Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 2 (2012), Pages 40–50.
OSCILLATION CRITERIA FOR A CLASS OF SECOND ORDER
NONLINEAR DIFFERENTIAL EQUATIONS WITH DAMPING
(COMMUNICATED BY HÜSEYIN BOR)
E. TUNÇ, H. AVCI
Abstract. In this paper, some oscillation criteria for solutions of a general
second order non-linear differential equations with damping of the form
′
a(t)Ψ (x (t)) k x′ (t)
+ p (t) k x′ (t) + q (t) f (x (t)) = 0,
are given. The results obtained extend some existing results in the literature
by using the refined integral averaging technique introduced by Rogovchenko
and Tuncay ([1], [2]).
1. Introduction
In this paper, we are concerned with the oscillation of solutions of the secondorder nonlinear differential equations with damping terms of the following form
′
(a(t)Ψ (x (t)) k (x′ (t))) + p (t) k (x′ (t)) + q (t) f (x (t)) = 0,
(1.1)
where t ≥ t0 ≥ 0, a(t), p(t), q(t) ∈ C ([t0 , ∞) ; R) and Ψ, k, f ∈ C(R, R). It is also
assumed that there are positive constants c, c1 , µ and γ such that the following
conditions are satisfied:
(C1) a(t) > 0 and xf (x) > 0 for all x 6= 0;
(C2) 0 < c ≤ Ψ (x) ≤ c1 for all x;
(C3) γ > 0 and k 2 (y) ≤ γyk(y) for all y ∈ R;
(C4) q(t) ≥ 0, f (x)
x ≥ µ > 0 for x 6= 0.
We recall that a function x : [t0 , t1 ) → R, t1 > t0 is called a solution of Eq.
(1.1) if x(t) satisfies Eq. (1.1) for all t ∈ [t0 , t1 ). In what follows, it will be always
assumed that solutions of Eq. (1.1) exist for any t0 ≥ 0. Furthermore, a solution
x(t) of Eq. (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise it is
called nonoscillatory. Finally, we say that Eq. (1.1) is oscillatory if all its solutions
are oscillatory.
2010 Mathematics Subject Classification. Primary 34C10; Secondary 34A30.
Key words and phrases. Nonlinear differential equations, Damping term, Second order,
Oscillation.
c
2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted March 9, 2012. Published April 21, 2012.
40
OSCILLATION CRITERIA FOR A CLASS OF SECOND ORDER NONLINEAR ...
41
In the last decades, there has been an increasing interest in obtaining sufficient
conditions for the oscillation of solutions for different classes of second order differential equations. Especially, by using the integral averaging technique and the
generalized Riccati technique, the oscillation problem for Eq. (1.1) and its special
cases such as the nonlinear equations with damping term
′
(r(t)x′ (t)) + p (t) x′ (t) + q (t) f (x (t)) = 0
and
(1.2)
′
(r(t)ψ(x (t))x′ (t)) + p (t) x′ (t) + q (t) f (x (t)) = 0.
(1.3)
has been studied extensively in recent years ( see, for example, [1]-[13] and the
references cited therein).
Following Philos [10], we define a family of functions P which will be used in the
rest of the article. For this purpose, let
D = {(t, s) : t ≥ s ≥ t0 } .
A function H ∈ C(D, R) is said to belong to the class P if
(i) H(t, t) = 0 for t ≥ t0 , H(t, s) > 0 for t > s ≥ t0 ;
(ii) H(t, s) has a continuous and nonpositive partial derivative on D with respect
to the second variable, and there is a function h ∈ C(D; [0, +∞)) such that
p
∂H
−
(t, s) = h(t, s) H(t, s) for all (t, s) ∈ D.
∂s
In this connection, in 2004, Wang [5] established oscillation criteria for Eq. (1.1).
We now state one of his main results for easier reference.
Theorem 1.1. ([5], Theorem 3.3). Let assumptions (C1)-(C4) be fulfilled. Let the
function H ∈ P, and suppose also that
H(t, s)
0 < inf lim inf
≤ ∞.
(1.4)
t→∞ H(t, t0 )
s≥t0
If there exist functions R, φ ∈ C ([t0 , ∞) ; R) and Φ ∈ C 1 ([t0 , ∞) ; (0, ∞)) such that
(aR) ∈ C 1 ([t0 , ∞) ; R) and
1
lim sup
t→∞ H(t, t0 )
Z∞
Zt
Φ(s)a(s)h22 (t, s)ds < ∞,
(1.5)
t0
φ2+ (s)
ds = ∞,
Φ(s)a(s)
t0
and for every T ≥ t0
1
lim sup
H(t,
T)
t→∞
Zt h
T
H(t, s)Q2 (s) −
i
c1 γ
Φ(s)a(s)h22 (t, s) ds ≥ φ(T ),
4
where
γ 1
1 p2 (t)
1
1
′
Q2 (t) = Φ(t) µq(t) −
−
− p(t)R(t) +
a(t)R2 (t) − (a(t)R(t)) ,
4 c c1 a(t)
c1
c1 γ
′
p
p(s)
Φ (s) 2R(s)
h2 (t, s) = h(t, s) − H(t, s)
+
−
Φ(s)
c1 γ
c1 a(s)
42
E. TUNÇ, H. AVCI
and
φ+ (s) = max {φ(s), 0} ,
then Eq. (1.1) is oscillatory.
We have two aims in this paper. The first aim is to remove the condition (1.5)
in Theorem 1.1 and to demonstrate this with an example. The second goal is to
extend the technique developed by Rogovchenko and Tuncay ([1], [2]) for (1.2) and
(1.3) to Eq. (1.1).
2. Main Results
Theorem 2.1. Suppose that (C1)-(C4) are satisfied. Suppose also that there exist
functions H ∈ P, g ∈ C 1 ([t0 , ∞) ; R) and χ ∈ C ([t0 , ∞) ; R) such that (1.4) holds
and for all t > t0 , all T ≥ t0 , and for some β > 1,
Zt
χ2+ (s)
ds = ∞,
a(s)v(s)
(2.1)
Zt βγc1
2
H(t, s)φ(s) −
a(s)v(s)h (t, s) ds ≥ χ(T ),
4
(2.2)
lim sup
t→∞
t0
and
1
lim sup
t→∞ H(t, T )
T
where
g 2 (t)
p(t)g(t)
φ(t) = v(t) µq(t) +
−
− g ′ (t) +
γc1 a(t)
c1 a(t)

and
v(t) = exp −
2
c1
Zt g(s)
p(s)
−
γa(s) 2a(s)
1
1
−
c1
c

ds ,
γp2 (t)
4a(t)
,
(2.3)
(2.4)
χ+ (s) = max (χ(s), 0) .
Then Eq. (1.1) is oscillatory.
Proof. Let x(t) be a non-oscillatory solution of Eq. (1.1). Then there exists
a T0 ≥ t0 such that x(t) 6= 0 for all t ≥ T0 . Without loss of generality, we may
assume that x(t) > 0 for all t ≥ T0 , for some T0 ≥ t0 . A similar argument holds
for the case when x(t) is eventually negative. As in [3], define a generalized Riccati
transformation by
a(t)Ψ (x (t)) k (x′ (t))
u(t) = v(t)
+ g(t) for all t ≥ T0 .
(2.5)
x (t)
Then differentiating (2.5) and using Eq. (1.1), we obtain
v ′ (t)
−p(t)k (x′ (t)) q(t)f (x(t)) a(t)Ψ (x (t)) k (x′ (t)) x′ (t)
′
′
u (t) =
u(t)+v(t)
−
−
+ g (t) .
v(t)
x(t)
x(t)
x2 (t)
OSCILLATION CRITERIA FOR A CLASS OF SECOND ORDER NONLINEAR ...
43
In view of (C1)-(C4), we conclude that for all t ≥ T0 ,
u′ (t)
h
≤ − γc2g(t)
+
1 a(t)
+v(t)
p(t)
c1 a(t)
i
k(x′ (t))
−p(t) x(t)
h
= − γc2g(t)
+
1 a(t)
p(t)
c1 a(t)
u(t)
− µq(t) −
i
a(t)Ψ(x(t))k2 (x′ (t))
γx2 (t)
+ g (t)
′
h
i
u(t)
1
u(t) + v(t) −p(t) a(t)Ψ(x(t))
−
g(t)
−
µq(t)
v(t)
2
u(t)
1
′
+v(t) − a(t)Ψ(x(t))
−
g(t)
+
g
(t)
γ
a(t)Ψ(x(t)) v(t)
[u(t)+ γ2 v(t)p(t)−v(t)g(t)]
= −µq(t)v(t) −
h
+
+v(t)g ′ (t) + − γc2g(t)
1 a(t)
u2 (t)
γc1 a(t)v(t)
≤ −µq(t)v(t) −
+v(t)g ′ (t) +
1
c
2
+
γv(t)p2 (t)
4a(t)Ψ(x(t))
−
v(t)g2 (t)
γc1 a(t)
γa(t)Ψ(x(t))v(t)
−
1
c1
p(t)
c1 a(t)
+
i
u(t)
p(t)g(t)v(t)
c1 a(t)
γv(t)p2 (t)
4a(t) .
Using (2.3) in the latter inequality, we have, for all t ≥ T0 ,
u′ (t) ≤ −φ(t) −
u2 (t)
γc1 a(t)v(t)
(2.6)
Multiplying both sides of (2.6) by H(t, s), integrating it with respect to s from T
to t, and using the properties of the function H(t, s), we get, for all t ≥ T ≥ T0,
Rt
H(t, s)φ(s)ds
T
≤−
Rt
H(t, s)u′ (s)ds −
T
Rt
T
= −H(t, s)u(s) |tT −
= H(t, T )u(T ) −
2
(s)
H(t, s) γc1ua(s)v(s)
ds
i
Rt h ∂H(t,s)
2
(s)
− ∂s u(s) + H(t, s) γc1ua(s)v(s)
ds
T
i
p
Rt h
2
(s)
h(t, s) H(t, s)u(s) + H(t, s) γc1ua(s)v(s)
ds.
T
(2.7)
Then, for any β > 1, (2.7) gives
Rt
H(t, s)φ(s)ds
≤ H(t, T )u(T ) −
T
Rt q
T
1
+ βγc
4
Rt
T
H(t,s)
βγc1 a(s)v(s) u(s)
a(s)v(s)h2 (t, s)ds −
Rt
T
+
1
2
2
p
βγc1 a(s)v(s)h(t, s) ds
(β−1)H(t,s) 2
βγc1 a(s)v(s) u (s)ds
(2.8)
44
E. TUNÇ, H. AVCI
and, for all t ≥ T ≥ T0 ,
Rt 2
1
H(t, s)φ(s) − βγc
4 a(s)v(s)h (t, s) ds ≤ H(t, T )u(T )
T
−
Rt q
T
H(t,s)
βγc1 a(s)v(s) u(s)
+
1
2
2
p
Rt
βγc1 a(s)v(s)h(t, s) ds −
T
(β−1)H(t,s) 2
βγc1 a(s)v(s) u (s)ds.
(2.9)
From (2.9),
1
H(t,T )
Rt H(t, s)φ(s) −
T
1
− H(t,T
)
Rt q
T
≤ u(T ) −
βγc1
2
4 a(s)v(s)h (t, s)
H(t,s)
βγc1 a(s)v(s) u(s)
1
H(t,T )
Rt
T
+
1
2
ds ≤ u(T ) −
1
H(t,T )
Rt
T
(β−1)H(t,s) 2
βγc1 a(s)v(s) u (s)ds
2
p
βγc1 a(s)v(s)h(t, s) ds
(β−1)H(t,s) 2
βγc1 a(s)v(s) u (s)ds.
Therefore, for all t > T ≥ T0 ,
1
lim sup H(t,T
)
t→∞
Rt H(t, s)φ(s) −
T
1
≤ u(T ) − lim inf H(t,T
)
t→∞
It follows from (2.2) that
Rt
T
βγc1
2
4 a(s)v(s)h (t, s)
ds
(β−1)H(t,s) 2
βγc1 a(s)v(s) u (s)ds.
1
u(T ) ≥ χ(T ) + lim inf
t→∞ H(t, T )
Zt
T
(2.10)
(β − 1)H(t, s) 2
u (s)ds,
βγc1 a(s)v(s)
for all T ≥ T0 and for any β > 1. This shows that
u(T ) ≥ χ(T ),
for all T ≥ T0
(2.11)
and
1
lim inf
t→∞ H(t, T0 )
Zt
H(t, s) 2
βγc1
u (s)ds ≤
(u(T0 ) − χ(T0 )) < ∞.
a(s)v(s)
(β − 1)
(2.12)
T0
We want to prove that
Z∞
u2 (s)
ds < ∞.
a(s)v(s)
(2.13)
Z∞
u2 (s)
ds = ∞.
a(s)v(s)
(2.14)
T0
Suppose to the contrary that
T0
OSCILLATION CRITERIA FOR A CLASS OF SECOND ORDER NONLINEAR ...
By (1.4), there exists a positive constant ρ such that
H(t, s)
inf lim inf
> ρ.
t→∞ H(t, t0 )
s≥t0
45
(2.15)
From (2.15),
H(t, s)
> ρ > 0,
H(t, t0 )
lim inf
t→∞
and there exists a T2 ≥ T1 such that H(t, T1 )/H(t, t0 ) ≥ ρ, for all t ≥ T2 . On the
other hand, by (2.14) for any positive number δ, there exists a T1 > T0 , such that,
for all t ≥ T1 ,
Zt
u2 (s)
δ
ds ≥ .
a(s)v(s)
ρ
T0
Using integration by parts, we obtain, for all t ≥ T1 ,
1
H(t,T0 )
Rt
T0
H(t,s) 2
a(s)v(s) u (s)ds
=
≥
=
1
H(t,T0 )
"
Rt h ∂H(t,s) i Rs
− ∂s
T0
δ
1
ρ H(t,T0 )
T0
Rt h
T1
δ H(t,T1 )
.
ρ H(t,T0 )
u2 (τ )
a(τ )v(τ ) dτ
#
i
− ∂H(t,s)
ds
∂s
This implies that
1
H(t, T0 )
Zt
H(t, s) 2
u (s)ds ≥ δ for all t ≥ T2 .
a(s)v(s)
T0
Since δ is an arbitrary positive constant,
lim inf
t→∞
1
H(t, T0 )
Zt
H(t, s) 2
u (s)ds = +∞,
a(s)v(s)
T0
which conradicts (2.12). Because of that, (2.13) holds, and from (2.11)
Z∞
T0
χ2+ (s)
ds ≤
a(s)v(s)
Z∞
u2 (s)
ds < +∞,
a(s)v(s)
T0
which contradicts (2.1). Therefore, Eq. (1.1) is oscillatory.
Following the classical ideas of Kamenev [4], we define H(t, s) as
n−1
H(t, s) = (t − s)
,
(t, s) ∈ D
where n is an integer and n > 2. Evidently, H ∈ P and
(n−3)/2
h(t, s) = (n − 1) (t − s)
,
(t, s) ∈ D.
Thus, by Theorem 2.1 we have the following oscillation result.
ds
46
E. TUNÇ, H. AVCI
Corollary 2.2. Let (C1)-(C4) hold. Suppose that there exist functions g ∈ C 1 ([t0 , ∞) ; R)
and χ ∈ C ([t0 , ∞) ; R) such that, for all T ≥ t0 , for some integer n > 2, and for
some β > 1,
!
Zt
2
βγc1 (n − 1)
n−3
1−n
n−1
a(s)v(s) (t − s)
ds ≥ χ(T )
lim sup t
(t − s)
φ(s) −
4
t→∞
T
and (2.1) holds, where φ(t) and v(t) are as in Theorem 2.1. Then Eq. (1.1) is
oscillatory.
Example 2.3. Consider the differential equation of the form
h ′
i′
′
−|x(t)|
x (t)
3 x (t)
t2 21 + e 2
+
2t
2
′
1+x (t)
1+x′2 (t)
+ 2 + 2t4 + 6t2 − 6t2 sin2 t x(t) 1 + x4 (t) = 0,
(2.16)
4
where x ∈ (−∞, ∞) and t ≥ 1. Since f (x)
x = 1 + x ≥ 1 = µ, c = 1/2, c1 = 1 the
assumptions (C1)-(C4) hold for γ = 1. Let us apply Corollary 2.2 with β = 2 and
g(t) = t3 , then v(t) = 1 and φ(t) = 2 + 3t2 − 6t2 sin2 t. A direct computation yields
with n = 3
Rt 2
n−3
lim sup t1−n
(t − s)n−1 φ(s) − βγc1 (n−1)
a(s)v(s)
(t
−
s)
ds
4
t→∞
T
= lim sup t12
t→∞
=
3
4
Rt
T
(t − s)2 2 + 3s2 − 6s2 sin2 s −
2·1·1·4 2
s
4
− 2T − 3T 2 sin T cos T − 32 T cos2 T + 32 T sin2 T +
The relation
χ2+ (t)
= O(t2 )
a(t)v(t)
ds
3
2
sin T cos T = χ(T ).
as t → ∞
implies that the condition (2.1) is satisfied. Therefore, Eq. (2.16) is oscillatory by
Corollary 2.2. Note that in this example
1
lim sup 2
t→∞ t
Zt
1
βγc1
1
a(s)v(s)h2 (t, s)ds = lim sup 2
4
t→∞ t
Zt
2·1·1 2
s · 1 · 4ds = ∞. (2.17)
4
1
(2.17) shows that we do not need to impose any condition similar to the condition
(1.5) in Theorem 1.1.
Theorem 2.4. Suppose that (C1)-(C4) and (2.1) are satisfied. Suppose also that
there exist functions H ∈ P, g ∈ C 1 ([t0 , ∞) ; R) and χ ∈ C ([t0 , ∞) ; R) such that
(1.4) holds and, for all T ≥ t0 , and for some β > 1,
1
lim inf
t→∞ H(t, T )
Zt βγc1
H(t, s)φ(s) −
a(s)v(s)h2 (t, s) ds ≥ χ(T )
4
T
where φ(t), v(t) and χ+ (t) are the same as in Theorem 2.1. Then Eq. (1.1) is
oscillatory.
OSCILLATION CRITERIA FOR A CLASS OF SECOND ORDER NONLINEAR ...
47
Proof. Since
χ(T ) ≤
lim inf
t→∞
1
H(t, T )
Zt βγc1
H(t, s)φ(s) −
a(s)v(s)h2 (t, s) ds
4
T
1
lim sup
H(t,
T)
t→∞
≤
Zt βγc1
2
H(t, s)φ(s) −
a(s)v(s)h (t, s) ds,
4
T
that Eq. (1.1) is oscillatory follows readily from Theorem 2.1.
From now on, we present a new set of oscillation theorems. We want to point
out that these theorems differ from Theorem 2.1 and 2.4. That is, they are neither
a special case nor a generalized form of Theorem 2.1 and 2.4.
Theorem 2.5. Let (C1)-(C4) hold. Suppose that there exists a function g ∈
C 1 ([t0 , ∞) ; R) such that, for some β ≥ 1 and for some H ∈ P
1
lim sup
t→∞ H(t, t0 )
Zt γp2 (s)v(s) βγc1
2
H(t, s)φ(s) − H(t, s)
−
a(s)v(s)h (t, s) ds = ∞,
2c1 a(s)
2
t0
(2.18)
where φ(s) is defined by (2.3) and

v(t) = exp −
2
c1
Then Eq. (1.1) is oscillatory.
Zt

g(s) 
ds .
γa(s)
(2.19)
Proof. Let x(t) be a non-oscillatory solution of the differential equation (1.1).
Then there exists a T0 ≥ t0 such that x(t) 6= 0 for all t ≥ T0 . Without loss of
generality, we may assume that x(t) > 0 for all t ≥ T0 . Define the function u(t) as
in (2.5), where v(t) is given by (2.19). Then differentiating (2.5) and using (1.1) ,
we have
u′ (t)
v ′ (t)
v(t)u(t)
−p(t)k( x′ (t))
+v(t)
x(t)
=
−
q(t)f (x(t))
x(t)
−
a(t)Ψ(x(t))k(x′ (t))x′ (t)
x2 (t)
+ g ′ (t) .
(2.20)
Using (C1)-(C4) in (2.20), we easily get
u′ (t) ≤ −φ(t) −
p(t)u(t)
u2 (t)
−
,
c1 a(t)
γc1 a(t)v(t)
(2.21)
where φ(t) is defined by (2.3). On the other hand, since the inequality
mz − nz 2 ≤
m2
n
− z 2 , n > 0, m, z ∈ R
2n
2
which holds for all n > 0 and all m, z ∈ R, we see from (2.21) that
φ(t) −
p2 (t)γv(t)
u2 (t)
≤ −u′ (t) −
2c1 a(t)
2γc1 a(t)v(t)
(2.22)
48
E. TUNÇ, H. AVCI
for all t ≥ T0 . Multiplying (2.22) by H(t, s) and integrating from T to t, we have
for some β ≥ 1 and for all t ≥ T ≥ T0,
Rt
2
H(t, s) φ(s) − γp2c(s)v(s)
ds
a(s)
1
T
≤ H(t, T )u(T ) −
Rt q
T
1
+ βγc
2
Rt
H(t,s)
2βγc1 a(s)v(s) u(s)
a(s)v(s)h2 (t, s)ds −
T
Rt
T
+
2
p
2βγc1 a(s)v(s)h(t, s) ds
1
2
(β−1)H(t,s) 2
2βγc1 a(s)v(s) u (s)ds.
This implies that, for all t ≥ T ≥ T0 ,
Rt 2
βγc1
2
H(t, s)φ(s) − H(t, s) γp2c(s)v(s)
−
a(s)v(s)h
(t,
s)
ds ≤ H(t, T )u(T )
a(s)
2
1
T
−
Rt
T
(β−1)H(t,s) 2
2βγc1 a(s)v(s) u (s)ds
−
Rt q
T
H(t,s)
2βγc1 a(s)v(s) u(s)
+
1
2
2
p
2βγc1 a(s)v(s)h(t, s) ds.
Using the properties of H(t, s), we see that for every t ≥ T0
Rt 2
βγc1
2
H(t, s)φ(s) − H(t, s) γp2c(s)v(s)
−
a(s)v(s)h
(t,
s)
ds
2
1 a(s)
T0
≤ H(t, T0 )u(T0 ) ≤ H(t, T0 ) |u(T0 )| ≤ H(t, t0 ) |u(T0 )| .
Therefore,
Rt t0
=
RT0 t0
+
2
−
H(t, s)φ(s) − H(t, s) γp2c(s)v(s)
1 a(s)
Rt T0
≤ H(t, t0 )
≤
RT0
t0
ds
βγc1
2
2 a(s)v(s)h (t, s)
#
for all t ≥ T0 . This gives
Rt 2
1
lim sup H(t,t
H(t, s)φ(s) − H(t, s) γp2c(s)v(s)
−
0)
1 a(s)
t→∞
ds
2
|φ(s)| ds + |u(T0 )|
t0
βγc1
2
2 a(s)v(s)h (t, s)
−
H(t, s)φ(s) − H(t, s) γp2c(s)v(s)
1 a(s)
T
R0
ds
2
H(t, s)φ(s) − H(t, s) γp2c(s)v(s)
−
1 a(s)
"
βγc1
2
2 a(s)v(s)h (t, s)
βγc1
2
2 a(s)v(s)h (t, s)
ds
|φ(s)| ds + |u(T0 )| < +∞,
t0
which contradicts with the assumption (2.18) of the theorem. This completes the
proof of Theorem 2.5.
Therefore, by Theorem 2.5 we have the following oscillation result.
OSCILLATION CRITERIA FOR A CLASS OF SECOND ORDER NONLINEAR ...
49
Corollary 2.6. Let (C1)-(C4) hold. Suppose that there exists a function g ∈
C 1 ([t0 , ∞) ; R) such that, for some integer n > 2 and some β ≥ 1
#
Zt "
2
γp2 (s)v(s)
βγc1 (n − 1)
n−3
1−n
n−1
lim sup t
(t − s)
φ(s) −
−
a(s)v(s) (t − s)
ds = ∞,
2c1 a(s)
2
t→∞
t0
(2.23)
where φ(t) and v(t) are as in Theorem 2.5. Then Eq. (1.1) is oscillatory.
Example 2.7. For t ≥ 1, consider the nonlinear differential equation
h
i
p
2
(t) x′ (t)
x′ (t)
1 + sin2 t 2+x
+
t
1 + sin2 t 1+x
2
2
′
′2 (t)
1+x (t) 1+x (t)
+ 2 + 38 t2 x(t) 1 +
= 0,
ds = ∞
1
2+x2 (t)
(2.24)
Obviously, for all x ∈ (−∞, ∞) one has 1 ≤ Ψ (x) ≤ 2 and f (x)/x ≥ 1 = µ. Let
g(t) = 0 and γ = 1, then v(t) = 1, and φ(t) = 2 + 14 t2 . Let us take n = 3, and for
any β ≥ 1,
i
Rt h
2
n−3
βγc1 (n−1)2
lim sup t1−n
(t − s)n−1 φ(s) − γp2c(s)v(s)
−
a(s)v(s)
(t
−
s)
ds
2
1 a(s)
t→∞
= lim sup t12
t→∞
1
Rt (t − s)2 × 2 − 4β 1 + sin2 s
1
Therefore, Eq. (2.24) is oscillatory by Corollary 2.6.
Theorem 2.8. Suppose that (C1)-(C4) are satisfied. Suppose also that there exist
functions H ∈ P, g ∈ C 1 ([t0 , ∞) ; R) and χ ∈ C ([t0 , ∞) ; R) such that (1.4) holds,
and for all t > t0 , any T ≥ t0 , and for some β > 1,
Zt 1
γp2 (s)v(s) βγc1
2
lim sup
H(t, s)φ(s) − H(t, s)
−
a(s)v(s)h (t, s) ds ≥ χ(T ),
2c1 a(s)
2
t→∞ H(t, T )
T
(2.25)
where φ(t) and v(t) are the same as in Theorem 2.5. If (2.1) is satisfied, Eq. (1.1)
is oscillatory.
Proof. The proof of this theorem is similar to that of the Theorem 2.1 and
hence it is omitted.
Theorem 2.9. Let all assumptions of Theorem 2.8 satisfied except that condition
(2.25) be replaced with
Zt 1
γp2 (s)v(s) βγc1
lim inf
H(t, s)φ(s) − H(t, s)
−
a(s)v(s)h2 (t, s) ds ≥ χ(T ).
t→∞ H(t, T )
2c1 a(s)
2
T
Then Eq. (1.1) is oscillatory.
Proof. By a similar argument to that in the proof of Theorem 2.4, one can
complete the proof of this theorem. Therefore, we omit the detailed proof for the
theorem.
Remark 2.10. If f (x) = x, then q(t) ≥ 0 is not necessary in the above Theorems.
50
E. TUNÇ, H. AVCI
Remark 2.11. If (2.5) is replaced by
a(t)Ψ (x (t)) k (x′ (t))
+ g(t) ,
u(t) = v(t)
f (x (t))
then, without putting any sign condition on q(t), we can obtain similar oscillation
results that are derived in the main results section of this paper for Eq. (1.1). But
in this case the assumption f ′ (x) ≥ σ > 0 is necessary.
Remark 2.12. When k(x′ ) = x′ , it is easy to see that Theorems 2.8 and 2.9 reduce
to Theorems 9 and 10 of [1] with γ = 1, respectively.
Acknowledgment: The authors would like to thank the referee for many
helpful corrections and suggestions.
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E. Tunç
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpas.a University,
60240, Tokat (Turkey)
E-mail address: ercantunc72@yahoo.com
H. Avcı
Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University,
Samsun (Turkey)