OSCILLATIONS OF FIRST ORDER DELAY AND ADVANCED
DIFFERENCE EQUATIONS WITH VARIABLE ARGUMENTS
by
I.P. Stavroulakis
Department of Mathematics
University of Ioannina
Email: [email protected]
1. Introduction
The problem of establishing su¢ cient conditions for the oscillation of all solutions
to the …rst-order delay di¤erence equation with constant arguments
x(n) + p(n)x(n
k) = 0;
n
0
(E)
has been the subject of many investigations. See, for example, [6, 8, 9, 12 17, 23,
26, 29, 34, 36, 37, 40, 43, 44, 48-58, 62, 64, 65] and the references cited therein. Recently this problem was also investigated for the delay di¤erence and the advanced
di¤erence equation with variable arguments of the form
x(n) + p(n)x( (n)) = 0,
n
0
(E1 )
and
rx(n)
p(n)x( (n) = 0, n
1, [ x(n)
p(n)x( (n) = 0, n
0];
(E2 )
where
denotes the forward di¤erence operator
x(n) = x(n + 1) x(n), r
denotes the backward di¤erence operator rx(n) = x(n) x(n 1), fp(n)g is a
sequence of nonnegative real numbers, f (n)g is a sequence of positive integers
such that (n) n 1; for all n 0, and f (n)g [f (n)g] is a sequence of positive
integers such that
(n)
n + 1 for all n
1,
[ (n)
n + 2 for all n
0] .
See, for example, [1 5, 7, 10, 18, 26, 41, 42, 63] and the references cited therein.
Strong interest in the equations (E1 ) and (E2 ) is motivated by the fact that they
represent the discrete analogues of the delay
x0 (t) + p(t)x( (t)) = 0,
t
t0 ,
(E1 )0
t
t0 ,
(E2 )0
and the advanced di¤erential equation
x0 (t)
p(t)x( (t)) = 0,
where p; ; 2 C([t0 ; 1); R+ ); R+ = [0; 1); (t); (t) are nondecreasing (t) < t
and (t) > t for t t0 [see, for example, 11, 19-27, 30-33, 35, 38, 39, 45-47, 59-61].
0
Key Words: Oscillation; delay, advanced di¤erence equations.
2010 Mathematics Subject Classi…cation: Primary 39A21, 39A10; Secondary 39A12.
1
2
2. Oscillation Criteria for Eq. (E1 )0
x0 (t) + p(t)x( (t)) = 0
1950, Myshkis
lim sup[t
t!1
(t)] < 1; lim inf [t
(t)] lim inf p(t) >
t!1
t!1
1972, Ladas, Lakshmikantham & Papadakis L := lim sup
1979 Ladas,1982 Koplatadze & Chanturija k := lim inf
Rt
(t)
Rt
(t)
1
e
p(s)ds > 1 (C1 )0
p(s)ds >
1
e
(C2 )0
In 1982 Ladas, S…cas and Stavroulakis and in 1984 Fukagai and Kusano established conditions (of the type (C1 )0 & (C2 )0 with oscillating coe¢ cient p (t) :
1988, Erbe & Zhang
1
e
0<k
1991 Jian Chao
and
L>1
1991 Kwong
p1
(1
L>
1 is
p
1 k
L>1
1990 Elbert & Stavroulakis
respectively, where
k2
2(1 k)
L>1
1992 Yu,Wang, Zhan & Qian
k2
4
L>1
ln
1 2k k2
2
1
)2
1 +1
1
the smaller root of the equation
1998 Philos & S…cas
L>1
1999 Jaroš & Stavroulakis
L>
ln
k2
2(1 k)
1 +1
L>
L > 2k +
ln
1
p
1+ 5 2
0:966
(C4 )0
0:892
(C5 )0
0:863
(C6 )0
0:845
(C7 )0
0:735
=e .
(C8 )0
p
1 k
1
2000 Kon,S…cas & Stavroulakis
2003 S…cas & Stavroulakis
k2
2
(C3 )0
0:709
1 2k k2
2
(C9 )0
2
(C10 )0
0:471
(C11 )0
0:459
1
1 +2k 1
1
1,
0:599
It is to be noted that as k ! 0, then all the previous conditions (C3 )0 -(C10 )0
reduce to the condition (C1 )0 , i.e. L > 1. However our condition (C11 )0 leads to
p
L > 3 1 t 0:732
Example. Consider the delay di¤erential equation
p
1
x0 (t) + px(t q sin2 t
) = 0;
pe
where p > 0, q > 0 and pq = 0:46 1e . Then k = 1e and L = 0:46. Thus,
according to (C11 )0 , all solutions of this equation oscillate. Observe, however, that
none of the conditions (C3 )0 -(C10 )0 is satis…ed for this equation.
3
3. Oscillation Criteria for Eq. (E)
In this section we study the delay di¤erence equation with constant argument
x(n) + p(n)x(n
k) = 0;
n = 0; 1; 2; ::::
(E)
where x(n) = x(n + 1) x(n); fp(n)g is a sequence of nonnegative real numbers
and k is a positive integer.
In 1981, Domshlak [12 ] was the …rst who studied this problem in the case where
k = 1: Then, in 1989, Erbe and Zhang [23] established that all solutions of Eq.(E)
are oscillatory if
kk
(2:1)
lim inf p(n) >
n!1
(k + 1)k+1
or
n
X
lim sup
p(i) > 1.
(C1 )
n!1
i=n k
In the same year, 1989, Ladas, Philos and S…cas [37]established the condition
lim inf
n!1
n
X1
p(i) >
i=n k
k
k+1
k+1
(C2 )
It is interesting to establish su¢ cient oscillation conditions for the equation (E)
in the case where neither (C1 ) nor (C2 ) is satis…ed.
In 1995, the following oscillation criterion was established by Stavroulakis [50]:
Theorem 2.1 ([50]) Assume that
0
n
X1
:= lim inf
n!1
k
k+1
p(i)
i=n k
and
k+1
2
0
lim sup p(n) > 1
(2:2)
4
n!1
then all solutions of Eq.(E) oscillate.
In 2004, the same author [51] improved the condition (2.2) as follows:
lim sup
n!1
or
n
X1
i=n k
lim sup
n!1
p(i) > 1
n
X1
p(i) > 1
2
0
(C3 )
4
k
0
(2:3)
i=n k
In 2006, Chatzarakis and Stavroulakis [6], established the following condition
lim sup
n!1
n
X1
i=n k
p(i) > 1
2
0
2(2
0)
Also, Chen and Yu [8] obtained the following oscillation condition
p
n
2
X
1
1 2 0
0
0
lim sup
p (i) > 1
:
2
n!1
i=n k
(2:4)
(C5 )
4
4. Oscillation Criteria for Eq. (E1 )
In this section we study the delay di¤erence equation with variable argument
x(n) + p(n)x( (n)) = 0, n = 0; 1; 2; :::,
(E1 )
where x(n) = x(n + 1) x(n); fp(n)g is a sequence of nonnegative real numbers
and f (n)g is a nondecreasing sequence of integers such that (n) n 1 for all
n 0 and limn!1 (n) = 1.
In 2008, Chatzarakis, Koplatadze and Stavroulakis [2] investigated for the …rst
time the oscillatory behaviour of Eq.(E1 ) in the case of a general delay argument
(n) and derived the following theorem.
Theorem 3.1 ([2])
If
n
X
lim sup
n!1
p(i) > 1
(D1 )
i= (n)
then all solutions of Eq. (E1 ) oscillate.
This result generalizes the oscillation criterion (C1 ): Also in the same year
Chatzarakis, Koplatadze and Stavroulakis [3] extended the oscillation criterion (C2 )
to the general case of Eq. (E1 ). More precisely, the following theorem has been
established in [3].
Theorem 3.2 ([3])
Assume that
n
X1
lim sup
n!1
p(i) < +1
(3:1)
i= (n)
and
n
X1
:= lim inf
n!1
p(i) >
i= (n)
1
:
e
(D2 )
Then all solutions of Eq.(E1 ) oscillate.
Remark 3.1 It should be mantioned that in the case of the delay di¤erential
equation
x0 (t) + p(t)x( (t)) = 0, t t0 ,
(E1 )0
it has been proved (see [35,30]) that each one of the conditions
lim sup
n!1
Zt
p(s)ds > 1
(C1 )0
1
:
e
(C2 )0
(t)
or
lim inf
n!1
Zt
p(s)ds >
(t)
0
implies that all solutions of Eq.(E1 ) oscillate. Therefore, the conditions (D1 ) and
(D2 ) are the discrete analogues of the conditions (C1 )0 and (C2 )0 and also the
analogues of the conditions (C1 ) and (C2 ) in the case of a general delay argument
(n):
5
As it has been mentioned above, it is interesting to …nd new su¢ cient conditions
for the oscillation of all solutions of (E1 ); in the case where neither (D1 ) nor (D2 )
is satis…ed.
In 2008, Chatzarakis, Koplatadze and Stavroulakis [2] and in 2008 and 2009,
Chatzarakis, Philos and Stavroulakis [4] and [5] derived the following conditions:
1
e.
Theorem 3.3 ([2,4,5]) (I) Assume that 0 <
conditions:
n
X
lim sup
p (j) > 1
1
n!1
or
lim sup
n!1
or
lim sup
n!1
p (j) > 1
j= (n)
p
1
1
2
p (j) > 1
j= (n)
p
1
1
1
2
implies that all solutions of Eq.(E1 ) oscillate.
1
(II) If 0 <
1
e and in addition, p(n)
lim sup
n!1
or if 0 <
6
n!1
n
X
n
X
j= (n)
p
1
p (j) > 1
j= (n)
p
4 2 and in addition, p(n)
lim sup
2
1
(3:2)
j= (n)
n
X
n
X
p
Then either one of the
p (j) > 1
1
2
4
3
2
p
1
p
1
2
(3:3)
2
2
(C5 )
for all large n; and
1
(3:4)
1
for all large n; and
p
4
12 +
2
(3:5)
then all solutions of Eq.(E1 ) are oscillatory.
Remark 3.4 In the case where the sequence f (n)g is not assumed to be
nondecreasing, de…ne (cf. [2, 3, 4, 5])
(n) = max f (s) : 0
s
n; s 2 Ng .
Clearly, the sequence of integers f (n)g is nondecreasing. In this case, Theorems
3.1, 3.2, 3.3, 3.4, and 3.5 can be formulated in a more general form. More precisely in
the conditions (D1 ); (D2 ); (3:2); (3:3); (C6 )0 , (3:4) and (3:5) the term (n) is replaced
by (n):
1
Remark 3.5 Observe the following:(i) When 0 <
e , it is easy to verify
that
p
p
p
2
p
1 2
1
1
1
1 2
1
p
>
>
> (1
1
)2
2
2
1
0
and therefore condition (C6 )p
is weaker than the conditions (3.4), (3.3) and (3.2).
(ii) When 0 <
6 4 2, it is easy to show that
p
p
1
1
2
2 3
4 12 + 2 >
1
1 2
4
2
and therefore in this case,
(C6 )0 and espep inequality (3.5) improves the inequality
0
cially, when = 6 4 2 ' 0:3431457; the lower bound in (C6 ) is 0.8929094 while
in (3.5) is 0.7573593.
6
5. Oscillation Criteria for Eq. (E2 )
In this section, we study the advanced di¤erence equation with variable argument
rx(n)
p(n)x( (n) = 0, n
1, [ x(n)
p(n)x( (n) = 0, n
0];
(E2 )
where r denotes the backward di¤erence operator rx(n) = x(n) x(n 1),
denotes the forward di¤erence operator
x(n) = x(n + 1) x(n), fp(n)g is a
sequence of nonnegative real numbers and f (n)g [f (n)g] is a sequence of positive
integers such that
(n)
n + 1 for all n
1,
[ (n)
n + 2 for all n
0] .
For Eq. (E2 ) very recently, Chatzarakis and Stavroulakis [7] established the
following theorems:
If
Theorem 4.1 ([7]) Assume that the sequence f (n)g [f (n)g] is nondecreasing.
lim sup
n!1
(n)
X
i=n
2
3
(n) 1
p(i) 4lim sup
n!1
then all solutions of Eq.(E2 ) oscillate.
X
p(i)5 > 1,
i=n
(A1 )
Theorem 4.2 ([7]) Assume that the sequence f (n)g [f (n)g] is nondecreasing,
and
2
3
(n)
(n) 1
X
X
lim inf
p (i) 4lim inf
p (i)5 = .
(4:1)
n!1
If 0 <
n!1
i=n+1
i=n+1
1, and
lim sup
n!1
(n)
X
i=n
2
(n) 1
p(i) 4lim sup
n!1
then all solutions of (E2 ) oscillate.
If 0 <
1=2, and
lim sup
n!1
(n)
X
i=n
2
(n) 1
p(i) 4lim sup
n!1
X
i=n
X
i=n
3
p(i)5 > 1
3
p(i)5 > 1
then all solutions of (E2 ) oscillate.
p
If 0 < < (3 5 5)=2; and in addition p(n)
2
3
(n)
(n) 1
X
X
lim sup
p(i) 4lim sup
p(i)5 > 1
n!1
n!1
i=n
i=n
p
1
p
1
1
2
1
p
p
3 1
1
2
1
1
,
2
(4:3)
for all large n, and
1
+
2
1 ,
p
4 2, and in addition p(n)
for all large n; and
2
2
3
(n)
(n) 1
p
X
X
1
4
lim sup
p(i) lim sup
p(i)5 > 1
2 3
4 12 +
4
n!1
n!1
i=n
i=n
or if 0 <
(4:2)
(4:4)
6
then all solutions of (E2 ) oscillate.
2
(4:5)
7
Remark 4.1. In the case where the sequence f (n)g [ (n)] is not assumed to
be nondecreasing, de…ne (cf. [7])
(n) = max f (s) : 1
s
n; s 2 Ng ,
[ (n) = max f (s) : 1
s
n; s 2 Ng] .
Clearly, the sequence of integers f (n)g [f (n)g] is nondecreasing. In this case,
Theorems 4.1 and 4.2 can be formulated in a more general form. More precisely,
in the conditions (A1 ); (4:2) (4:3), (4:4) and (4:5) the term (n) [ (n)] is replaced
by (n) [ (n)] :
Remark 4.2. Observe the following:
When
! 0, then the conditions (4.3) and (4.5) reduce to
2
3
(n)
(n) 1
X
X
lim sup
p(i) 4lim sup
p(i)5 > 1,
n!1
i=n
n!1
i=n
that is, to the condition (A1 ). However, when 0 <
1=2, then we have
p
p
1
2
1 2 > 1
1
,
1
2
which means that the condition (4.3)
p improves the condition
p
In the case where 0 <
6 4 2, (and since 1
1
that
p
1
1
p
4 12 + 2 >
2 3
4
3 1
+
which means that the condition (4.5) improves the condition
(4.2).
> =2), we can show
2
(4.4).
1 ,
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