.1
Proof. Without loss of generality, we assume thatxnis eventually positive; that is,xn >0 for n > n0>0. From1.1, we clearly see thatΔyn<0, then we know that eitheryn>0 oryn<0
eventually holds.
Ifyn< yN1 <0 forn > N1> n0, then we have
Δxnf
n, yn
≤fn, yN
<0, 2.2
summing up fromN1ton, and by1.2ofH3, we get
xn−xN1≤
n−1
sN1
fs, yN
−→ −∞ n−→ ∞. 2.3
This contradiction completes the proof of the lemma.
Theorem 2.2. Iff and gare strongly sublinear (i.e.,0 < αβ < 1), then a necessary and sufficient condition for1.1to oscillate is that
∞
nn1
g
n, n−1
sn0
fs, c
∞ 2.4
for everyc >0, wheren1> n0.
Proof. Sufficiency. If1.1has a nonoscillatory solutionxn, then without loss of generality, we
assume thatxnis eventually positive. Then, byLemma 2.1, forn0sufficiently large,
yn>0, Δxn>0, Δyn<0, forn≥n0. 2.5
Since{yn}is decreasing, hence there existsc >0 such that
yn≤yN≤c, forN≥n0. 2.6
Summing up
fromsn0ton−1, we obtain
xn−xn0
n−1
sn0
fs, ys
≥yαn n−1
sn0
fs, ys yαs ≥
yαn n−1
sn0
fs, c cα ,
xn≥c−αyαn n−1
sn0
fs, c,
2.8
and so
gn, xn≥g
n, c−αyαn n−1
sn0
fs, c
. 2.9
Therefore
−Δyngn, xn≥g
n, c−αyαn n−1
sn0
fs, c
. 2.10
Since
c−αyαn n−1
sn0
fs, c≤ n−1
sn0
fs, c, 2.11
we have
−Δyn ≥g
n, c−αyαn n−1
sn0
fs, c
g n, c −αyα
n
n−1 sn0fs, c
c−αyα n
n−1 sn0fs, c
β
c−αyαn n−1
sn0
fs, c
β
≥ g n,
n−1
sn0fs, c
n−1 sn0fs, c
β
c−αyαn n−1
sn0
fs, c
β
yn
αβ c−αβg
n, n−1
sn0
fs, c
,
−Δyn
yn
αβ ≥c −αβg
n, n−1
sn0
fs, c
,
2.12
and letn1> n0, we have
n−1
in1
−Δyi
yiαβ
≥c−αβ n−1
in1
g
i, i−1
sn0
fs, c
From
yn
yn1 1
uαβdu
1
ξαβ
yn−yn1≥ −Δyn
ynαβ, yn1< ξ < yn, 2.14
we get
∞
in1
yi
yi1 1
uαβdu≥ ∞
in1
−Δyi
yiαβ ≥c −αβ∞
in1
g
i, i−1
sn0
fs, c
,
∞
in1
g
i, i−1
sn0
fs, c
≤cαβ
yn
1
c
1
uαβdu <∞,
2.15
which leads to a contradiction. Necessity. If
∞
nn1
g
n, n−1
sn0
fs, c
<∞ 2.16
for somec >0, then there existM > n0>0 andc/2> d >0, such that
∞
nM g
n, n−1
sn0
fs, c
< d. 2.17
LetXbe the Banach space of all the real-valued sequences{xn}with the norm
xsup
n≥M
|xn|
n−1
sn0fs, c
, 2.18
letΨbe the subset ofXdefined by
Ψ
{xn} ∈X :
n−1
sn0
fs, d≤xn≤ n−1
sn0
fs,2d
2.19
and letF:Ψ → Xbe the operator defined by
Fxn n−1
sn0
f
s, d ∞
is gi, xi
Then the mappingF satisfies the assumptions of Knaster’s fixed-point theoremsee
11, page 8:FmapsΨinto itself andFis increasing. The latter statement is easy to see, and
the former statement follows from
Fxn≥ n−1
nn0
fs, d,
Fxn n−1
sn0
f
s, d ∞
is gi, xi
≤ n−1
sn0
f
s, d ∞
is g
i, i−1
sn0
fs,2d
≤ n−1
sn0
f
s, d ∞
is g
i, i−1
sn0
fs, c
≤ n−1
sn0
fs,2d
2.21
for any{xn} ∈ Ψ. From Knaster’s fixed-point theorem, we know that there exists{xn} ∈ Ψ
such thatxn Fxn. Let
ynd ∞
sn
gs, xs, 2.22
then limn→ ∞yndandΔyn −gn, xn. On the other hand, we have
xn Fxn n−1
in0
fi, yi
. 2.23
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 ofTheorem 2.2is completed.
Example 2.3. Considering the difference system,
Δxn 2n11/3y1n/3,
Δyn −n2n3
1n2x
5/3
n , n≥n0.
2.24
Here α 1/3, β 5/3, and f and g are strongly sublinear. It is easy to verify that the
conditions of Theorem 2.2are satisfied and hence all solutions are oscillatory. In fact, we
clearly see that the sequence{xn, yn} {−1n,−1n1/n1} is such a solution for
the difference system.
Example 2.4. Considering the difference system,
Δxn
n n1
1/3 y1n/3,
Δyn− 1
n5/3
1
nn1x
5/3
n , n≥n0.
Hereα 1/3,β 5/3, andf andg are strong sublinear. We clearly see that the conditions ofTheorem 2.2are not satisfied and hence there exists a nonoscillatory solution. In fact, the
sequence{xn, yn}{n,n1/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 for1.1to oscillate is that
∞
n1 f
n, ∞
sn gs, c
∞ 2.26
for everyc >0.
Proof. Sufficiency. If1.1has a nonoscillatory solutionxn, then without loss of generality, we
may assume thatxn is eventually positive. Then byLemma 2.1, we have forN1sufficiently
large,
xn >0, Δxn>0, Δyn<0, forn > N1. 2.27
Since{xn}is increasing, hence there existsc >0 such that
xn≥yN≥c, forN > n0. 2.28
Summing up
Δyn−gn, xn 2.29
fromsnto∞, we have
−yn≤y∞−yn− ∞
sn
gs, xs,
yn≥ ∞
sn
gs, xs≥ ∞
sn1 gs, xs
xβs
xβs ≥xβn1 ∞
sn1 gs, c
cβ c −βxβ
n1 ∞
sn1 gs, c.
2.30
Therefore
Δxn fn, yn≥f
n, c−βxβn1 ∞
sn1 gs, c
Fromxn≥candc−βxβn1
∞
sn1gs, c≥∞sn1gs, c, we get
Δxn≥ f n, c
−βxβ n1
∞
sn1gs, c
c−βxβn1∞sn1gs, cα
c−βxβn1 ∞
sn1 gs, c
α
≥ f
n,∞sn1gs, c
∞
sn1gs, cα
c−βxβn1 ∞
sn1 gs, c
α
xαβn1c−αβf
n, ∞
sn1 gs, c
,
Δxn xαβn1 ≥
c−αβf
n, ∞
sn1 gs, c
,
∞
in Δxi xαβi1 ≥
c−αβ ∞
in f
i, ∞
si1 gs, c
.
2.32
But
xn1
xn
1
uαβdu
1
ξαβxn1−xn≥ Δxn xn1αβ
, xn< ξ < xn1,
∞
sn
xs1
xs
1
uαβdu≥ ∞
sn Δxs xs1αβ
≥c−αβ ∞
sn f
s, ∞
ts1 gt, c . 2.33 Therefore ∞
sN1
f
s, ∞
ts1 gt, c
≤cαβ
∞
xN1
1
uαβdu <∞, 2.34
which leads to a contradiction. Necessity. If
∞ n1 f n, ∞
sn gs, c
<∞ 2.35
for somec >0, then there existsM >0 large enough, such that
∞
nM f
n, ∞
sn gs, c
< c
LetXbe the set of all bounded and real-valued sequences{xn}with the norm
xsup
n≥M|xn| 2.37
andΨbe the subset ofXdefined by
Ψ {xn} ∈X: c
2 ≤xn≤c
, 2.38
thenΨis a bounded, convex, and closed subset ofX. LetF:Ψ → Xbe the operator defined
by
Fxnc− ∞
in f
i, ∞
si
gs, xs
. 2.39
ThenFmapsΨintoΨ. In fact, if{xn} ∈Ψ, then
c≥Fxn≥c− ∞
in f
i, ∞
si gs, c
≥c−
∞
iM f
i, ∞
si gs, c
≥ c
2.
2.40
Next, we show that F is continuous. Let{xnj} be a convergent sequence in Ψ such that
limi→ ∞xnj−xn 0, then from thatΨis closed{xn} ∈ Ψand the definition ofF, we
have
Fxj
n−Fxn
∞
in f
i, ∞
si g s, xjs
−∞
in f
i, ∞
si
gs, xs
≤∞
in
f
i, ∞
si g s, xjs
−f
i, ∞
si
gs, xs.
2.41
Since∞infi,∞sigs, xs <∞infi,∞sigs, c< ∞, now from the continuity offand
g together with the well-known Lebesgue’s dominated convergence theoremsee 11, page
Finally, we show thatFΨis precompact. Let{xm} ∈Ψ,{xn} ∈Ψ, then for large enough
m > nwe have
|Fxm−Fxn|
∞
im f
i, ∞
si
gs, xs
−∞
in f
i, ∞
si
gs, xs
m in f
i, ∞
si
gs, xs
≤m
in f
i, ∞
si gs, c
< ε
2.42
for anyε > 0. From Schauder’s fixed-point theoremsee 11, we know that there exists
{xn} ∈Ψsuch thatxn Fxn.
Let
yn ∞
sn
gs, xs, 2.43
then limn→ ∞yn0 andΔyn−gn, xn. On the other hand, we have
xn Fxnc− ∞
sn
fs, ys. 2.44
Therefore, limn→ ∞xn candΔxn fn, yn,which leads to a contradiction. The proof of
Theorem 2.5is completed.
Example 2.6. Considering the difference system,
Δxn23n1y3n,
Δyn−3 2nx
3
n, n≥n0.
2.45
Here α 3, β 3, andf 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 y2n/3,
Δyn− 1
n5/3
1
nn1x
5/3
n , n≥n0.
Hereα 2/3,β 5/3, andf andg are strong sublinear. However, it is easy to see that the
conditions ofTheorem 2.5are not satisfied and hence there exists a nonoscillatory solution.
In fact, the sequence{xn, yn}{n,n1/n}is such a solution.
If we setfn, yn bnynα, gn, xn anxβn, then the difference system1.1is reduced
to1.5. From Theorems2.2and2.5, we get the following results for1.5.
Corollary 2.8. If0< αβ <1, then every solution of 1.5oscillation if and only if
∞
nn1
an
n−1
sn0
bs
β
∞, 2.47
wheren1> n0.
Corollary 2.9. Ifαβ >1, then every solution of 1.5oscillation if and only if
∞
n1 bn
∞
sn as
α
∞. 2.48
Remark 2.10. It is easy to see that Theorems A and B are the special cases of our Corollaries
2.8and2.9, 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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