Mathematische Nachrichten, 20 December 2008
Belohorec-type Oscillation Theorem for Second
Order Sublinear Dynamic Equations on Time Scales
Lynn Erbe∗1 , Baoguo Jia∗∗2 , and Allan Peterson∗∗∗1
1
2
1
University of Nebraska-Lincoln, Lincoln, Nebraska 68588 0130, USA
Zhongshan University, Guangzhou 51025, China
University of Nebraska-Lincoln, Lincoln, Nebraska 68588 0130, USA
Key words oscillation; Emden-Fowler equation; sublinear
MSC (2000) 34K11 ; 39A10 ; 39A99.
Consider the Emden-Fowler sublinear dynamic equation
x∆∆ (t) + p(t)xα (σ(t)) = 0,
(0.1)
where p ∈ C(T, R), where T is a time scale, 0 < α < 1, α is the quotient of odd positive integers. When p(t) is
allowed to take on negative values, we obtain a Belohorec-type oscillation theorem for (0.1). As an application,
we get that the sublinear difference equation
∆2 x(n) + p(n)xα (n + 1) = 0,
(0.2)
is oscillatory, if
∞
X
nα p(n) = ∞,
and the sublinear q-difference equation
x∆∆ (t) + p(t)xα (qt) = 0.
(0.3)
where t ∈ q N0 , q > 1, is oscillatory, if
Z
∞
tα p(t)∆t = ∞.
1
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1
Introduction
Consider the second order sublinear dynamic equation
x∆∆ + p(t)xα (σ(t)) = 0,
(1.1)
where p ∈ C(T, R), α > 0 is the quotient of odd positive integers and where T is a time scale with sup T = ∞.
Equation (1.1) is called superlinear if α > 1 and sublinear if 0 < α < 1. We call an equation oscillatory if all its
continuable solutions are oscillatory.
When T = R, the dynamic equation (1.1) is the second order nonlinear differential equation
x00 (t) + p(t)xα (t) = 0.
(1.2)
∗
e-mail: [email protected], Phone: 01 402 472 3408, Fax: 01 402 472 8466
e-mail: [email protected], Phone: 86 20 84034316, Fax: 86 20 84111696
∗∗∗ email: [email protected] Phone: 01 402 472 7225, Fax: 01 402 472 8466
∗∗
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2
Erbe, Jia, and Peterson: Belohorec-type Oscillation Theorem
When T = N0 , the dynamic equation (1.1) is the second order nonlinear difference equation
∆2 x(n) + p(t)xα (n + 1) = 0.
(1.3)
When p(t) is nonnegative, stronger oscillation results exist for the nonlinear equation (1.2) when α 6= 1,
notably the following:
Theorem A (Atkinson [1]). Let α > 1. Then (1.2) is oscillatory if and only if
Z ∞
(1.4)
tp(t) dt = ∞.
Theorem B (Belohorec [5]). Let 0 < α < 1. Then (1.2) is oscillatory if and only if
Z ∞
tα p(t) dt = ∞.
(1.5)
When p(t) is allowed to take on negative values, for α > 1, Kiguradze [17] proved that (1.4) is sufficient for
the differential equation (1.2) to be oscillatory and for 0 < α < 1 Belohorec [6] proved that (1.5) is sufficient for
the differential equation (1.2) to be oscillatory. These results have been further extended by Kwong and Wong
[18].
When p(n) is nonnegative, J. W. Hooker and W. T. Patula [13, Theorem 4.1], and A. Mingarelli [19], respectively proved the following:
Theorem C Let α > 1. Then (1.3) is oscillatory if and only if
∞
X
np(n) = ∞.
(1.6)
1
Theorem D Let 0 < α < 1. Then (1.3) is oscillatory if and only if
∞
X
nα p(n) = ∞.
(1.7)
1
For the case α > 1 and when p(n) is allowed to take on negative values, we proved in [2] that (1.6) is sufficient
for the difference equation (1.3) to be oscillatory.
In this paper, we prove that when p(n) is allowed to take on negative values, for 0 < α < 1, (1.7) is sufficient
for the oscillation of the difference equation (1.3). In particular, we get that the sublinear difference equation
1
b(−1)n α
∆2 x(n) + β+1 +
x (n + 1) = 0,
n
nβ
is oscillatory, if 0 ≤ β ≤ α and is nonoscillatory if β > α. We note that the transformation used here is different
from that in [6], which can not be applied to more general time scales. Moreover, the technique of proof in the
superlinear and sublinear cases are essentially different and also differ substantially from the case T = R.
For completeness, (see [8] and [9] for elementary results for the time scale calculus), we recall some basic
results for dynamic equations and the calculus on time scales. Let T be a time scale (i.e., a closed nonempty
subset of R) with sup T = ∞. The forward jump operator is defined by
σ(t) = inf{s ∈ T : s > t},
and the backward jump operator is defined by
ρ(t) = sup{s ∈ T : s < t},
where sup ∅ = inf T, where ∅ denotes the empty set. If σ(t) > t, we say t is right-scattered, while if ρ(t) < t
we say t is left-scattered. If σ(t) = t we say t is right-dense, while if ρ(t) = t and t 6= inf T we say t is left-dense.
Given a time scale interval [c, d]T := {t ∈ T : c ≤ t ≤ d} in T the notation [c, d]κ T denotes the interval [c, d]T
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in case ρ(d) = d and denotes the interval [c, d)T in case ρ(d) < d. The graininess function µ for a time scale T
is defined by µ(t) = σ(t) − t, and for any function f : T → R the notation f σ (t) denotes f (σ(t)). We say that
x : T → R is differentiable at t ∈ T provided
x∆ (t) := lim
s→t
x(t) − x(s)
,
t−s
exists when σ(t) = t (here by s → t it is understood that s approaches t in the time scale) and when x is
continuous at t and σ(t) > t
x(σ(t)) − x(t)
x∆ (t) :=
.
µ(t)
Note that if T = R , then the delta derivative is just the standard derivative, and when T = Z the delta derivative is
just the forward difference operator. Hence our results contain the discrete and continuous cases as special cases
and generalize these results to arbitrary time scales (for example the time scale q N0 := {1, q, q 2 , · · · } which is
very important in quantum theory [14]).
2
Main Theorem
In the case when T is such that µ(t) is not eventually identically zero, we define the set of all right–scattered
points by T̂ := {t ∈ T : µ(t) > 0} and note that T̂ is necessarily countable. We let χ denote the characteristic
function of T̂. The following condition, which will be needed later, imposes a lower bound on the graininess
function µ(t), for t ∈ T̂. More precisely, we introduce the following: (see [10]).
Condition (C): We say that T satisfies condition (C) if there is an M > 0 such that
χ(t) ≤ M µ(t), t ∈ T.
We note that if T satisfies condition (C), then the subset Ť of T defined by
Ť = {t ∈ T| t > 0 is right-scattered or left-scattered}
(2.1)
is also necessarily countable and, of course, T̂ ⊂ Ť. So we can suppose that
Ť = {ti ∈ T| 0 < t1 < t2 < · · · < tn < · · · }.
(2.2)
Remark 2.1 To clarify the arguments below, we let A := {n ∈ N : (tn−1 , tn ) ⊂ T} so that we can write
T = Ť ∪ [∪n∈A (tn−1 , tn )] . Also, note that if T satisfies condition (C), then σ(t) is differentiable.
Theorem 2.2 Assume that T satisfies condition (C) and let Ť be given by (2.2). If there exists a real number
β, 0 ≤ β ≤ 1 such that
Z
∞
σ αβ (t)p(t)∆t = ∞,
t1
then (1.1) is oscillatory.
P r o o f. In this proof we use the notation (tβ )∆ |σ(t) to mean (tβ )∆ evaluated at σ(t). For the sake of
contradiction, assume that (1.1) is nonoscillatory. Then without loss of generality there is a solution x(t)
of (1.1) and a T ∈ T with x(t) > 0, for all t ∈ [T, ∞)T . Making the substitution x(t) = tβ u(t) in
(1.1), where 0 ≤ β ≤ 1 and noticing (see Remark 2.1) that x∆ (t) = (tβ )∆ u(σ(t)) + tβ u∆ (t), x∆∆ (t) =
(tβ )∆∆ u(σ(t)) + (tβ )∆ |σ(t) (u(σ(t)))∆ + (tβ )∆ u∆ (t) + (σ(t))β u∆∆ (t), we get that
(tβ )∆∆ u(σ(t)) + (tβ )∆ |σ(t) (u(σ(t)))∆ + (tβ )∆ u∆ (t)
+
β ∆∆
(σ(t)) u
(t) + σ
αβ
(2.3)
α
(t)p(t)u (σ(t)) = 0.
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Erbe, Jia, and Peterson: Belohorec-type Oscillation Theorem
1
, integrating from T to t, and using an integration by parts formula we
Multiplying both sides of (2.3) by uα (σ(t))
get
Z t β ∆∆
Z t β ∆
(s ) |σ(s) (u(σ(s)))∆
(s ) u(σ(s))
∆s
+
∆s
α
u (σ(s))
uα (σ(s))
T
T
β ∆ t
Z t β ∆ ∆
s u (s)
(s ) u (s)
+
∆s
+
α
u (σ(s))
uα (s) T
T
Z t
Z t
∆
sβ
∆
u
(s)∆s
+
(2.4)
(σ(s))αβ p(s)∆s = 0.
−
uα (s)
T
T
Next, using the quotient rule and then Pötzsche’s chain rule [8, Theorem 1.90] gives
Z t β ∆
s
u∆ (s)∆s
α (s)
u
T
Z t β α
Z t β ∆ ∆
s [u (s)]∆ u∆ (s)
(s ) u (s)
∆s
−
∆s
=
uα (σ(s))
uα (s)uα (σ(s))
T
T
Z t
Z t β ∆ ∆
Z
1
sβ [u∆ (s)]2
(s ) u (s)
∆s
−
α
[uh (s)]α−1 dh∆s
=
α
α
uα (σ(s))
T u (s)u (σ(s)) 0
T
Z t
(sβ )∆ ∆
≤
u (s)∆s,
α
T u (σ(s))
where we used the fact that uh (s) := u(s) + hµ(s)u∆ (s) = (1 − h)u(s) + huσ (s) ≥ 0. Using this last inequality
in (2.4) we get
Z t β ∆∆
Z t β ∆
(s ) |σ(s) (u(σ(s)))∆
(s ) u(σ(s))
∆s
+
∆s
(2.5)
uα (σ(s))
uα (σ(s))
T
T
Z t
tβ u∆ (t) T β u∆ (T )
−
+
(σ(s))αβ p(s)∆s ≤ 0.
+ α
u (t)
uα (T )
T
Note that
Z
t
T
(sβ )∆ |σ(s) (u(σ(s)))∆
∆s
uα (σ(s))
(2.6)
(tβ )∆ u(σ(t)) (tβ )∆ |T u(σ(T ))
−
uα (t)
uα (T )
Z
t
(sβ )∆∆ u(σ(s))
(uα (s))∆ (sβ )∆ u(σ(s))
∆s
+
∆s.
uα (σ(s))
uα (s)uα (σ(s))
T
=
Z
t
−
T
Let us define A :=
T β u∆ (T )
uα (T ) ,
B :=
(tβ )∆ |T u(σ(T ))
.
uα (T )
Then we get from (2.5) and (2.6) that
Z t α
(u (s))∆ (sβ )∆ u(σ(s))
(tβ )∆ u(σ(t))
−
B
+
∆s
(2.7)
uα (t)
uα (s)uα (σ(s))
T
Z t
tβ u∆ (t)
+ α
−A+
(σ(s))αβ p(s)∆s ≤ 0.
u (t)
T
R1
Note that (tβ )∆ = β 0 [(1 − h)t + hσ(t)]β−1 dh > 0, for t > 0. So the first term of (2.7) is positive. From (2.7),
we get that
Z
t
−B +
T
(uα (s))∆ (sβ )∆ u(σ(s))
tβ u∆ (t)
∆s
+
−A+
uα (s)uα (σ(s))
uα (t)
Z
t
(σ(s))αβ p(s)∆s ≤ 0.
(2.8)
T
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Since 0 < β ≤ 1, one can use the Pötzsche chain rule to show that (tβ )∆ is nonincreasing. Using the second
mean value theorem [9, Theorem 5.45], we get that for each t ∈ [T, ∞)T
Z t α
(u (s))∆ (sβ )∆ u(σ(s))
∆s = Λ(t)(sβ )∆ |T ,
(2.9)
α (s)uα (σ(s))
u
T
where mu ≤ Λ(t) ≤ Mu , and where mu and Mu denote the infimum and supremum, respectively, of the function
R t (uα (s))∆ u(σ(s))
∆s.
T uα (s)uα (σ(s))
Next, we will show that
Z t α
α
(u (s))∆ u(σ(s))
∆s ≥
[−u1−α (T )].
(2.10)
α (s)uα (σ(s))
u
1
−
α
T
Assume first that t = ti−1 < ti = σ(t). We get that
Z ti
1
(uα (s))∆ u(σ(s))
1
Li (u) :=
∆s = α
−
u(ti ).
α
α
u (ti−1 ) uα (ti )
ti−1 u (s)u (σ(s))
Setting v(ti ) :=
1
uα (ti ) ,
Li (u) :=
(2.11)
we have that
v(ti−1 ) − v(ti )
1
v α (ti )
.
(2.12)
We consider the two possible cases v(ti−1 ) ≥ v(ti ) and v(ti−1 ) < v(ti ). First if v(ti−1 ) ≥ v(ti ) we have that
Z v(ti−1 )
v(ti−1 ) − v(ti )
1
(2.13)
≥
1
1 ds
v α (ti )
sα
v(ti )
1
1
α
=
[v 1− α (ti ) − v 1− α (ti−1 )].
1−α
On the other hand if v(ti−1 ) < v(ti ), then
Z v(ti )
v(ti ) − v(ti−1 )
1
≤
1
1 ds
α
v (ti )
v(ti−1 ) s α
1
1
α
[v 1− α (ti−1 ) − v 1− α (ti )],
=
1−α
which implies that
v(ti−1 ) − v(ti )
1
α
v (ti )
≥
1
1
α
[v 1− α (ti ) − v 1− α (ti−1 )].
1−α
Hence, whenever t = ti−1 < ti = σ(t), we have that
Li (u) ≥
1
1
α
[v 1− α (ti ) − v 1− α (ti−1 )].
1−α
(2.14)
If the real interval [ti−1 , ti ] ⊂ T, it is easy to see that
Z ti
Z ti
αu0 (s)
(uα (s))∆ u(σ(s))
Li (u) :=
∆s
=
ds
(2.15)
α
α
α
ti−1 u (s)
ti−1 u (s)u (σ(s))
α
=
[u1−α (ti ) − u1−α (ti−1 )]
1−α
1
1
α
=
[v 1− α (ti ) − v 1− α (ti−1 )].
1−α
Note that since T satisfies condition (C), we have from (2.14), (2.15) and the additivity of the integral that
Z t α
1
1
(u (s))∆ u(σ(s))
α
∆s ≥
[v 1− α (t) − v 1− α (T )]
(2.16)
α (s)uα (σ(s))
u
1
−
α
T
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Erbe, Jia, and Peterson: Belohorec-type Oscillation Theorem
α
α
[u1−α (t) − u1−α (T )] ≥
[−u1−α (T )].
1−α
1−α
Therefore, we get that (2.10) holds. From (2.8), (2.9), (2.10), we get that
Z t
tβ u∆ (t)
−αu1−α (T )
β ∆
+ α
−A+
−B + (s ) |T
(σ(s))αβ p(s)∆s ≤ 0,
1−α
u (t)
T
R ∞ αβ
Since t1 σ (t)p(t)∆t = ∞, from (2.17), there exists T1 ≥ T such that for t ≥ T1 , we have
=
(2.17)
tβ u∆ (t)
≤ −1.
uα (t)
Dividing both sides of this last inequality by tβ and integrating from T1 to t, we get, using inequality (2.11) in
[3],
Z t ∆
Z t
u (s)
1
1
[u1−α (t) − u1−α (T1 )] ≤
∆s
≤
−
∆s.
(2.18)
α (s)
β
1−α
u
s
T1
T1
R∞
Since T1 s1β ∆s = ∞ for β ≤ 1 (see [9, Theorem 5.68]) we get u1−α (t) < 0, for large t, which is a contradiction.
Thus, equation (1.1) is oscillatory.
As a consequence of Theorem 2.2, it follows that (1.3) is oscillatory if
∞
X
(n + 1)αβ p(n) = ∞
for some 0 < β ≤ 1. We would like to show that in Theorem 2.2, the assumption that
Z ∞
(σ(t))αβ p(t)∆t = ∞
t1
can be replaced by
Z
∞
tαβ p(t)∆t = ∞
t1
(where 0 < β ≤ 1). This would then imply, in particular, that the condition (1.7) implies oscillation of all
solutions of (1.3), which is the desired improvement of the Hooker–Patula–Mingarelli result mentioned earlier.
In order to extend Theorem 2.2 in this manner, as in [2], we shall make an additional assumption. As in [2] we
introduce the function r(t) defined by
t,
if t is right–dense and left–scattered
r(t) =
(2.19)
ρ(t),
otherwise.
It is clear that r(σ(t)) = t for any time scale T satisfying condition (C). The additional assumption which will
be needed is a monotonicity assumption on the expression (rβ (t))∆ . That is, we shall assume that (rβ (t))∆
is nonincreasing. Clearly, if T = R, T = Z0 or T = q N0 , it is easy to see that (rβ (t))∆ = (ρβ (t))∆ is
nonincreasing. We note, as was established in [2], that this need not hold (i.e., one can find an example [2,
Example 2.3] where (ρβ (t))∆ is not monotone).
Theorem 2.3 Assume that T satisfies condition (C) and let
Ť = {ti ∈ T| 0 < t1 < t2 < · · · < tn < · · · }.
If there exists a real number β, 0 ≤ β ≤ 1 such that the delta derivative (rβ (t))∆ is nonincreasing, where the
function r(t) is defined in (2.19), and
Z ∞
tαβ p(t)∆t = ∞,
t1
then (1.1) is oscillatory.
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P r o o f. Assume that (1.1) is nonoscillatory. Then without loss of generality there is a solution x(t) of (1.1)
and a T ∈ T with x(t) > 0, for all t ∈ [T, ∞)T . Making the substitution x(t) = rβ (t)u(t) and noticing that
x∆ (t)
(rβ (t))∆ u(σ(t)) + rβ (t)u∆ (t)
=
x∆∆ (t)
=
(rβ (t))∆∆ u(σ(t)) + (rβ (t))∆ |σ(t) [u(σ(t))]∆
+
(rβ (t))∆ u∆ (t) + rβ (σ(t))u∆∆ (t)
=
(rβ (t))∆∆ u(σ(t)) + (rβ (t))∆ |σ(t) [u(σ(t))]∆
+
(rβ (t))∆ u∆ (t) + tβ u∆∆ (t)
we get from (1.1) that
(rβ (t))∆∆ u(σ(t))
+
(rβ )∆ (σ(t))[u(σ(t))]∆ + (rβ (t))∆ u∆ (t)
+ tβ u∆∆ (t) + tαβ p(t)uα (σ(t)) = 0.
1
uα (σ(t))
Multiplying by
Z
t
and integrating from T to t we get
(rβ (s))∆∆ u(σ(s))
∆s +
uα (σ(s))
T
Z
t
+
T
Z
(rβ (s))∆ u∆ (s)
∆s +
uα (σ(s))
t
Z
T
t
Z
(rβ )∆ (σ(s))[u(σ(s))]∆
∆s
uα (σ(s))
(2.20)
sβ u∆∆ (s)
∆s
uσ (s)
T
t
sαβ p(s)∆s = 0.
+
T
But, using r(σ(t)) = t, and then integrating by parts we get that the fourth term in (2.20) is given by
Z t β ∆∆
Z t β
r (σ(s))u∆∆ (s)
s u (s)
∆s
=
∆s
α
uα (σ(s))
T
T u (σ(s))
β
t
Z t β
Z t β∆
r (s)u∆ (s)
[r ] (s)u∆ (s)
r (s)[uα (s)]∆ u∆ (s)
=
−
∆s
+
∆s.
α
α
u (s)
u (σ(s))
uα (s)uα (σ(s)))
T
T
T
Using the Pötzsche chain rule we get the third term on the last line satisfies
Z t β
r (s)[uα (s)]∆ u∆ (s)
∆s
uα (s)uα (σ(s)))
T
R
Z t β
1
r (s) 0 [u(s) + hµ(s)u∆ (s)]α−1 dh[u∆ (s)]2
= α
uα (s)uα (σ(s))
T
R
Z t β
1
r (s) 0 [(1 − h)u(s) + huσ (s)]α−1 dh[u∆ (s)]2
= α
≥ 0,
uα (s)uα (σ(s))
T
which implies that the fourth term in (2.20) satisfies the inequality
β
t
Z t β ∆∆
Z t β ∆
s u (s)
(r ) (s)u∆ (s)
r (s)u∆ (s)
∆s
≥
∆s.
−
α
α
u (s)
uα (σ(s))
T u (σ(s))
T
T
(2.21)
Using this last inequality we get from (2.20) that
Z
t
T
(rβ (s))∆∆ u(σ(s))
∆s +
uα (σ(s))
Z
t
T
rβ (t)u∆ (t) rβ (T )u∆ (T )
+
−
+
uα (t)
uα (T )
(rβ )∆ (σ(s))[u(σ(s))]∆
∆s
uα (σ(s))
Z
(2.22)
t
sαβ p(s)∆s ≤ 0.
T
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8
Erbe, Jia, and Peterson: Belohorec-type Oscillation Theorem
Note that
Z
t
T
(rβ )∆ (σ(s))[u(σ(s))]∆
∆s
uα (σ(s))
(2.23)
(rβ (t))∆ u(σ(t)) (rβ )∆ (T )u(σ(T ))
−
uα (t)
uα (T )
Z
t
(rβ (s))∆∆ u(σ(s))
(uα (s))∆ (rβ (s))∆ u(σ(s))
∆s +
∆s.
α
u (σ(s))
uα (s)uα (σ(s))
T
=
Z
t
−
T
(r β )∆ (T ))u(σ(T ))
,
uα (T )
β
∆
)u (T )
B := r (T
. From (2.22), (2.23), we get that
uα (T )
Z t α
rβ (t)u(t)
(u (s))∆ (rβ (s))∆ u(σ(s))
−
B
+
∆s
(2.24)
α
u (t)
uα (s)uα (σ(s))
T
Z t
(rβ (t))∆ u(σ(t))
+
−
A
+
sαβ p(s)∆s ≤ 0.
uα (t)
T
R1
Note that r∆ (t) ≥ 0 and rσ (t) = t, so (rβ (t))∆ = β 0 [(1 − h)r(t) + ht]β−1 dhr∆ (t) ≥ 0, for t > 0. So
Let A :=
(r β (t))∆ u(σ(t))
uα (t)
≥ 0, and from (2.24), we get that
Z t
Z t α
rβ (t)u(t)
(u (s))∆ (rβ (s))∆ u(σ(s))
−
B
+
∆s
−
A
+
sαβ p(s)∆s ≤ 0.
uα (t)
uα (s)uα (σ(s))
T
T
(2.25)
Since (rβ (s))∆ is nonincreasing, by the second mean value theorem [9, Theorem 5.45]), we get that for each
t ∈ [T, ∞)T
Z t α
(u (s))∆ (rβ (s))∆ u(σ(s))
(2.26)
∆s = (rβ (s))∆ |T Λ(t)
uα (s)uα (σ(s))
T
where mu ≤ Λ(t) ≤ Mu , and where mu and Mu denote the infimum and supremum, respectively, of the function
R t (uα (s))∆ u(σ(s))
∆s.
T uα (s)uα (σ(s))
From (2.10), we get that
Z t α
(u (s))∆ u(σ(s))
α
∆s ≥
[−u1−α (T )].
(2.27)
α (s)uα (σ(s))
u
1
−
α
T
Therefore from (2.25), (2.26), we get that
rβ (t)u(t)
αu1−α (T )
β
∆
−
B
−
(r
(s))
|
−A+
T
uα (t)
1−α
Since
R∞
t1
Z
t
sαβ p(s)∆s ≤ 0.
(2.28)
T
σ αβ (t)p(t)∆t = ∞, from (2.28), there exists T1 ≥ T such that for t ≥ T1 , we have
rβ (t)(u(t))∆
≤ −1.
uα (t)
Integrating from T1 to t, using an inequality of [3] and [9, Theorem 5.68] and noticing that r(t) ≤ t, we get that,
as t → ∞
u1−α (t) − u1−α (T1 )
1−α
Z t ∆
u (s)
∆s
≤
α
T1 u (s)
Z t
1
≤ −
∆s
β
T1 r (s)
Z t
1
≤ −
∆s → −∞.
β
s
T1
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9
Therefore u1−α (t) < 0, for large t, which is a contradiction. Thus equation (1.1) is oscillatory.
When T = Z and T = q N0 , q > 1, we get the following corollaries. Corollary 2.4 shows that with no sign
assumption on p(n), the condition
∞
X
nα p(n) = ∞
is sufficient for the oscillation of the difference equation (1.3).
Corollary 2.4 Assume T = Z. If
∞
X
nα p(n) = ∞
then (1.3) is oscillatory.
Corollary 2.5 Assume T = q N0 . If
∞
Z
tα p(t)∆t = ∞
1
then the q-difference equation x∆∆ (t) + p(t)xα (t) = 0, is oscillatory.
Remark 2.6 By using the idea in Theorem 2.2, we can give a simple proof of Belohorec’s Theorem (see [6]).
Consider the differential equation
x00 + p(t)xα (t) = 0,
0 < α < 1,
(2.29)
R∞
where 1 tαβ p(t)dt = ∞, for some 0 ≤ β ≤ 1. Suppose x(t) > 0 for t ≥ T > 1 is a nonoscillatory solution.
Let x(t) = tβ u(t), Then from (2.29), we get
β(β − 1)tβ−2 u(t) + 2βtβ−1 u0 (t) + tβ u00 (t) + tαβ p(t)uα (t) = 0.
Multiplying
1
uα (t)
and integrating from T to t, we get
Z
t
β(β − 1)
sβ−2
T
Z t
u(s)
u0 (s)
ds
+
2β
sβ−1 α ds
α
u (s)
u (s)
T
Z t
Z t
00
β u (s)
sαβ p(s)ds = 0.
+
s α ds +
u (s)
T
T
(2.30)
Note that
Z
t
sβ
T
u00 (s)
ds
uα (s)
Z
=
tβ u0 (t) T β u0 (T )
− α
−
uα (t)
u (T )
Z
≥
tβ u0 (t) T β u0 (T )
− α
−
uα (t)
u (T )
t
T
t
T
βsβ−1 u0 (s)
ds +
uα (s)
Z
t
T
αsβ [u0 (s)]2
ds
u1+α (s)
βsβ−1 u0 (s)
ds.
uα (s)
So by (2.30), we get that
Z t
u(s)
u0 (s)
ds
+
β
sβ−1 α ds
α
u (s)
u (s)
T
T
Z t
β 0
β 0
t u (t) T u (T )
− α
+
sαβ p(s)ds ≤ 0.
uα (t)
u (T )
T
Z
β(β − 1)
+
t
sβ−2
(2.31)
Applying integration by parts to the second term in (2.31), we get that
Z t
u0 (s)
βtβ−1 u(t) βT β−1 u(T )
β
sβ−1 α ds =
−
u (s)
uα (t)
uα (T )
T
Z t β−2
Z t
1 u0 (s)
s
u(s)
− β(β − 1)
ds
+
αβ
ds.
1−β uα (s)
uα (s)
T s
T
(2.32)
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10
Erbe, Jia, and Peterson: Belohorec-type Oscillation Theorem
1
But, since 0 ≤ β ≤ 1, s1−β
is nonincreasing, we have by the (continuous) second mean value theorem [4,
Exercise 30 N, page 236] there is a point ξ ∈ (T, t), which could depend on t, such that
Z ξ 0
u0 (s)
u (s)
1
ds
=
ds
1−β uα (s)
1−β
α (s)
s
T
u
T
T
1−α ξ
u
(s)
u1−α (T )
1
≥−
= 1−β
.
T
1−α T
(1 − α)T 1−β
Z
t
1
(2.33)
Then from (2.31), (2.32), and (2.33) there is a constant K such that
βtβ−1 u(t) tβ u0 (t)
+ α
≤K−
uα (t)
u (t)
Since
R∞
Z
t
sαβ p(s)ds.
(2.34)
T
tαβ p(t)dt = ∞, it follows from (2.34), that there exists T1 ∈ T such that for t ∈ [T1 , ∞)T ,
u0 (t)
1
≤ − β.
α
u (t)
t
Integrating from T1 to t, we get that as t → ∞
1
[u1−α (t) − u1−α (T1 )] ≤ −
1−α
Z
t
T1
1
ds → −∞.
sβ
Therefore u1−α (t) < 0, for large t, which is a contradiction. Thus, equation (2.29) is oscillatory.
3
Example
Consider the sublinear difference equation
(−1)n α
1
2
x (n + 1) = 0
∆ x(n) + β+1 + b β
n
n
(3.1)
where β ≥ 0, and b is any real number. Since
∞
X
nα
n=1
1
nβ+1
+b
(−1)n
= ∞,
nβ
(3.2)
for 0 ≤ β ≤ α, by Corollary 2.4, (3.1) is oscillatory.
Note that we can also use Theorem 2.2 to obtain the fact that (3.1) is oscillatory, since (taking α = 1)
∞
X
=
=
=
=
(−1)n
+b β
(n + 1)
nβ+1
n
n=1
∞
β
X
1
1
n
+ b(−1)
1+
n
n
n=1
∞ X
β
1
1
1+ +o
+ b(−1)n
n
n
n
n=1
∞
X 1
(−1)n
1
+ b(−1)n + bβ
+O
2
n
n
n
n=1
∞.
αβ
1
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11
In the following, using [11, Theorem 2.1], we will prove that when β > α, (3.1) has a nonoscillatory solution,
which shows sharpness of the oscillation result. By [7] and [11], we have
∞
X
1
1
1
=
+
o
.
β+1
β
β
i
βn
n
n
∞
X
1
(−1)i
(−1)n
=
+
o
.
β
β
β
i
2n
n
n
So
|
where A is constant. Let F (n) =
A+
,
nβ
∞ X
1
(−1)i
A
+
| ≤ β,
β+1
β
i
i
n
n
then we have
P (n) ≤ F (n + 1).
At the end of Example 1 in [11], we proved that
nα |(n + 1)−β − n−β | ∼
Using this we obtain
Z
β
n1−α+β
.
∞
tα |F ∆ (t)|∆ < ∞.
1
By [11, Theorem 2.1], we get that (3.1) has a nonoscillatory solution.
References
[1] F. V. Atkinson, On second order nonlinear oscillations, Pacific J. Math. 5 643–64(1955).
[2] Jia Baoguo, Lynn Erbe and Allan Peterson, Kiguradze-type oscillation theorem for second order superlinear dynamic
equation on time scales, submitted for publication.
[3] Jia Baoguo, Lynn Erbe and Allan Peterson, Oscillation of sublinear Emden-Fowler Dynamic equation on time scales, J.
Difference Equ. Appl. to appear.
[4] Robert G. Bartle, The Elements of Real Analysis, Second Edition, John Wiley and Sons, New York, 1976.
[5] S. Belohorec, Oscillatory solutions of certain nonlinear differential equations of second order, Mat. Fyz. Časopis Sloven
Akad. Vied. 11 250–255(1961). (Czech)
[6] S. Belohorec, Two remarks on the properties of solutions of a nonlinear differential equation, Acta Fac. Rerum Natur.
Univ. Comenian. Math. Publ. 22 19–26(1969).
[7] Martin Bohner, Lynn Erbe, and Allan Peterson, Oscillation for nonlinear second order dynamic equations on a time
scale, J. Math. Anal. Appl. 301 491–50(2005).
[8] M. Bohner and A. Peterson, Dynamic Equation on Time Scales: An Introduction with Applications, Birkhäuser, Boston,
2001.
[9] M. Bohner and A. Peterson, editors, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
[10] Lynn Erbe, Oscillation criteria for second order linear equations on a time scale, Canad. Appl. Math. Quart. 9 346–
35(2001).
[11] Lynn Erbe, Jia Baoguo and Allan Peterson, Nonoscillation for second order sublinear dynamic equations on time scales,
submitted for publication.
[12] P. Hartman, On nonoscillatory linear differential equations of second order, Amer. J. Math. 4 489–500(1952).
[13] J. W. Hooker and W. T. Patula, A second order nonlinear difference equation: Oscillation and asymptotic behavior, J.
Math. Anal. Appl. 91 9–29(1983).
[14] Victor Kac and Pokman Cheung, Quantum Calculus, Universitext, Springer, New York, 2001.
[15] Walter Kelley and Allan Peterson, Difference Equations: An Introduction with Applications, Harcourt/Academic Press,
Second Edition, San Diego, 2001.
[16] Walter Kelley and Allan Peterson, The Theory of Differential Equations Classical and Qualitative, Pearson Education,
Inc., Upper Saddle River, New Jersey, 2004.
[17] I. T. Kiguradze, A note on the oscillation of solution of the equation u00 + a(t)un sgn u = 0, Casopis Pest. Mat. 92
343–350(1967).
[18] Man Kam Kwong and J. S. W. Wong, On an oscillation theorem of Belohorec, SIAM J. Math. Anal. 14 44–46(1983).
[19] A. B. Mingarelli, Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lecture Notes
in Mathematics 989, Springer Verlag, Berlin, 1983.
[20] J. S. W. Wong, Oscillation criteria for second order nonlinear differential equations involving general means, J. Math.
Anal. Appl. 24 489–505(2000).
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