Electronic Journal of Differential Equations, Vol. 2010(2010), No. 131, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATION OF SOLUTIONS FOR THIRD ORDER
FUNCTIONAL DYNAMIC EQUATIONS
ELMETWALLY M. ELABBASY, TAHER S. HASSAN
Abstract. In this article we study the oscillation of solutions to the third
order nonlinear functional dynamic equation
L3 (x(t)) +
n
X
pi (t)Ψk αk i(x(hi (t))) = 0,
i=0
on an arbitrary time scale T. Here
L0 (x(t)) = x(t),
Lk (x(t)) =
“ [L
k−1 x(t)]
ak (t)
∆ ”γ k
k
,
k = 1, 2, 3
with a1 , a2 positive rd-continuous functions on T and a3 ≡ 1; the functions pi
are nonnegative rd-continuous on T and not all pi (t) vanish in a neighborhood
of infinity; Ψk c(u) = |u|c−1 u, c > 0. Our main results extend known results
and are illustrated by examples.
1. Introduction
The theory of time scales, which has recently received a lot of attention, was
introduced by Stefan Hilger in his PhD dissertation [30], written under the direction
of Bernd Aulbach. Since then a rapidly expanding body of literature has sought to
unify, extend, and generalize ideas from discrete calculus, quantum calculus, and
continuous calculus to arbitrary time scale calculus. Recall that a time scale T is a
nonempty, closed subset of the reals, and the cases when this time scale is the reals or
the integers represent the classical theories of differential and of difference equations.
Many other interesting time scales exist, and they give rise to many applications (see
[7]). Not only does the new theory of the so-called “dynamic equations” unify the
theories of differential equations and difference equations, but also extends these
classical cases to cases “in between”, e.g., to the so-called q-difference equations
when T =q N0 (which has important applications in quantum theory [31]) and can
be applied on different types of time scales like T =hZ, T = N20 and T = Hn the
space of harmonic numbers. In this work a knowledge and understanding of time
scales and time scale notation is assumed; for an excellent introduction to the
calculus on time scales, see Hilger [30], and Bohner and Peterson [7, 8].
2000 Mathematics Subject Classification. 34K11, 39A10, 39A99.
Key words and phrases. Oscillation; third order; functional dynamic equations; time scales.
c
2010
Texas State University - San Marcos.
Submitted April 13, 2010. Published September 14, 2010.
1
2
E. M. ELABBASY, T. S. HASSAN
EJDE-2010/131
We are concerned with the oscillatory behavior of solutions to the third order
nonlinear functional dynamic equation
n
X
L3 (x(t)) +
pi (t)Ψk αi (x(hi (t))) = 0,
(1.1)
i=0
on an arbitrary time scale T, where L0 (x(t)) = x(t),
Lk (x(t)) = (
[Lk−1 x(t)]∆ γk k
) ,
ak (t)
k = 1, 2, 3,
with a1 , a2 are positive rd-continuous functions on T and a3 ≡ 1, and γk k, k = 1, 2
are quotients of odd positive integers, γk 3 = 1 and αk 0 = γ1 γk 2. We also assume
Ψk c(u) = |u|c−1 u, c > 0 and pi , i = 0, 1, 2, . . . , n are nonnegative rd-continuous
functions on T such that not all pi (t) vanish in a neighborhood of infinity. The
functions hi : T → T satisfy limt→∞ hi (t) = ∞, i = 0, 1, 2, . . . , n. Since we are
interested in the oscillatory behavior of solutions near infinity, we assume that
sup T = ∞, and define the time scale interval [t0 , ∞)T by [t0 , ∞)T := [t0 , ∞) ∩ T.
1
By a solution of (1.1) we mean a nontrivial real-valued function L0 x ∈ Crd
[Tx , ∞)T ,
1
1
Tx ≥ t0 which has the property that L1 x ∈ Crd [Tx , ∞)T , L2 x ∈ Crd [Tx , ∞)T and
x(t) satisfies equation (1.1) on [Tx , ∞)T , where Crd is the space of rd-continuous
functions. The solutions vanishing identically in some neighborhood of infinity will
be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it
is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.
Throughout the paper, we define, for i = 0, 1, . . . , n and k = 1, 2,
Z s
1/γ 1
A(s, u) := a1 (s)A2 k (s, u), Ak (s, u) :=
ak (u)∆u,
u
Z ∞
1/γ 1
ϕk i(t, u) := A1 (hi (t), u)φi,2 k (t), φk i, k(t) :=
ak (s)∆s,
hi (t)
h(t) := min{t, hi (t); i = 0, 1, . . . , n},
∆
(Φ (t))+ := max{0, Φ∆ (t)}.
In the previous few years, there has been an increasing interest in obtaining
sufficient conditions for the oscillation/nonoscillation of solutions of different classes
of dynamic equations, we refer the reader to the papers [1, 2, 3, 4, 6, 9, 10, 11, 12, 13,
14, 28, 29, 15, 16, 17, 18, 23, 25, 27] and the references cited therein. Regarding third
order dynamic equations, [19, 20, 26, 21, 22] considered the third order dynamic
equation (1.1), in particular case and under quite restrictive conditions, for example,
Erbe et al [19] studied the dynamic equation (1.1), when n = 0, γ1 = γk 2 = 1, αk 0 =
1 and h0 (t) ≡ t and established sufficient conditions which ensure the solution of
equation (1.1) is either oscillatory or tends to zero and Erbe et al [20] extended
the results which established in [19], when γk 2 ≥ 1 is the quotient of odd positive
integers. Also, Hassan [26] generalized the results which were established in [20, 19],
for the equation (1.1), when n = 0, γk 1 = 1, αk 0 = γk 2, γ2 > 0 is the quotient of
odd positive integers, and h0 (t) ≤ t and h∆
0 (t) ≥ 0, for t ∈ T and h0 ◦ σ = σ ◦ h0 .
A number of sufficient conditions for oscillation were obtained for the cases when,
for k = 1, 2,
Z
∞
ak (t)∆t = ∞.
t0
Erbe, Hassan and Peterson [21] solved this problem for the third order functional
dynamic equation (1.1), when n = 0, γk 1 = 1, αk 0 = γk 2, γk 2 > 0 is the quotient
EJDE-2010/131
OSCILLATION OF SOLUTIONS
3
of odd positive integers and for both cases, for k = 1, 2,
Z ∞
ak (t)∆t = ∞,
(1.2)
t0
and
Z
∞
ak (t)∆t < ∞.
(1.3)
t0
Recently, Erbe, Hassan and Peterson [22] extended previous results for the third
order dynamic equation (1.1) under some restrictive conditions, n = 2 and h ◦ σ =
σ ◦ h.
The fact that the condition h ◦ σ = σ ◦ h is not satisfied for some time scales,
see [15]. The purpose of this paper is to extend the oscillation criteria which are
established by [22], for the more general third order functional dynamic equation
with mixed arguments (1.1), for several terms n and without restrictive condition
h◦σ = σ◦h and for both of the cases (1.2) and (1.3). We will still assume γ1 , γk 2 > 0
are the quotient of odd positive integers and, hence our results will improve and
extend results in the [19, 20, 26, 21, 22], and many known results on nonlinear
oscillation.
2. Main Results
Throughout this paper, we assume that αk 1 > αk 2 > · · · > αk m > αk 0 >
αk m + 1 > · · · > αk n > 0. Before stating our main results, we begin with the
following lemmas which will play an important role in the proof of our main results.
Lemma 2.1. For each n-tuple (αk 1, αk 2, . . . , αk n), there exists (ηk 1, ηk 2, . . . , ηn )
with 0 < ηk i < 1 satisfying
n
X
αk iηk i = αk 0,
i=1
n
X
ηi = 1.
(2.1)
i=1
The proof of the above lemma is the same as [29, Lemma 2.1].
Lemma 2.2. Let a1 be nondecreasing and delta differentiable on [t0 , ∞)T and x be
a positive solution of (1.1) such that
x∆ (t) > 0
and
(L1 (x(t)))∆ > 0,
for t ≥ t0 .
(2.2)
Then, if
n Z
X
i=0
∞
ki
pi (t)hα
i (t)∆t = ∞,
(2.3)
t0
we have
x∆∆ (t) > 0,
x(t) ∆
<0
t
on [t0 , ∞)T .
Proof. Let x be as in the statement of this lemma. Since
(L1 (x(t)))∆ =
((x∆ (t))γk 1 )∆
(x∆ (t))γ1 (aγ1k 1 (t))∆
−
> 0,
aγ11 σ (t)
aγ1k 1 (t)aγ1k 1σ (t)
for t ≥ t0 .
(2.4)
4
E. M. ELABBASY, T. S. HASSAN
EJDE-2010/131
Using the Pötzsche chain rule [7, Theorem 1.90], we have
Z 1
((x∆ (t))γk 1 )∆ = γk 1
[x∆ (t) + hµ(t)x∆∆ (t)]γk 1−1 dh x∆∆ (t)
0
= γk 1x∆∆ (t)
(2.5)
1
Z
[hx∆σ (t) + (1 − h)x∆ (t)]γk 1−1 dh.
0
and
(aγ1k 1 (t))∆ = γk 1
Z
1
γ1 −1
[a1 (t) + hµ(t)a∆
dh a∆
1 (t)]
1 (t)
0
=
γk 1a∆
1 (t)
Z
(2.6)
1
[haσ1 (t)
γk 1−1
+ (1 − h)a1 (t)]
dh.
0
Using that a1 is a nondecreasing and x is increasing, we obtain, from (2.4), (2.5)
∆
< 0. To see
and (2.6) that x∆∆ (t) > 0 on [t0 , ∞)T . Next, we show that x(t)
t
∆
∆
∆∆
this, let U (t) := x(t) − tx (t), then U (t) = −σ(t)x (t) < 0 for t ∈ [t0 , ∞)T .
This implies that U (t) is strictly decreasing on [t0 , ∞)T . We claim U (t) > 0 on
[t0 , ∞)T . Assume not, then there exists t1 ≥ t0 such that U (t) < 0 on [t1 , ∞)T .
Therefore,
x(t) ∆
tx∆ (t) − x(t)
U (t)
=
=−
> 0,
t
tσ(t)
tσ(t)
for t ∈ [t1 , ∞)T .
(2.7)
Pick t2 ∈ [t1 , ∞)T so that hi (t) ≥ t1 , for t ≥ t2 and i = 0, 1, . . . , n. Then, for t ≥ t2
and i = 0, 1, . . . , n, we have from (2.7) that
x(t1 )
x(hi (t))
≥
=: c > 0,
hi (t)
t1
for t ≥ t2 and i = 0, 1, . . . , n,
so x(hi (t)) ≥ chi (t) for t ≥ t2 and i = 0, 1, . . . , n. Now by integrating both sides of
the dynamic equation (1.1) from t2 to t, we have
n Z t
X
L2 (x(t2 )) − L2 (x(t)) =
pi (s)Ψk αi (x(hi (s)))∆s.
i=0
t2
This implies, from (2.2) that
Z t
n Z t
n
X
X
αk i
i
L2 (x(t2 )) ≥
pi (s)Ψk αk i(x(hi (s)))∆s ≥
c
pi (s)hα
i (s)∆s.
i=0
t2
i=0
(2.8)
t2
Letting t → ∞ we obtain a contradiction to assumption (2.3). Hence U (t) =
x(t) − tx∆ (t) > 0 on [t0 , ∞)T . Consequently,
U (t)
x(t) ∆
=−
< 0,
t
tσ(t)
t ∈ [t0 , ∞)T .
This completes the proof of Lemma 2.2.
(2.9)
First, we establish oscillation criteria for (1.1) when (1.2) holds.
Theorem 2.3. Let αk 0 = γk 1γk 2 and hi (t), i = 0, 1, 2, . . . , n be nondecreasing
functions on [t0 , ∞)T . Assume that (1.2) and
Z ∞
n
Z ∞
X
1/γk 2 1/γk 1
a1 (t)
a2 (s)
Pi (s)
∆s
∆t = ∞,
(2.10)
t0
t
i=0
EJDE-2010/131
OSCILLATION OF SOLUTIONS
5
R∞
where Pi (s) := s pi (u)∆u, i = 0, 1, . . . , n. Furthermore, suppose that, for all
sufficiently large T1 , there is T > T1 such that h(T ) > T1 and
Z h(t)
αk 0
lim sup Q1 (t)
A(s, T1 )∆s
> 1,
(2.11)
t→∞
T1
where
Q1 (t) := P0 (t) +
n
Y
(ηk i−1 Pi (t))ηk i .
i=1
Then every solution of (1.1) is either oscillatory or tends to zero.
Proof. Assume (1.1) has a non-oscillatory solution x on [t0 , ∞)T . Then, without
loss of generality, there is a t1 ∈ [t0 , ∞)T , sufficiently large, such that x(t) > 0 and
x(hi (t)) > 0 on [t1 , ∞)T , for i = 0, 1, 2, . . . , n and not all of the pi (t)’s are identically
zero on [t1 , ∞)T . From (1.1), we have
L3 (x(t)) = −
n
X
pi (t)Ψk αi (x(hi (t))) < 0,
(2.12)
i=0
on [t1 , ∞)T . Then L2 (x(t)) is strictly decreasing on [t1 , ∞)T . Therefore, x∆ (t) and
(L1 (x(t)))∆ are eventually of one sign. Then, there exists a sufficiently large t2 ≥ t1
so that x∆ (t) and L∆
1 (x(t)) are of fixed sign for t ≥ t2 . Therefore, we consider the
following four cases:
(i) x∆ (t) < 0 and (L1 (x(t)))∆ < 0;
(ii) x∆ (t) > 0 and (L1 (x(t)))∆ < 0;
(iii) x∆ (t) < 0 and (L1 (x(t)))∆ > 0;
(iv) x∆ (t) > 0 and (L1 (x(t)))∆ > 0, on [t2 , ∞)T .
For case (i), we have
Z t
Z t
1/γ1
1/γ1
x(t) = x(t2 ) +
a1 (s)L1 (x(s))∆s ≤ x(t2 ) + L1 (x(t2 ))
a1 (s)∆s.
t2
t2
Hence by (1.2), we have limt→∞ x(t) = −∞, which contradicts the fact that x is a
positive solution of (1.1). For the case (ii), from (2.12) we have
Z t
1/γ 2
L1 (x(t)) = L1 (x(t2 )) +
a2 (s)L2 k (x(s))∆s
t2
1/γk 2
≤ L1 (x(t2 )) + L2
Z
t
(x(t2 ))
a2 (s)∆s.
t2
Then, by (1.2), we have limt→∞ L1 (x(t)) = −∞, which contradicts x∆ (t) > 0, for
t ≥ t2 . For the case (iii), we have limt→∞ x(t) = c0 ≥ 0 and limt→∞ L1 (x(t)) =
c1 ≤ 0. If we assume c0 > 0, then Ψk αk i(x(hi (t))) ≥ K, for i = 0, 1, 2, . . . , n and
i
t ≥ t3 ≥ t2 , where K := min{cα
0 : i = 0, 1, 2, . . . , n}. Integrating equation (1.1)
from t to ∞, we obtain
n Z ∞
X
−L2 (x(t)) < −
pi (s)Ψk αk i(x(hi (s)))∆s
i=0 t
n Z ∞
X
≤ −c0
i=0
t
pi (s)∆s = −c0
n
X
i=0
Pi (t),
6
E. M. ELABBASY, T. S. HASSAN
EJDE-2010/131
which implies
n
X
1/γk 2
−(L1 (x(t))) < −C0 a2 (t)
Pi (t)
,
∆
i=0
where C0 := K
1/γk 2
≥ 0. Integrating this inequality from t to ∞, we obtain
Z ∞
n
X
1/γ2
L1 (x(t)) < −C0
a2 (s)
Pi (s)
∆s + c1
t
Z
i=0
∞
≤ −C0
n
X
1/γk 2
a2 (s)
Pi (s)
∆s.
t
i=0
Finally, integrating the last inequality from t3 to t, we obtain
Z t
n
Z ∞
1/γk 1
X
∆s + x(t3 ).
x(t) < −c0
a1 (s)
a2 (u)(
Pi (u))1/γ2 ∆u
t3
s
i=0
Hence by (2.10), we have limt→∞ x(t) = −∞, which contradicts the fact that x
is a positive solution of (1.1). Thus, we conclude that limt→∞ x(t) = 0. For the
case (iv), integrating both sides of the dynamic equation (1.1) from t to ∞ and
then using the facts that x(t) is strictly increasing and hi (t), i = 0, 1, 2, . . . , n are
nondecreasing, we obtain
n
X
Ψk αk i(x(hi (t)))Pi (t) ≤ L2 (x(t)).
(2.13)
i=0
Since L2 (x(t)) is strictly decreasing on [t2 , ∞)T , we obtain
Z t
1/γ 2
L1 (x(t)) > L1 (x(t)) − L1 (x(t2 )) =
a2 (s)L2 k (x(s))∆s
t2
1/γk 2
≥ L2
Z
t
(x(t))
1/γk 2
a2 (s)∆s = L2
(x(t))A2 (t, t2 ).
t2
Hence
1/γk 1
x∆ (t) ≥ a1 (t)A2
1/α0
(t, t2 )L2
(x(t)).
Similarly, we see that
1/αk 0
x(t) ≥ L2
Z
t
(x(t))
1/γk 1
a1 (s)A2
(s, t2 )∆s,
t2
and so
Z t
αk 0
xαk 0 (t) ≥ L2 (x(t))
A(s, t2 )∆s
.
(2.14)
t2
Pick t3 > t2 , sufficiently large, so that h(t) > t2 , for t ≥ t3 . Then from (2.14), for
t ≥ t3 , we obtain
Z h(t)
αk 0
Ψk αk 0(x(h(t))) ≥ L2 (x(h(t)))
A(s, t2 )∆s
t2
(2.15)
Z h(t)
αk 0
≥ L2 (x(t))
A(s, t2 )∆s
t2
.
EJDE-2010/131
OSCILLATION OF SOLUTIONS
7
Using (2.15) in (2.13), for t ≥ t3 , we find that
n
X
Z
Ψk αk i(x(hi (t)))Pi (t) ≤ Ψαk 0 (x(h(t)))
h(t)
−αk 0
A(s, t2 )∆s
,
t2
i=0
∆
which yields from the fact x (t) > 0 on [t2 , ∞)T that
n
Z h(t)
−αk 0
X
P0 (t) +
Ψk αk i − α0 (x(h(t)))Pi (t) <
A(s, t2 )∆s
.
(2.16)
t2
i=1
Using the arithmetic-geometric mean inequality see [5, Page 17]
n
X
ηk iui ≥
i=1
n
Y
uηi i ,
where ui ≥ 0.
i=1
From Lemma 2.1, for t ≥ T , we obtain
n
X
Ψk αk i − αk 0(x(h(t)))Pi (t) =
n
X
ηk i(ηk i−1 Ψk αk i − α0 (x(h(t)))Pi (t))
i=1
i=1
≥
n
Y
(ηk i−1 Pi (t))ηk i Ψk ηk i(αi − αk 0)(x(h(t)))
i=1
n
(2.1) Y
=
(ηk i−1 Pi (t))ηk i ,
i=1
(2.17)
Using (2.17) in (2.16), we have
Z h(t)
αk 0
Q1 (t)
A(s, t2 )∆s
< 1,
for t ≥ t3 .
t2
Then
Z
lim sup Q1 (t)
t→∞
h(t)
αk 0
A(s, t2 )∆s
≤ 1,
t2
which leads to a contradiction to (2.11).
Theorem 2.4. Assume that (1.2) and (2.10) hold. Furthermore, suppose that, for
all sufficiently large T ∈ [t0 , ∞)T , such that hi (T ) > t0 , i = 0, 1, . . . , n,
n Z ∞
X
ki
pi (t)Aα
(2.18)
1 (hi (t), t0 )∆t = ∞.
i=0
t0
Then every solution of equation (1.1) is either oscillatory or tends to zero.
Proof. Assume (1.1) has a non-oscillatory solution x on [t0 , ∞)T . Then, without
loss of generality, there is a t1 ∈ [t0 , ∞)T , sufficiently large, such that x(t) > 0 and
x(hi (t)) > 0 on [t1 , ∞)T , for i = 0, 1, 2, . . . , n and not all pi (t) are identically zero on
[t1 , ∞)T . Therefore, as in the proof of Theorem 2.3, we obtain there exists t2 ≥ t1
so that
(L2 (x(t)))∆ < 0, (L1 (x(t)))∆ > 0, on [t2 , ∞)T ,
and either L1 (x(t)) > 0 on [t2 , ∞)T or limt→∞ x(t) = 0. Assume L1 (x(t)) > 0 on
[t2 , ∞)T . Since (L1 (x(t)))∆ > 0 on [t2 , ∞)T , then
L1 (x(t)) ≥ L1 (x(t2 )) =: c > 0.
8
E. M. ELABBASY, T. S. HASSAN
Thus
Z
EJDE-2010/131
t
x(t) ≥ x(t) − x(t2 ) ≥ C
a1 (s)∆s,
t2
where C := c1/γk 1 . Pick t3 ≥ t2 such that hi (t) > t2 , for t ≥ t3 and i = 0, 1, . . . , n,
then, for i = 0, 1, . . . , n,
x(hi (t)) > CA1 (hi (t), t2 )
on [t3 , ∞)T .
(2.19)
It follows from (1.1) and (2.19) that
−L3 (x(t)) =
n
X
pi (t)Ψk αi (x(hi (t))) ≥
i=0
n
X
pi (t)[CA1 (hi (t), t2 )]αk i .
i=0
Integrating both sides of the last inequality from t3 to t, we have
n Z t
X
pi (s)[CA1 (hi (s), t2 )]αk i ∆s + L2 (x(t))
L2 (x(t3 )) >
>
i=0 t3
n Z t
X
i=0
pi (s)[CA1 (hi (s), t2 )]αk i ∆s,
t3
which contradicts (2.18). This completes the proof.
Example 2.5. Consider the third order dynamic equation (1.1), for t0 ≥ 1, where
0 < γk 1 ≤ 1 and γk 2 = γk1 1 are the quotient of odd positive integers and αk i,
i = 0, 1, . . . , n are positive constants and we assume hi (t) ≤ tγk 2 , i = 0, 1, . . . , n.
Let
1
1
a1 (t) = 1, a2 (t) = γ 1 , pi (t) =
, i = 0, 1, . . . , n.
t k
thi (t)
It is clear that conditions (1.2) hold, since
Z ∞
Z ∞
∆t
= ∞, for 0 < γk 1 ≤ 1,
∆t = ∞,
γk 1
t
t0
t0
by [8, Example 5.60]. Note that
Z ∞
Z ∞
Z ∞
Z ∞
∆u
∆u
1
−1 ∆
1 1
pi (u)∆u =
≥
≥
∆u =
,
γ
2+1
γ
2
k
uhi (u)
u
γk 2 s
u
γk 2 sγ2
s
s
s
Z
Z ∞
Z ∞
n
X
∆s
γk 0 ∞ −1 ∆
γk 0
a2 (s)(
Pi (s))1/γk 2 ∆s ≥ γk 0
≥
( γ 1 ) ∆s =
,
γk 1+1
k
s
γ
1
s
γ
t γk 1
k
1
t
t
t
i=0
1/γk 2
and
where γk 0 := ( n+1
γk 2 )
Z ∞
Z
n
Z ∞
1/γk 1
X
a1 (t)
a2 (s)(
Pi (s))1/γk 2 ∆s
∆t ≥ γ
t0
t
i=0
∞
t0
∆t
= ∞,
t
1/γ1
where γ := (γk 0/γk 1
, so that condition (2.10) holds. To apply Theorem 2.4,
it remains to prove that condition (2.18) holds. To see this, note that
Z ∞
n Z ∞
h Z hi (t)
iαi
X
1
t0 a1 (s)∆s ∆t ≥
pi (t)
−
∆t = ∞,
t
th
0 (t)
t0
t0
i=0 t0
R∞
for those time scales, where t0 th∆t
< ∞. Then, by Theorem 2.4, every solution
0 (t)
of equation (1.1) is either oscillatory or tends to zero.
EJDE-2010/131
OSCILLATION OF SOLUTIONS
9
By using Lemma 2.2, we obtain the following oscillation criterion for (1.1).
Theorem 2.6. Let αk 0 = γk 1γk 2 and a1 be nondecreasing and delta differentiable
on [t0 , ∞)T . Assume that (1.2), (2.3) and (2.10) hold. Furthermore, suppose that
there exists a positive ∆−differentiable function φ(t) and, for all sufficiently large
T1 , there is T > T1 such that
Z th
i
((φ∆ (s))+ )αk 0+1
∆s = ∞,
(2.20)
φ(s)Q2 (s) −
lim sup
(αk 0 + 1)αk 0+1 (φ(s)A(s, T1 ))αk 0
t→∞
T
where
Q2 (t) := p0 (t)H0 (t) +
n
Y
(ηk i−1 pi (t)Hi (t))ηk i ,
i=1
and, for i = 0, 1, . . . , n,
(
1,
Hi (t) :=
( hit(t) )αk i ,
hi (t) ≥ t
hi (t) ≤ t.
Then every solution of (1.1) is either oscillatory or tends to zero.
Proof. Assume (1.1) has a non-oscillatory solution x on [t0 , ∞)T . Then, without
loss of generality, there is a t1 ∈ [t0 , ∞)T , sufficiently large, such that x(t) > 0 and
x(hi (t)) > 0 on [t1 , ∞)T , for i = 0, 1, 2, . . . , n and not all of the pi (t)’s are identically
zero on [t1 , ∞)T . Therefore, as in the proof of Theorem 2.3, we obtain there exists
t2 ≥ t1 so that
(L2 (x(t)))∆ < 0,
(L1 (x(t)))∆ > 0,
on [t2 , ∞)T ,
and either L1 (x(t)) > 0 on [t2 , ∞)T or limt→∞ x(t) = 0. Assume L1 (x(t)) > 0 on
[t2 , ∞)T . Consider the Riccati substitution
w(t) = φ(t)
L2 (x(t))
.
xα0 (t)
By the product rule and then the quotient rule
φ(t)
φ(t) ∆ σ
(L2 (x(t)))∆ + α 0
L2 (x(t))
xαk 0 (t)
x k (t)
L3 (x(t)) φ∆ (t)
φ(t)(xαk 0 (t))∆ σ
= φ(t) α 0
+
−
L (x(t)).
x k (t)
xαk 0σ (t) xαk 0 (t)xαk 0σ (t) 2
w∆ (t) =
(2.21)
From (2.21) and the definition of w(t), we have, for t ≥ t2
w∆ (t) = −φ(t)
n
X
i=0
pi (t)
xαk i (hi (t)) φ∆ (t) σ
φ(t)(xαk 0 (t))∆ σ
+ σ w (t) − σ
w (t).
α
0
k
x (t)
φ (t)
φ (t)xαk 0 (t)
Now, for a fixed i and let t be a fixed point in [t2 , ∞)T . Then either hi (t) ≤ t or
hi (t) ≥ t. First, consider the case when hi (t) ≥ t. Then, by using the fact that
x is strictly increasing, we obtain x(hi (t)) ≥ x(t). Next, consider the case when
hi (t) ≤ t. Then, in view of Lemma 2.2, we obtain x(hi (t)) ≥ hit(t) x(t). It follows
from the definition of Hi (t) that, for t ≥ t2 ,
w∆ (t) ≤ −φ(t)
n
X
i=0
pi (t)Hi (t)xαk i−αk 0 (t) +
φ(t)(xαk 0 (t))∆ σ
φ∆ (t) σ
w
(t)
−
w (t).
φσ (t)
φσ (t)xαk 0 (t)
10
E. M. ELABBASY, T. S. HASSAN
EJDE-2010/131
As in the proof of Theorem 2.3, we obtain, for t ≥ t2 ,
w∆ (t) ≤ −φ(t)Q2 (t) +
φ∆ (t) σ
φ(t)(xαk 0 (t))∆ σ
w
w (t).
(t)
−
φσ (t)
φσ (t)xαk 0 (t)
Then, by the Pötzsche chain rule [7, Theorem 1.90], we obtain
Z 1
(xαk 0 (t))∆ = αk 0
[x(t) + hµ(t)x∆ (t)]αk 0−1 dh x∆ (t)
0
Z
1
= αk 0
[(1 − h)x(t) + hxσ (t)]αk 0−1 dh x∆ (t)
0
(
αk 0x(αk 0−1)σ (t) x∆ (t), 0 < αk 0 ≤ 1
≥
αk 0xαk 0−1 (t) x∆ (t),
αk 0 ≥ 1.
If 0 < αk 0 ≤ 1, we have
w∆ (t) ≤ −φ(t)Q2 (t) +
αk 0φ(t)wσ (t) x∆ (t) xσ (t) αk 0
φ∆ (t) σ
w
(t)
−
,
φσ (t)
φσ (t)
xσ (t) x(t)
whereas if αk 0 ≥ 1, we have
w∆ (t) ≤ −φ(t)Q2 (t) +
αk 0φ(t)wσ (t) x∆ (t) xσ (t)
φ∆ (t) σ
w
(t)
−
.
φσ (t)
φσ (t)
xσ (t) x(t)
Using that x(t) is strictly increasing on [t2 , ∞)T , we obtain that, for αk 0 > 0,
w∆ (t) ≤ −φ(t)Q2 (t) +
φ∆ (t) σ
αk 0φ(t)wσ (t) x∆ (t)
w
(t)
−
.
φσ (t)
φσ (t)
xσ (t)
(2.22)
Then using that L2 (x(t)) is strictly decreasing on [t2 , ∞)T , we obtain that, for
t ≥ t2 ,
Z t
1/γ 2
L1 (x(t)) > L1 (x(t)) − L1 (x(t2 )) =
a2 (s)L2 k (x(s))∆s
t2
(2.23)
Z t
1/γk 2
σ/γk 2
≥ L2
(x(t))
a2 (s)∆s ≥ L2
(x(t))A2 (t, t2 ).
t2
From (2.22) and (2.23), we obtain
w∆ (t) ≤ −φ(t)Q2 (t) +
where α :=
αk 0+1
αk 0 .
X α :=
αk 0φ(t)A(t, t2 ) ασ
(φ∆ (t))+ σ
w (t) −
w (t),
φσ (t)
φασ (t)
(2.24)
Define X ≥ 0 and Y ≥ 0 by
αk 0φ(t)A(t, t2 ) ασ
w (t),
φασ (t)
Y α−1 :=
(φ∆ (t))+
.
α(α0 φ(t)A(t, t2 ))1/α
Then, using the inequality, see [24],
αXY α−1 − X α ≤ (α − 1)Y α ,
(2.25)
we obtain
(φ∆ (t))+ σ
αk 0φ(t)A(t, t2 ) ασ
((φ∆ (t))+ )αk 0+1
.
w (t) −
w (t) ≤
σ
ασ
φ (t)
φ (t)
(αk 0 + 1)αk 0+1 (φ(t)A(t, t2 ))αk 0
From the above inequality and (2.25), we obtain
w∆ (t) ≤ −φ(t)Q2 (t) +
((φ∆ (t))+ )αk 0+1
.
(αk 0 + 1)α0 +1 (φ(t)A(t, t2 ))αk 0
EJDE-2010/131
OSCILLATION OF SOLUTIONS
11
Integrating both sides from t2 to t,
Z th
i
((φ∆ (s))+ )αk 0+1
φ(s)Q2 (s) −
∆s ≤ w(t2 ) − w(t) ≤ w(t2 ),
(αk 0 + 1)α0 +1 (φ(s)A(s, t2 ))αk 0
t2
which leads to a contradiction to (2.20).
Example 2.7. Consider the third order nonlinear dynamic equation (1.1), for
t0 ≥ 1, where γk 1 and γk 2 are the quotient of odd positive integers and αk i,
i = 0, 1, . . . , n are positive constants with αk 1 > αk 2 > · · · > αk m > αk 0 >
αm+1 > · · · > αk n > 0 and also, we assume hi (t) ≥ t, i = 0, 1, . . . , n such that
(2.3) holds. Note that since hi (t) ≥ t, i = 0, 1, . . . , n it follows that Hi (t) ≡ 1,
i = 0, 1, . . . , n. Let
1
a1 (t) = t1/αk 0 , a2 (t) = 1−1/γ 2 ,
k
t
βk 0
βk i
p0 (t) = α 0+2 , pi (t) = η i+2 , i = 1, 2, . . . , n.
t k
t k
where ηk i, i = 1, 2, . . . , n and βk i, i = 0, 1, . . . , n are positive constants such that
αk 0 ≥ ηk i, i = 1, 2, . . . , n. It is clear that conditions (1.2) hold, since
Z ∞
Z ∞
a1 (t)∆t =
t1/αk 0 ∆t = ∞,
t0
and
Z
t0
∞
Z
∞
∆t
= ∞,
1−1/γk 2
t
t0
t0
by [7, Example 5.60]. Therefore, we can find T > T1 such that A2 (s, T1 ) ≥ 1, for
s ≥ T . Also, using the Pötzsche chain rule,
Z ∞
Z ∞
∆u
P0 (s) =
p0 (u)∆u = β0
αk 0+2
u
s
s
Z ∞
βk 0
−1 ∆
βk 0
1
≥
( α 0+1 ) ∆u =
,
α
k
k
αk 0 + 1 s u
αk 0 + 1 s 0+1
and
Z ∞
Z ∞
∆u
Pi (s) =
pi (u)∆u = βi
uηk i+2
s
s
Z ∞
βk i
−1
≥
( η i+1 )∆ ∆u
ηk i + 1 s u k
βk i
1
=
ηk i + 1 sηk i+1
βk i
1
≥
, i = 1, 2, . . . , n.
αk 0 + 1 sαk 0+1
Hence
n
Z ∞
X
1/γk 2 1/γk 1
Z ∞ ∆s 1/γk 1
a2 (s)
Pi (s)
∆s
≥β
sγk 1+1
t
t
i=0
Z ∞ −1
1/γk 1
β
≥
( γ 1 )∆ ∆s
1/γk 1
s k
γk 1
t
β
1
=
,
1/γ 1
γk 1 k t
a2 (t)∆t =
12
E. M. ELABBASY, T. S. HASSAN
1
where β := ( αk 0+1
Z
EJDE-2010/131
Pn
βk i)1/αk 0 . Then
n
∞
Z ∞
1/γk 1
X
a1 (t)
a2 (s)(
Pi (s))1/γk 2 ∆s
∆t
i=0
t0
t
β
=
1/γ 1
γk 1 k
Z
i=0
∞
1
t1−1/αk 0
t0
∆t = ∞,
so that condition (2.10) holds. Let us take φ(t) = tαk 0+1 , then, by the Pötzsche
chain rule
Z 1
φ∆ (t) = (αk 0 + 1)
(t + hµ(t))α0 dh ≤ (αk 0 + 1)(σ(t))αk 0 .
0
Now, we assume T is a time scale satisfying σ(t) ≤ kt, for some k > 0, t ≥ Tk > T .
Note that
Z t
((φ∆ (s))+ )αk 0+1
lim sup
[φ(s)Q2 (s) −
]∆s
(αk 0 + 1)αk 0+1 (φ(s)A(s, T1 ))αk 0
t→∞
T
Z t
βk 0 k αk 0(αk 0+1)
≥ lim sup
[
−
]∆s
s
s
t→∞
Tk
Z t
∆s
≥ (βk 0 − k αk 0(αk 0+1) ) lim sup
= ∞,
t→∞
Tk s
if βk 0 > k αk 0(αk 0+1) and hence (2.20) holds. We conclude that if [T, ∞)T is a time
scale where σ(t) ≤ kt, for some k > 0, t ≥ Tk , then, by Theorem 2.6, every solution
of (1.1) is either oscillatory or tends to zero if βk 0 > k αk 0(αk 0+1) .
In the following, we assume that (1.3) holds and establish sufficient conditions
which ensure that every solution x(t) of (1.1) is either oscillatory or tends to zero.
By Theorems 2.3 and 2.6 and [22, Theorem 2.1], we obtain the following oscillation
criteria for equation (1.1).
Corollary 2.8. Let αk 0 = γk 1γk 2 and hi (t), i = 0, 1, 2, . . . , n be nondecreasing
functions on [t0 , ∞)T . Assume that (1.3) and (2.10) hold. Furthermore, suppose
that, for all sufficiently large T1 ∈ [t0 , ∞)T , there is T > T1 such that hi (T ) > T1 ,
i = 1, 2, . . . , n,
Z ∞
n Z s
hZ t
i1/γk 1
X
a1 (t)
a2 (s)[
pi (u)φk i, 1αk i (u)∆u]1/γk 2 ∆s
∆t = ∞, (2.26)
T
T
i=1
and
Z
∞
T
T
n Z t
hX
i1/γ2
ki
a2 (t)
pi (u)ϕα
(u,
T
)∆u
∆t = ∞.
1
i
i=1
(2.27)
T
If condition (2.11) holds, then every solution of (1.1) is either oscillatory or tends
to zero.
Corollary 2.9. Assume that (1.3), (2.10), (2.26) and (2.27) hold. If (2.18) holds,
then every solution of (1.1) is either oscillatory or tends to zero.
Corollary 2.10. Let αk 0 = γk 1γk 2 and a1 be nondecreasing and delta differentiable
on [t0 , ∞)T . Assume that (1.3), (2.3), (2.10), (2.26) and (2.27) hold. If (2.20)
holds, then every solution of (1.1) is either oscillatory or tends to zero.
EJDE-2010/131
OSCILLATION OF SOLUTIONS
13
Remark 2.11. If (2.10) is not satisfied, we have sufficient conditions which ensure
that every solution x(t) of (1.1) oscillates or limt→∞ x(t) exists as a finite number.
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Elmetwally M. Elabbasy
Department of Mathematics, Faculty of Science, Mansoura University
Mansoura, 35516, Egypt
E-mail address: emelabbasy@mans.edu.eg
URL: http://www.mans.edu.eg/pcvs/10805/
Taher S. Hassan
Department of Mathematics, Faculty of Science, Mansoura University
Mansoura, 35516, Egypt
E-mail address: tshassan@mans.edu.eg
URL: http://www.mans.edu.eg/pcvs/30140/