Oscillation of certain fourth-order functional differential equations
Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the form $$\frac{d^2}{dt^2} \left(a(t) \left(\frac{d^2x(t)}{dt^2}\right)^{α} \right) + q(t)f(x[g(t)])=0, \quad α>0,$$ are established.
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| author | Agarwal, P. Grace, S. R. O’Regan, D. Агарвал, Р. П. Грасе, С. Р. О'Реган, Д. |
| author_facet | Agarwal, P. Grace, S. R. O’Regan, D. Агарвал, Р. П. Грасе, С. Р. О'Реган, Д. |
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| description | Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the form
$$\frac{d^2}{dt^2} \left(a(t) \left(\frac{d^2x(t)}{dt^2}\right)^{α} \right) + q(t)f(x[g(t)])=0, \quad α>0,$$
are established. |
| first_indexed | 2026-03-24T02:40:05Z |
| format | Article |
| fulltext |
UDC 517.9
R. P. Agarwal (Florida Inst. Technol., USA),
S. R. Grace (Cairo Univ., Orman, Giza, Egypt),
D. O’Regan (Nat. Univ. Ireland, Galway, Ireland)
OSCILLATION OF CERTAIN FOURTH ORDER
FUNCTIONAL DIFFERENTIAL EQUATIONS
KOLYVANNQ DEQKYX FUNKCIONAL\NYX
DYFERENCIAL\NYX RIVNQN\ ÇETVERTOHO PORQDKU
Some new criteria for the oscillation of fourth order nonlinear functional differential equations of the form
d2
dt2
(
a(t)
(
d2x(t)
dt2
)α)
+ q(t)f
(
x[g(t)]
)
= 0, α > 0,
are established.
Vstanovleno deqki novi kryteri] kolyvannq nelinijnyx funkcional\nyx dyferencial\nyx rivnqn\
vyhlqdu
d2
dt2
(
a(t)
(
d2x(t)
dt2
)α)
+ q(t)f
(
x[g(t)]
)
= 0, α > 0.
1. Introduction. In this paper we are concerned with the oscillatory behavior of fourth
order nonlinear differential equations of the type
d2
dt2
(
a(t)
(
d2x(t)
dt2
)α)
+ q(t)f(x[g(t)]) = 0, (1.1)
where
(i) a(t), q(t) ∈ C([t0,∞), R
+ = (0,∞)),
(ii) g(t) ∈ C([t0,∞), R = (−∞,∞)) and limt→∞ g(t) = ∞,
(iii) f ∈ C(R,R) and xf(x) > 0 for x �= 0, and
(iv) α is the ratio of two positive odd integers.
In what follows we shall assume that
∞∫
a−1/α(s)ds = ∞. (1.2)
By a solution of equation (1.1), we mean a function x ∈ C2([tx,∞),R) such that
a(t)
(
x′′(t)
)α ∈ C2
(
[tx,∞),R
)
and satisfies the equation at every point t ≥ tx ≥ t0 ≥ 0.
Here, we are concerned with proper solutions of equation (1.1), that is, those solutions
x(t) which satisfy sup
{∣∣x(t)
∣∣ : t ≥ T
}
> 0 for every T ≥ tx. Such a solution is said to
be oscillatory if it has an infinite sequence of zeros clustering at infinity and nonoscillatory
if it has at most a finite number of zeros in its interval of existence.
We introduce the notation Li, i = 0, 1, 2, 3, 4, for the lower order derivatives associ-
ated with the operator L4x(t) =
d2
dt2
(
a(t)
(
d2x(t)
dt2
)α)
:
L0x(t) = x(t), L1x(t) =
d
dt
L0x(t), L2x(t) = a(t)
(
d
dt
L1x(t)
)α
,
L3x(t) =
d
dt
L2x(t), L4x(t) =
d
dt
L3x(t).
(1.3)
c© R. P. AGARWAL, S. R. GRACE, D. O’REGAN, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3 291
292 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
The classical Atkinson – Belohorec oscillation results [7] for the Emden – Fowler dif-
ferential equation
x′′(t) + q(t)
∣∣x(t)
∣∣γ sgnx(t) = 0, (1.4)
where 0 < γ �= 1 is a constant and q(t) ∈ C
(
[t0,∞),R+
)
has been studied and gen-
eralized in various directions in the literature. One of the remarkable extensions of the
oscillation due to Atkinson – Belohorec is for nonlinear differential equations of the type(∣∣x′(t)
∣∣α sgnx′(t)
)′
+ q(t)
∣∣x(t)
∣∣β sgnx(t) = 0, (1.5)
where α, β > 0 are constants and q(t) ∈ C
(
[t0,∞),R+
)
and was carried out by Elbert
etc. [9] and Kusano etc. [13]. For related results the reader is referred to our book [5], and
[1 – 4, 6] and the references cited therin.
Our main objective is to present a systematic study on the oscillation of equation (1.1)
and establish some new oscillation criteria. In Section 2, we shall give the proof of an im-
portant lemma which is useful throughout this paper. Also, we present oscillation results
when f satisfies the condition f1−1/α(x)f ′(x) ≥ k > 0 for x �= 0, or f(x) sgnx ≥ |x|β
for x �= 0, where β is the ratio of two positive odd integers, β > α, β = α and
β < α. Results that involve comparison with linear and half–linear differential equa-
tions are studied. Section 3 is devoted to the study of equation (1.1) when f satisfies
either
∫ ±∞
du/f1/α(u) < ∞, or
∫
±0
du/f1/α(u) < ∞. In Section 4 we give neces-
sary and sufficient conditions for the oscillation of all bounded and unbounded solutions
of equation (1.1) when f(x) sgnx ≥ |x|β for x �= 0. In Section 5 we give a comparison
result which allows us to extend our results to certain neutral differential equations and
also, when f need not be a monotonic function. The obtained results are new, and extend
and improve those known in the literature for the equation (1.5).
2. Oscillation and comparison results. Before we state our results, we shall need the
following preliminaries: If x(t) is an eventually positive solution of equation (1.1), then
L4x(t) ≤ 0 eventually, and since condition (1.2) holds, it follows that Lix(t), i = 1, 2, 3,
are eventually of constant sign. We distinguish the following two cases:
(I) Lix(t) > 0, i = 0, 1, 2, 3 and L4x(t) ≤ 0 eventually,
(II) L0x(t) > 0, L1x(t) > 0, L2x(t) < 0, L3x(t) > 0 and L4x(t) ≤ 0 eventually.
Let (I) hold. Since L3x(t) > 0 is decreasing (say) for t ≥ t0 ≥ 0, we have
L2x(t) − L2x(t0) =
t∫
t0
L3x(s)ds,
or
a(t)
(
d
dt
L1x(t)
)α
≥ (t− t0)L3x(t) for t ≥ t0,
or
x′′(t) ≥
(
t− t0
a(t)
)1/α
L
1/α
3 x(t) for t ≥ t0. (2.1)
Integrating (2.1) from t0 to t and using (I) and the decreasing property of L3x(t) on
[t0,∞), we have
x′(t) ≥
t∫
t0
(
u− t0
a(u)
)1/α
du
L
1/α
3 x(t), t ≥ t0, (2.2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 293
and
x(t) ≥
t∫
t0
(t− u)
(
u− t0
a(u)
)1/α
du
L
1/α
3 x(t), t ≥ t0. (2.3)
Let (II) hold. Then for t ≥ u ≥ t0 and the decreasing property of L3x(t) > 0, we
obtain
L2x(t) − L2x(u) =
t∫
u
L3x(τ)dτ,
or
−a(u)
(
x′′(u)
)α ≥ (t− u)L3x(t),
or
−x′′(u) ≥
(
t− u
a(u)
)1/α
L
1/α
3 x(t), t ≥ u ≥ t0. (2.4)
Integrating (2.4) from λt to t ≥ t0 for some λ, 0 < λ < 1, using (II) and the decreasing
property of L3x(t), t ≥ t0, we get
x′(λt) ≥
t∫
λt
(
t− u
a(u)
)1/α
du
L
1/α
3 x(t) (2.5)
and for t ≥ T/λ ≥ t0,
x(t) ≥ x(λt) ≥
t∫
T
(λt− u)
(
t− u
a(u)
)1/α
du
L
1/α
3 x(t). (2.6)
For t ≥ T/λ ≥ t0 and for some constant λ, 0 < λ < 1, we let
h(t, T ; a;λ) = min
λt∫
T
(
u− T
a(u)
)1/α
du,
t∫
λt
(
t− u
a(u)
)1/α
du
,
H(t, T ; a;λ) = min
t∫
T
(t− u)
(
u− T
a(u)
)1/α
du,
λt∫
T
(λt− u)
(
t− u
a(u)
)1/α
du
.
Combining the above results, we are ready to state the following interesting lemma.
Lemma 2.1. Let x(t) be a positive solution of equation (1.1) for t ≥ t0. Then for
some constant λ, 0 < λ < 1, and all large t ≥ T/λ ≥ t0,
x′(λt) ≥ h(t, T ; a;λ)L1/α
3 x(t) (2.7)
and
x(t) ≥ x(λt) ≥ H(t, T ; a;λ)L1/α
3 x(t). (2.8)
We shall also need the following lemma given in [12].
Lemma 2.2. If X and Y are nonnegative, then
Xλ + (λ− 1)Y λ − λXY λ−1 ≥ 0, λ > 1,
where equality holds if and only if X = Y.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
294 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
For t ≥ t0, we define
g∗(t, t0; a) =
t∫
t0
s∫
t0
(
u
a(u)
)1/α
duds.
We shall also assume that
f1/α−1(x)f ′(x) ≥ k > 0 for x �= 0, k is a real constant (2.9)
and there exists a function σ(t) ∈ C1
(
[t0,∞),R+
)
such that
σ(t) = inf
{
t, g(t)
}
, σ′(t) > 0 for t ≥ t0 and lim
t→∞
σ(t) = ∞. (2.10)
Our first result is embodied in the following theorem.
Theorem 2.1. Let conditions (1.2), (2.9) and (2.10) hold. If there exist a function
ρ(t) ∈ C1
(
[t0,∞),R+
)
and a constant λ, 0 < λ < 1, such that for σ(t) > T/λ, for
some T ≥ t0,
lim sup
t→∞
t∫
T
[
ρ(s)q(s) − 1
(λk)α
αα
(1 + α)1+α
(ρ′(s))α+1
[ρ(s)σ′(s)h(σ(s), t0; a;λ)]α
]
ds = ∞,
(2.11)
where h is as in Lemma 2.1, then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. From equation (1.1), we see that L4x(t) ≤ 0 for t ≥ t0 and so Lix(t),
i = 1, 2, 3, are eventually of one sign, and either (I) or (II) holds. In view of Lemma 2.1,
there exist a t1 ≥ t0 and a λ, 0 < λ < 1, such that
x′(λt) ≥ h(t, t1; a;λ)L1/α
3 x(t) for t ≥ t1/λ. (2.12)
We define
w(t) = ρ(t)
L3x(t)
f
(
x[λσ(t)]
) , t ≥ t2 ≥ t1. (2.13)
Then for t ≥ t2, we have
w′(t) = ρ(t)
(L3x(t))′
f(x[λσ(t)])
+ ρ′(t)
L3x(t)
f(x[λσ(t)])
−
−ρ(t)
L3x(t)f ′(x[λσ(t)])x′[λσ(t)]λσ′(t)
f2(x[λσ(t)])
=
= −ρ(t)q(t)
f(x[g(t)])
f(x[λσ(t)])
+
ρ′(t)
ρ(t)
w(t) −
−λρ(t)σ′(t)
f ′(x[λσ(t)])
f1−1/α(x[λσ(t)])
L3x(t)x′[λσ(t)]
f1+1/α(x[λσ(t)])
. (2.14)
There exists a t2 ≥ t1 such that σ(t) > t1/λ and
x′[λσ(t)
]
≥ h(σ(t), t1; a;λ)L1/α
3 x(t) for t ≥ t2. (2.15)
Using (2.9) and (2.15) and the fact that x(t) is increasing for t ≥ t2 in (2.14), we obtain
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t)− λkρ−1/α(t)σ′(t)h(σ(t), t1; a;λ)w1+1/α(t), t ≥ t2.
(2.16)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 295
Set
X =
(
λkρ−1/α(t)σ′(t)h(σ(t), t1; a;λ)
)α/(α+1)
w(t), λ =
α + 1
α
> 1,
and
Y =
(
α
α + 1
)α (
ρ′(t)
ρ(t)
)α [(
λkρ−1/α(t)σ′(t)h(σ(t), t1; a;λ)
)−α/(α+1)
]α
in Lemma 2.2 to conclude that for t ≥ t2,
ρ′(t)
ρ(t)
w(t) − λkρ−1/α(t)σ′(t)h(σ(t), t1; a;λ)w1+1/α(t) ≤
≤ 1
(λk)α
αα
(1 + α)1+α
(ρ′(t))α+1[
ρ(t)σ′(t)h(σ(t), t1; a;λ)
]α .
Thus, we have
w′(t) ≤ −ρ(t)q(t)+
1
(λk)α
αα
(1 + α)1+α
(ρ′(t))α+1[
ρ(t)σ′(t)h(σ(t), t1; a;λ)
]α , t ≥ t2. (2.17)
Integrating (2.17) from t2 to t, we get
0 < w(t) ≤
≤ w(t2) −
t∫
t2
[
ρ(s)q(s) − 1
(λk)α
αα
(1 + α)1+α
(ρ′(s))α+1
[ρ(s)σ′(s)h(σ(s), t1; a;λ)]α
]
ds.
Taking the lim sup of both sides of the above inequality as t → ∞ and applying condi-
tion (2.12), we obtain w(t) → −∞ as t → ∞, which is a contradiction. This completes
the proof.
Theorem 2.2. Let α ≥ 1, conditions (1.2) and (2.10) hold, and
f(x) sgnx ≥ |x|β for x �= 0, (2.18)
where β is the ratio of two positive odd integers. If there exist a function ρ(t) ∈ C1
(
[t0,∞),
R
+
)
and a constant λ, 0 < λ < 1, such that
lim sup
t→∞
t∫
T
[
ρ(s)q(s) − (ρ′(s))2
4λβσ′(s)ρ(s)h(σ(s), t0; a;λ)Hα−1(σ(s), t0; a;λ)C(s)
]
ds =
= ∞, (2.19)
where h and H are as in Lemma 2.1 and σ(t) > T/λ > t0, and
C(t) =
c1, c1 is any positive constant, when β > α,
1, when β = α,
c2g
β−α
∗ (t, t0; a), c2 is any positive constant, when β < α,
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 > 0. Proceeding as in the proof of Theorem 2.1, there exists a T > T ≥ t0 such
that for σ(T ) > T/λ and some λ ∈ (0, 1), we have
x′[λσ(t)] ≥ h(σ(t), T ; a;λ)L1/α
3 x(t), t ≥ T, (2.20)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
296 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
and
x(t) ≥ x
[
σ(t)
]
≥ x
[
λσ(t)
]
≥ H
(
σ(t), T ; a;λ
)
L
1/α
3 x(t), t ≥ T. (2.21)
Next, there exists a constant b > 0 such that
L3x(t) ≤ b for t ≥ T.
Integrating this inequality from T to t, one can easily find that there exist a constant
b1 > 0 and a T1 ≥ T such that
x
[
λσ(t)
]
≤ x(t) ≤ b1g∗(t, T ; a) for t ≥ T1. (2.22)
We define the function w(t) as in (2.13) and proceed as in the proof of Theorem 2.1 to
obtain (2.14) with f(x) replaced by xβ . Using (2.20) and (2.21) in (2.14), for t ≥ T we
get
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
−λβ
σ′(t)
ρ(t)
h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)xβ−α[λσ(t)]w2(t). (2.23)
Now, we need to consider the following three cases:
Case 1. If β > α, then there exist a constant b1 > 0 and a T2 ≥ T such that
x[λσ(t)] ≥ b1 for t ≥ T2. (2.24)
Thus, the inequality (2.23) becomes
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
−λβbβ−α
1
σ′(t)
ρ(t)
h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)w2(t), t ≥ T2. (2.25)
Case 2. If β = α, then inequality (2.23) becomes
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
−λβ
σ′(t)
ρ(t)
h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)w2(t), t ≥ T. (2.26)
Case 3. If β < α, then by (2.22) we get
xβ−α[λσ(t)] ≥ γgβ−α
∗ (t, T ; a), γ = bβ−α
1 for t ≥ T1 (2.27)
and inequality (2.23) takes the form
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
−λβγ
σ′(t)
ρ(t)
gβ−α
∗ (t, T ; a)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)w2(t), t ≥ T. (2.28)
Let T ∗ = max{T, T1, T2}, so that we can combine inequalities (2.25), (2.26) and (2.28),
to obtain for t ≥ T ∗,
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 297
−λβ
σ′(t)
ρ(t)
C(t)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)w2(t) = (2.29)
= −ρ(t)q(t) −
[√
λβ
σ′(t)
ρ(t)
C(t)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)w(t) −
− ρ′(t)
2ρ(t)
√
λβ σ′(t)
ρ(t) C(t)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)
]2
+
+
(ρ′(t))2
4λβσ′(t)ρ(t)C(t)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)
≤
≤ −
[
ρ(t)q(t) − (ρ′(t))2
4λβσ′(t)ρ(t)C(t)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)
]
. (2.30)
Integrating (2.30) from T ∗ to t, we have
0 < w(t) ≤ w(T ∗) −
−
t∫
T∗
[
ρ(s)q(s) − (ρ′(s))2
4λβσ′(s)ρ(s)C(s)h(σ(s), T ; a;λ)Hα−1(σ(s), T ; a;λ)
]
ds.
Taking the lim sup of both sides of the above inequality as t → ∞ and applying condi-
tion (2.19), we see that w(t) → −∞ as t → ∞, which is a contradiction. This completes
the proof.
Next, we have the following result for equation (1.1) when 0 < α ≤ 1.
Theorem 2.3. Let 0 < α ≤ 1, conditions (1.2), (2.10) and (2.18) hold, and assume
that there exist a function ρ(t) ∈ C1
(
[t0,∞),R+
)
and a constant λ, 0 < λ < 1, such
that for σ(t) > T/λ for some T ≥ t0,
lim sup
t→∞
t∫
T
[
ρ(s)q(s) − (ρ′(s))2Q1−1/α(s)
4λβσ′(s)h(σ(s), t0; a;λ)C̃(s)
]
ds = ∞, (2.31)
where h is as in Lemma 2.1, Q(t) =
∫ ∞
t
q(s)ds, and
C̃(t) =
c1, c1 is any positive constant, if β > α,
1, if β = α,
c2g
β/α−1
∗ (t, t0; a), c2 is any positive constant, if β < α,
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 > 0. Define the function w(t) by (2.13) with f(x) = xβ and proceed as in the
proof of Theorems 2.1 and 2.2 to obtain (2.14), (2.20) and (2.22) for t ≥ T. Using (2.20)
in (2.14) one obtains
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
−λβσ′(t)ρ−1/α(t)w2(t)w1/α−1(t)h(σ(t), T ; a;λ)xβ/α−1[λσ(t)], t ≥ T. (2.32)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
298 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
It is easy to see that
w(t) ≥ ρ(t)Q(t) for t ≥ T.
Using this inequality in (2.32), we obtain
w′(t) ≤ −ρ(t)q(t) +
ρ′(t)
ρ(t)
w(t) −
−λβ
σ′(t)
ρ(t)
Q1/α−1(t)h(σ(t), T ; a;λ)w2(t)xβ/α−1[λσ(t)], t ≥ T. (2.33)
The rest of the proof is similar to that of Theorem 2.2 and hence omitted.
Our next results involve comparison with related linear and half-linear second order
differential equations, so that known oscillation theorems from the literature can be em-
ployed directly. To obtain these comparison criteria we need the following lemmas given
in [5].
Lemma 2.3. The half-linear differential equation(
a(t)
(
x′(t)
)α)′ + q(t)xα(t) = 0, (2.34)
where a, q and α are as in equation (1.1) is nonoscillatory if and only if there exist a
number T ≥ t0 and a function v(t) ∈ C1
(
[t0,∞),R
)
which satisfies the inequality
v′(t) + αa−1/α(t)|v(t)|1+1/α + q(t) ≤ 0 on [T,∞).
Lemma 2.4. Let h(t) ∈ C
(
[T,∞),R+
)
, T ≥ t0. If there exists a function v(t) ∈
∈ C1
(
[T,∞),R
)
such that
v′(t) + h(t)v2(t) + q(t) ≤ 0 for every t ≥ T,
then the second order linear differential equation(
1
h(t)
x′(t)
)′
+ q(t)x(t) = 0
is nonoscillatory.
First, we relate the oscillation of equation (1.1) to that of half-linear equations of
type (2.34).
Theorem 2.4. Let the hypotheses of Theorem 2.1 hold with ρ(t) = 1 and condi-
tion (2.11) is replaced by: the half-linear second order equation
(A(t)(y′(t))α)′ + q(t)yα(t) = 0 (2.35)
is oscillatory, where
A(t) =
(
λk
α
σ′(t)h(σ(t), t0; a;λ)
)−α
.
Then the conclusion of Theorem 2.1 holds.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. Proceed as in the proof of Theorem 2.1 with ρ(t) = 1 to obtain (2.16) which
takes the form
w′(t) ≤ −q(t) − λkσ′(t)h(σ(t), t1; a;λ)w1+1/α(t) for t ≥ t2.
Applying Lemma 2.3 to the above inequality, we conclude that the equation (2.35) is
nonoscillatory, which is a contradiction and completes the proof.
In the following results we shall compare the oscillation of equation (1.1) with that of
linear second order ordinary differential equations.
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 299
Theorem 2.5. Let the hypotheses of Theorem 2.2 hold with ρ(t) = 1 and condi-
tion (2.19) is replaced by: the linear second order equation(
1
r(t)
z′(t)
)′
+ q(t)z(t) = 0 (2.36)
is oscillatory, where
r(t) = λβσ′(t)C(t)h(σ(t), t0; a;λ)Hα−1(σ(t), t0; a;λ).
Then the conclusion of Theorem 2.2 holds.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. Proceed as in the proof of Theorem 2.2 with ρ(t) = 1 to obtain (2.29) which
takes the form
w′(t) ≤ −q(t) − λβσ′(t)C(t)h(σ(t), T ; a;λ)Hα−1(σ(t), T ; a;λ)w2(t), t ≥ T ∗.
Applying Lemma 2.4 to the above inequality, we find that the equation (2.36) is nonoscil-
latory, a contradiction and the proof is complete.
Theorem 2.6. Let the hypotheses of Theorem 2.3 hold with ρ(t) = 1 and condi-
tion (2.31) is replaced by: the linear second order equation(
1
b(t)
x′(t)
)′
+ q(t)x(t) = 0 (2.37)
is oscillatory, where
b(t) = λβσ′(t)Q1/α−1(t)h(σ(t), t0; a;λ)C̃(t).
Then the conclusion of Theorem 2.3 holds.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 > 0. Proceed as in the proof of Theorem 2.3 with ρ(t) = 1 to obtain the inequal-
ity (2.33) which takes the form
w′(t) ≤ −q(t) − λβσ′(t)C̃(t)Q1/α−1(t)h(σ(t), T ; a;λ)w2(t), t ≥ T.
The rest of the proof is similar to that of Theorem 2.5 and hence omitted.
Next, we have the following comparison results.
Theorem 2.7. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0 and (2.18) hold.
If the first order delay equation
y′(t) + q(t)Hβ(σ(t), T ; a;λ)yβ/α[σ(t)] = 0 (2.38)
for some T ≥ t0 and λ ∈ (0, 1) is oscillatory, then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. As in the proof of Theorem 2.2 we obtain the inequality (2.22) which takes
the form
x[σ(t)] ≥ H(σ(t), T ; a;λ)L1/α
3 x[σ(t)] (2.39)
for some T ≥ t0 and a constant λ ∈ (0, 1). Now, using (2.18) and (2.39) in equation (1.1),
we find
L4x(t) + q(t)Hβ(σ(t), T ; a;λ)Lβ/α
3 x
[
σ(t)
]
≤ 0, t ≥ T.
Letting y(t) = L3x(t) in the above inequality, we get
y′(t) + q(t)Hβ
(
σ(t), T ; a;λ
)
yβ/α
[
σ(t)
]
≤ 0 for t ≥ T. (2.40)
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300 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
Integrating (2.40) from t ≥ T to u and letting u → ∞, we find
y(t) ≥
∞∫
t
q(s)Hβ(σ(s), T ; a;λ)yβ/α
[
σ(s)
]
ds, t ≥ T.
As in [16] it is easy to conclude that there exists a positive solution y(t) of the equa-
tion (2.38) with limt→∞ y(t) = 0, a contradiction to the fact that equation (2.38) is
oscillatory. This completes the proof.
The following corollary is immediate.
Corollary 2.1. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0 and (2.18) hold.
If
lim inf
t→∞
t∫
σ(t)
q(s)Hβ(σ(s), T ; a;λ)ds >
1
e
when α = β,
or
∞∫
q(s)Hβ(σ(s), T ; a;λ)ds = ∞ when α < β
for some t ≥ T and a constant λ ∈ (0, 1), then equation (1.1) is oscillatory.
Theorem 2.8. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0 and (2.18) hold.
If the second order equation
y′′(t) + Q∗(t)yβ/α[σ(t)] = 0 (2.41)
is oscillatory, where
Q∗(t) =
1
a(t)
∞∫
t
∞∫
s
q(u)duds
1/α
,
then all bounded solutions of equation (1.1) are oscillatory.
Proof. Let x(t) be a bounded nonoscillatory solution of equation (1.1), say, x(t) > 0
for t ≥ t0 ≥ 0. In this case x(t) satisfies (II).
Integrating equation (1.1) from t ≥ t0 to u, using conditions (2.10) and (2.18) and
letting u → ∞, we obtain
L3x(t) ≥ xβ
[
σ(t)
] ∞∫
t
q(s)ds.
Once again, integrating this inequality from t ≥ t0 to u and letting u → ∞, we have
−x′′(t) ≥ Q∗(t)xβ/α[σ(t)],
or
x′′(t) + Q∗(t)xβ/α[σ(t)] ≤ 0 for t ≥ t1 ≥ t0.
By applying a known comparison criterion (see [16]), one can easily see that equa-
tion (2.41) has an eventually positive bounded solution, which contradicts the hypothesis
and completes the proof.
The following corollary is now obvious.
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 301
Corollary 2.2. Let conditions (1.2), (2.12) and (2.18) hold. If
(i) σ′(t) ≥ 0 for t ≥ t0 and
∫ ∞
σ(s)Q∗(s)ds = ∞ when β > α,
(ii) σ′(t) > 0 for t ≥ t0 and there is a function ρ(t) ∈ C1
(
[t0,∞),R+
)
such that
∞∫ [
ρ(s)Q∗(s) − (ρ′(s))2
4ρ(s)
]
ds = ∞ when β = α,
(iii) σ′(t) ≥ 0 for t ≥ t0,
∞∫
(σ(s))β/αQ∗(s)ds = ∞ when β < α,
where Q∗ is as in Theorem 2.8, then all bounded solutions of equation (1.1) are oscilla-
tory.
3. Further oscillation results. In this section we shall present some sufficient condi-
tions for the oscillation of equation (1.1) when f(x) satisfies conditions of the type
±∞∫
du
f1/α(u)
< ∞, (3.1)
or ∫
±0
du
f(u1/α)
< ∞. (3.2)
Theorem 3.1. Let conditions (1.2), (2.10) and (3.1) hold. Moreover, assume that
there exists a function ρ(t) ∈ C1([t0,∞),R+) such that
ρ′(t) ≥ 0 and
(
(ρ′(t))1/α
σ′(t)h(σ(t), t0; a;λ)
)′
≤ 0 (3.3)
for all large t ≥ t0 and some constant λ ∈ (0, 1). If
∞∫
ρ(s)q(s)ds = ∞, (3.4)
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. As in the proof of Theorem 2.1, we define w(t) as in (2.13) and obtain (2.14)
and (2.15) for t ≥ t3. Now
w′(t) ≤ −ρ(t)q(t) + ρ′(t)
L3x(t)
f(x[λσ(t)])
for t ≥ T ≥ t3. (3.5)
Using (2.15) in (3.5), we find
w′(t) ≤ −ρ(t)q(t) +
(
(ρ′(t))1/α
λσ′(t)h(σ(t), t1; a;λ)
λx′[λσ(t)]σ′(t)
f1/α(x[λσ(t)])
)α
for t ≥ T.
Now by the second Bonnet mean value theorem for a fixed t ≥ T and for some ξ ∈ [T, t],
we have
t∫
T
(
(ρ′(s))1/α
λσ′(s)h(σ(s), t1; a;λ)
) (
λx′[λσ(s)]σ′(s)
f1/α(x[λσ(s)])
)
ds =
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302 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
=
(
(ρ′(T ))1/α
λσ′(T )h(σ(T ), t1; a;λ)
) ξ∫
T
λx′[λσ(s)]σ′(s)
f1/α(x[λσ(s)])
ds =
=
(
(ρ′(T ))1/α
λσ′(T )h(σ(T ), t1; a;λ)
) x[λσ(ξ)]∫
x[λσ(T )]
du
f1/α(u)
≤
≤
(
(ρ′(T ))1/α
λσ′(T )h(σ(T ), t1; a;λ)
) ∞∫
x[λσ(T )]
du
f1/α(u)
= M, (3.6)
where M is a positive constant. Using (3.6) in (3.5) and integrating from T to t, we obtain
t∫
T
ρ(s)q(s)ds ≤ −w(t) + w(T ) + Mα.
Letting t → ∞ in the above inequality, we arrive at a contradiction to condition (3.4).
This completes the proof.
Theorem 3.2. Let condition (3.3) in Theorem 3.1 be replaced by: for all large t ≥ t0
and some constant λ ∈ (0, 1),
ρ′(t) ≥ 0 and
∞∫ ∣∣∣∣∣
(
(ρ′(s))1/α
σ′(s)h(σ(s), t0; a;λ)
)′∣∣∣∣∣ ds < ∞. (3.7)
Then the conclusion of Theorem 3.1 holds.
Next, we present the following oscillation criteria for equation (1.1) when
Q(t) =
∞∫
t
q(s)ds < ∞. (3.8)
Theorem 3.3. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0, (3.1) and (3.8)
hold. If for all large t ≥ t0 and some constant λ ∈ (0, 1),
∞∫
h(σ(s), t0; a;λ)σ′(s)Q1/α(s)ds = ∞, (3.9)
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1) and assume that x(t) >
> 0 for t ≥ t0 ≥ 0. Define w(t) as in (2.13) with ρ(t) = 1. Then, we obtain
t∫
t2
q(s)ds ≤ L3x(t)
f(x[λσ(t)])
and hence for any t ≥ t2,
Q(t) ≤ L3x(t)
f(x[λσ(t)])
,
or
Q1/α(t) ≤ L
1/α
3 x(t)
f1/α(x[λσ(t)])
. (3.10)
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 303
Proceeding as in the proof of Theorem 2.1, we obtain (2.15) for t ≥ t3. Using (2.15) in
(3.10), we get
λh(σ(t), t1; a;λ)σ′(t)Q1/α(t) ≤ λx′[λσ(t)]σ′(t)
f1/α(x[λσ(t)])
for t ≥ t3. (3.11)
Integrating (3.11) from t3 to t, we obtain
λ
t∫
t3
h(σ(s), t1; a;λ)σ′(s)Q1/α(s)ds ≤
≤
t∫
t3
λx′[λσ(s)]σ′(s)
f1/α(x[λσ(s)])
ds =
=
x[λσ(t)]∫
x[λσ(t3)]
du
f1/α(u)
≤
∞∫
x[λσ(t3)]
du
f1/α(u)
< ∞,
which contradicts condition (3.9). This completes the proof.
Theorem 3.4. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0, (3.1) and (3.8)
hold and suppose that
f ′(x)f (1−α)/α(x) = γ(x), (3.12)
where γ(x) is a positive nondecreasing function for x �= 0. If for all large T ≥ t0 ≥ 0,
some constant λ ∈ (0, 1) and every constant c > 0,
∞∫
h(σ(s), t0; a;λ)σ′(s) ×
×
Q(s) + c
∞∫
s
σ′(u)h(σ(u), t0; a;λ)Q(1+α)/α(u)du
ds = ∞, (3.13)
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1) and assume that x(t) >
> 0 for t ≥ t0 ≥ 0. We define w(t) as in (2.13) with ρ(t) = 1 and as in the proof of
Theorem 2.1, we obtain (2.14) which takes the form
w′(t) ≤
≤ −q(t) − λσ′(t)f ′(x[λσ(t)])f (1−α)/α(x[λσ(t)])
L3x(t)x′[λσ(t)]
f1+1/α(x[λσ(t)])
for t ≥ t2.
(3.14)
Using (2.15) in (3.14), we get
w′(t) ≤ −q(t) − λσ′(t)h(σ(t), t1; a;λ)γ(x[λσ(t)])w1+1/α(t) for t ≥ t2. (3.15)
Since x(t) is increasing on [t2,∞) and γ(x) is a nondecreasing function, there exists a
constant m and a T ≥ t2 such that
x[λσ(t)] ≥ m for t ≥ T. (3.16)
Using (3.16) in (3.15), we find
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304 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
w′(t) ≤ −q(t) − λσ′(t)γ(m)h(σ(t), t1; a;λ)w1+1/α(t) for t ≥ T. (3.17)
Integrating (3.17) from t to u ≥ t and letting u → ∞, we have
L3x(t) ≥
≥ f
(
x
[
λσ(t)
])Q(t) + λγ(m)
∞∫
t
σ′(s)h(σ(s), t1; a;λ)w1+1/α(s)ds
for t ≥ T.
(3.18)
Clearly, we have
w(t) ≥ Q(t) for t ≥ T. (3.19)
Using (3.19) in (3.18), we obtain
x′[λσ(t)](λσ′(t))
f1/α(x[λσ(t)])
≥ λh(σ(t), t1; a;λ) ×
×
Q(t) + λγ(m)
∞∫
t
σ′(s)h(σ(s), t1; a;λ)Q1+1/α(s)ds
1/α
for t ≥ T.
Integrating the above inequality from T to t and using condition (3.1), we obtain a con-
tradiction to condition (3.13). This completes the proof.
Corollary 3.1. Let condition (3.12) in Theorem 3.4 be replaced by
f ′(x)f (1−α)/α(x) ≥ k > 0 for x �= 0, (3.20)
where k is a constant and let c = k in condition (3.13). Then the conclusion of Theo-
rem 3.4 holds.
Theorem 3.5. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0 and (3.2) hold,
and assume that f satisfies
−f(−xy) ≥ f(xy) ≥ f(x)f(y) for xy > 0. (3.21)
If for all large t ≥ t0 and some constant λ ∈ (0, 1),
∞∫
q(s)f(H(σ(s), t0; a;λ))ds = ∞, (3.22)
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. Proceeding as in Theorem 2.2 we obtain the inequality (2.21), which takes
the form
x[g(t)] ≥ H(σ(t), t0; a;λ)L1/α
3 x(t) for t ≥ T ≥ t0 and some constant λ ∈ (0, 1).
(3.23)
Using condition (3.21) and inequality (3.23) in equation (1.1), we get
− d
dt
L3x(t) ≥ q(t)f(x[g(t)]) ≥ q(t)f
(
H(σ(t), t0; a;λ)L1/α
3 x(t)
)
≥
≥ q(t)f(H(σ(t), t0; a;α))f
(
L
1/α
3 x(t)
)
for t ≥ T.
Substituting u(t) for L3x(t), t ≥ T we have
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 305
−du(t)
dt
≥ q(t)f(H(σ(t), t0; a;λ))f
(
u1/α(t)
)
for t ≥ t0. (3.24)
Dividing both sides of (3.24) by f(u1/α(t)) and integrating from T to t, we obtain
t∫
T
q(s)f(H(σ(s), t0; a;λ))ds ≤
T∫
t
u′(s)ds
f(u1/α(s))
=
u(T )∫
u(t)
du
f(u1/α)
.
Letting t → ∞, we conclude
∞∫
T
q(s)f(H(σ(s), t0; a;λ))ds ≤
u(T )∫
0
du
f(u1/α)
< ∞,
which contradicts condition (3.22) and completes the proof.
Theorem 3.6. Let conditions (1.2), (2.10) with σ′(t) ≥ 0 for t ≥ t0 and (2.18) with
β < α and assume that
0 < Q(t) =
∞∫
t
q(s)ds < ∞. (3.25)
If for all large t ≥ t0, some constant λ ∈ (0, 1) and every constant c > 0,
lim sup
t→∞
H(σ(t), t0; a;λ)
Q(t) + c
∞∫
t
h(σ(s), t0; a;λ)σ′(s)Q1+1/β(s)ds
1/β
= ∞,
(3.26)
then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. Define, w(t) = L3x(t)/xβ [λσ(t)] for t ≥ t1 ≥ t0. Then for t ≥ t1, we get
w′(t) ≤ −q(t) − βλσ′(t)L−1/α
3 x(t)x′[λσ(t)
]
w1+1/α(t)x(β−α)/α
[
λσ(t)
]
.
As in the proof of Theorem 2.1 we obtain the inequality (2.15) for t ≥ t2. Using (2.15) in
the above inequality, we have
w′(t) ≤ −q(t) − βλσ′(t)h(σ(t), t1; a;λ)w1+1/α(t)x(β−α)/α
[
λσ(t)
]
for t ≥ t2.
(3.27)
Integrating (3.27) from t ≥ t2 to u ≥ t and letting u → ∞, we get
L3x(t) ≥ xβ [λσ(t)] ×
×
Q(t) + λβ
∞∫
t
h(σ(s), t1; a;λ)σ′(s)w1+1/α(s)x(β−α)/α[λσ(s)]ds
, (3.28)
and hence Q(t) ≤ w(t) for t ≥ t2.
There exist a constant b > 0 and a t3 ≥ t2 such that
L3x(t) ≤ b for t ≥ t3. (3.29)
Using (3.29) in (3.28), we obtain
x(β−α)/α[λσ(t)] ≥ b(β−α)/αβQ(α−β)/αβ(t) for t ≥ t3. (3.30)
Next, we proceed as in the proof of Theorem 2.2 and obtain (2.21) with T = t1 for
t ≥ t3 ≥ t2. Now, for t ≥ t3,
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306 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
L
1/β
3 x(t) ≥ H(σ(t), t1; a;λ)L1/α
3 x(t) ×
×
Q(t) + λβb(β−α)/αβ
∞∫
t
h(σ(s), t1; a;λ)σ′(s)Q1+1/β(s)ds
1/β
,
or
b(α−β)/αβ ≥ L
(α−β)/αβ
3 x(t) ≥
≥ H(σ(t), t1; a;λ)
Q(t) + λβb(β−α)/αβ
∞∫
t
h(σ(s), t1; a;λ)σ′(s)Q1+1/β(s)ds
1/β
.
Taking the lim sup of this inequality as t → ∞ we arrive at a contradiction to condition
(3.26) and complete the proof.
The following result is concerned with the oscillation of advanced equation (1.1), i.e.,
when g(t) ≥ t for t ≥ t0. We shall need the following lemma due to Werbowski [17].
Lemma 3.1. Consider the integrodifferential inequality with deviating argument
y′(t) ≥
∞∫
t
Q(t, s)y[g(s)]ds, (3.31)
where Q ∈ C(R+ × R
+,R+) and g(t) ∈ C(R+,R+), g(t) ≥ t for t ≥ t0 ≥ 0. If
lim inf
t→∞
g(t)∫
t
∞∫
s
Q(s, u)duds >
1
e
,
then inequality (3.31) has no eventually positive solution.
Theorem 3.7. Let conditions (1.2), (2.18) with β = α and (3.25) hold, and assume
that g(t) ≥ t and g′(t) ≥ 0 for t ≥ t0. If
lim inf
t→∞
g(t)∫
t
P (s)ds >
1
e
, (3.32)
where
P (t) = min
∞∫
t
1
a(s)
∞∫
s
Q(τ)dτ
1/α
ds, Q1/α(t)h(t, t0; a;λ)
for all large t ≥ t0 and some constant λ ∈ (0, 1), then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. There exists a t1 ≥ t0 such that x(t) satisfies Case (I), or Case (II) for t ≥ t1.
Now we consider:
Case (I). Integrating equation (1.1) from t ≥ t1 to u ≥ t and letting u → ∞, we
obtain
L3x(t) ≥
∞∫
t
q(s)ds
f(x[g(t)]) = Q(t)f(x[g(t)]). (3.33)
Once again, we integrate the above inequality from t ≥ t1 to u ≥ t and let u → ∞, we
find
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 307
−x′′(t) ≥
1
a(t)
∞∫
t
Q(s)ds
1/α
f1/α(x[g(t)]) ≥
≥
1
a(t)
∞∫
t
Q(s)ds
1/α
x[g(t)] for t ≥ t1.
Integrating the above inequality from t ≥ t1 to u and letting u → ∞, we have
x′(t) ≥
∞∫
t
1
a(s)
∞∫
s
Q(τ)dτ
1/α
x[g(s)]ds. (3.34)
Inequality (3.34) in view of condition (3.32) and Lemma 3.1 has no eventually positive
solution, which is a contradiction.
Case (II). Proceeding as in Theorem 2.1 and Case (I) we obtain (2.12) and (3.33) for
t ≥ T ≥ t1. Now, using (2.12) in (3.33) and the fact that x′(t) is increasing on [t1,∞),
we get
x′(t) ≥ x′(λt) ≥ h(t, t1; a;λ)L1/α
3 x(t) ≥
≥ h(t, t1; a;λ)Q1/α(t)f1/α(x[g(t)]) ≥
≥ h(t, t1; a;λ)Q1/α(t)x[g(t)] for t ≥ T. (3.35)
Inequality (3.35) in view of condition (3.32) and a result in [15] has no eventually positive
solution, a contradiction. This completes the proof.
4. Necessary and sufficient conditions. In this section we shall establish some
necessary and sufficient conditions for the oscillation of a special case of equation (1.1),
namely, the equation
L4x(t) + q(t)xβ [g(t)] = 0, (4.1)
where β is the ratio of two positive odd integers.
The following theorem is concerned with a necessary and sufficient condition for the
oscillation of all unbounded solutions of the sublinear equation (4.1), i.e., when β < α.
Theorem 4.1. Let condition (1.2) hold and g(t) ≤ t and g′(t) ≥ 0 for t ≥ t0 and let
β < α. All unbounded solutions of equation (4.1) are oscillatory if and only if
∞∫
q(s)Hβ
1 (g(s), T ; a)ds = ∞ (4.2)
for all large T ≥ t0, where
H1(t, T ; a) =
t∫
T
(t− u)
(
u− T
a(u)
)1/α
du.
Proof. Let x(t) be an unbounded nonoscillatory solution of equation (4.1), say,
x(t) > 0 for t ≥ t0 ≥ 0. Clearly, x(t) satisfies Case (I). Now, the proof of the “if”
part is similar to that of Theorem 3.5 and hence omitted. To prove the “only if” part it
suffices to assume that
∞∫
q(s)Hβ
1 (g(s), T ; a)ds < ∞, (4.3)
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308 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
and show the existence of a nonoscillatory solution of equation (4.1).
Let c > 0 be an arbitrary constant and choose T1 > T ≥ t0 sufficiently large so that
∞∫
T1
q(s)Hβ
1 (g(s), T ; a)ds < c1−β/α. (4.4)
Define the set X by
X =
{
x ∈ C[T1,∞) : c1H1(t, T ; a) ≤ x(t) ≤ c2H1(t, T ; a), t ≥ T1
}
,
where c1 = (c/2)1/α and c2 = (2c)1/α.
Clearly, X is a closed convex subset of the locally convex space C[T1,∞) of continu-
ous functions on [T1,∞) equipped with the topology of uniform convergence on compact
subintervals of [T1,∞). Next, let S be a mapping defined on X as follows: For x ∈ X,
(Sx)(t) =
=
t∫
T1
(t− s)
1
a(s)
c(s− T1) +
s∫
T1
∞∫
u
q(τ)xβ [g(τ)]dτdu
1/α
ds for t ≥ T1.
(4.5)
Clearly, S is well-defined and continuous on X. It can be shown without any difficulty
that S maps X into itself and S(X) is relatively compact in C[T1,∞). Therefore, by the
Schauder – Tychonoff fixed point theorem, S has a fixed element x in X, which satisfies
x(t) =
t∫
T1
(t− s)
1
a(s)
c(s− T1) +
s∫
T1
∞∫
u
q(τ)xβ [g(τ)]dτdu
1/α
ds, t ≥ T1.
Differentiation shows that x = x(t) is a positive solution of equation (4.1) on [T,∞) such
that limt→∞ x(t)/H1(t, T ; a) = γ > 0, where γ is a constant. For more details, we refer
the reader to [6].
Theorem 4.1 can be reformulated as follows:
Theorem 4.1′. Let condition (1.2) hold, and g(t) ≤ t and g′(t) ≥ 0 for t ≥ t0 and let
β < α. Equation (4.1) has a nonoscillatory solution x(t) such that limt→∞ x(t)/H1(t,
T ; a) = nonzero constant, and t ≥ T (large) ≥ t0 if and only if
∞∫
q(s)Hβ
1 (g(s), T ; a)ds < ∞, (4.6)
where H1 is as in Theorem 5.1.
Next, we present the following necessary and sufficient condition for the oscillation
of all bounded solutions of the superlinear equation (4.1), i.e., with β > α.
Theorem 4.2. Let condition (1.2) hold, g(t) ≤ t and g′(t) ≥ 0 for t ≥ t0 and β > α.
All bounded solutions of equation (4.1) are oscillatory if and only if
∞∫
s
1
a(s)
∞∫
s
∞∫
u
q(τ)dτdu
1/α
ds = ∞. (4.7)
Proof. Let x(t) be a bounded nonoscillatory solution of equation (4.1), say, x(t) > 0
for t ≥ t0 ≥ 0. Clearly, x(t) satisfies Case (II). The proof of the “if” part is similar to that
of Theorem 2.8 and Corollary 2.2(i) and hence omitted.
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 309
The “only if” part of the theorem is proved as follows: Let c > 0 be given arbitrarily
and choose a large T ≥ t0 such that
∞∫
T
s
1
a(s)
∞∫
s
∞∫
u
q(τ)dτdu
1/α
ds ≤ 1
2
c1−β/α.
We define the set Y and the mapping F by
Y =
{
x ∈ C[T,∞) :
c
2
≤ x[g(t)] ≤ c, t ≥ T
}
and
Fx(t) = c−
∞∫
t
∞∫
s
1
a(u)
∞∫
u
∞∫
v
q(τ)xβ [g(τ)]dτdv
1/α
duds
respectively. Now, it is easy to prove that F maps Y into itself and F is a continuous map-
ping. Also, F (Y ) is relatively compact in C[T,∞). Therefore, by Schauder – Tychnoff
fixed point theorem there exists an element x ∈ Y such that x = Fx. It is clear that
this fixed point x = x(t) is a positive solution of equation (4.1) on [T,∞) such that
x(∞) = c. This completes the proof.
Once again we can reformulate Theorem 4.2 as follows:
Theorem 4.2′. Let condition (1.2) hold, g(t) ≤ t and g′(t) ≥ 0 for t ≥ t0, and β >
> α. Equation (4.1) has a nonoscillatory solution x(t) such that limt→∞ x(t) = nonzero
constant, if and only if
∞∫
s
1
a(s)
∞∫
s
∞∫
u
q(τ)dτdu
1/α
ds < ∞.
Remark 4.1. 1. It is easy to see that the results obtained for equation (4.1) can be
easily extended to equation (1.1).
2. We note that if equation (4.1) has a bounded eventually positive solution x(t), then
x(t) satisfies (II) and so there exist a constant c > 0 and a t1 ≥ t0 such that
c
2
≤ x[g(t)] ≤ c for t ≥ t1.
In this case, we see that all bounded solutions of equation (4.1) are oscillatory if
∞∫ ∞∫
s
Q∗(u)duds = ∞,
where Q∗(t) is as in Theorem 2.8. The details are easy and left to the reader.
5. Comparison and extensions. In this section we shall obtain a comparison result
which is useful in extending our previous results to the neutral equations of the form
L4(x(t) + p(t)x[τ(t)]) + q(t)f(x[g(t)]) = 0, (5.1)
where the operator L4 and the functions q, g and f are as in equation (1.1), and p(t),
τ(t) ∈ C([t0,∞),R) and limt→∞ τ(t) = ∞.
Now, we present the following comparison result.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
310 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
Theorem 5.1. Let condition (1.2) hold. If the inequality
L4x(t) + q(t)f(x[g(t)]) ≤ 0 (≥ 0) (5.2)
has an eventually positive (negative) solution, then equation (1.1) also has eventually
positive (negative) solution.
Proof. Let x(t) be an eventually positive solution of inequality (5.2). It is easy to see
that there exists a t0 ≥ 0 such that x(t) satisfies either (I), or (II) for t ≥ t0. Integrating
inequality (5.2) from t ≥ t0 to u ≥ t and letting u → ∞, we have
L3x(t) ≥
∞∫
t
q(s)f(x[g(s)])ds.
Integrating again from t ≥ t0 to u ≥ t, we obtain
L2x(u) − L2x(t) ≥
u∫
t
∞∫
s1
q(s)f(x[g(s)])dsds1. (5.3)
Now, we distinguish the two cases:
Case (I). Since Lix(t) > 0, i = 0, 1, 2, and t ≥ t0, we replace t by t0 and u by t in
(5.3), to obtain
L2x(t) ≥
t∫
t0
∞∫
s1
q(s)f(x[g(s)])dsds1,
or
x′′(t) ≥
1
a(t)
t∫
t0
∞∫
s1
q(s)f(x[g(s)])dsds1
1/α
and hence
x(t) ≥ x(t0) +
t∫
t0
s3∫
t0
1
a(s2)
s2∫
t0
∞∫
s1
q(s)f(x[g(s)])dsds1
1/α
ds2ds3 =:
=: c + Φ(t, x[g(t)]), (5.4)
where x(t0) = c.
Now, it is easy to show the existence of a positive solution to the integral equation
w(t) = c + Φ(t, w[g(t)]) for t ≥ t0.
We define the sequence {wn(t)}, n = 0, 1, 2, . . . , such that w0(t) = x(t) for t ≥ t0,
wn+1(t) =
{
c + Φ(t, wn[g(t)]) for t ≥ t0,
c for t ≤ t0.
Then one can easily see that wn(t) is well defined and
0 ≤ wn(t) ≤ x(t), c ≤ wn+1(t) ≤ wn(t).
Thus, by the Lebesgue monotone convergence theorem there exists w(t) such that w(t) =
= limn→∞ wn(t) for t ≥ t0, and
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OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 311
w(t) = c + Φ(t, w[g(t)]) for t ≥ t0.
If we differentiate (5.4) four times, we obtain equation (1.1).
Case (II). Since L2x(t) < 0 for t ≥ t0, we let u → ∞ in (5.3) to have
−x′′(t) ≥
1
a(t)
∞∫
t
∞∫
s1
q(s)f(x[g(s)])dsds1
1/α
and hence
x(t) ≥ x(t0) +
t∫
t0
∞∫
s3
1
a(s2)
∞∫
s2
∞∫
s1
q(s)f(x[g(s)])dsds1
1/α
ds2ds3 =:
=: c + Ψ(t, x[g(t)]),
where x(t0) = c. The rest of the proof is similar to that of Case (I) and hence omitted.
This completes the proof.
Next, we shall employ Theorem 5.1 to extend the obtained results to the neutral equa-
tion (5.1). In fact, we have the following comparison theorem.
Theorem 5.2. Let conditions (1.2) and (3.21) hold. Moreover, assume that 0 ≤
≤ p(t) ≤ 1, τ(t) < t, τ ′(t) > 0, g(t) ≤ t and g′(t) ≥ 0 for t ≥ t0 and p(t) �≡ 1
eventually. If the equation
L4x(t) + q(t)f(1 − p[g(t)])f(x[g(t)]) = 0 (5.5)
is oscillatory, then equation (5.1) is oscillatory.
Theorem 5.3. Let conditions (1.2) and (3.21) hold and assume that p(t) ≥ 1, τ(t) >
> t, τ ′(t) > 0, τ−1 ◦ g(t) ≤ t, (τ−1 ◦ g(t))′ ≥ 0 for t ≥ t0 and p(t) �≡ 1 eventually. If
the equation
L4x(t) + q(t)f(P [g(t)])f(x[τ−1 ◦ g(t)]) = 0 (5.6)
is oscillatory, where
P (t) =
1
p[τ−1(t)]
(
1 − 1
p[τ−1 ◦ τ−1(t)]
)
,
then equation (5.1) is oscillatory.
Proof of Theorems 5.2 and 5.3. Let x(t) be a nonoscillatory solution of equa-
tion (5.1), say, x(t) > 0 for t ≥ t0 ≥ 0. Define
y(t) = x(t) + p(t)x[τ(t)], t ≥ t0.
Then for t ≥ t0,
L4y(t) + q(t)f(x[g(t)]) = 0. (5.7)
It is easy to check that there exists a t1 ≥ t0 such that y′(t) > 0 for t ≥ t1. Now, by using
the hypotheses of Theorem 5.2, we find
x(t) = y(t) − p(t)x
[
τ(t)
]
= y(t) − p(t)
[
y[τ(t)] − p[τ(t)]x
[
τ ◦ τ(t)
]]
≥
≥ y(t) − p(t)y
[
τ(t)
]
≥
(
1 − p(t)
)
y(t) for t ≥ t1. (5.8)
Using (5.8) and condition (3.21) in equation (5.7), we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
312 R. P. AGARWAL, S. R. GRACE, D. O’REGAN
L4y(t) + q(t)f(1 − p
[
g(t)
])
f
(
y[g(t)]
)
≤ 0 for t ≥ t1. (5.9)
Next, by using the hypotheses of Theorem 5.3, we find
x(t) =
1
p[τ−1(t)]
(
y[τ−1(t)] − x[τ−1(t)]
)
=
=
y[τ−1(t)]
p[τ−1(t)]
− 1
p[τ−1(t)]
(
y[τ−1 ◦ τ−1(t)]
p[τ−1 ◦ τ−1(t)]
− x[τ−1 ◦ τ−1(t)]
p[τ−1 ◦ τ−1(t)]
)
≥
≥ y[τ−1(t)]
p[τ−1(t)]
− y[τ−1 ◦ τ−1(t)]
p[τ−1(t)]p[τ−1 ◦ τ−1(t)]
≥
≥ 1
p[τ−1(t)]
(
1 − 1
p[τ−1 ◦ τ−1(t)]
)
y[τ−1(t)] = P (t)y[τ−1(t)] for t ≥ t1.
(5.10)
Using (5.10) and condition (3.21) in equation (5.7), we get
L4y(t) + q(t)f
(
P [g(t)]
)
f
(
y
[
τ−1 ◦ g(t)
])
≤ 0 for t ≥ t1.
Inequalities (5.9) and (5.10) have positive solutions and hence equations (5.5) and (5.6)
have also positive solutions which contradicts the hypotheses and completes the proof.
Finally, we shall extend the results of this paper to equations (1.1) and (5.1) when the
function f need not be a monotone function. For this, we let
Rt0 =
(−∞,−t0] ∪ [t0,∞) if t0 > 0,
(−∞, 0) ∪ (0,∞) if t0 = 0,
C(R) =
{
f : R → R is continuous and xf(x) > 0 for x �= 0
}
and
CB(Rt0) =
{
f ∈ C(R) : f is of bounded variation on every interval [a, b] ⊆ Rt0
}
.
We shall need the following Lemma [14].
Lemma 5.1. Suppose t0 > 0 and f ∈ C(R). Then f ∈ CB(Rt0) if and only if
f(x) = H(x)G(x) for all x ∈ Rt0 , where G : Rt0 → (0,∞) is nondecreasing on
(−∞, t0) and nonincreasing on (t0,∞) and H : Rt0 → R is nondecreasing on Rt0 .
Theorem 5.4. Let condition (1.2) hold and assume that f ∈ C(Rt0), t0 ≥ 0, and
let G and H be a pair of continuous components with H being the nondecreasing one.
In addition, let g(t) ≤ t and g′(t) ≥ 0 for t ≥ t0. If for every constant k > 0 and all
sufficiently large T ≥ t0, the equation
L4y(t) + q(t)G(kg∗(g(t), T ; a))H(y[g(t)]) = 0
is oscillatory, then equation (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for
t ≥ t0 ≥ 0. As in the proof of Theorem 2.2 we obtain (2.22) for t ≥ T1 ≥ T. Now,
L4x(t) + q(t)G
(
b1g∗(g(t), T ; a)
)
H
(
x[g(t)]
)
≤ L4x(t) + q(t)G
(
x[g(t)]
)
H
(
x[g(t)]
)
=
= L4x(t) + q(t)f
(
x[g(t)]
)
= 0, t ≥ T1.
The rest of the proof is similar to that of Theorems 5.2 and 5.3 and hence omitted.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
OSCILLATION OF CERTAIN FOURTH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 313
6. General remarks.
1. The results of this paper are presented in a form which is essentially new and are
of higher degree of generality.
2. The results of this paper can be extended to higher order functional differential
equations of the form
(a(t)(x(n)(t))α)(n) + δq(t)f(x[g(t)]) = 0,
where δ = ±1, n > 0 is an integer.
3. The results of this paper can be extended to forced functional differential equations
of the type (
a(t)
(
x(n)(t)
)α
)(n)
+ q(t)f(x[g(t)]) = e(t),
where e(t) ∈ C([t0,∞),R).
4. When a(t) ≡ 1 for t ≥ t0, one can easily see that
h(t, t0; 1; 1/2) = c1t
1+1/α for all large t
and
H(t, t0; 1; 1/2) = c2t
2+1/α for all large t,
where c1 and c2 are positive constants and can be calculated easily. Here, we omit the
details.
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Received 03.11.2006
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:40:05Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/f5/f1abb3c5704443d6617abedc00551bf5.pdf |
| spelling | umjimathkievua-article-33082020-03-18T19:51:00Z Oscillation of certain fourth-order functional differential equations Коливання деяких функціональних диференціальних рівнянь четвертого порядку Agarwal, P. Grace, S. R. O’Regan, D. Агарвал, Р. П. Грасе, С. Р. О'Реган, Д. Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the form $$\frac{d^2}{dt^2} \left(a(t) \left(\frac{d^2x(t)}{dt^2}\right)^{α} \right) + q(t)f(x[g(t)])=0, \quad α>0,$$ are established. Встановлено деякі нові критерії коливання нелінійних функціональних диференціальних рівнянь вигляду $$\frac{d^2}{dt^2} \left(a(t) \left(\frac{d^2x(t)}{dt^2}\right)^{α} \right) + q(t)f(x[g(t)])=0, \quad α>0.$$ Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3308 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 291–313 Український математичний журнал; Том 59 № 3 (2007); 291–313 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3308/3361 https://umj.imath.kiev.ua/index.php/umj/article/view/3308/3362 Copyright (c) 2007 Agarwal P.; Grace S. R.; O’Regan D. |
| spellingShingle | Agarwal, P. Grace, S. R. O’Regan, D. Агарвал, Р. П. Грасе, С. Р. О'Реган, Д. Oscillation of certain fourth-order functional differential equations |
| title | Oscillation of certain fourth-order functional differential equations |
| title_alt | Коливання деяких функціональних диференціальних рівнянь четвертого порядку |
| title_full | Oscillation of certain fourth-order functional differential equations |
| title_fullStr | Oscillation of certain fourth-order functional differential equations |
| title_full_unstemmed | Oscillation of certain fourth-order functional differential equations |
| title_short | Oscillation of certain fourth-order functional differential equations |
| title_sort | oscillation of certain fourth-order functional differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3308 |
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