PORTUGALIAE MATHEMATICA
Vol. 61 Fasc. 3 – 2004
Nova Série
OSCILLATION OF
THIRD-ORDER DIFFERENCE EQUATIONS
S.H. Saker
Recommended by Carlos Rocha
Abstract: In this paper we will study the oscillatory properties of third order
difference equations. By means of the Reccati transformation techniques we will establish
some sufficient conditions which are sufficient for all solutions to be oscillatory or tend
to zero.
1 – Introduction
In recent years, the oscillation theory and the asymptotic behavior of difference equations and their applications have been and still are receiving intensive
attention. In fact, in the last few years several monographs and hundreds of
research papers have been written, see for example the monographs [1, 2, 4].
Compared to the second order difference equations, the study of third order difference equations has received considerably less attention in the literature, even
though such equations arise in the study of Economics, Mathematical Biology,
and other areas of mathematics where discrete models are used (see for example [3]). Some recent results on third order difference equations can be found in
[5-10].
In this paper we shall consider the third order difference equation
(1.1)
∆3 Vn + Pn Vn+1 = 0 ,
n ≥ n0 ,
where Pn > 0 for n ≥ n0 and ∆ denotes the forward difference operator ∆Vn =
Vn+1 − Vn for any sequence {Vn } of real numbers.
Received : January 12, 2003; Revised : March 10, 2003.
250
S.H. SAKER
By a solution of (1.1) we mean a nontrivial real sequence {Vn } that is defined
for n > n0 and satisfies equation (1.1) for n≥ n0 . A solution {Vn } of (1.1) is said
to be oscillatory, if it is neither eventually positive nor eventually negative, and
nonoscillatory otherwise.
A number of dynamical behavior of solutions of difference equations are possible; here we will only be concerned with conditions which are sufficient for all
solutions of (1.1) to be oscillatory or tend to zero as n → ∞.
2 – Main results
In this section, by using the Reccati transformation techniques we establish
some new conditions which are sufficient for all solutions of (1.1) to be oscillatory
or tend to zero as n → ∞.
Theorem 2.1. Assume that
∞
X
(2.1)
l=n3


l−1 X
t−1
X
t=n3 s=n2

Ps  = ∞ ,
and there exists a positive sequence {ρn }∞
n=n0 such that,
(2.2)
lim sup
n→∞
n
X
l=n2
(∆ρl )2
ρl P l −
= ∞,
4ρl (l − n1 )
"
#
for n2 > n1 .
Then every solution {Vn } of Eq.(1.1) oscillates or limn→∞ Vn = 0.
Proof: Let {Vn } be a nonoscillatory solution of (1.1). Without loss of
generality we may assume that Vn > 0 for n > n1 where n1 ≥ n0 is chosen so
large. From (1.1) we have ∆3 Vn ≤ 0 for n > n1 . Then {Vn }, {∆Vn } and {∆2 Vn }
are monotone and eventually of one sign. We claim ∆2 Vn > 0. Suppose to the
contrary that ∆2 Vn ≤ 0 for n > n2 for n2 > n1 . Since ∆2 Vn is nonincreasing
there exists a negative constant C and n3 > n2 such that ∆2 Vn ≤ C for n > n3 .
Summing from n3 to n − 1, we obtain
∆Vn ≤ ∆Vn3 + C(n − 1 − n3 ) .
Letting n→ ∞, then ∆Vn → −∞ . Thus, there is an integer n4 > n3 such that
for n > n4 , ∆Vn ≤ ∆Vn4 < 0. Summing from n4 to n − 1 we obtain
Vn − Vn4 ≤ C(n − 1 − n4 ) ,
OSCILLATION OF THIRD-ORDER DIFFERENCE EQUATIONS
251
this implies that Vn → −∞ as n→ ∞ which is a contradiction with the fact that
Vn is positive. Then ∆2 Vn > 0. Therefore, there are only the following two cases
for n > n1 sufficiently large:
(I) Vn > 0, ∆Vn > 0, ∆2 Vn > 0.
(II) Vn > 0, ∆Vn < 0, ∆2 Vn > 0.
First we consider the Case (I): Define wn by the Reccati substitution
wn = ρ n
(2.3)
∆2 Vn
,
Vn+1
n > n1
we have wn > 0 and
∆wn = ∆2 Vn+1 ∆
·
ρn ∆3 Vn
ρn
,
+
Vn+1
Vn+1
¸
this and (1.1), imply that
(2.4)
∆wn ≤ −ρn Pn +
∆ρn
ρn ∆2 Vn ∆(Vn+1 )
wn+1 −
,
ρn+1
Vn+1 Vn+2
From the Case (I) we have Vn+2 ≥ Vn+1 , then from (2.4) we obtain
(2.5)
∆wn ≤ −ρn Pn +
ρn ∆2 Vn+1 ∆(Vn+1 )
∆ρn
wn+1 −
.
2
ρn+1
Vn+2
Also From the Case (I) and Eq.(1.1) we have
(2.6)
∆Vn = ∆Vn1 +
n−1
X
∆2 Vs > (n − 1 − n1 ) ∆2 Vn ,
n > n1 + 1 .
s=n1
This implies that
(2.7)
∆Vn+1 > (n − n1 )∆2 Vn+1 ,
n > n2 = n1 + 1 ,
Substituting from (2.7) in (2.8), we obtain
(2.8)
∆wn ≤ −ρn Pn +
ρn (n − n1 ) (∆2 Vn+1 )2
∆ρn
wn+1 −
.
2
ρn+1
Vn+2
From (2.3) and (2.8) we obtain
(2.9)
∆wn ≤ −ρn Pn +
∆ρn
ρn (n − n1 ) 2
wn+1 .
wn+1 −
ρn+1
ρ2n+1
252
S.H. SAKER
By completing the square we have
(∆ρn )2
−
∆wn ≤ −ρn Pn +
4ρn (n − n1 )
"p
(∆ρn )2
.
< − ρn P n −
4ρn (n − n1 )
#
"
Then, we have
(2.10)
∆ρn
ρn (n − n1 )
wn+1 − p
ρn+1
2 ρn (n − n1 )
#2
(∆ρn )2
∆wn < − ρn Pn −
.
4ρn (n − n1 )
#
"
Summing (3.11) from n2 to n, we obtain
−wn2 < wn+1 − wn2 < −
n
X
l=n2
which yields
(2.11)
n
X
l=n2
"
"
(∆ρl )2
ρl P l −
,
4ρl (l − n1 )
#
(∆ρl )2
ρl P l −
< c1 ,
4ρl (l − n1 )
#
for all large n, and this is contrary to (2.2). Next we assume that the Case (II)
holds. Since {Vn } is positive and decreasing it follows that limn→∞ Vn = b > 0.
Now we claim that b = 0. If not then Vn → b > 0 as n→ ∞, and hence there
exists n2 ≥ n1 such that Vn+1 ≥ b. Therefore from (1.1) we have
(2.12)
∆3 Vn + P n b ≤ 0 ,
n ≥ n2 ,
Define the sequence un = ∆2 Vn for n≥ n2 . Then we have
∆un ≤ −Pn b .
Summing the last inequality from n2 to n − 1, we have
(2.13)
un ≤ u n2 − b
n−1
X
Ps ,
s=n2
From (2.2), by choosing ρn = 1 we have
∞
P
Pn =∞, and then from (2.13) it is
n=n0
possible to choose an integer n3 sufficiently large such that for all n ≥ n3
un ≤ −
X
b n−1
Ps ,
2 s=n2
OSCILLATION OF THIRD-ORDER DIFFERENCE EQUATIONS
and hence
∆2 Vn ≤ −
253
X
b n−1
Ps .
2 s=n2
Summing the last inequality from n3 to n − 1 we obtain
∆Vn ≤ ∆Vn3
X
b n−1
−
2 t=n3
à t−1
X
!
Ps .
s=n2
Since ∆Vn < 0 for n > n0 , the last inequality implies that
X
b n−1
∆Vn ≤ −
2 t=n3
à t−1
X
s=n2
!
Ps .
Summing from n3 to n − 1 we have
Vn ≤ V n 3


l−1 X
t−1
X X
b n−1

Ps  .
−
2 l=n t=n3 s=n2
3
Condition (2.1) implies that Vn → −∞ as n → ∞ which is a contradiction with
the fact that Vn is positive. Then b = 0 and this completes the proof.
Remark 2.1. From Theorem 2.1, we can obtain different conditions for
oscillation of all solutions of (1.1) by different choices of {ρn }. Let ρn = nλ ,
n ≥ n0 and λ ≥ 1 is a constant. Hence we have the following results.
Corollary 2.1. Assume that all the assumptions of Theorem 2.1 hold, except
that the condition (2.2) is replaced by

³
´2 
λ − sλ
n
(s
+
1)
X
 λ

lim sup
 = ∞,
s P s −
λ
n→∞
s=n2
4 s (s − n1 )
for n2 > n1 .
Then, every solution {Vn } of Eq.(1.1) oscillates or limn→∞ Vn = 0.
Theorem 2.2. Assume that (2.1) holds. Let {ρn }∞
n=n0 be a positive sequence.
Furthermore, we assume that there exists a double sequence {Hm,n : m ≥ n ≥ 0}
such that
(i) Hm,m = 0 for m ≥ 0,
(ii) Hm,n > 0 for m > n ≥ 0,
(iii) ∆2 Hm,n = Hm,n+1 − Hm,n ≤ 0 for m ≥ n ≥ 0.
254
S.H. SAKER
If
(2.14)
lim sup
m→∞
1
Hm,n2
m−1
X
n=n2
"
(ρn+1 )2
∆ρn q
Hm,n ρn Pn −
hm,n −
Hm,n
4ρn (n − n1 )
ρn+1
µ
¶2 #
= ∞.
for n2 > n1 , where
∆2 Hm,n
hm,n = − p
,
Hm,n
m>n≥0.
Then, every solution {V n } of Eq.(1.1) oscillates or limn→∞ Vn = 0.
Proof: Proceeding as in Theorem 2.1, we assume that Eq.(1.1) has a nonoscillatory solution, say Vn > 0 for all n ≥ n1 . From the proof of Theorem 2.1 there
are two possible cases. First, we consider the case Case (I). Defining again {w n }
by (2.3), then from Theorem 2.1, we have wn > 0 and (2.9) holds. From (2.9) we
have for n ≥ n2
−
ρn
∆ρn
2
wn+1
.
wn+1 −
ρn Pn ≤ −∆wn +
ρn+1
(ρn+1 )2
(2.15)
Therefore, we have
(2.16)
m−1
X
Hm,n ρn Pn ≤ −
n=n2
m−1
X
Hm,n ∆wn +
m−1
X
Hm,n
n=n2
−
m−1
X
Hm,n
n=n2
−
n=n2
ρn
(ρn+1 )2
∆ρn
wn+1
ρn+1
2
wn+1
.
which yields after summing by parts
m−1
X
Hm,n ρn Pn ≤ Hm,n2 wn2 +
m−1
X
wn+1 ∆2 Hm,n
n=n2
n=n2
+
m−1
X
n=n2
Hm,n
−
m−1
X
ρn
∆ρn
2
wn+1 −
Hm,n
2 wn+1
ρn+1
(ρ
)
n+1
n=n2
OSCILLATION OF THIRD-ORDER DIFFERENCE EQUATIONS
255
hence
m−1
X
Hm,n ρn Pn =
n=n2
m−1
X
= Hm,n2 wn2 −
−
m−1
X
Hm,n
n=n2
hm,n
n=n2
−
ρn
(ρn+1 )2
q
Hm,n wn+1 +
m−1
X
Hm,n
n=n2
∆ρn
wn+1
ρn+1
2
wn+1
= Hm,n2 wn2
−
m−1
X
n=n2
r



Hm,n
−
ρn
ρn+1
wn+1 + r
2
ρn+1
q
hm,n Hm,n −
−
Hm,n ρ n
X (ρn+1 )2
1 m−1
∆ρn q
+
h
−
Hm,n
m,n
4 n=n2 −
ρn+1
ρn
µ
µ
¶2
2

∆ρn
Hm,n 

ρn+1
¶
.
Then,
m−1
X
n=n2
"
∆ρn q
(ρn+1 )2
Hm,n
hm,n −
Hm,n ρn Pn −
4ρn
ρn+1
µ
¶2 #
< Hm,n2 wn2 .
which implies that
lim sup
m→∞
1
Hm,n2
m−1
X
n=n2

Hm,n ρn Pn −
(ρn+1 )2
−
4 ρn
µ
∆ρn q
Hm,n
hm,n −
ρn+1
¶2

 < wn < ∞ ,
2
which contradicts (2.14). If the Case (II) holds we are then back to the proof
of the second case of Theorem 2.1 to prove that limn→∞ Vn = 0. The proof is
complete.
Remark 2.2. By choosing the sequence {Hm,n } in appropriate ways, we
can derive several oscillation criteria for (1.1). For instance, let us consider the
double sequence {Hm,n } defined by
Hm,n = (m − n)λ ,
or
Hm,n
m+1
= log
n+1
µ
¶λ
λ ≥ 1, m ≥ n ≥ 0 ,
,
λ ≥ 1, m ≥ n ≥ 0 ,
256
S.H. SAKER
or
Hm,n = (m − n)(λ) ,
λ > 2, m ≥ n ≥ 0 .
where (m − n)(λ) = (m − n)(m − n + 1) · · · (m − n + λ − 1) and
∆2 (m − n)(λ) = (m − n − 1)(λ) − (m − n)(λ) = −λ(m − n)(λ−1) .
Then Hm,m = 0 for m ≥ 0 and Hm,n > 0 and ∆2 Hm,n ≤ 0 for m > n ≥ 0.
Hence we have the following results.
Corollary 2.2. Assume that all the assumptions of Theorem 2.2 hold, except
that the condition (2.14) is replaced by
X
λ−2
ρ2n+1
1 m−1
∆ρn q
λ(m−n) 2 −
(m−n)λ
lim sup λ
(m−n)λ ρn Pn −
m→∞
m n=n2
4ρn (n−n1 )
ρn+1
"
µ
¶2 #
= ∞.
Then, every solution {V n } of Eq.(1.1) oscillates or limn→∞ Vn = 0.
Corollary 2.3. Assume that all the assumptions of Theorem 2.2 hold, except
that the condition (2.14) is replaced by
lim sup ³
m→∞
−
1
log(m + 1)
ρ2n+1
4ρn (n − n1 )
´λ


m−1
X
n=n2

µ
¶
m+1 λ

ρn P n −
 log
n+1
m+1
λ
log
n+1
n+1
µ
¶λ−2
2
−
∆ρn
ρn+1
s
µ
log
m+1
n+1
¶λ
2 

 = ∞.
Then, every solution {V n } of Eq.(1.1) oscillates or limn→∞ Vn = 0.
Corollary 2.4. Assume that all the assumptions of Theorem 2.2 hold, except
that the condition (2.14) is replaced by
µ
¶
X
ρ2n+1
1 m−1
λ
∆ρn 2
(λ)
lim sup (λ)
= ∞.
(m−n)
−
ρn P n −
m→∞
4ρn (n−n1 ) m−n+λ−1 ρn+1
m n=n2
"
Then, every solution {V n } of Eq.(1.1) oscillates or limn→∞ Vn = 0.
#
OSCILLATION OF THIRD-ORDER DIFFERENCE EQUATIONS
257
Remark 2.3. Our results can be extended to nonlinear difference equations
of the form
∆3 Vn + Pn f (Vn+1 ) = 0 , n ≥ n0 .
where f : R → R is continuous such that uf (u) > 0 for u 6= 0 and f (u)/u > K > 0
except that the term Pn is replaced by KPn .
ACKNOWLEDGEMENT – The author would like to thank the referee for his (her)
helpful comments.
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S.H. Saker,
Mathematics Department, Faculty of Science,
Mansoura University, Mansoura, 35516 – EGYPT
E-mail: shsaker@mum.mans.edu.eg