Oscillation criteria for higher order nonlinear functional differential equations with advanced argument
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irk-123456789-1770402021-02-11T01:28:02Z Oscillation criteria for higher order nonlinear functional differential equations with advanced argument Koplatadze, R. 2013 Article Oscillation criteria for higher order nonlinear functional differential equations with advanced argument / R. Koplatadze // Нелінійні коливання. — 2013. — Т. 16, № 1. — С. 44-64. — Бібліогр.: 8 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177040 517.9 en Нелінійні коливання Інститут математики НАН України |
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Article |
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Koplatadze, R. |
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Koplatadze, R. Oscillation criteria for higher order nonlinear functional differential equations with advanced argument Нелінійні коливання |
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Koplatadze, R. |
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Koplatadze, R. |
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Oscillation criteria for higher order nonlinear functional differential equations with advanced argument |
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Oscillation criteria for higher order nonlinear functional differential equations with advanced argument |
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Oscillation criteria for higher order nonlinear functional differential equations with advanced argument |
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Oscillation criteria for higher order nonlinear functional differential equations with advanced argument |
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Oscillation criteria for higher order nonlinear functional differential equations with advanced argument |
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oscillation criteria for higher order nonlinear functional differential equations with advanced argument |
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Інститут математики НАН України |
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2013 |
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Oscillation criteria for higher order nonlinear functional differential equations with advanced argument / R. Koplatadze // Нелінійні коливання. — 2013. — Т. 16, № 1. — С. 44-64. — Бібліогр.: 8 назв. — англ. |
| series |
Нелінійні коливання |
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AT koplatadzer oscillationcriteriaforhigherordernonlinearfunctionaldifferentialequationswithadvancedargument |
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2025-07-15T15:00:03Z |
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2025-07-15T15:00:03Z |
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1837725494396583936 |
| fulltext |
UDC 517.9
OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR
FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ADVANCED
ARGUMENT*
КРИТЕРIЇ ОСЦИЛЯЦIЇ ДЛЯ НЕЛIНIЙНИХ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ВИЩОГО ПОРЯДКУ З ВИПЕРЕДЖЕНИМ АРГУМЕНТОМ
R. Koplatadze
Tbilisi State Univ.
University Str., 2, Tbilisi, 0186, Georgia
e-mail: r_koplatadze@yahoo.com
In the paper the differential equation
u(n)(t) + p(t) |u(σ(t))|µ(t)signu(σ(t)) = 0
is considered, where p ∈ Lloc(R+;R+), µ ∈ C(R+; (0,+∞)), σ ∈ C(R+;R+) and σ(t) ≥ t for
t ∈ R+. We say that the equation is almost linear if the condition limt→+∞ µ(t) = 1 is fulfilled, while
if lim supt→+∞ µ(t) 6= 1 or lim inft→+∞ µ(t) 6= 1, then the equation is an essentially nonlinear differential
equation. In case of almost linear differential equations oscillatory properties have been extensively studi-
ed. In this paper new sufficient (necessary and sufficient) conditions are established for a general class of
essentially nonlinear functional differential equations to have Property A.
Розглядається диференцiальне рiвняння
u(n)(t) + p(t)|u(σ(t))|µ(t)signu(σ(t)) = 0,
де p ∈ Lloc (R+;R+), µ ∈ C(R+; (0,+∞)), σ ∈ C(R+;R+) та σ(t) ≥ t для t ∈ R+. Будемо
казати, що рiвняння є майже лiнiйним, якщо виконується умова limt→+∞ µ(t) = 1, i суттєво
нелiнiйним, якщо lim supt→+∞ µ(t) 6= 1 або lim inft→+∞ µ(t) 6= 1. У випадку майже лiнiйного ди-
ференцiального рiвняння коливнi властивостi було широко вивчено. У цiй роботi встановлено
новi достатнi (необхiднi та достатнi) умови для того, щоб загальний клас суттєво нелiнiйних
функцiонально-диференцiальних рiвнянь задовольнив властивостi A.
1. Introduction. This work deals with investigation of oscillatory properties of solutions of a
functional differential equation of the form
u(n)(t) + p(t)|u(σ(t))|µ(t)signu(σ(t)) = 0, (1.1)
where
p ∈ Lloc(R+;R), µ ∈ C(R+; (0,+∞)),
(1.2)
σ ∈ C(R+;R+), and σ(t) ≥ t for t ∈ R+.
∗ Was supported by Sh. Rustaveli National Science Foundation (Georgia). Grant No. GNSF/ST09-81-3-101.
c© R. Koplatadze, 2013
44 ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 45
It will always be assumed that the condition
p(t) ≥ 0 for t ∈ R+ (1.3)
is fulfilled.
Let t0 ∈ R+. A function u : [t0,+∞) → R is said to be a proper solution of (1.1) if
it is locally absolutely continuous along with its derivatives up to the order n − 1 inclusive,
sup{|u(s)| : s ≥ t} > 0 for t ≥ t0 and it satisfies (1.1) almost everywhere on [t0,+∞).
A proper solution u : [t0,+∞) → R of the equation (1.1) is said to be oscillatory if it has a
sequence of zeros tending to +∞. Otherwise the solution u is said to be nonoscillatory.
Definition 1.1. We say that the equation (1.1) has Property A if any of its proper solutions is
oscillatory when n is even, and is either oscillatory or satisfies
|u(i)(t)| ↓ 0, as t ↑ +∞, i = 0, . . . , n− 1, (1.4)
when n is odd.
Definition 1.2. We say that the equation (1.1) is almost linear if the condition
lim
t→+∞
µ(t) = 1
holds, while if
lim sup
t→+∞
µ(t) 6= 1 or lim inf
t→+∞
µ(t) 6= 1,
then we say that the equation is an essentially nonlinear differential equation.
Oscillatory properties of almost linear equations are studied well enough in [1 – 5]. In the
present paper essentially nonlinear differential equations of the type (1.1) are considered with
one of the following conditions being satisfied:
µ(t) ≤ λ (λ ∈ (0, 1)) for t ∈ R+, (1.5)
or
µ(t) ≥ λ (λ ∈ (0, 1)) for t ∈ R+. (1.6)
In the present paper, under conditions (1.5) and (1.6), sufficient (necessary and sufficient)
conditions are established for the equation (1.1) to have Property A. Of the obtained results,
some are specific to generalized equations and do not have analogous for the classical (Emden –
Fowler) equations. Analogous results for Emden – Fowler equations are given in the paper [6].
2. Some auxiliary lemmas. In the sequel, C̃loc ([t0,+∞)) will denote the set of all functions
u : [t0,+∞) → R absolutely continuous on any finite subinterval of [t0,+∞) along with their
derivatives of order up to and including n− 1.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
46 R. KOPLATADZE
Lemma 2.1 [7]. Let u ∈ C̃n−1loc ([t0,+∞)), u(t) > 0, u(n)(t) ≤ 0 for t ≥ t0, and n(n)(t) 6≡ 0
in any neighborhood of +∞. Then there exist t1 ≥ t0 and ` ∈ {0, . . . , n − 1} such that l + n is
odd and
u(i)(t) > 0 for t ≥ t1, i = 0, . . . , `− 1,
(2.1`)
(−1)i+`u(i)(t) > 0 for t ≥ t1, i = `, . . . , n− 1.
Remark 2.1. If n is odd and ` = 0, then in (2.10) it is meant that only the second inequalities
are fulfilled.
Lemma 2.2 [7]. Let u ∈ C̃n−1loc ([t0,+∞)) and (2.1`) be fulfilled for some ` ∈ {0, . . . , n − 1}
with l + n odd. Then
+∞∫
t0
tn−`−1|u(n)(t)| dt < +∞. (2.2)
If, moreover,
+∞∫
t0
tn−`|u(n)(t)| dt = +∞, (2.3)
then there exists t∗ > t0 such that
u(i)(t)
t`−i
↓, u(i)(t)
t`−i−1
↑, i = 0, . . . , `− 1, (2.4i)
u(t) ≥ t`−1
`!
u(`−1)(t) for t ≥ t∗, (2.5)
and
u(`−1)(t) ≥ t
(n− `)!
+∞∫
t
sn−`−1|u(n)(s)| ds+ 1
(n− `)!
t∫
t∗
sn−`|u(n)(s)| ds for t ≥ t∗. (2.6)
3. Necessary conditions for the existence of solutions of type (2.1`). The results of this
section play an important role in establishing sufficient conditions for the equation (1.1) to
have Property A.
Definition 3.1. Let t0 ∈ R+. By U`,t0 we denote the set of all proper solutions of the equation
(1.1) satisfying the condition (2.1`).
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 47
Theorem 3.1. Let the conditions (1.2), (1.3) and (1.5) be fulfilled, ` ∈ {1, . . . , n−1} with `+n
odd and let
+∞∫
0
tn−`(σ(t))(`−1)µ(t)p(t) dt = +∞, (3.1`)
+∞∫
0
tn−`−1(σ(t))`µ(t)p(t) dt = +∞. (3.2`)
If, moreover, U`,t0 6= ∅ for some t0 ∈ R+, then for any k ∈ N and δ ∈ [0, λ], and σ∗ ∈
∈ C([t0,+∞)) such that
t ≤ σ∗(t) ≤ σ(t) for t ≥ t0 (3.3)
we have
+∞∫
0
tn−`−1+λ−δ(σ∗(t))
µ(t)−λ(σ(t))(`−1)µ(t)(ρ`,k(σ∗(t)))
δdt < +∞, (3.4)
where
ρ`,1(t) =
1− λ
`!(n− 1)!
t∫
0
+∞∫
s
ξn−`−1+µ(ξ)−λ(σ(ξ))(`−1)µ(ξ)p(ξ)dξ ds
1
1−λ
, (3.5`)
ρ`,i(t) =
1
`!(n− `)!
t∫
0
+∞∫
s
ξn−`−1(σ(ξ))(`−1)µ(ξ) (ρ`,i−1(σ(ξ)))
µ(ξ) p(ξ) dξ ds, i= 2, . . . , k. (3.6`)
Proof. Let t0 ∈ R+ and U`,t0 6= ∅. By definition of the set U`,t0 (see Definition 3.1), the
equation (1.1) has a proper solution u ∈ U`,t0 satisfying the condition (2.1`).By (1.1), (2.1`) and
(3.1`) it is clear that the condition (2.3) holds. Thus by Lemma 2.2 the conditions (2.4i) – (2.6)
are fulfilled and
u(`−1)(t) ≥ t
(n− `)!
+∞∫
t
sn−`−1(u(σ(s)))µ(s)p(s) ds+
+
1
(n− `)!
t∫
t∗
sn−`(u(σ(s)))µ(s)p(s) ds for t ≥ t∗. (3.7)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
48 R. KOPLATADZE
According to (2.5) from (3.7) we get
u(`−1)(t) ≥ t
(n− `)!
+∞∫
t
sn−`−1(u(σ(s)))µ(s)p(s) ds−
− 1
(n− `)!
t∫
t∗
sd
+∞∫
s
ξn−`−1(u(σ(ξ)))µ(ξ)p(ξ) dξ ≥
≥ 1
(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1p(ξ)(u(σ(ξ)))µ(ξ) dξ ds ≥
≥ 1
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1(σ(ξ))(`−1)µ(ξ)(u(`−1)(σ(ξ)))µ(ξ)p(ξ) dξ ds. (3.8)
Therefore, by (1.2) and (2.4`−1), from (3.8) we have
u(`−1)(t) ≥ 1
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1+µ(ξ)(σ(ξ))(`−1)µ(ξ)
(
u(`−1)(ξ)
ξ
)µ(ξ)
p (ξ) dξ ds for t ≥ t∗.
(3.9)
On the other hand, by (2.4`−1) and (3.2`) it is obvious that
u`−1(t)
t
↓ 0 as t ↑ +∞. (3.10)
By (3.10), without loss of generality we can assume that u(`−1)(t)/t ≤ 1 for t ≥ t∗. Since
0 < µ(t) ≤ λ < 1, from (3.9) we have
u(`−1)(t) ≥ 1
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1−λ+µ(ξ)(σ(ξ))(`−1)µ(ξ)p(ξ)(u(`−1)(ξ))λ dξ ds. (3.11)
By (2.4`−1), it is obvious that
x′(t) ≥ (u(`−1)(t))λ
`!(n− `)!
+∞∫
t
ξn−`−1−λ+µ(ξ)(σ(ξ))(`−1)µ(ξ)p(ξ) dξ, (3.12)
where
x(t) =
1
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1−λ+µ(ξ)(σ(ξ))(`−1)µ(ξ)p(ξ)(u(`−1)(ξ))λ dξ ds. (3.13)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 49
Thus, according to (3.11) and (3.13), from (3.12) we get
x′(t) ≥ xλ(t)
`!(n− `)!
+∞∫
t
ξn−`−1−λ+µ(ξ)(σ(ξ))(`−1)µ(ξ)p(ξ) dξ for t ≥ t∗.
Therefore,
x(t) ≥
1− λ
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1−λ+µ(ξ)(σ(ξ))(`−1)µ(ξ)p(ξ) dξ ds
1
1−λ
for t ≥ t∗.
Hence, according to (3.11) and (3.13), we have
u(`−1)(t) ≥ ρt∗,`,1(t) for t ≥ t∗, (3.14)
where
ρt∗,`,1(t) =
1− λ
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1−λ+µ(ξ)(σ(ξ))(`−1)µ(ξ)p(ξ) dξ ds
1
1−λ
. (3.15)
Thus by (3.8), (3.13) and (3.14) we get
u(`−1)(t) ≥ ρt∗,`,k(t) for t ≥ t∗, (3.16)
where
ρt∗,`,k(t) =
1
`!(n− `)!
t∫
t∗
+∞∫
s
ξn−`−1(σ(ξ))(`−1)µ(ξ)×
× (ρt∗,`,k−1(σ(ξ)))
µ(ξ) p(ξ) dξ ds, k = 2, 3, . . . . (3.17)
On the other hand, by (1.2), (2.1`), (2.5) and (3.3) from (3.7) we have
u(`−1)(t) ≥ t
`!(n− `)!
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)(u(`−1)(σ(s)))µ(s)p(s) ds ≥
≥ t
`!(n− `)!
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)(u(`−1)(σ∗(s)))
µ(s)p(s) ds =
=
t
`!(n− `)!
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)p(s)
(
u(`−1)(σ∗(s))
σ∗(s)
)µ(s)
ds.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
50 R. KOPLATADZE
Consequently, by (2.4`−1), (1.5), (3.10) and (3.3), for any δ ∈ [0, λ]
u(`−1)(t) ≥ t
`!(n− `)!
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λ×
× p(s)(u(`−1)(σ∗(s)))δ(u`−1)(s))λ−δ ds.
Therefore, according to (3.16), we have
u(`−1)(t) ≥ t
`!(n− `)!
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λp(s)×
× (ρt∗,`,k(σ∗(s))
δ(u(`−1)(s))λ−δ ds for t ≥ t∗, k = 1, 2, . . . . (3.18)
If δ = λ, then from (3.18)
+∞∫
t∗
sn−`−1(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λp(s)(ρt∗,`,k(σ∗(s)))
λ ds ≤
≤ `!(n− `)! u
(`−1)(t∗)
t∗
≤ `!(n− `)!. (3.19)
Let δ ∈ [0, λ). Then from (3.18)
(u(`−1)(t))λ−δ ≥ tλ−δ
(`!(n− `)!)λ−δ
( +∞∫
t
sn−`−1(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λp(s)×
× (ρt∗,`,k(σ∗(s))
δ(u`−1)(s))λ−δ ds
)λ−δ
for t ≥ t∗, k = 1, 2, . . . .
Thus we have
ϕ(t)(∫ +∞
t ϕ(s) ds
)λ−δ ≥ 1
(`!(n− `)!)λ−δ
tn−`−1+λ−δ(σ(t))(`−1)µ(t)(σ∗(t))
µ(t)−λ×
× (ρt∗,`,k(σ∗(t))
δp(t) for t ≥ t∗, k = 1, 2, . . . ,
where
ϕ(t) = tn−`−1(σ(t))(`−1)µ(t)(σ∗(t))
µ(t)−λ(ρt∗,`,k(σ∗(t))
δ(u(`−1)(t))λ−δp(t).
From the last inequality we get
−
y(t)∫
y(t∗)
ds
sλ−δ
≥ 1
(`!(n− `)!)λ−δ
t∫
t∗
sn−`−1+λ−δ(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λ(ρt∗,`,k(σ∗(s)))
δp(s) ds,
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 51
where
y(t) =
+∞∫
t
ϕ(s) ds. (3.20)
Therefore
t∫
t∗
sn−`−1+λ−δ(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λ(ρt∗,`,k(σ∗(s)))
δp(s) ds ≤
≤ (n− `)!)λ−δ
y(t∗)∫
0
ds
sλ−δ
. (3.21)
By (3.20), without loss of generality we can assume that y(t∗) ≤ 1. Thus from (3.21) we have
t∫
t∗
sn−`−1+λ−δ(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λ(ρt∗,`,k(σ∗(s)))
δp(s) ds ≤
≤ (`!(n− `)!)λ−δ
1∫
0
ds
sλ−δ
=
(`!(n− `)!)λ−δ
1− λ+ δ
for t ≥ t∗.
Passing to limit in the latter inequality, we obtain
+∞∫
t∗
sn−`−1+λ−δ(σ(s))(`−1)µ(s)(σ∗(s))
µ(s)−λ(ρt∗,`,k(σ∗(s))
δp(s) ds < +∞. (3.22)
Therefore, since
lim
t→+∞
ρ`,k(t)
ρt∗,`,k(t)
= 1, k = 1, 2, . . . ,
by (3.22) and (3.19) it is obvious that for any δ ∈ [0, λ] and k ∈ N (3.4) holds, which proves the
validity of the theorem.
Analogously we can prove the following theorem.
Theorem 3.2. Let the conditions (1.2), (1.3), (3.1`), (3.2`) and (1.6) be fulfilled, ` ∈ {1, . . . , n−
−1} with `+ n odd and U`,t0 6= ∅ for some t0 ∈ R+. Then for any k ∈ N and δ ∈ [0, λ]
+∞∫
0
tn−`−1+δ(σ(t))(`−1)µ(t)(ρ̃`,k(σ(t)))
µ(t)−δp(t) dt < +∞, (3.23)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 1
52 R. KOPLATADZE
where
ρ̃`,1(t) =
1− λ
`!(n− `)!
t∫
0
+∞∫
s
ξn−`−1(σ(ξ))(`−1)µ(ξ)p(ξ) dξ ds
1
1−λ
, (3.24`)
ρ̃`,i(t) =
1
`!(n− `)!
t∫
0
+∞∫
s
ξn−`−1(σ(ξ))(`−1)µ(ξ)(ρ̃`,i−1(σ(ξ)))
µ(ξ)p(ξ) dξ ds, i = 2, . . . , k.
(3.25`)
4. Sufficient conditions for nonexistence of solutions of the type (2.1`).
Theorem 4.1. Let the conditions (1.2), (1.5), (3.1`) and (3.2`) be fulfilled, where ` ∈ {1, . . . , n−
−1} with `+n odd, and let there exist δ ∈ [0, λ], k ∈ N and σ∗ ∈ C(R+) satisfying the condition
(3.3) such that
+∞∫
0
tn−`−1+λ−δ(σ∗(t))
µ(t)−λ(σ(t))(`−1)µ(t)(ρ`,k(σ∗(t)))
δp(t) dt = +∞ (4.1`)
holds. Then for any t0 ∈ R+ we have U`,t0 = ∅, where ρ`,k is defined by (3.5`) and (3.6`).
Proof. Assume the contrary. Let there exist t0 ∈ R+ such that U`,t0 6= ∅ (see Definiti-
on 3.1). Then the equation (1.1) has a proper solution u : [t0,+∞) → R satisfying the condi-
tion (2.1`). Since the conditions of Theorem 3.1 are fulfilled, for any δ ∈ [0, λ], k ∈ N and
σ∗ ∈ C(R+) satisfying the condition (3.3) the condition (3.4) holds, which contradicts (4.1`).
The obtained contradiction proves the validity of the theorem.
Using Theorem 3.2, analogously we can prove the following theorem.
Theorem 4.2. Let the conditions (1.2), (1.3), (1.6), (3.1`) and (3.2`) be fulfilled, where ` ∈
∈ {1, . . . , n− 1} with `+ n odd, and let there exist δ ∈ [0, λ] and k ∈ N such that
+∞∫
0
tn−`−1+δ(σ(t))(`−1)µ(t)(ρ̃`,k(σ(t)))
µ(t)−δp(t) dt = +∞. (4.2`)
Then for any t0 ∈ R+ we have U`,t0 = ∅, where ρ̃`,k is defined by (3.24`) and (3.25`).
Corollary 4.1. Let the conditions (1.2), (1.3), (1.5) and (3.2`) be fulfilled, where ` ∈ {1, . . . , n−
−1} with `+ n odd, and let for some γ ∈ (0, 1)
lim inf
t→+∞
tγ
+∞∫
t
sn−`−1+µ(s)−λ(σ(s))(`−1)µ(s)p(s) ds > 0. (4.3`)
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OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 53
If, moreover, there exist δ ∈ [0, λ] and σ∗ ∈ C(R+) satisfying the condition (3.3) such that
+∞∫
0
tn−`−1+λ−δ(σ∗(t))
µ(t)−λ+ δ(1−γ)
1−λ (σ(t))(`−1)µ(t)p(t) dt = +∞ (4.4`)
holds, then for any t0 ∈ R+ we have U`,t0 = ∅.
Proof. Clearly the condition (3.1`) is fulfilled by virtue of (4.3`). On the other hand, accor-
ding to (3.5`) and (4.3`), there exist c > 0 and t1 ∈ [t0,+∞) such that
ρ`,1(t) ≥ c t
1−γ
1−λ for t ≥ t1.
Therefore from (4.4`) it follows (4.1`) with k = 1. Thus all conditions of Theorems 4.1 are
fulfilled, which proves the corollary.
Analogously we can prove the following corollary.
Corollary 4.2. Let the conditions (1.2), (1.3), (1.5) and (3.2`) be fulfilled, where ` ∈ {1, . . . , n−
−1} with `+ n odd and
lim inf
t→+∞
t
+∞∫
t
sn−`−1+µ(s)−λ(σ(s))(`−1)µ(s)p(s) ds > 0. (4.5`)
If, moreover, there exist δ ∈ [0, λ] and σ∗ ∈ C(R+) satisfying the condition (3.3) such that
+∞∫
0
tn−`−1+λ−δ(σ∗(t))
µ(t)−λ(σ(t))(`−1)µ(t)(ln(1 + σ∗(t)))
δ
1−λ p(t) dt = +∞ (4.6`)
holds, then for any t0 ∈ R+ we have U`,t0 = ∅.
Corollary 4.3. Let the conditions (1.2), (1.3), (1.6) and (3.2`) be fulfilled, where ` ∈ {1, . . . , n−
−1} with `+ n odd, and let for some γ ∈ (0, 1)
lim inf
t→+∞
tγ
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)p(s) ds > 0. (4.7`)
If, moreover, there exists δ ∈ [0, λ] such that
+∞∫
0
tn−`−1+δ(σ(t))(`−1)µ(t)+
(µ(t)−δ)(1−γ)
1−λ p(t) dt = +∞, (4.8`)
then for any t0 ∈ R+ we have U`,t0 = ∅.
Proof. According to (4.7`) and (3.24`) there exist c > 0 and t1 ∈ [t0,+∞) such that
ρ̃`,1(t) ≥ c t
1−γ
1−λ for t ≥ t1.
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54 R. KOPLATADZE
Therefore, from (4.8`) it follows (4.2`) with k = 1. Thus all the conditions of Theorems 4.2 hold,
which proves the corollary.
Corollary 4.4. Let the conditions (1.2), (1.3), (1.6) and (3.2`) be fulfilled, where ` ∈ {1, . . . , n−
−1} with `+ n odd, and let
lim inf
t→+∞
t
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)p(s) ds > 0. (4.9`)
If, moreover, for some δ ∈ [0, λ] the condition
+∞∫
0
tn−`−1+δ(σ(t))(`−1)µ(t)(ln(1 + σ(t)))
µ(t)−δ
1−λ p(t) dt = +∞ (4.10`)
holds, then for any t0 ∈ R+ we have U`,t0 = ∅.
Corollary 4.5. Let the conditions (1.2), (1.3), (1.6), (3.2`) and (4.7`) be fulfilled, where ` ∈
∈ {1, . . . , n− 1} with `+ n odd, and let there exist α ∈ (1,+∞) such that
lim inf
t→+∞
σ(t)
tα
> 0. (4.11)
If, moreover, either
αλ ≥ 1 (4.12)
or, if αλ < 1, for some ε > 0,
+∞∫
0
tn−`−1+µ(t)(
α(1−γ)
1−αλ −ε)(σ(t))(`−1)µ(t)p(t) dt = +∞, (4.13`)
then for any t0 ∈ R+ we have U`,t0 = ∅.
Proof. It suffices to show that the condition (4.2`) is satisfied for δ = 0 and for some k ∈ N .
Indeed, according to (4.7`) and (4.11), there exist α > 1, c > 0, γ ∈ (0, 1) and t1 ∈ [t0,+∞)
such that
tγ
+∞∫
t
sn−`−1(σ(s))(`−1)µ(s)p(s) ds ≥ c for t ≥ t1 (4.14)
and
σ(t) ≥ c tα for t ≥ t1. (4.15)
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OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 55
Choose k0 ∈ N and c∗ ∈ (1,+∞) such that
(1− γ)(k0 − 1) ≥ 1
λ
when αλ ≥ 1, (4.16)
if ε > 0, then
1 + αλ+ . . .+ (αλ)k0−2 ≥ 1
1− αλ
− ε
α(1− γ)
when αλ < 1 (4.17)
and for any k ∈ {1, . . . , k0}
cλ
k
∗
(
c
2`!(n− `)!(1− γ)(1 + αλ+ . . .+ (αλ)k−2
)1+λ+...+λk−2
≥ 1. (4.18)
According to (4.14) and (3.24`) it is obvious that limt→+∞ ρ̃`,1(t) = +∞. Therefore without
loss of generality we can assume that ρ̃`,1(t) ≥ c∗ for t ≥ t1. Thus, by (4.14), from (3.25`) we
get
ρ̃`,2(t) ≥
1
`!(n− `)!
t∫
t1
+∞∫
s
ξn−`−1(σ(ξ))(`−1)µ(ξ)p(ξ) dξ ds ≥
≥ cλ∗c
`!(n− `)!
t∫
t1
s−γ ds =
cλ∗c
`!(n− `)!(1− γ)
(t1−γ − t1−γ1 ).
Choose t2 > t1 such that
ρ̃`,2(t) ≥
cλ∗c t
1−γ
2`!(n− `)!(1− γ)
for t ≥ t2.
Then by (1.6), (4.14), (4.15) and (4.18), from (3.25`) we have
ρ̃`,3(t) ≥ cλ
2
∗
(
c
2`!(n− `)!(1− γ)(1 + αλ)
)1+λ
t(1−γ)(1+αλ) for t ≥ t3,
where t3 > t2 is a sufficiently large number. Therefore, for k0 ∈ N there exists tk0 ∈ R+ such
that
ρ̃`,k0(t) ≥ cλ
k0−1
∗
(
c
2`!(n− `)!(1− γ)(1 + αλ+ . . .+ (αλ)k0−2)
)1+λ+...+λk0−2
×
× t(1−γ)(1+αλ+...+(αλ)k0−2) for t ≥ tk0 . (4.19)
Assume that (4.12) is fulfilled. Then, according to (1.6), (4.14), (4.16) and (4.19) it is obvious
that, if δ = 0 for k = k0, (4.2`) holds. In the case, where (4.12) holds, the validity of the
theorem has been already proved.
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56 R. KOPLATADZE
Assume now that αλ < 1 and for some ε > 0 (4.13`) is fulfilled. Then, by (4.17) from (4.19)
we have
(ρ̃`,k0(σ(t)))
µ(t) ≥ c1t
µ(t)(
α(1−γ)
1−αλ −ε) for t ≥ tk0 ,
where c1 > 0. Consequently, according to (4.13`), it is obvious that (4.2`) holds with if δ = 0
and k = k0.
The theorem is proved.
In a similar manner we can prove the following corollary.
Corollary 4.6. Let the conditions (1.2), (1.3), (1.6), (3.2`) and (4.9`) be fulfilled, where ` ∈
∈ {1, . . . , n− 1} with `+ n odd, and let there exist α > 0 such that
lim inf
t→+∞
t−α lnσ(t) > 0. (4.20)
Then for any t0 ∈ R+ we have U`,t0 = ∅h.
5. Differential equations with property A.
Theorem 5.1. Let the conditions (1.2), (1.3), (1.5) be fulfilled and (3.1`) and (3.2`) hold for any
` ∈ {1, . . . , n− 1} with `+ n odd. Let, moreover, there exist δ ∈ [0, λ], k ∈ N and σ∗ ∈ C(R+)
satisfying the condition (3.3) such that (4.1`) holds. If, moreover,
+∞∫
0
tn−1p(t) dt = +∞ (5.1)
when n is odd, then the equation (1.1) has Property A, where ρ`,k is defined by (3.5`) and (3.6`).
Proof. Let the equation (1.1) have a proper nonoscillatory solution u : [t0,+∞)→ (0,+∞)
(the case u(t) < 0 is similar). Then by (1.2), (1.3) and Lemma 2.1 there exists ` ∈ {0, 1, . . . , n−
1} such that ` + n is odd and the condition (2.1`) holds. Since the conditions of Theorem 4.1
are fulfilled for any ` ∈ {1, . . . , n − 1} with ` + n odd, we have ` 6∈ {1, . . . , n − 1}. Therefore,
n is odd and ` = 0. Show that the condition (1.4) holds. If that is not the case, then there exists
c ∈ (0, 1) such that u(t) ≥ c for sufficiently large t. According to (2.10) and (1.5) we have
n−1∑
i=0
(n− i− 1)! ti1|u(i)(t1)| ≥
t∫
t1
sn−1p(s)cµ(s) ds ≥ cλ
t∫
t1
sn−1p(s) ds for t ≥ t1, (5.2)
where t1 is a sufficiently large number. The inequality (5.2) contradicts the condition (5.1).
Therefore the equation (1.1) has Property A.
Theorem 5.2. Let the conditions (1.2), (1.3), (1.6) be fulfilled and for any ` ∈ {1, . . . , n − 1}
with `+ n odd (3.1`), (3.2`) and for some δ ∈ [0, λ] and k ∈ N, (4.2`) holds. If, moreover,
lim sup
t→+∞
µ(t) < +∞ (5.3)
and (5.1) holds when n is odd, then the equation (1.1) has Property A, where ρ̃`,k is defined by
(3.24`) and (3.25`).
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OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 57
Proof. The proof of the theorem is analogous to that of Theorem 5.1. We have just to use
Theorem 4.2 instead of Theorem 4.1, and change λ by µ = sup{µ(t) : t ∈ R+} in the inequali-
ty (5.2).
Theorem 5.3. Let the conditions (1.2), (1.3), (1.5), (5.1) and
lim inf
t→+∞
(σ(t))µ(t)
t
> 0 (5.4)
be fulfilled. If, moreover, there exist δ ∈ [0, λ], k ∈ N and σ∗ ∈ C(R+) satisfying the condition
(3.3) such that for even n (for odd n) (4.11) ((4.12)) holds, then the equation (1.1) has Property A,
where ρ1,k (ρ2,k) is defined by (3.51) and (3.61) ((3.52) and (3.62)).
Proof. To prove the theorem it suffices to show that the conditions of Theorem 5.1 are
fulfilled. Indeed, according to (4.11) and (5.4) ((4.12) and (5.4)) it is obvious that (4.1`) holds for
any ` ∈ {1, . . . , n− 1} with `+n odd. Thus according to (5.1) all the conditions of Theorem 5.1
are fulfilled, which proves the validity of the theorem.
Using Theorem 5.2, the next theorem can be proved similarly.
Theorem 5.4. Let the conditions (1.2), (1.3), (1.6) and (5.4) hold. If, moreover, there exist
δ ∈ [0, λ] and k ∈ N such that for even n (for odd n) (4.21) ((4.22), (5.1) and (5.3)) holds, then
the equation (1.1) has Property A, where ρ̃1,k (ρ̃2,k) is defined by (3.241) and (3.251) ((3.242) and
(3.252)).
Corollary 5.1. Let the conditions (1.2), (1.3), (1.5) and (5.4) be fulfilled. If, moreover,
+∞∫
0
tn−2+µ(t)p(t) dt = +∞ (5.5)
for even n, and
+∞∫
0
tn−3+µ(t)(σ(t))µ(t)p(t) dt = +∞ (5.6)
for odd n, then the equation (1.1) has Property A.
Proof. It suffices to note that by (1.5), (5,4), (5.5) and (5.6) all the conditions of Theorem 5.3
are fulfilled with σ∗(t) = t and δ = 0.
Corollary 5.2. Let the conditions (1.2), (1.3), (1.5), (5.1) and (5.4) be fulfilled. Let, moreover,
for some k ∈ N
+∞∫
0
tn−2(σ(t))µ(t)−λ(ρ1,k(σ(t)))
λp(t) dt = +∞ (5.7)
hold when n is even, and
+∞∫
0
tn−3(σ(t))2µ(t)−λ(ρ2,k(σ(t)))
λp(t)dt = +∞ (5.8)
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58 R. KOPLATADZE
hold when n is odd. Then the equation (1.1) has Property A, where ρ1,k (ρ2,k) is defined by (3.51)
and (3.61) ((3.52) and (3.62)).
Proof. It suffices to note that by (1.5), (5.4), (5.7) and (5.8) all the conditions of Theorem 5.3
are fulfilled with σ∗(t) = σ(t) and δ = λ.
Corollary 5.3. Let the conditions (1.2), (1.3), (1.6), (5.1) and (5.4) be fulfilled. Let, moreover,
for some k ∈ N
+∞∫
0
tn−2(ρ̃1,k(σ(t)))
µ(t)p(t) dt = +∞ (5.9)
hold when n is even, and
+∞∫
0
tn−3(σ(t))µ(t)(ρ̃2,k(σ(t)))
µ(t)p(t) dt = +∞ (5.10)
hold when n is odd, then the equation (1.1) has Property A, where ρ̃1,k (ρ̃2,k) is defined by (3.241)
and (3.251) ((3.242) and (3.252)).
Proof. It suffices to note that by (1.6), (5.4), (5.9) and (5.10) all the conditions of Theorem
5.4 are fulfilled with δ = 0.
Theorem 5.5. Let the conditions (1.2), (1.3), (3.2n−1), (1.5) and
lim sup
t→+∞
(σ(t))µ(t)
t
< +∞ (5.11)
hold. If, moreover, there exist δ ∈ [0, λ], k ∈ N and σ∗ ∈ C(R+) satisfying the condition (3.3)
such that (4.1n−1) holds, then the equation (1.1) has Property A,where ρn−1 is defined by (3.5n−1)
and (3.6n−1).
Proof. By virtue of (1.2), (1.3), (1.5), (3.2n−1), (4.1n−1) and (5.11), the conditions of Theo-
rem 5.1 are obviously satisfied. Therefore according to that theorem the equation (1.1) has
Property A.
The validity of Theorem 5.6 below is proved similarly.
Theorem 5.6. Let the conditions (1.2), (1.3), (1.6), (3.2n−1) and (5.11) be fulfilled. If, moreover,
there exist δ ∈ [0, λ] and k ∈ N such that (4.2n−1) holds, then the equation (1.1) has Property A,
where ρ̃n−1,k is defined by (3.24n−1) and (3.25n−1).
Corollary 5.4. Let the conditions (1.2), (1.3), (1.5) and (5.11) be fulfilled. If, moreover,
+∞∫
0
tλ(σ(t))(n−1)µ(t)−λp(t) dt = +∞, (5.12)
then the equation (1.1) has Property A.
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OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 59
Proof. It suffices to note that by (1.2), (1.3), (1.5), (5.11) and (5.12) all the conditions of
Theorem 5.5 are fulfilled with δ = 0 and σ∗(t) = σ(t).
Corollary 5.5. Let the conditions (1.2), (1.3), (1.5) and (5.11) be fulfilled. If, moreover,
+∞∫
0
tµ(t)(σ(t))(n−2)µ(t)p(t) dt = +∞ (5.13)
holds, then the equation (1.1) has Property A.
Proof. According to Theorem 5.5, it suffices to note that by (5.13) the condition (4.1n−1)
holds with δ = 0 and σ∗(t) ≡ t.
Corollary 5.6. Let the conditions (1.2), (1.3), (1.5), (3.2n−1) and (5.11) be fulfilled. If, moreover,
for some k ∈ N
+∞∫
0
(σ(t))(n−1)µ(t)−λ(ρn−1,k(σ(t)))
λp(t) dt = +∞ (5.14)
holds. Then the equation (1.1) has Property A, where ρn−1,k is defined by (3.5n−1) and (3.6n−1).
Proof. Since from (5.14) it follows (4.1n−1) with δ = λ and σ∗(t) ≡ σ(t), the validity of the
corollary follows from Theorem 5.5.
Analogously we can prove the following corollary.
Corollary 5.7. Let the conditions (1.2), (1.3), (1.6), (5.11) and (3.2n−1) be fulfilled and for
some k ∈ N
+∞∫
0
(σ(t))(n−2)µ(t)(ρ̃n−1,k(σ(t)))
µ(t)p(t)dt = +∞. (5.15)
Then the equation (1.1) has Property A, where ρ̃n−1,k is defined by (3.24n−1) and (3.25n−1).
Theorem 5.7. Let the conditions (1.2), (1.3), (1.5), (5.4) hold and for some γ ∈ (0, 1) (4.31)
((4.32) and (5.1)) are fulfilled when n is even (when n is odd). If, moreover, there exist δ ∈ [0, λ]
and σ∗ ∈ C(R+) satisfying (3.3) such that the condition (4.41) ((4.42)) holds, then the equation
(1.1) has Property A.
Proof. According to (5.4) and (4.31) ((4.32)) it is obvious that for any ` ∈ {1, . . . , n − 1}
with ` + n odd the condition (4.3`) holds. Assume that the equation (1.1) has a nonoscillatory
solution u : [t0,+∞) → (0,+∞) satisfying the condition (2.1`). Then, by Corollary 4.1, ` 6∈
6∈ {1, . . . , n − 1}. Therefore n is odd and ` = 0. In this case by (5.1) we can show that (1.4)
holds. Therefore, the equation (1.1) has Property A.
Corollary 5.8. Let the conditions (1.2), (1.3), (1.5), (5.1) and (5.4) hold and for some γ ∈
∈ (0, 1) (4.31) ((4.32) and (5.1)) be fulfilled when n is even (when n is odd). If, moreover,
+∞∫
0
tn−2−λ+µ(t)+
λ(1−γ)
1−λ p(t) dt = +∞ (5.16)
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60 R. KOPLATADZE
holds for even n and
+∞∫
0
tn−3+λ+µ(t)+
λ(1−γ)
1−λ (σ(t))µ(t)p(t) dt = +∞ (5.17)
hold for odd n, then the equation (1.1) has Property A.
Proof. It suffices to note that by (5.16) and (5.17) the conditions (4.41) and (4.42) are satisfied
with δ = λ and σ∗(t) ≡ t.
Analogously to Theorem 5.7 we can prove the following theorem.
Theorem 5.8. Let the conditions (1.2), (1.3), (1.5), (5.4) hold and (4.51) ((4.52) and (5.1)) be
fulfilled for even n (for odd n). If, moreover, there exist δ ∈ [0, λ] and σ∗ ∈ C(R+) satisfying
(3.3) such that the condition (4.61) ((4.62)) holds, then the equation (1.1) has Property A.
Corollary 5.9. Let the conditions (1.2), (1.3), (1.5), (5.4) hold and (4.51) ((4.32) and (5.1)) be
fulfilled for even n (for odd n). If, moreover,
+∞∫
0
tn−2(σ(t))µ(t)−λ(ln(1 + σ(t)))
λ
1−λ p(t) dt = +∞ (5.18)
for even n and
+∞∫
0
tn−3(σ(t))2µ(t)−λ(ln(1 + σ(t)))
λ
1−λ p(t) dt = +∞ (5.19)
for odd n, then the equation (1.1) has Property A.
Proof. It suffices to note that by (5.18) and (5.19) the conditions (4.61) and (4.62) hold with
δ = λ and σ∗(t) = σ(t).
Theorem 5.9. Let the conditions (1.2), (1.3), (1.5), (5.11) and (4.3n−1) be fulfilled. If, moreover,
there exist δ ∈ [0, λ] and σ∗ ∈ C(R+) satisfying (3.3) such that the condition (4.4n−1) holds, then
the equation (1.1) has Property A.
Corollary 5.10. Let the conditions (1.2), (1.3), (1.5), (5.11) hold, for some γ ∈ (0, 1), (4.3n−1)
be fulfilled and
+∞∫
0
(σ(t))(n−1)µ(t)−λ+
λ(1−γ)
1−λ p(t) dt = +∞. (5.20)
Then the equation (1.1) has Property A.
Proof. It suffices to note that by (5.20) the condition (4.4n−1) holds with δ = λ and σ∗(t) =
= σ(t).
Theorem 5.10. Let the conditions (1.2), (1.3), (1.6), (5.4) hold and for some γ ∈ (0, 1) (4.71)
((4.72) and (5.1)) be fulfilled for even n (for odd n). If, moreover, there exists δ ∈ [0, λ] such that
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OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 61
(4.81) holds when n is even and (4.82) and (4.51) hold when n is odd, then the equation (1.1) has
Property A.
The theorem can be proved similarly to Theorem 5.7.
Corollary 5.11. Let the conditions (1.2), (1.3), (1.6), (5.4) hold and (4.71) ((4.72) and (5.1))
be fulfilled for even n (for odd n). If, moreover,
+∞∫
0
tn−2(σ(t))
µ(t)(1−γ)
1−λ p(t) dt = +∞ (5.21)
when n is even and
+∞∫
0
tn−3(σ(t))
µ(t)(2−λ−γ)
1−λ p(t) dt = +∞ (5.22)
when n is odd, then the equation (1.1) has Property A.
Proof. According to Theorem 5.10 it suffices to note that by (5.21) and (5.22) the conditions
(4.81) and (4.82) hold with δ = 0.
Using Theorem 5.6, similarly to Theorem 5.7 one can prove the following theorem.
Theorem 5.11. Let the conditions (1.2), (1.3), (1.6), (5.11) and (4.7n−1) be fulfilled and for
some δ ∈ [0, λ] (4.8n−1) hold. Then the equation (1.1) has Property A.
Corollary 5.12. Let the conditions (1.2), (1.3), (1.6), (5.11) and (4.7n−1) be fulfilled and
+∞∫
0
(σ(t))µ(t)(n−2+
1−γ
1−λ)p(t) dt = +∞. (5.23)
Then the equation (1.1) has Property A.
Proof. According to Theorem 5.11 it suffices to note that by (5.23) the condition (4.8n−1)
holds with δ = 0.
Theorem 5.12. Let the conditions (1.2), (1.3), (1.6), (5.1), (5.4) be fulfilled and (4.71) ((4.72))
hold for even n (for odd n). If, moreover, there exists α ∈ (1,+∞) such that (4.11) holds, then
for the equation (1.1) to have Property A it is sufficient that at least one the conditions (4.12) or,
if αλ < 1, (4.131) ((4.132)) holds for even n (for odd n).
Proof. According to (4.71), (4.131) and (5.4) ((5.72), (5.4) and (4.132)) it is obvious that for
any ` ∈ {1, . . . , n − 1} with ` + n odd (3.2`) and (3.1`) hold. Assume that the equation (1.1)
has a nonoscillatory solution u : [t0,+∞) → (0,+∞) satisfying the condition (2.1`). Then by
Corollary 4.5, ` 6∈ {1, . . . , n−1}. Therefore n is odd and ` = 0. In this case by (5.1) it is obvious
that (1.4) holds. Therefore the equation (1.1) has Property A.
Using Corollaries 4.5 and 4.6, in a similar manner we can prove Theorems 5.13 and 5.14
below.
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62 R. KOPLATADZE
Theorem 5.13. Let the conditions (1.2), (1.3), (1.6), (5.11) and (4.7n−1) be fulfilled. If, moreover,
there exist α ∈ (1,+∞) such that (4.11) holds, then for the equation (1.1) to have Property A it
is sufficient that at least one of the conditions (4.12) or, if αλ < 1, (4.13n−1) holds.
Theorem 5.14. Let the conditions (1.2), (1.3), (1.6), (5.4) be fulfilled and (4.91) ((4.92) and
(5.1)) hold for even n (for odd n). If, moreover, there exists α > 0 such that (4.20) holds, then the
equation (1.1) has Property A.
6. Necessary and sufficient conditions.
Theorem 6.1. Let the conditions (1.2), (1.3) and (1.5) be fulfilled and
lim sup
t→+∞
σ(t)
t
< +∞. (6.1)
Then the condition
+∞∫
0
t(n−1)µ(t)p(t) dt = +∞ (6.2)
is necessary and sufficient for the equation (1.1) to have Property A.
Proof. Necessity. Assume that the equation (1.1) has Property A and
+∞∫
0
t(n−1)µ(t)p(t) dt < +∞. (6.3)
By (1.5), (6.1) and (6.3)
+∞∫
0
(σ(t))(n−1)µ(t)p(t) dt < +∞.
Therefore, following Lemma 4.1 [7], there exists c 6= 0 such that the equation (1.1) has a proper
solution u : [t0,∞) → R satisfying the condition limt→+∞ u
(n−1)(t) = c. But this contradicts
the fact that the equation (1.1) has Property A.
Sufficiency. By (6.1) and (6.2) it is obvious that the condition (5.12) holds. Therefore the
sufficiency follows from Corollary 5.4.
From Theorem 6.1, when µ(t) ≡ λ (λ ∈ (0, 1)) and σ(t) ≡ t, follows a theorem of Ličko
and Švec [8].
Corollary 6.1. Let the conditions (1.2), (1.3), (1.6) and (6.1) be fulfilled and
lim sup
t→+∞
tµ(t) < +∞.
Then the condition
+∞∫
0
p(t) dt = +∞
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OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS . . . 63
is necessary and sufficient for the equation (1.1) to have Property A.
Remark 6.1. Note that a necessary and sufficient condition of this kind, which does not
depend on the order of the equation, is given for the first time.
Theorem 6.2. Let n be odd, the conditions (1.2), (1.3) and (1.5) be fulfilled and
lim inf
t→+∞
σ(t)
t
2−µ(t)
µ(t)
> 0. (6.4)
Then the condition (5.1) is necessary and sufficient for the equation (1.1) to have Property A.
Proof. Necessity. Assume that the equation (1.1) has Property A and
+∞∫
0
tn−1p(t) dt < +∞. (6.5)
According to (6.5), by Lemma 4.1 [7] there exists c 6= 0 such that the equation (1.1) has a proper
solution u : [t0,∞) → R satisfying the condition limt→+∞ u(t) = c. But this contradicts the
fact that the equation (1.1) has Property A.
Sufficiency. According to (1.5) and (6.4) it is obvious that the condition (5.4) holds. On the
other hand, by (5.1) and (6.4) the condition (5.6) holds. Thus, since n is odd, all the conditions
of Corollary 5.1 are fulfilled, i.e., the equation (1.1) has Property A.
Corollary 6.2. Let n be odd, the conditions (1.2), (1.3), (1.5) be fulfilled and
lim
t→+∞
µ(t) = λ (λ ∈ (0, 1)), lim inf
t→+∞
tµ(t)−λ > 0, lim inf
t→+∞
σ(t)
t
2−λ
λ
> 0. (6.6)
Then the condition (5.1) is necessary and sufficient for the equation (1.1) to have Property A.
Remark 6.2. The condition (6.6) defines a set of the functions σ for which the condition (5.1)
is necessary and sufficient. It turns out that the number
2− λ
λ
is optimal. Indeed, let ε > 0,
λ ∈ (1/(1 + ε), 1) and γ ∈ (1, 2). Consider the differential equation (1.1) with
p(t) = −γ(γ − 1) . . . (γ − n+ 1)t−n+γ(1−µ(t)(
2−λ
λ
−ε)),
σ(t) = t
2−λ
λ
−ε, t ≥ 1, lim
t→+∞
µ(t) = λ.
It is obvious that the condition (5.1) is fulfilled and
lim inf
t→+∞
σ(t)
t
2−λ
λ
= 0, and lim inf
t→+∞
σ(t)
t
2−λ
λ
−ε
> 0.
On the other hand, for odd n, u(t) = tγ is a solution of equation (1.1). Therefore, when n is
odd, the equation (1.1) does not have Property A.
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64 R. KOPLATADZE
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Received 29.12.11
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