Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales
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Інститут математики НАН України
2013
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| Цитувати: | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales / Yong Zhou, Yonghong Lan // Нелінійні коливання. — 2013. — Т. 16, № 2. — С. 191-206. — Бібліогр.: 20 назв. — англ. |
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| author | Yong Zhou Yonghong Lan |
| author_facet | Yong Zhou Yonghong Lan |
| citation_txt | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales / Yong Zhou, Yonghong Lan // Нелінійні коливання. — 2013. — Т. 16, № 2. — С. 191-206. — Бібліогр.: 20 назв. — англ. |
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UDC 517.9
CLASSIFICATION AND EXISTENCE OF NONOSCILLATORY SOLUTIONS
OF SECOND-ORDER NEUTRAL DELAY DYNAMIC EQUATIONS
ON TIME SCALES*
КЛАСИФIКАЦIЯ ТА IСНУВАННЯ НЕКОЛИВНИХ РОЗВ’ЯЗКIВ
ДИНАМIЧНИХ РIВНЯНЬ ДРУГОГО ПОРЯДКУ
З НЕЙТРАЛЬНИМ ЗАПIЗНЕННЯМ НА ЧАСОВIЙ ШКАЛI
Yong Zhou, Yonghong Lan
Xiangtan Univ.
Hunan 411105, P. R. China
e-mail: yzhou@xtu.edu.cn
In this paper, we give a classification of nonoscillatory solutions of the second-order neutral delay dynamic
equation on time scales
[x(t)− c(t)x(t− τ)]∆∆ + f(t, x(g1(t)), . . . , x(gm(t))) = 0, t ∈ T.
Some existence results for each kind of nonoscillatory solutions are also established.
Наведено класифiкацiю неколивних розв’язкiв динамiчних рiвнянь другого порядку з нейтраль-
ним запiзненням на часовiй шкалi
[x(t)− c(t)x(t− τ)]∆∆ + f(t, x(g1(t)), . . . , x(gm(t))) = 0, t ∈ T,
а також доведено iснування неколивних розв’язкiв кожного типу.
1. Introduction. The theory of time scales was introduced by Hilger [10] in 1988 in order to
unify continuous and discrete analysis. Recently, the study of dynamic equations on time scales
has received a lot of attention. The general idea is to prove a result for a dynamic equation
where the domain of the unknown function is a so-called time scale, which is a special case of
a measure chain. By choosing the time scale to be the set of real numbers, the general results
yields a result concerning a differential equation. On the other hand, by choosing the time
scale to be the set of integers, the same general result yields a result for difference equations.
However, since there are many other time scales than just the set of real numbers or the set
of integers, one has a much more general results. The monographs by Bohner and Peterson
[4, 5] and the survey on dynamic equations on time scales by Agarwal, Bohner, O’Regan and
Peterson [2] summarized some important work in this area. Recently, the oscillation of dynamic
equations on time scales has received much attention (see [6 – 8, 12 – 18]). But the classification
and existence of nonoscillatory solutions of the delay dynamic equations on time scales received
much less attention.
∗ Project supported by National Natural Science Foundation of P.R. China (11271309), the Specialized Research
Fund for the Doctoral Program of Higher Education (20114301110001) and Hunan Provincial Natural Science
Foundation of China (12JJ2001).
c© Yong Zhou, Yonghong Lan, 2013
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 2 191
192 YONG ZHOU, YONGHONG LAN
In this paper, we consider the second-order neutral delay dynamic equation on time scales
[x(t)− c(t)x(t− τ)]∆∆ + f(t, x(g1(t)), . . . , x(gm(t))) = 0, (1.1)
where t ∈ [t0,∞) = T0 ⊆ T. With respect to (1.1), throughout we shall assume the following:
(H1) τ > 0, c(t) ∈ Crd(T0,R+),R+ = [0,∞), there exists δ ∈ (0, 1] such that c(t) ≤ 1− δ
for t ∈ T0.
(H2) gi ∈ Crd(T0,R+), and limt→∞ gi(t) = ∞, i = 1, 2, . . . ,m.
(H3) f : T0 ×Rm → R is right-dense continuous on T0 and continuous with respect to the
last m arguments, y1f(t, y1, . . . , ym) > 0 for y1yi > 0, i = 2, . . . ,m. Moreover,
|f(t, x1, . . . , xm)| ≥ |f(t, y1, . . . , ym)|
when |yi| ≤ |xi| and xiyi > 0, i = 1, 2, . . . ,m.
For convenience, we set
y(t) = x(t)− c(t)x(t− τ). (1.2)
In Section 3, we will study the existence and asymptotic behavior of nonoscillatory soluti-
ons of equation (1.1). More precisely, we give a classification of nonoscillatory solutions of
equation (1.1) according to their asymptotic behavior. Moreover, we established some existence
results for each kind of nonoscillatory solutions of equation (1.1). In particular, we obtain two
necessary and sufficient conditions for the existence of nonoscillatory solutions of (1.1).
2. Preliminaries. To understand the delay dynamic equations on time scales we need some
preliminary definitions (see [4]).
Let T be a time scale (i.e., a closed subset of the real numbers R) with supT = ∞. We
assume throughout that T has the topology that it inherits from the standard topology on the
real numbers R.
Definition 2.1. For t ∈ T we define the forward jump operator σ : T → T by
σ(t) := inf{s ∈ T : s > t}
while the backwards jump operator ρ : T → T is defined by
ρ(t) := sup{s ∈ T : s < t}.
If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t we say that t is left-scattered. Points
that are right-scattered and left-scattered at the same time are called isolated. Also, if t < supT
and σ(t) = t, then t is called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense.
Points that are right-dense and left-dense at same time are called dense.
Definition 2.2. Define the interval in T
[a, b] := {t ∈ T such that a ≤ t ≤ b}.
Open intervals and half-open intervals etc. are defined accordingly. Note that [a, b]K = [a, b]
if b is left-dense and [a, b]K = [a, b) = [a, ρ(b)] if b is left-scattered.
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CLASSIFICATION AND EXISTENCE OF NONOSCILLATORY SOLUTIONS OF SECOND-ORDER NEUTRAL DELAY . . . 193
Definition 2.3. Assume f : T → R and let t ∈ T (if t = supT assume t is not left-scattered),
then we define f∆(t) to be the number (provided it exists) with the property that given any ε > 0,
there is a neighborhood U of t (i.e., U = (t− δ, t+ δ) ∩ T for some δ > 0) such that
|[f(σ(t))− f(s)]− f∆(t)[σ(t)− s]| ≤ ε|σ(t)− s|, for all s ∈ U.
We call f∆(t) the delta (or Hilger) derivative of f at t.
It can be shown that if f : T → R is continuous at t ∈ T and t is right-scattered, then
f∆(t) =
f(σ(t))− f(t)
µ(t)
.
If t is right-dense, then
f∆(t) = lim
s→t
f(σ(t))− f(s)
t− s
.
Lemma 2.1. Assume g : T → R to be differentiable and g∆(t) ≥ 0. Then g(t) is nondecrea-
sing.
Definition 2.4. We say f : T → R is right-dense continuous on T provided it is continuous at
all right-dense points and at points that are left-dense and right-scattered we just assume the left
hand limit exists (and is finite). We denote this by f ∈ Crd(T,R).
Lemma 2.2. Assume f : T → R to be differentiable at t, then f is continuous at t.
Definition 2.5. If F∆(t) = f(t), then we define an integral by
t∫
a
f(τ)∆τ := F (t)− F (a).
Definition 2.6. If a ∈ T and f ∈ Crd([a,∞),R), then we define the improper integral by
∞∫
a
f(t)∆t := lim
b→∞
b∫
a
f(t)∆t
provided this limit exists, and we say that the improper integral converges in this case. If this limit
does not exist, then we say that the improper integral diverges.
Lemma 2.3. Let a ∈ TK , b ∈ T and assume f : T × TK → R is continuous at (t, t), where
t ∈ TK with t > a. Also assume that f∆(t, ·) is rd-continuous on [a, σ(t)]. Suppose that for each
ε > 0 there exists a neighborhood U of t, independent of τ ∈ [a, σ(t)], such that
|f(σ(t), τ)− f(s, τ)− f∆(t, τ)(σ(t)− s)| ≤ ε|σ(t)− s|, for all s ∈ U,
where f∆ denotes the derivative of f with respect to the first variable. Then
(i) g(t) :=
∫ t
a
f(t, τ)∆τ implies g∆(t) =
∫ t
a
f∆(t, τ)∆τ + f(σ(t), t);
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 2
194 YONG ZHOU, YONGHONG LAN
(ii) h(t) :=
∫ b
t
f(t, τ)∆τ implies h∆(t) =
∫ b
t
f∆(t, τ)∆τ − f(σ(t), t).
Definition 2.7. A solution of (1.1) is said to be oscillatory if it is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory.
3. Main results. First we show some lemmas which will be useful for the main results of this
section.
Lemma 3.1. Let x(t) be an eventually positive (or negative) solution of (1.1). If limt→∞ x(t) =
= 0, then y(t) is eventually negative (or positive) and limt→∞ y(t) = 0. If limt→∞ x(t) = 0 fails,
then y(t) is eventually positive (or negative).
Proof. Let x(t) be an eventually positive solution of (1.1). From (1.1), y∆∆(t) < 0 eventually.
Thus y∆(t) is decreasing and y∆(t) > 0 or y∆(t) < 0 eventually. Also, y(t) > 0 or y(t) < 0
eventually. If limt→∞ x(t) = 0, from (1.2) we have limt→∞ y(t) = 0. Since y(t) is monotonic, so
limt→∞ y
∆(t) = 0, which implies that y∆(t) > 0. Therefore, y(t) < 0 eventually. If
limt→∞ x(t) = 0 fail, then lim supt→∞ x(t) > 0. We show that y(t) > 0 eventually. If not,
then y(t) < 0 eventually. If x(t) is unbounded, then there exists a sequence {tn} such that
limn→∞ tn = ∞, x(tn) = maxt0≤t≤tn x(t) and limn→∞ x(tn) = ∞. From (1.2), we have
y(tn) = x(tn)− c(tn)x(tn − τ) ≥ x(tn)(1− c(tn)). (3.1)
Thus limn→∞ y(tn) = ∞, which is a contradiction. If x(t) is bounded, then there exists a
sequence {tn} such that limn→∞ tn = ∞ and limn→∞ x(tn) = lim supt→∞ x(t). Since the
sequences {c(tn)} and {x(tn − τ)} are bounded, there exist convergent subsequences. Without
loss of generality, we may assume that limn→∞ x(tn − τ) and limn→∞ c(tn) exist. Hence
0 ≥ lim
n→∞
y(tn) = lim
n→∞
(x(tn)− c(tn)x(tn − τ)) ≥ lim sup
t→∞
x(t)(1− lim
n→∞
c(tn)) > 0
which is a contradiction again. Therefore, y(t) > 0 eventually. A similar proof can be given if
x(t) < 0 eventually.
Lemma 3.2. Assume that lim
t→∞
c(t) = c ∈ [0, 1), and x(t) is an eventually positive (or negati-
ve) solution of (1.1). If limt→∞ y(t) = a ∈ R, then limt→∞ x(t) =
a
1− c
. If limt→∞ y(t) = ∞
(or−∞), then limt→∞ x(t) = ∞ (or −∞).
Proof. Let x(t) be an eventually positive solution of (1.1). Then x(t) ≥ y(t) eventually. If
limt→∞ y(t) = ∞, then limt→∞ x(t) = ∞. Now we consider the case that limt→∞ y(t) = a ∈
∈ R. Thus y(t) is bounded which implies that x(t) is bounded (see (3.1)). Therefore, there exists
a sequence {tn} such that limn→∞ tn = ∞ and limn→∞ x(tn) = lim supt→∞ x(t). As before,
without loss of generality, we may assume that limn→∞ c(tn) and limn→∞ x(tn− τ) exist. Hence
a = lim
n→∞
y(tn) = lim
n→∞
x(tn)− lim
n→∞
c(tn) lim
n→∞
x(tn − τ) ≥ lim sup
t→∞
x(t)(1− c)
i.e.,
a
1− c
≥ lim sup
t→∞
x(t). (3.2)
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CLASSIFICATION AND EXISTENCE OF NONOSCILLATORY SOLUTIONS OF SECOND-ORDER NEUTRAL DELAY . . . 195
On the other hand, there exists {t′n} such that limn→∞ x(t′n) = lim inft→∞ x(t). Without loss of
generality, we assume that limn→∞ c(t
′
n) and limn→∞ x(t′n − τ) exist. Hence
a = lim
n→∞
y(t′n) = lim
n→∞
x(t′n)− lim
n→∞
c(t′n) lim
n→∞
x(t′n − τ) ≤ lim inf
t→∞
x(t)(1− c)
or
a
1− c
≤ lim inf
t→∞
x(t). (3.3)
Combining (3.2) and (3.3) we obtain limt→∞ x(t) =
a
1− c
. A similar proof can be given if
x(t) < 0.
We are now ready to prove the following results.
Theorem 3.1. Assume that limt→∞ c(t) = c ∈ [0, 1). Let x(t) be a nonoscillatory solution of
(1.1). Let E denote the set of all nonoscillatory solution of (1.1), and define
E(0, 0, 0) = {x(t) ∈ E : lim
t→∞
x(t) = 0, lim
t→∞
y(t) = 0, lim
t→∞
y∆(t) = 0},
E(b, a, 0) = {x(t) ∈ E : lim
t→∞
x(t) = b =
a
1− c
, lim
t→∞
y(t) = a, lim
t→∞
y∆(t) = 0},
E(∞,∞, 0) = {x(t) ∈ E : lim
t→∞
x(t) = ∞, lim
t→∞
y(t) = ∞, lim
t→∞
y∆(t) = 0},
E(∞,∞, d) = {x(t) ∈ E : lim
t→∞
x(t) = ∞, lim
t→∞
y(t) = ∞, lim
t→∞
y∆(t) = d 6= 0}.
Then
E = E(0, 0, 0) ∪ E(b, a, 0) ∪ E(∞,∞, 0) ∪ E(∞,∞, d).
Proof. Without loss of generality, let x(t) be an eventually positive solution of (1.1). If
limt→∞ x(t) = 0, by Lemma 3.1, limt→∞ y(t) = 0 and limt→∞ y
∆(t) = 0, i.e., x(t) ∈ E(0, 0, 0).
If limt→∞ x(t) = 0 fails, then by Lemma 3.1, y(t) > 0 eventually, and it is easy to see that
y∆(t) > 0, y∆∆(t) < 0 eventually. If limt→∞ y(t) = a > 0 exists, then limt→∞ y
∆(t) = 0, by
Lemma 3.2, and we have limt→∞ x(t) =
a
1− c
= b, i.e., x(t) ∈ E(b, a, 0). If limt→∞ y(t) = ∞,
then by Lemma 3.2 limt→∞ x(t) = ∞. Since y∆∆(t) < 0 and y∆(t) > 0, we have
limt→∞ y
∆(t) = d, where d = 0 or d > 0. Then either x(t) ∈ E(∞,∞, 0), or x(t) ∈
∈ E(∞,∞, d).
In the following we shall show some existence results for each kind of nonoscillatory soluti-
on of Eq. (1.1).
Theorem 3.2. Assume that limt→∞ c(t) = c ∈ [0, 1). Then Eq.(1.1) has a nonoscillatory
solution x(t) ∈ E(b, a, 0)(b 6= 0, a 6= 0) if and only if
∞∫
t0
σ(u)|f(u, b1, . . . , b1)|∆u < ∞ for some b1 6= 0. (3.4)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 2
196 YONG ZHOU, YONGHONG LAN
Proof. Necessity. Without loss of generality, let x(t) ∈ E(b, a, 0) be an eventually positive
solution of (1.1). By Theorem 3.1 we know that b > 0, a > 0. From (1.1) and (1.2) we have
y∆∆(t) = −f(t, x(g1(t)), . . . , x(gm(t))).
Integrating it from s to∞ for s ≥ t0 we obtain
y∆(s) =
∞∫
s
f(u, x(g1(u)), . . . , x(gm(u)))∆u.
Integrating it from t1 to t for t1 sufficiently large, we get
y(t) = y(t1) +
t∫
t1
(σ(u)− t1)f(u, x(g1(u)), . . . , x(gm(u)))∆u+
+
∞∫
t
(t− t1)f(u, x(g1(u)), . . . , x(gm(u)))∆u.
Since limu→∞ x(gi(u)) = b > 0, i = 1, 2, . . . ,m, there exists an t1 ≥ t0 such that x(gi(u)) ≥ b
2
for t ≥ t1. Hence we have
t∫
t1
(σ(u)− t1)
∣∣∣∣f (u, b2 , . . . , b2
)∣∣∣∣∆u < y(t)− y(t1)
which implies that (3.4) holds.
Sufficiency. Set b1 > 0 andA > 0 so thatA < (1−c)b1. From (3.4) there exists a sufficiently
large t1 so that for t ≥ t1 we have t− τ ≥ t0 and gi(t) ≥ t0, i = 1, 2, . . . ,m, and
A
b1
+ c(t) +
1
b1
∞∫
t1
σ(u)f(u, b1, . . . , b1)∆u ≤ 1. (3.5)
Let X denote the Banach space of all bounded rd-continuous functions x(t) on [t0,∞) with the
norm ‖x(t)‖ = supt≥t0 |x(t)| < ∞. Define a set Ω by
Ω = {x(t) ∈ X|0 ≤ x(t) ≤ b1, t ≥ t0}
and an operator S on Ω by
(Sx)(t) =
A+ c(t)x(t− τ)
+
∫ t
t1
σ(u)f(u, x(g1(u)), . . . , x(gm(u)))∆u+
+
∫ ∞
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u, if t ≥ t1,
(Sx)(t1), if t0 ≤ t < t1.
(3.6)
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CLASSIFICATION AND EXISTENCE OF NONOSCILLATORY SOLUTIONS OF SECOND-ORDER NEUTRAL DELAY . . . 197
Clearly, for x(t) ∈ Ω,
(Sx)(t) ≤ A+ c(t)b1 +
t∫
t1
σ(u)f(u, b1, . . . , b1)∆u+
∞∫
t
σ(u)f(u, b1, . . . , b1)∆u ≤
≤ A+ c(t)b1 +
∞∫
t1
σ(u)f(u, b1, . . . , b1)∆u ≤ b1, t ≥ t1,
and
(Sx)(t) = (Sx)(t1) ≤ b1, t0 ≤ t ≤ t1,
i.e., SΩ ⊂ Ω.
Define a series of sequences {xk(t)}, k ∈ N0, as
x0(t) = 0,
(3.7)
xk(t) = (Sxk−1)(t), k ∈ N, t ≥ t0.
By induction, we can prove that
0 ≤ xk(t) ≤ xk+1(t) ≤ b1, t ≥ t0, k ∈ N0.
Then there exists x(t) ⊂ Ω such that limt→∞ xk(t) = x(t), t ≥ t0.
In the following, we shall show that
lim
k→∞
∞∫
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u =
∞∫
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u.
In fact, by (3.4), for any ε > 0 there exists t1 ≥ t0 such that
∞∫
t1
σ(u)f(u, b1, . . . , b1)∆u < ε.
Thus, for t2 ≥ t1 we get∣∣∣∣∣∣
t2∫
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u−
∞∫
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u
∣∣∣∣∣∣ =
=
∣∣∣∣∣∣
∞∫
t2
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u
∣∣∣∣∣∣ ≤
∞∫
t2
σ(u)f(u, xk(g1(u)), . . . , xk(gm(u)))∆u ≤
≤
∞∫
t2
σ(u)f(u, b1, . . . , b1)∆u < ε.
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198 YONG ZHOU, YONGHONG LAN
Hence,
∫ t2
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u →
∫ ∞
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u uni-
formly for k ∈ N as t2 → ∞. Therefore,
lim
k→∞
∞∫
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u =
= lim
k→∞
lim
t2→∞
t2∫
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u =
= lim
t2→∞
lim
k→∞
t2∫
t
tf(u, xk(g1(u)), . . . , xk(gm(u)))∆u =
= lim
t2→∞
t2∫
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u =
=
∞∫
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u.
Let k → ∞. Then (3.7) gives
x(t) =
A+ c(t)x(t− τ) +
∫ t
t1
σ(u)f(u, x(g1(u)), . . . , x(gm(u)))∆u+
+
∫ ∞
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u, if t ≥ t1,
x(t1), if t0 ≤ t < t1.
Clearly, x(t) > 0 on [t0,∞). Therefore, x(t) is a positive solution of (1.1). Since 0 < A ≤ x(t) ≤
≤ b1, from Theorem 3.1, x(t) ∈ E(b, a, 0).
Theorem 3.3. Assume that limt→∞ c(t) = c ∈ [0, 1). Then Eq.(1.1) has a nonoscillatory
solution x(t) ∈ E(∞,∞, d) (d 6= 0) if and only if
∞∫
t0
|f(u, hg1(u), . . . , hgm(u))|∆u < ∞ for some h 6= 0. (3.8)
Proof. Necessity. Without loss of generality, let x(t) ∈ E(∞,∞, d) be an eventually positive
solution of (1.1). From Theorem 3.1, we have that d > 0. From (1.1) and (1.2) we have
y∆∆(t) + f(t, x(g1(t)), . . . , x(gm(t))) = 0.
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Integrating it from t1 to t, we get
y∆(t)− y∆(t1) +
t∫
t1
f(u, x(g1(u)), . . . , x(gm(u)))∆u = 0.
Since limt→∞ y
∆(t) = d > 0, we obtain
∞∫
t1
f(u, x(g1(u)), . . . , x(gm(u)))∆u < ∞ (3.9)
and there exist d1 > 0 and t2 ≥ t1 such that y(t) ≥ d1t for t ≥ t2. Therefore,
∞∫
t1
f(u, x(g1(u)), . . . , x(gm(u)))∆u ≥
∞∫
t1
f(u, y(g1(u)), . . . , y(gm(u)))∆u ≥
≥
∞∫
t1
f(u, d1g1(u), . . . , d1gm(u))∆u. (3.10)
Choosing h = d1 and combining (3.9) and (3.10), we get
∞∫
t1
f(u, hg1(u), . . . , hgm(u))∆u < ∞.
Sufficiency. Set h > 0. Let d > 0, B > 0. From (3.8) there exists a sufficiently large t1 so
that for t ≥ t1 we have t− τ ≥ t0 and gi(t) ≥ t0, i = 1, 2, . . . ,m, and
d
h
+
B
th
+ c(t) +
1
h
∞∫
t1
f(u, hg1(u), . . . , hgm(u))∆u < 1. (3.11)
Define a set Ω by
Ω = {z(t) ∈ X|d ≤ z(t) ≤ h}
and a operator S on Ω by
(Sz)(t) =
d+
B
t
+ c(t)
t− τ
t
z(t− τ)+
+
1
t
∫ t
t1
σ(u)f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u+
+
∫ ∞
t
f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u, if t ≥ t1,
(Sz)(t), if t0 ≤ t < t1.
(3.12)
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200 YONG ZHOU, YONGHONG LAN
Clearly, for z(t) ∈ Ω
(Sz)(t) ≤ d+
B
t
+ c(t)h+
1
t
t∫
t1
σ(u)f(u, hg1(u), . . . , hgm(u))∆u+
+
∞∫
t
f(u, hg1(u), . . . , hgm(u))∆u ≤ d+
B
t
+ c(t)h+
+
∞∫
t1
f(u, hg1(u), . . . , hgm(u))∆u < h, t ≥ t1,
and
(Sz)(t) = (Sz)(t1) ≤ h, t0 ≤ t < t1.
It is easy to see that (Sz)(t) ≥ d for t ≥ t0. Hence, TΩ ⊂ Ω. Define a series of sequences
{zk(t)}, k ∈ N, by
z0(t) = d, zk(t) = (Szk−1)(t), t ≥ t0, k ∈ N0.
We can prove that
d ≤ zk(t) ≤ zk+1(t) ≤ h, t ≥ t0, k ∈ N0.
Then there exists z(t) ∈ Ω such that lim
k→∞
zk(t) = z(t), t ≥ t0 and d ≤ z(t) ≤ h. Clearly,
z(t) = (Sz)(t) (t ≥ t0), i.e.,
z(t) =
d+
B
t
+ c(t)
t− τ
t
z(t− τ)+
+
1
t
∫ t
t1
σ(u)f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u+
+
∞∫
t
f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u, if t ≥ t1,
z(t1), if t0 ≤ t < t1.
Let x(t) = tz(t), t ≥ t0. Then we have
x(t) =
dt+B + c(t)x(t− τ)+
+
∫ t
t1
σ(u)f(u, x(g1(u)), . . . , x(gm(u)))∆u+
+
∫ ∞
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u, if t ≥ t1,
x(t1), if t0 ≤ t < t1.
(3.13)
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Hence, x(t) is a positive solution of (1.1). On the other hand, x(t) ≥ y(t) ≥ dt + B. Hence
limt→∞ x(t) = ∞ and limt→∞ y(t) = ∞. From (3.13), we have
y∆(t) = d+
∞∫
t
f(u, x(g1(u)), . . . , x(gm(u)))∆u =
= d+
∞∫
t
f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u ≤
≤ d+
∞∫
t
f(u, hg1(u), . . . , hgm(u))∆u.
Hence, limt→∞ y
∆(t) = d. Therefore, x(t) ∈ E(∞,∞, d).
Theorem 3.4. Assume that limt→∞ c(t) = c ∈ [0, 1). Further, assume that
∞∫
t0
|f(u, hg1(u), . . . , hgm(u))|∆u < ∞ for some h 6= 0 (3.14)
and
∞∫
t0
σ(u)|f(u, b1, . . . , b1)|∆u = ∞ for some b1 6= 0, (3.15)
where b1h > 0. Then Eq. (1.1) has a nonoscillatory solution x(t) ∈ E(∞,∞, 0).
Proof. Without loss of generality, assume that h > 0 and b1 > 0. From (3.14) there exists a
sufficiently large t1 so that for t ≥ t1 we have t− τ ≥ t0 and gi(σ(t)) ≥ t0, i = 1, 2, . . . ,m, and
b1
th
+ c(t) +
1
h
∞∫
t1
f(u, hg1(u), . . . , hgm(u))∆u < 1. (3.16)
Define a set Ω by
Ω = {z(t) ∈ X|0 ≤ z(t) ≤ h, t ≥ t0}
and an operator S on Ω by
(Sz)(t) =
b1
t
+ c(t)
t− τ
t
z(t− τ)+
+
1
t
∫ t
t1
σ(u)f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u+
+
∫ ∞
t
f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u, if t ≥ t1,
(Sz)(t), if t0 ≤ t < t1.
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202 YONG ZHOU, YONGHONG LAN
Clearly, for z(t) ∈ Ω,
(Sz)(t) ≤ b1
t
+ c(t)h+
1
t
t∫
t1
σ(u)f(u, hg1(u), . . . , hgm(u))∆u+
+
∞∫
t
f(u, hg1(u), . . . , hgm(u))∆u ≤ b1
t
+ c(t)h+
+
∞∫
t1
f(u, hg1(u), . . . , hgm(u))∆u ≤ h, t ≥ t1,
and (Sz)(t) = (Sz)(t1) ≤ h, t0 ≤ t < t1, i.e., SΩ ⊂ Ω.
Define a series of sequences {zk(t)}, k ∈ N, by
z0(t) = 0, zk(t) = (Szk−1)(t), t ≥ t0, k ∈ N0. (3.17)
By induction, we can prove that
0 ≤ zk(t) ≤ zk+1(t) ≤ h, t ≥ t0, k ∈ N.
Then there exists z(t) ∈ Ω such that limk→∞ zk(t) = z(t), t ≥ t0.
Clearly, z(t) = (Sz)(t), t ≥ t0, i.e.,
z(t) =
b1
t
+ c(t)
t− τ
t
z(t− τ)+
+
1
t
∫ t
t1
σ(u)f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u+
+
∫ ∞
t
f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u, if t ≥ t1,
z(t1), if t0 ≤ t < t1.
Let x(t) = tz(t), t ≥ t0. Then we have
x(t) =
b1 + c(t)x(t− τ)+
+
∫ t
t1
σ(u)f(u, x(g1(u)), . . . , x(gm(u)))∆u+
+
∫ ∞
t
tf(u, x(g1(u)), . . . , x(gm(u)))∆u, if t ≥ t1,
x(t1), if t0 ≤ t < t1.
(3.18)
Hence, x(t) is a positive solution of (1.1). On the other hand, from (3.18), we have x(t) ≥ b1
and that
x(t) ≥ y(t) = x(t)− c(t)x(t− τ) ≥
t∫
t1
σ(u)f(u, b1, . . . , b1)∆u
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CLASSIFICATION AND EXISTENCE OF NONOSCILLATORY SOLUTIONS OF SECOND-ORDER NEUTRAL DELAY . . . 203
which together with (3.15) imply limt→∞ x(t) = ∞ and limt→∞ y(t) = ∞. By (3.18), we get
y∆(t) =
∞∫
t
f(u, x(g1(u)), . . . , x(gm(u)))∆u =
=
∞∫
t
f(u, g1(u)z(g1(u)), . . . , gm(u)z(gm(u)))∆u ≤
≤
∞∫
t
f(u, hg1(u), . . . , hgm(u))∆u.
Hence
0 ≤ lim
t→∞
y∆(t) ≤ lim
t→∞
∞∫
t
f(u, hg1(u), . . . , hgm(u))∆u = 0,
i.e., limt→∞ y
∆(t) = 0. Therefore, x(t) ∈ E(∞,∞, 0).
Theorem 3.5. Assume that limt→∞ c(t) = c ∈ [0, 1). Further assume that there exists d > 0
such that
∞∫
t0
f(u, d1, . . . , d1)∆u = ∞ for any d1 ∈ (0, d]. (3.19)
Then every solution x(t) of Eq. (1.1) either oscillates or {x(t)} ∈ E(0, 0, 0).
Proof. Let x(t) be an eventually positive solution of (1.1). By Lemma 3.1, if limt→∞ x(t) =
= 0, then limt→∞ y(t) = 0 and so limt→∞ y
∆(t) = 0.Hence, x(t) ∈ E(0, 0, 0). If limt→∞ x(t) =
= 0 fails, then y(t) > 0 eventually. Since y∆∆(t) < 0,we have y∆(t) > 0, eventually. Therefore,
there exists d ∈ (0, d] such that x(t) ≥ y(t) ≥ d. From (1.1) and (1.2), we have
y∆∆(t) = −f(t, x(g1(t)), . . . , x(gm(t))).
Integrating it from t0 to t, we obtain
y∆(t)− y∆(t0) = −
t∫
t0
f(u, x(g1(u)), . . . , x(gm(u)))∆u ≤ −
t∫
t0
f(u, d, . . . , d)∆u.
Let t → ∞. Then we get
∫ ∞
t0
f(u, d, . . . , d)∆u < ∞ which contradicts (3.19) and completes
the proof.
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204 YONG ZHOU, YONGHONG LAN
The above results can be extended to the second order neutral equation
[x(t)− c(t)x(t− τ)]∆∆ = f(t, x(g1(t)), . . . , x(gm(t))). (3.20)
With respect to equation (3.20), we assume that conditions (H1), (H2) and (H3) hold. By the
same argument, we have the following theorems.
Theorem 3.6. Assume that limt→∞ c(t) = c ∈ [0, 1). Let x(t) be a nonoscillatory solution of
(3.20). Let S denote the set of all nonoscillatory solution of (3.20), and define
E(0, 0, 0) = {x(t) ∈ E : lim
t→∞
x(t) = 0, lim
t→∞
y(t) = 0, lim
t→∞
y∆(t) = 0},
E(b, a, 0) =
{
x(t) ∈ E : lim
t→∞
x(t) = b =
a
1− c
, lim
t→∞
y(t) = a, lim
t→∞
y∆(t) = 0
}
,
E(∞,∞, d) = {x(t) ∈ E : lim
t→∞
x(t) = ∞, lim
t→∞
y(t) = ∞, lim
t→∞
y∆(t) = d 6= 0},
E(∞,∞,∞) = {x(t) ∈ E : lim
t→∞
x(t) = ∞, lim
t→∞
y(t) = ∞, lim
t→∞
y∆(t) = ∞}.
Then
E = E(0, 0, 0) ∪ E(b, a, 0) ∪ E(∞,∞, d) ∪ E(∞,∞,∞).
Theorem 3.7. Assume that limt→∞ c(t) = c ∈ [0, 1). Further, assume that
∞∫
t0
σ(u)|f(u, b1, . . . , b1)|∆u < ∞ for some b1 6= 0.
Then Eq. (3.20) has a nonoscillatory solution x(t) ∈ E(b, a, 0)(b 6= 0, a 6= 0).
Theorem 3.8. Assume that limt→∞ c(t) = c ∈ [0, 1). Then the following statements are true.
(i) If Eq. (3.20) has a nonoscillatory solution x(t) ∈ E(∞,∞, d), d 6= 0, then
∞∫
t0
|f(u, hg1(u), . . . , hgm(u))|∆u < ∞, for some h 6= 0.
(ii) If
∞∫
t0
σ(u)|f(u, hg1(u), . . . , hgm(u))|∆u < ∞, for some h 6= 0,
then Eq. (3.20) has a nonoscillatory solution x(t) ∈ E(∞,∞, d), d 6= 0.
Theorem 3.9. Assume that limt→∞ c(t) = c ∈ [0, 1). Further assume that there exists d > 0
such that
∞∫
t0
f(u, d1, . . . , d1)∆u = ∞ for any d1 ∈ (0, d].
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CLASSIFICATION AND EXISTENCE OF NONOSCILLATORY SOLUTIONS OF SECOND-ORDER NEUTRAL DELAY . . . 205
Then every solution x(t) of Eq. (1.1) either oscillates or x(t) ∈ E(0, 0, 0), or x(t) ∈ E(∞,∞,∞).
In the following, we shall give some examples.
Example 3.1. In the case where T = R, we consider the second order differential equation(
x(t)− 1
2
x(t− τ)
)′′
+
2(t− 1)3 − t3
(t− 1)6
x3(t) = 0. (3.21)
(3.4) becomes
∞∫
t0
u|f(u, b1, . . . , b1)| du < ∞ for some b1 6= 0. (3.22)
It is easy to see that (3.22) holds. Therefore, (3.21) has a nonoscillatory solution x(t) ∈ E(b, a, 0),
b 6= 0, a 6= 0. In fact, x(t) = 1− 1
t
is such a solution, where a =
1
2
, b = 1.
Example 3.2. In the case where T = N, we consider the second order delay difference
equation
∆2
(
xn −
1
4
xn−1
)
+
2−n−3
(n− 1− 2−n+1)5
x5
n−1 = 0, n ≥ 2, (3.23)
for which (3.8) becomes
∞∑
j=n0
|f(j, hg1(j), . . . , hgm(j))| < ∞. (3.24)
It is easy to see that (3.24) is satisfied. In fact, the sequence xn =
{
n− 1
2n
}
is a nonoscillatory
solution of (3.23) which belongs to the class S
(
∞,∞, 3
4
)
.
Example 3.3. In the case where T = hZ = {hk|k ∈ Z} for h > 1, we consider the second
order delay dynamic equation[
x(t)− 1
2
x(t− h)
]∆∆
+
(1
2 − 2−h)(2−h − 1)2
h2
x(t− h) = 0 (3.25)
for which condition (3.19) of Theorem 3.5 is satisfied. In fact, x(t) =
1
2t
is a nonoscillatory
solution of (3.25) which belongs to the class S(0, 0, 0).
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Received 23.12.12
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 2
|
| id | nasplib_isofts_kiev_ua-123456789-177113 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T13:18:27Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Yong Zhou Yonghong Lan 2021-02-10T13:14:13Z 2021-02-10T13:14:13Z 2013 Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales / Yong Zhou, Yonghong Lan // Нелінійні коливання. — 2013. — Т. 16, № 2. — С. 191-206. — Бібліогр.: 20 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177113 517.9 Project supported by National Natural Science Foundation of P.R. China (11271309), the Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Hunan Provincial Natural Science Foundation of China (12JJ2001). en Інститут математики НАН України Нелінійні коливання Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales Класифiкацiя та iснування неколивних розв’язкiв динамiчних рiвнянь другого порядку з нейтральним запiзненням на часовiй шкалi Классификация и существование неколебательных решений динамических уравнений второго порядка с нейтральным запаздыванием на временной шкале Article published earlier |
| spellingShingle | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales Yong Zhou Yonghong Lan |
| title | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales |
| title_alt | Класифiкацiя та iснування неколивних розв’язкiв динамiчних рiвнянь другого порядку з нейтральним запiзненням на часовiй шкалi Классификация и существование неколебательных решений динамических уравнений второго порядка с нейтральным запаздыванием на временной шкале |
| title_full | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales |
| title_fullStr | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales |
| title_full_unstemmed | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales |
| title_short | Classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales |
| title_sort | classification and existence of non-oscillatory solutions of second-order neutral delay dynamic equations on time scales |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/177113 |
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