Mem. Differential Equations Math. Phys. 24(2001), 125–135
R. Koplatadze
PROPERTY A OF HIGH ORDER LINEAR DIFFERENTIAL
EQUATIONS WITH SEVERAL DEVIATIONS
(Reported on April 30, 2001)
1. Introduction
Consider the equation
u(n) (t) +
m
X
pi (t)u(τi (t)) = 0,
(1.1)
i=1
where n ≥ 2, n, m ∈ N , pi ∈ Lloc (R+ ; R+ ), τi ∈ C(R+ ; R+ ) and lim τi (t) = +∞
t→∞
(i = 1, . . . , m).
Oscillatory properties of higher order ordinary differential equations are studied well
enough (see e.g. [1,2]) Analogous questions for the equation (1.1) are studied in [3,4] (see
also references therein). Using comparison theorems presented in [5] and earlier results,
we can derive effective criteria for the equation (1.1) have Property A (for definition of
Property A see [5]). In the case m = 1 some of the results of the present paper are given
in [6].
Let l; j ∈ N ; αi ; ci ∈ (0, +∞) (i = 1, . . . , j). Below we will make use of the following
notation
Ml,j
( P
j
c αλ
i=1 i i
= inf Qn−1
i=0
|λ − i|
: λ ∈ (l − 1, l)
)
.
(1.2l,j )
2. Effective criteria for Property A
Theorem 2.1. Let m = k, τi (t) ≤ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , m), and
one of the following two conditions be fulfilled:
1) τi (t) ≥ αi t (i = 1, . . . , m),
lim inf t
t→+∞
Z+∞
τin−2 (s)pi (s) ds ≥ ci αin−2 (i = 1, . . . , m)
(2.1)
t
and Mn−1,m > 1;
2) τi (t) ≤ αi t (i = 1, . . . , m),
lim inf t
t→+∞
Z+∞
s−1 τin−1 (s)pi (s) ds ≥ ci αin−1 (i = 1, . . . , m)
(2.2)
t
2000 Mathematics Subject Classification. 34K15.
Key words and phrases. Advanced differential equation, Property A, comparison
theorem.
126
and Mn−1,m > 1, where Mn−1,m is defined by (1.2n−1,m ).
Then the equation (1.1) has Property A.
Theorem 2.2. Let k = m, τi (t) ≥ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , m) one
of the following two conditions be fulfilled:
1) τi (t) ≥ αi t (i = 1, . . . , m),
(i)
lim inf t
t→+∞
Z+∞
sn−2 pi (s) ds ≥ ci (i = 1, . . . , m)
t
and M1,m > 1 if n is even,
(ii) (2.1) holds,
lim inf t
t→+∞
Z+∞
sn−3 τi (s)pi (s) ds ≥ ci αin−1 (i = 1, . . . , m),
t
Z+∞
tn−1
t
m
X
pi (t) dt = +∞,
(2.3)
i=1
M2,m > 1 and Mn−1,m > 1 if n is odd;
2) τi (t) ≤ αi t (i = 1, . . . , m),
(i) (2.3) holds,
lim inf t
t→+∞
Z+∞
sn−3 τi (s)pi (s) ds ≥ ci αi (i = 1, . . . , m)
t
and Mi,m > 1 if n is even,
(ii) (2.2), (2.3) hold,
lim inf t
t→+∞
Z+∞
sn−4 τi2 (s)pi (s) ds ≥ ci αin−1 (i = 1, . . . , m),
t
M2,m > 1 and Mn−1,m > 1 if n is odd, where Mn−1,m and M2,m are definmed by
(1.22,m ) and (1.2n−1,m ).
Then the equation (1.1) has Property A.
Theorem 2.3. Let m = 1, α ∈ (0, 1), τi (t) ≤ t for t ∈ R+ and one of the following
two conditions be fulfilled:
τ1 (t)
1) lim inf
> 0 and lim inf tα
t→+∞
tα
t→+∞
Z+∞
τ1n−2 (s)p1 (s) ds > 0;
t
2) lim inf
t→+∞
τ1 (t)
< +∞ and lim inf tα
tα
t→+∞
Z+∞
s−α τ1n−1 (s)p1 (s) ds > 0.
t
Then the equation (1.1) has Property A.
127
Theorem 2.4. Let k = m, τi (t) ≤ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , m) and
one of the following two conditions be fulfilled:
1) τi (t) ≥ αi t (i = 1, . . . , m),
lim inf t−1
t→+∞
Zt
s2 τin−2 (s)pi (s) ds ≥ ci αin−2 (i = 1, . . . , m)
0
and Mn−1,m > 1;
2) τi (t) ≤ αi t (i = 1, . . . , m),
lim inf t−1
t→+∞
Zt
sτin−1 (s)pi (s) ds ≥ ci αin−1 (i = 1, . . . , m)
(2.4)
0
and Mn−1,m > 1, where Mn−1,m is defined by (2.2n−1,m ).
Then the equation (1.1) has Property A
Theorem 2.5. Let k = m, τi (t) ≥ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , m) one
of the following two conditions be fulfilled:
1) τi (t) ≥ αi t (i = 1, . . . , m),
(i) (2.3) holds,
lim inf t
−1
t→+∞
Zt
sn−1 τi (s)pi (s) ds ≥ ci αi (i = 1, . . . , m)
0
and M1 m > 1 if n is even,
(ii) (2.3), (2.4) hold,
lim inf t
−1
t→+∞
Zt
sn−2 τi2 (s)pi (s) ds ≥ ci α2i (i = 1, . . . , m),
0
M2,m > 1 and Mn−1,m > 1 if n is odd;
2) τi (t) ≤ αi t (i = 1, . . . , m),
(i) (2.1) holds,
lim inf t
−1
t→+∞
Zt
sn−1 τi (s)pi (s) ds > ci αi (i = 1, . . . , m)
0
and M1,m > 1 if n is even,
(ii) (2.3), (2.4) hold,
lim inf t−1
t→+∞
Zt
sn−2 τi2 (s)pi (s) ds ≥ ci α2i (i = 1, . . . , m),
0
M2,m > 1 and Mn−1,m > 1, for l = 2, n − 1 if n is odd, where M2,m and Mn−1,m > 1
are defined by (1.22,m ) ((1.2n−1,m ).
Then the equation (1.1) has Property A.
128
Below we will make use of the following notation
τ∗ (t) = min{τi (t) : i = 1, . . . , m}, τ ∗ (t) = max{τi (t) : i = 1, . . . , m}
Let αi ∈ (0, +∞) (i = 1, . . . , k), α∗ = min{αi : i = 1, . . . , }, α∗ = max{αi : i = 1, . . . , k}.
Theorem 2.6. Let τ ∗ (t) ≤ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , k) and one of
the following eight conditions be fulfilled:
1) τ∗ (t) ≥ α∗ t,
Z+∞X
m
lim inf t
t→+∞
t
pi (s)τin−2 (s) ds ≥
i=1
k
X
ci αin−2
(2.5)
i=1
and Mn−1,k > 1;
2) τ∗ (t) ≥ α∗ t,
lim inf t
t→+∞
Z+∞X
m
t
pi (s)τin−2 (s) ds ≥
1
α∗
k
X
ci αin−1
(2.6)
i=1
i=1
and Mn−1,k > 1;
3) τ∗ (t) ≤ α∗ t,
lim inf t
t→+∞
Z+∞
τ∗ (s)
s
m
X
pi (s)τin−2 (s) ds ≥ α∗
i=1
t
k
X
ci αin−2
(2.7)
ci αin−2
(2.8)
i=1
and Mn−1,k > 1;
4) τ∗ (t) ≤ α∗ t,
lim inf t
t→+∞
Z+∞
s−1 τ∗ (s)
m
X
pi (s)τin−2 (s) ds ≥
i=1
t
k
X
i=1
and Mn−1,k > 1;
5) τ ∗ (t) ≥ α∗ t,
lim inf t
t→+∞
Z+∞
1
τ ∗ (s)
t
m
X
pi (s)τin−1 (s) ds ≥
i=1
k
X
ci αin−1
(2.9)
i=1
and Mn−1,k > 1;
6) τ ∗ (t) ≥ α∗ t,
lim inf t
t→+∞
Z+∞
t
and Mn−1,k > 1;
1
τ∗ (s)
m
X
i=1
pi (s)τin−1 (s) ds ≥
1
α∗
k
X
i=1
ci αin−1
(2.10)
129
7) τ ∗ (t) ≤ α∗ t,
lim inf t
t→+∞
Z+∞
s−1
m
X
k
X
pi (s)τin−1 (s) ds ≥
i=1
t
ci αin−2
(2.11)
ci αin−1
(2.12)
i=1
and Mn−1,k > 1;
8) τ ∗ (t) ≤ α∗ t,
lim inf t
t→+∞
Z+∞
s−1
m
X
k
X
pi (s)τin−1 (s) ds ≥
i=1
t
i=1
and Mn−1,k > 1.
Then the equation (1.1) has Property A, where the constant Mn−1,k is defined by
(1.2n−1,k ).
Theorem 2.7. Let τ∗ (t) ≥ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , k) and one of
the following eight conditions be fulfilled:
1) τ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−2
t
m
X
pi (s) ds ≥
i=1
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.5) hold, M2,k > 1, Mn−1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3
m
X
pi (s)τi (s) ds ≥
i=1
t
k
X
c i αi
i=1
if n is odd;
2) τ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−2
t
m
X
pi (s) ds ≥
1
α∗
i=1
k
X
c i αi
i=1
if n is even,
(ii) (2.3), (2.6) hold, M2,k > 1, Mn−1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3
t
if n is odd;
3) τ∗ (t) ≤ α∗ t and
m
X
i=1
pi (s)τi (s) ds ≥
k
X
i=1
c i αi
130
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3 τ∗ (s)
m
X
pi (s) ds ≥ α∗
i=1
t
k
X
c i αi
i=1
if n is even,
(ii) (2.3), (2.7) hold, M2,k > 1, Mn−1,k > 1 and
Z+∞
m
X
sn−4 τ∗ (s)
lim inf t
t→+∞
pi (s)τi (s) ds ≥
i=1
t
k
X
c i αi
i=1
if n is odd;
4) τ∗ (t) ≤ α∗ t and
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3 τ∗ (s)
m
X
pi (s) ds ≥
i=1
t
k
X
c i αi
i=1
if n is even,
(ii) (2.3), (2.8) hold, M2,k > 1, Mn−1,k > 1 and
Z+∞
sn−4 τ∗ (s)
lim inf t
t→+∞
m
X
pi (s)τi (s) ds ≥
i=1
t
k
X
ci α2i
i=1
if n is odd;
5) τ ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
f racsn−2 τ∗ (s)
m
X
pi (s)τi (s) ds ≥
i=1
t
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.9) hold, M2,k > 1, Mn−1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3
τ ∗ (s)
t
m
X
pi (s)τi2 (s) ds ≥
i=1
k
X
c i αi
i=1
if n is odd;
6) τ ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
t
if n is even,
sn−2
τ ∗ (s)
m
X
i=1
pi (s)τi (s) ds ≥
1
α∗
k
X
i=1
c i αi
131
(ii) (2.3), (2.10) hold, M2,k > 1, Mn−1,k > 1 and
Z+∞
lim inf t
t→+∞
sn−3
τ ∗ (s)
t
m
X
pi (s)τi2 (s) ds ≥
1
α∗
i=1
k
X
ci α2i
i=1
if n is odd;
7) τ ∗ (t) ≤ α∗ t and
(i) M1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3
m
X
pi (s)τi (s) ds ≥ α∗
i=1
t
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.11) hold, M2,k > 1, Mn−1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3
m
X
k
X
pi (s)τi2 (s) ds ≥
t
c i αi
i=1
i=1
if n is odd;
8) τ ∗ (t) ≤ α∗ t and
(i) Mn−1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−3
m
X
pi (s)τi (s) ds ≥
i=1
t
k
X
c i αi
i=1
if n is even,
(ii) (2.3), (2.12) hold, M2,k > 1, Mn−1,k > 1 and
lim inf t
t→+∞
Z+∞
sn−4
m
X
pi (s)τi2 (s) ds ≥
i=1
t
k
X
ci α2i
i=1
if n is odd, where M1,k , M2,k and Mn−1,k are defined by (1.21,k ), (1.22,k ) and (1.2n−1,k ).
Then the equation (1.1) has Property A, where the constants Ml,k are defined by
(1.2lk ).
Theorem 2.8. Let τ ∗ (t) ≤ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , k) and one of
the following eight conditions be fulfilled:
1) τ∗ (t) ≤ α∗ t,
1
lim inf
t→+∞ t
Zt X
m
s
pi (s)τ∗ (s)τin−2 (s) ds ≥ α∗
i=1
0
k
X
ci αin−2
(2.13)
i=1
and Mn−1,k > 1;
2) τ∗ (t) ≤ α∗ t,
1
lim inf
t→+∞ t
Zt
0
sτ∗ (s)
m
X
i=1
pi (s)τin−2 (s) ds ≥
k
X
i=1
ci αin−1
(2.14)
132
and Mn−1,k > 1;
3) τ∗ (t) ≥ α∗ t,
lim inf
t→+∞
Zt
1
t
s2
m
X
pi (s)τin−2 (s) ds ≥
i=1
0
k
X
ci αin−2
(2.15)
i=1
and Mn−1,k > 1;
4) τ∗ (t) ≥ α∗ t,
1
lim inf
t→+∞ t
Zt
s2
m
X
pi (s)τin−2 (s) ds ≥
1
α∗
i=1
0
k
X
ci αin−1
(2.16)
i=1
and Mn−1,k > 1;
5) τ ∗ (t) ≤ α∗ t,
1
lim inf
t→+∞ t
Zt X
m
s
pi (s)τin−1 (s) ds ≥ α∗
i=1
0
k
X
ci αin−2
(2.17)
c i αi
(2.18)
ci αin−2
(2.19)
i=1
and Mn−1,k > 1;
6) τ ∗ (t) ≤ α∗ t,
1
lim inf
t→+∞ t
Zt X
m
s
pi (s)τin−1 (s) ds ≥
i=1
0
k
X
i=1
and Mn−1,k > 1;
7) τ ∗ (t) ≥ α∗ t,
1
lim inf
t→+∞ t
Zt
s2
0
m
X
pi (s)
i=1
τin−1
τ ∗ (s)
ds ≥
k
X
i=1
and Mn−1,k > 1;
8) τ ∗ (t) ≤ α∗ t,
lim inf
t→+∞
1
t
Zt
0
s2
m
X
i=1
pi (s)
τin−1 (s)
τ ∗ (s)
ds ≥
k
X
ci αin−1
(2.20)
i=1
and Mn−1,k > 1.
Then the equation (1.1) has Property A, where the constant Mn−1,k is defined by
(1.2n−1,k ).
Theorem 2.9. Let τ∗ (t) ≥ t for t ∈ R+ , αi ; ci ∈ (0, +∞) (i = 1, . . . , k) and one of
the following eight conditions be fulfilled:
1) τ∗ (t) ≤ α∗ t and
133
(i) M1,k > 1 and
Zt
1
lim inf
t→+∞ t
sn−1 τ∗ (s)
m
X
pi (s) ds ≥ α∗
i=1
0
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.13) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn−2 τ∗ (s)
m
X
pi (s)τi (s) ds ≥ α∗
i=1
0
k
X
c i αi
i=1
if n is odd;
2) τ∗ (t) ≤ α∗ t and
(i) M1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn−1
m
X
pi (s)τi (s) ds ≥
i=1
0
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.14) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn−2
m
X
k
X
pi (s)τ∗ (s)τi (s) ds ≥
i=1
0
ci α2i
i=1
if n is odd;
3) τ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn
m
X
pi (s) ds ≥
i=1
0
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.15) hold, M2,k > 1, Mn−1,k > 1 and
lim inf
t→+∞
1
t
Zt
sn−1
m
X
pi (s)τi (s) ds ≥
i=1
0
k
X
c i αi
i=1
if n is odd;
4) τ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
1
lim inf
t→+∞ t
Zt
0
if n is even,
sn
m
X
i=1
pi (s) ds ≥
1
α∗
k
X
i=1
c i αi
134
(ii) (2.3), (2.16) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn−1
m
X
pi (s)τi (s) ds ≥
k
X
1
α∗
i=1
0
ci α2i
i=1
if n is odd;
5) τ ∗ (t) ≤ α∗ t and
(i) M1,k > 1 and
Zt
1
lim inf
t→+∞ t
m
X
sn−1
pi (s)τi (s) ds ≥ α∗
i=1
0
k
X
ci
i=1
if n is even,
(ii) (2.3), (2.17) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn−2
m
X
pi (s)τi2 (s) ds ≥ α∗
i=1
0
k
X
c i αi
i=1
if n is odd;
6) τ ∗ (t) ≤ α∗ t and
(i) M1,k > 1 and
1
lim inf
t→+∞ t
Zt
sn−1
m
X
pi (s)τi (s) ds ≥
i=1
0
k
X
c i αi
i=1
if n is even,
(ii) (2.3), (2.18) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
m
X
sn−2
pi (s)τi2 (s) ds ≥
i=1
0
k
X
c i αi
i=1
if n is odd;
7) τ ∗ (t) ≥ α∗ t and
(i) M1,k > 1 and
1
lim inf
t→+∞ t
Zt
0
sn
m
X
k
pi (s)
i=1
X
τi (s)
ds
≥
ci
τ ∗ (s)
i=1
if n is even,
(ii) (2.3), (2.19) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
0
if n is odd;
8) τ ∗ (t) ≥ α∗ t and
sn−1
τ ∗ (s)
m
X
i=1
pi (s)τi2 (s) ds ≥
k
X
i=1
c i αi
135
(i) M1,k > 1 and
1
lim inf
t→+∞ t
Zt
0
sn
m
X
k
pi (s)
i=1
X
τi (s)
ds
≥
c i αi
τ ∗ (s)
i=1
if n is even,
(ii) (2.3), (2.20) hold, M2,k > 1, Mn−1,k > 1 and
1
lim inf
t→+∞ t
Zt
0
sn−1
m
X
i=1
pi (s)
τi2 (s)
τ ∗ (s)
ds ≥
k
X
ci α2i
i=1
if n is odd, where M1,k , M2,k and Mn−1,k are defined by (1.21,k ), (1.22,k ) and (1.2n−1,k ).
Then the equation (1.1) has Property A, where the constant Mlk is defined by (1.2lk ).
Acknowledgement
This work was supported by a Research Grant of the Greek Ministry of Development
in the framework of Bilateral S&T Cooperation between the Hellenic Republic and the
Republic of Georgia
References
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Author’s address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, Aleksidze St., Tbilisi 380093
Georgia