Oscillation Criteria for Nonlinear Damped Dynamic Equations on Time ...
Oscillation Criteria for Nonlinear Damped Dynamic Equations on Time Scales Lynn Erbe, Taher S. Hassan, and Allan Peterson
Abstract. We present new oscillation criteria for the second order nonlinear damped delay dynamic equation (r(t)(x∆ (t))γ )∆ + p(t)(x∆σ (t))γ + q(t)f (x(τ (t))) = 0. Our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation. Our results are illustrated with examples.
1. Introduction In this paper, we are concerned with oscillation behavior of the second order nonlinear damped delay dynamic equation (1.1)
(r(t)(x∆ (t))γ )∆ + p(t)(x∆σ (t))γ + q(t)f (x(τ (t))) = 0,
on an arbitrary time scale T, where γ is a quotient of odd positive integers, r, p and q are positive rd−continuous functions on T, and the so-called delay function τ : T → T satisfies τ (t) ≤ t for t ∈ T and lim t→∞ τ (t) = ∞. The function f ∈ C (R, R) is assume to satisfy uf (u) > 0 and fu(u) γ ≥ K, for u 6= 0 and for some K > 0. Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that sup T = ∞, and define the time scale interval [t0 , ∞)T by [t0 , ∞)T := [t0 , ∞) ∩ T. By a solution of (1.1) we mean a 1 [Tx , ∞), Tx ≥ t0 which has the property that nontrivial real–valued function x ∈ Crd γ ∆ 1 r(t) x (t) ∈ Crd [Tx , ∞) and satisfies equation (1.1) on [Tx , ∞), where Crd is the space of rd−continuous functions. The solutions vanishing 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. Recently there has been a large number of papers devoted to second order nonlinear dynamic equations on time scales. For example. Agarwal et al [1] considered the second order delay dynamic equation (1.2)
x∆∆ (t) + q(t)x(τ (t)) = 0,
2000 Mathematics Subject Classification. 34K11, 39A10, 39A99. Key words and phrases. Oscillation, delay nonlinear dynamic equations, time scales. 1
2
L. ERBE, T. S. HASSAN, AND A. PETERSON
and established some sufficient conditions for oscillation of (1.2). Zhang et al [19] study the oscillation of the second order nonlinear delay dynamic equation (1.3)
x∆∆ (t) + q(t)f (x(t − τ )) = 0,
and the second order nonlinear dynamic equation (1.4)
x∆∆ (t) + q(t)f (x(σ(t))) = 0,
and established the equivalence of the oscillation of (1.3) and (1.4), from which they obtained some oscillation criteria and comparison theorems for (1.3). Sahiner [15] considered the second order nonlinear delay dynamic equation (1.5)
x∆∆ (t) + q(t)f (x(τ (t))) = 0,
and obtained some sufficient conditions for oscillation of (1.5) by means of the Riccati transformation technique. Erbe et al [10] extended Sahiner’s result to the second order nonlinear delay dynamic equation (r(t)x∆ (t))∆ + q(t)f (x(τ (t))) = 0, Erbe et al [8] considered the pair of second order dynamic equations (1.6)
(r(t)(x∆ (t))γ )∆ + q(t)xγ (t) = 0, (r(t)(x∆ (t))γ )∆ + q(t)xγ (σ(t)) = 0,
and established some necessary and sufficient conditions for nonoscillation of HilleKneser type. Saker [16] examined oscillation for half-linear dynamic equations (1.6), where γ > 1 is an odd positive integer and Agarwal et al [2] studied oscillation for the same equation (1.6), where γ > 1 is the quotient of odd positive integers. These results can not be applied when 0 < γ ≤ 1. Very recently Erbe et al [6] and [7] considered the half-linear delay dynamic equation (r(t)(x∆ (t))γ )∆ + q(t)xγ (τ (t)) = 0, for the cases γ ≥ 1 and 0 < γ ≤ 1 respectively. In addition, it was assumed that r∆ (t) ≥ 0,
(1.7) and Z
∞
τ γ (t)q(t)∆t = ∞.
(1.8) t0
Both of the two cases Z
∞
(1.9)
∆t 1
t0
= ∞,
r γ (t)
and Z
∞
(1.10)
∆t 1
t0
< ∞,
r γ (t)
were considered. Therefore it is of great interest to study the equation (1.1) without necessarily assuming the conditions (1.7)-(1.10). We will still assume γ > 0 is the quotient of odd positive integers and, hence our results will improve and extend many known results on half-linear oscillation.
NONLINEAR DAMPED DYNAMIC EQUATION
Note that in the special case when T = R, then Z b Z σ(t) = t, µ(t) = 0, g ∆ (t) = g 0 (t), g(t)∆t = a
3
b
g(t)dt,
a
and (1.1) becomes the second order nonlinear damped delay differential equation 0
(r(t)(x0 (t))γ ) + p(t)(x0 (t))γ + q(t)f (x(τ (t))) = 0.
(1.11)
The oscillation of equation (1.11) when r (t) = 1, p(t) = 0 and f (x(τ (t))) = xγ (τ (t)), was studied by Agarwal et al [3] and proved that if γ Z ∞ τ (s) lim sup tγ q(s) ds > 1, s t→∞ t then every solution of (1.11) oscillates. When T = Z, then Z b b−1 X σ(t) = t + 1, µ(t) = 1, g ∆ (t) = ∆g(t), g(t)∆t = g(t), a
t=a
and (1.1) becomes the second order nonlinear damped delay difference equation (1.12)
∆(r(t)(∆x(t))γ ) + p(t)(∆x(t))γ + q(t)f (x(τ (t))) = 0.
The oscillation of equation (1.12) when r (t) = 1, p(t) = 0 and f (x(τ (t))) = x(t), was studied by Thandapani et al [17] where γ > 0, p(t) is a positive sequence, and it was shown that every solution of (1.12) is oscillatory, if ∞ X
q(n) = ∞.
n=n0
We will see that our results not only unify some of the known oscillation results for differential and difference equations but can be applied to other cases to determine the oscillatory behavior. Note that, if T =hZ, h > 0, then σ(t) = t + h, µ(t) = h, y ∆ (t) = ∆h y(t) = b−a−h h
b
Z
g(t)∆t = a
y(t + h) − y(t) , h
X
g(a + kh)h,
k=0
and (1.1) becomes the second order nonlinear damped delay difference equation (1.13)
∆h (r(t)(∆h x(t))γ ) + p(t)(∆h x(t))γ + q(t)f (x(τ (t))) = 0.
If T =q N0 = {t : t = q k , k ∈ N0 , q > 1}, then σ(t) = qt, µ(t) = (q − 1)t, x∆ (t) = ∆q x(t) = (x(q t) − x(t))/(q − 1) t, Z ∞ ∞ X g(t)∆t = g(q k )µ(q k ), t0
k=n0
n0
where t0 = q , and (1.1) becomes the second order q−nonlinear damped delay difference equation (1.14)
∆q (r(t)(∆q x(t))γ ) + p(t)(∆q x(t))γ + q(t)f (x(τ (t))) = 0.
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L. ERBE, T. S. HASSAN, AND A. PETERSON
If T = N20 := {n2 : n ∈ N0 }, then
√ y(( t + 1)2 ) − y(t) √ σ(t) = ( t + 1) , µ(t) = 1 + 2 t, ∆N y(t) = , 1+2 t and (1.1) becomes the second order nonlinear damped delay difference equation √
(1.15)
√
2
∆N (r(t)(∆N x(t))γ ) + p(t)(∆N x(t))γ + q(t)f (x(τ (t))) = 0.
If T = {Hn : n ∈ N0 } where Hn denotes the set of numbers defined by H0 = 0, Hn =
n X 1 , n ∈ N, k
k=1
then 1 , y ∆ (t) = ∆Hn y(Hn ) = (n + 1)∆y(Hn ), n+1 and (1.1) becomes the second order nonlinear damped delay difference equation σ(Hn ) = Hn+1 , µ(Hn ) =
(1.16) ∆Hn (r(Hn )(∆Hn x(Hn ))γ )+p(Hn )(∆Hn x(Hn ))γ +q(Hn )f (x(τ (Hn ))) = 0. If
√ T = T2 = { n : n ∈ N0 },
then
√ p p x( t2 + 1) − x(t) ∆ 2 2 √ σ(t) = t + 1, µ(t) = t + 1 − t, x (t) = ∆2 x(t) = , t2 + 1 − t and (1.1) becomes the second order nonlinear delay damped difference equation (1.17)
∆2 (r(t)(∆2 x(t))γ ) + p(t)(∆2 x(t))γ + q(t)f (x(τ (t))) = 0.
We recall that for a discrete time scale, we have Z b X g(t)µ(t). g(t)∆t = a
t∈[a,b)T
We will utilize in the following a Riccati transformation technique to establish oscillation criteria for (1.1), where γ is the quotient of odd positive integers. These results improve and generalize the results that have been established in [2], [6], [7], and [16] and our results are essentially new for equations (1.11)–(1.17). Also, interesting examples that illustrate the importance of our results are included in section 3 below. 2. Main Results Throughout this paper, we let and d− (t) := max {0, −d(t)} ,
d+ (t) := max{0, d(t)} R a ∆s u
θ(a, b; u) := R b u
1
r γ (s) ∆s 1 rγ
,
α(t, u) := θ(τ (t), σ(t); u),
(s)
1 P (t) := σ (δ ∆ (t)rσ (t) − δ(t)p(t)), r (t)
a(t) := e
p(t) r σ (t)
(t, t0 ),
NONLINEAR DAMPED DYNAMIC EQUATION
5
and ∞
∆s
τ (t)
R γ (s)
Z R(t) := a(t)r(t),
Q(t) := a(t)q(t),
β(t) :=
1
.
The first two Lemmas give some sufficient conditions in order that a positive solution of (1.1) is eventually increasing. Lemma 2.1. Assume that Z
∞
(2.1)
∆t
= ∞,
1
t0
R γ (t)
holds and (1.1) has a positive solution x on [t0 , ∞)T . Then there exists a T ∈ [t0 , ∞)T , sufficiently large, so that γ x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T . Proof. Pick t1 ∈ [t0 , ∞)T such that x(τ (t)) > 0 on [t1 , ∞)T . From (1.1), we have γ (2.2) (r(t) x∆ (t) )∆ + p(t)(x∆σ (t))γ = −q(t)f (x(τ (t))) < 0. For t ∈ [t1 , ∞)T . we can write the left hand side of (2.2) in the form γ γ p(t) (r(t) x∆ (t) )∆ a(t) + σ (r(t) x∆ (t) )σ a(t) < 0, r (t) which implies
γ ∆ R(t) x∆ (t) < 0.
Now, we claim that x∆ (t) > 0 on [t1 , ∞)T . If not, then there exists t2 ≥ t1 such γ that x∆ (t2 ) < 0. Using the fact that R(t) x∆ (t) is decreasing, we obtain, for t ∈ [t2 , ∞)T R(t)(x∆ (t))γ < c := R(t2 )(x∆ (t2 ))γ < 0. Integrating from t2 to t, we find that x(t) < x(t2 ) + c
1 γ
Z
t
∆s 1
t2
,
R γ (s)
for t ≥ t2 . Condition (2.1) implies that x(t) is eventually negative, which is a ∆ contradiction. γ ∆Therefore, x (t) > 0 on [t1 , ∞)T and hence, from (1.1), we get ∆ (r(t) x (t) ) < 0 on [t1 , ∞)T . The proof is complete. Lemma 2.2. Assume that γ1 Z ∞ Z t 1 γ Q(s)β (s)∆s ∆t = ∞, (2.3) R(t) t0 t0 holds and (1.1) has a positive solution x on [t0 , ∞)T . Then there exists a T ∈ [t0 , ∞)T , sufficiently large, so that γ x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T .
6
L. ERBE, T. S. HASSAN, AND A. PETERSON
Proof. As in the proof of Lemma 2.1, assume there is a t2 ≥ t0 such that x∆ (t) < 0 on γ [t2 , ∞)T . Pick t3 ≥ t2 so that τ (t) ≥ t2 , for t ≥ t3 . Using the fact that R(t) x∆ (t) is decreasing, we obtain γ 1 Z ∞ (R(s) x∆ (s) ) γ ∆s −x(τ (t)) < x (∞) − x(τ (t)) = 1 τ (t) R γ (s) Z γ 1 ∞ ∆s ≤ (R(τ (t)) x∆ (τ (t)) ) γ 1 τ (t) R γ (s) Z ∞ γ 1 ∆s ≤ (R(t2 ) x∆ (t2 ) ) γ = Lβ(t), 1 τ (t) R γ (s) γ 1 where L := (R(t2 ) x∆ (t2 ) ) γ < 0. From (1.1), we get, for t ≥ t3 γ (R(t) x∆ (t) )∆ = −Q(t)f (x(τ (t))) ≤ −KQ(t)xγ (τ (t)) ≤ KLγ Q(t)β γ (t), Hence, for t ≥ t3 , we have R(t)(x∆ (t))γ
≤ R(t3 )(x∆ (t3 ))γ + KLγ
Z
t
Q(u)β γ (u)∆u
t3
≤ KLγ
Z
t
Q(u)β γ (u)∆u.
t3
It follows from this last inequality that γ1 Z s Z t 1 1 γ γ Q(u)β (u)∆u ∆s. x(t) − x(t3 ) ≤ K L t3 R(s) t3 Hence by (2.3), we have limt→∞ x(t) = −∞, which contradicts the fact that x is ∆ a positive solution γ ∆of (1.1). Thus x (t) > 0 on [t1 , ∞)T and hence, from (1.1), we ∆ get (r(t) x (t) ) < 0 on [t1 , ∞)T . Lemma 2.3. Assume that γ1 Z t Z ∞ 1 Q(s)∆s ∆t = ∞, (2.4) R(t) t0 t0 holds and (1.1) has a positive solution x on [t0 , ∞)T . Then either there exists a T ∈ [t0 , ∞)T , sufficiently large, so that γ x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T , or limt→∞ x(t) = 0. Proof. As in the proof of Lemma 2.1, assume there is a t2 ≥ t0 such that x∆ (t) < 0 on [t2 , ∞)T . Thus, we get limt→∞ x(t) = l ≥ 0. If we assume l > 0, then x(τ (t)) ≥ l, for t ≥ t3 . From (1.1), we get γ (R(t) x∆ (t) )∆ = −Q(t)xγ (τ (t)) ≤
−lγ Q(t),
NONLINEAR DAMPED DYNAMIC EQUATION
7
Hence, for t ≥ t3 , we have ∆
γ
R(t)(x (t))
∆
γ
≤ R(t3 )(x (t3 )) − l
γ
Z
t
Q(u)∆u t3
≤
−lγ
Z
t
Q(u)∆u. t3
It follows from this last inequality that γ1 Z t Z s 1 x(t) − x(t3 ) ≤ −l Q(u)∆u ∆s. t3 R(s) t3 Hence by (2.4), we have limt→∞ x(t) = −∞, which contradicts the fact that x is a positive solution of (1.1). Thus, l = 0. This completes the proof. Lemma 2.4. Assume that there exists T ≥ t0 , sufficiently large, such that γ x(t) > 0, x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T . Then x(τ (t)) > α (t, T ) xσ (t) , for t ≥ T1 ≥ T. γ Proof. Since r(t) x∆ (t) is strictly decreasing on [T, ∞). We can choose T1 ≥ T so that τ (t) ≥ T, for t ≥ T1 . Then, we have that γ 1 Z σ(t) (r(s) x∆ (s) ) γ xσ (t) − x(τ (t)) = ∆s 1 τ (t) r γ (s) Z γ 1 σ(t) ∆s , ≤ (r(τ (t)) x∆ (τ (t)) ) γ 1 τ (t) r γ (s) and so γ 1 Z (r(τ (t)) x∆ (τ (t)) ) γ σ(t) ∆s xσ (t) (2.5) ≤1+ . 1 x(τ (t)) x(τ (t)) τ (t) r γ (s) Also, we see that γ 1 Z τ (t) (r(s) x∆ (s) ) γ x(τ (t)) > x(τ (t)) − x (T ) = ∆s 1 T r γ (s) Z γ 1 τ (t) ∆s ≥ (r(τ (t)) x∆ (τ (t)) ) γ , 1 T r γ (s) and hence !−1 γ 1 Z τ (t) (r(τ (t)) x∆ (τ (t)) ) γ ∆s < . (2.6) 1 x(τ (t)) T r γ (s) Therefore, (2.5) and (2.6) imply Z σ(t) ∆s xσ (t) < 1 x(τ (t)) T r γ (s)
Z
τ (t)
∆s 1
T
!−1 ,
r γ (s)
and hence we get the desired inequality x(τ (t)) > α (t, T ) xσ (t) , This completes the proof.
for t ≥ T1 ≥ T.
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L. ERBE, T. S. HASSAN, AND A. PETERSON
¿From Lemmas 2.1-2.2, we get the following oscillation criteria for equation (1.1). Theorem 2.1. Assume one of the conditions (2.1) or (2.3) holds. Furthermore, suppose that there exists a positive ∆−differentiable function δ(t) such that, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 γ (2.7) lim sup Kα (s, T ) δ(s)q(s) − ∆s = ∞. (γ + 1)γ+1 δ γ (s) t→∞ T Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume (1.1) has a nonoscillatory solution on [t0 , ∞)T . Then, without loss of generality, there is a T ∈ [t0 , ∞)T , sufficiently large, so that x(t) satisfies the conclusions of Lemmas 2.1-2.2 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . In particular, we have γ x(τ (t)) > 0, x∆ (t) > 0, (r(t) x∆ (t) )∆ < 0, for t ≥ T. Consider the generalized Riccati substitution ∆ γ x (t) . (2.8) w(t) = δ(t)r(t) x(t) By the product rule and then the quotient rule σ ∆ r(t)(x∆ (t))γ r(t)(x∆ (t))γ + δ(t) w∆ (t) = δ ∆ (t) xγ (t) xγ (t) σ (r(t)(x∆ (t))γ )∆ r(t)(x∆ (t))γ r(t)(x∆ (t))γ (xγ (t))∆ + δ(t) = δ ∆ (t) − δ(t) . xγ (t) xγσ (t) xγ (t)xγσ (t) ¿From (1.1) and the definition of w(t) and P (t), we have γ x(τ (t)) r(t)(x∆ (t))γ (xγ (t))∆ P (t) σ ∆ − δ(t) . w (t) ≤ σ w (t) − Kδ(t)q(t) δ (t) xσ (t) xγ (t)xγσ (t) Using the fact that r(t)(x∆ (t))γ is strictly decreasing and the definition of w(t), we obtain γ P (t) x(τ (t)) δ(t)wσ (t) (xγ (t))∆ w∆ (t) ≤ σ wσ (t) − Kδ(t)q(t) − . σ δ (t) x (t) δ σ (t) xγ (t) ¿From Lemma 2.4, we get (2.9)
w∆ (t) ≤
P (t) σ δ(t)wσ (t) (xγ (t))∆ γ w (t) − Kα (t, T )δ(t)q(t) − . δ σ (t) δ σ (t) xγ (t)
By the P¨ otzsche chain rule ([4, Theorem 1.90]), we obtain Z 1 γ−1 ∆ (xγ (t)) = γ x(t) + hµ(t)x∆ (t) dh x∆ (t) 0 1
Z = γ ≥
γ−1
[(1 − h) x(t) + hxσ (t)]
dh x∆ (t)
0 γ−1
γ (x(t))
x∆ (t),
γ>1 .
γ−1
γ (xσ (t))
x∆ (t),
0 1, we have that (2.11)
w∆ (t) ≤
γδ(t)wσ (t) x∆ (t) xσ (t) P (t) σ γ w (t) − Kα (t, T ) δ(t)q(t) − . δ σ (t) δ σ (t) xσ (t) x(t)
Using the fact that x(t) is strictly increasing and r(t)(x∆ (t))γ is strictly decreasing, we get that σ γ1 r (t) σ ∆ (x∆ (t))σ . (2.12) x (t) ≥ x(t), x (t) ≥ r(t) ¿From (2.10), (2.11) and (2.12), we obtain (2.13) where λ :=
w∆ (t) ≤ γ+1 γ .
P+ (t) σ γδ(t)(wσ (t))λ . w (t) − Kαγ (t, T ) δ(t)q(t) − 1 σ δ (t) (δ σ (t))λ r γ (t)
Define A > 0 and B > 0 by λ
A :=
γδ(t)(wσ (t))λ 1
(δ σ (t))λ r γ (t)
1
,
B
λ−1
:=
1
(r γ (t)) λ P+ (t) 1
1
λγ λ (δ(t)) λ
.
Then, using the inequality λAB λ−1 − Aλ ≤ (λ − 1)B λ ,
(2.14) we get that
γδ(t)(wσ (t))λ P+ (t) σ w (t) − 1 δ σ (t) (δ σ (t))λ r γ (t)
= λAB λ−1 − Aλ ≤ (λ − 1)B λ r(t)(P+ (t))γ+1 = . (γ + 1)γ+1 δ γ (t)
¿From this last inequality and (2.13) we get w∆ (t) ≤
r(t)(P+ (t))γ+1 − Kδ(t)αγ (t, T ) q(t) (γ + 1)γ+1 δ γ (t)
Integrating both sides from T to t we get Z t r(s)(P+ (s))γ+1 γ ∆s ≤ w(T ) − w(t) ≤ w(T ), Kα (s, T ) δ(s)q(s) − (γ + 1)γ+1 δ γ (s) T which leads to a contradiction to (2.7).
By choosing δ(t) = 1 and δ(t) = t, t ≥ t0 in Theorem 2.1 we have the following oscillation results. Corollary 2.1. Assume one of the conditions (2.1) or (2.3) holds and, for all sufficiently large T, Z ∞ αγ (t, T ) q(t)∆t = ∞. (2.15) T
Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T .
10
L. ERBE, T. S. HASSAN, AND A. PETERSON
Corollary 2.2. Assume one of the conditions (2.1) or (2.3) holds and, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 γ (2.16) lim sup ∆s = ∞, Ksα (s, T ) q(s) − (γ + 1)γ+1 sγ t→∞ T where P (t) = 1 − [t0 , ∞)T
tp(t) r σ (t) .
Then every solution of equation (1.1) is oscillatory on
We are now ready to state and prove Philos-type oscillation criteria for the equation (1.1). Theorem 2.2. Assume one of the conditions (2.1) or (2.3) holds. Furthermore, suppose that there exist functions H, h ∈ Crd (D, R), where D ≡ {(t, s) : t ≥ s ≥ t0 } such that (2.17)
H (t, t) = 0,
t ≥ t0 , H (t, s) > 0,
t > s ≥ t0 ,
and H has a nonpositive continuous ∆-partial derivative H ∆s (t, s) with respect to the second variable and satisfies H ∆s (t, s) + H (t, s)
(2.18)
γ h (t, s) P (s) =− σ (H (t, s)) γ+1 , σ δ (s) δ (s)
and, for all sufficiently large T, (2.19) # γ+1 Z t" 1 (h (t, s)) r (s) − ∆s = ∞, lim sup Kαγ (s, T ) δ (s) q (s) H (t, s) − (γ + 1)γ+1 δ γ (s) t→∞ H (t, T ) T where δ(t) is a positive ∆−differentiable function. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume (1.1) has a nonoscillatory solution on [t0 , ∞)T . Then, without loss of generality, there is a T ∈ [t0 , ∞)T , sufficiently large, so that x(t) satisfies the conclusions of Lemmas 2.1-2.2 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . In particular, we have γ x(τ (t)) > 0, x∆ (t) > 0, (r(t) x∆ (t) )∆ < 0, for t ≥ T. We define w (t) also, as in Theorem 2.1. From (2.13) with P+ (t) replaced by P (t), we have Kαγ (t, T )δ(t)q(t) ≤ −w∆ (t) +
(2.20)
P (t) σ γδ(t) w (t) − 1 (wσ (t))λ . σ λ δ σ (t) γ r (t)(δ (t))
Multiplying both sides of (2.20), with t replaced by s, by H (t, s), integrating with respect to s from T to t, t ≥ T, Z t Z t γ H (t, s) Kα (s, T )δ(s)q(s)∆s ≤ − H (t, s) w∆ (s) ∆s T
Z
t
+
H (t, s) T
T
P (s) σ w (s) ∆s − δ σ (s)
Z
t
H (t, s) T
γδ(s) 1 γ
r (s)(δ σ (s))λ
(wσ (s))λ ∆s.
NONLINEAR DAMPED DYNAMIC EQUATION
11
Integrating by parts and using (2.17) and (2.18), we obtain Z t H (t, s) Kαγ (s, T )δ(s)q(s)∆s T Z t ≤ H (t, T ) w (T ) + H ∆s (t, s) wσ (s) ∆s T
Z
t
+
H (t, s) T
P (s) σ w (s) ∆s − δ σ (s)
Z
t
H (t, s) T
γδ(s) 1 γ
r (s)(δ σ (s))λ
(wσ (s))λ ∆s
≤ H (t, T ) w (T ) Z t" + T
# 1 γδ(s) h (t, s) σ σ λ (H (t, s)) λ w (s) − H (t, s) 1 (w (s)) ∆s − σ δ (s) r γ (s)(δ σ (s))λ ≤ H (t, T ) w (T )
Z t" (2.21)
+ T
# 1 h− (t, s) γδ(s) (wσ (s))λ ∆s. (H (t, s)) λ wσ (s) − H (t, s) 1 δ σ (s) r γ (s)(δ σ (s))λ
Again, define A > 0 and B > 0 by Aλ :=
γH (t, s) δ(s)(wσ (s))λ 1
r γ (s)(δ σ (s))λ
1
,
B λ−1 :=
h− (t, s) r γ+1 (s) 1
,
λ (γδ(s)) λ
and using the inequality (2.14), we obtain 1 γH (t, s) δ(s)(wσ (s))λ h− (t, s) σ λ (H (t, s)) w (s) − = λAB λ−1 − Aλ 1 δ σ (s) r γ (s)(δ σ (s))λ
hγ+1 (t, s) r (s) − . (γ + 1)γ+1 δ γ (t) ¿From this last inequality and (2.21), we have # γ+1 Z t" (h− (t, s)) r (s) γ Kα (s, T ) δ (s) q (s) H (t, s) − ∆s ≤ H (t, T ) w (T ) , (γ + 1)γ+1 δ γ (s) T ≤ (λ − 1)B λ =
and this implies that # γ+1 Z t" (h− (t, s)) r (s) 1 γ Kα (s, T ) δ (s) q (s) H (t, s) − ∆s ≤ w (T ) , H (t, T ) T (γ + 1)γ+1 δ γ (s) which contradicts assumption (2.19). This completes the proof.
Also, by Lemma 2.3, we obtain another oscillation criterion for the equation (1.1) as in Theorems 2.1 and 2.2 and Corollaries 2.1 and 2.2 as follows. Corollary 2.3. Assume that (2.4) and (2.7) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 2.4. Assume that (2.4) and (2.15) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 2.5. Assume that (2.4) and (2.16) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero.
12
L. ERBE, T. S. HASSAN, AND A. PETERSON
Corollary 2.6. Assume that (2.4), (2.17), (2.18) and (2.19) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero. Remark 2.1. Note that conditions (1.7) and (1.8) are not assumed to hold and condition (1.9) is also not necessary in order that (1.1) be oscillatory, in contrast to the results of [6] and [7]. We introduce the following notation, for all sufficiently large T, Z ∞ Z 1 t sγ+1 tγ Q∗ (s)∆s, Q∗ (s)∆s, q∗ := lim inf p∗ := lim inf t→∞ t T r(s) t→∞ r(t) σ(t) tγ wσ (t) tγ wσ (t) , R∗ := lim sup , t→∞ r (t) r(t) t→∞ t . Note that where Q∗ (t) = Kαγ (t, T )q (t) , and assume that l := lim inf t→∞ σ(t) 0 ≤ l ≤ 1. In order for the definition of p∗ to make sense we assume that Z ∞ (2.22) Q∗ (s)∆s < ∞. r∗ := lim inf
t0
Theorem 2.3. Assume (2.1) holds and r(t) is a (delta) differentiable function with r∆ (t) ≥ 0 and (2.22) holds. Furthermore, assume l > 0 and (2.23)
p∗ >
lγ 2 (γ
γγ , + 1)γ+1
or (2.24)
p∗ + q ∗ >
1 lγ(γ+1)
.
Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume (1.1) has a nonoscillatory solution on [t0 , ∞)T . Then, without loss of generality, there is a T ∈ [t0 , ∞)T , sufficiently large, so that x(t) satisfies the conclusions of Lemmas 2.1-2.2 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . In particular, we have γ x(τ (t)) > 0, x∆ (t) > 0, (r(t) x∆ (t) )∆ < 0, for t ≥ T. We define w (t) also, as in Theorem 2.1 by putting δ(t) = 1. Note in this case P+ (t) = 0. From (2.13), we have γ+1 γ (2.25) w∆ (t) ≤ −Q∗ (t) − 1 (wσ (t)) γ ≤ 0, for t ∈ [T, ∞)T . γ r (t) γ First, we assume (2.23) holds. It follows from (2.8) and r(t) x∆ (t) is strictly decreasing that !−γ ∆ γ Z t x (t) ∆s w(t) = r(t) < , for t ∈ [T, ∞)T . 1 x(t) t0 r γ (s) Rt Since (2.1) implies t0 ∆s = ∞, we have that limt→∞ w(t) = 0. Integrating (2.25) 1 r γ (s)
from σ(t) to ∞ and using limt→∞ w(t) = 0, we have 1 Z ∞ Z ∞ (wσ (s)) γ wσ (s) (2.26) wσ (t) ≥ Q∗ (s)∆s + γ ∆s. 1 σ(t) σ(t) r γ (s)
NONLINEAR DAMPED DYNAMIC EQUATION
13
It follows from (2.26) that tγ wσ (t) tγ ≥ r(t) r(t)
(2.27)
∞
Z
Q∗ (s)∆s + γ σ(t)
tγ r(t)
1
∞
(wσ (s)) γ wσ (s)
σ(t)
r γ (s)
Z
1
∆s.
Let > 0, then by the definition of p∗ and r∗ we can pick t1 ∈ [T, ∞)T , sufficiently large, so that Z ∞ tγ tγ wσ (t) (2.28) ≥ r∗ − , Q∗ (s)∆s ≥ p∗ − , and r(t) σ(t) r(t) for t ∈ [t1 , ∞)T . From (2.27) and (2.28) and using the fact r∆ (t) ≥ 0, we get that 1 Z ∞ tγ wσ (t) s (wσ (s)) γ sγ wσ (s) tγ ∆s ≥ (p∗ − ) + γ 1 r(t) r(t) σ(t) sγ+1 r γ (s) γ Z ∞ γr(s) 1+ γ1 t ∆s ≥ (p∗ − ) + (r∗ − ) r(t) σ(t) sγ+1 Z ∞ γ 1+ 1 ≥ (p∗ − ) + (r∗ − ) γ tγ (2.29) ∆s. γ+1 σ(t) s Using the P¨ otzsche chain rule ([4, Theorem 1.90]), we get ∆ Z 1 −1 1 dh = γ γ+1 sγ [s + hµ(s)] 0 Z 1 γ dh ≤ sγ+1 0 γ (2.30) = . sγ+1 Then from (2.29) and (2.30), we have γ tγ wσ (t) t 1+ γ1 ≥ (p∗ − ) + (r∗ − ) . r(t) σ(t) Taking the lim inf of both sides as t → ∞ we get that 1+ γ1 γ
r∗ ≥ p∗ − + (r∗ − )
l .
Since > 0 is arbitrary, we get 1+ γ1 γ
p∗ ≤ r∗ − r∗
(2.31)
l .
Using the inequality Bu − Au
γ+1 γ
≤
γγ B γ+1 γ+1 (γ + 1) Aγ
with B = 1 and A = lγ we get that p∗ ≤
γγ , l (γ + 1)γ+1 γ2
which contradicts (2.23). Next, we assume (2.24) holds. Multiplying both sides of γ+1 (2.25) by tr(t) , and integrating from T to t (t ≥ T ) we get Z
t
T
sγ+1 ∆ w (s)∆s ≤ − r(s)
Z
t
T
sγ+1 Q∗ (s)∆s − γ r(s)
Z t T
sγ wσ (s) r(s)
γ+1 γ ∆s.
14
L. ERBE, T. S. HASSAN, AND A. PETERSON
Using integration by parts, we obtain Z t γ+1 Z t γ+1 ∆ tγ+1 w(t) s T γ+1 w(T ) s wσ (s)∆s − ≤ + Q∗ (s)∆s r(t) r(T ) r(s) T r(s) T Z t γ σ γ+1 s w (s) γ ∆s. −γ r(s) T By the quotient rule and applying the P¨otzsche chain rule, γ+1 ∆ s (sγ+1 )∆ sγ+1 r∆ (s) = − σ r(s) r (s) r(s)rσ (s) (γ + 1)σ γ (s) ≤ rσ (s) (γ + 1)σ γ (s) (2.32) ≤ . r(s) Hence tγ+1 w(t) r(t)
γ Z t γ+1 Z t σ (s)wσ (s) T γ+1 w(T ) s ∆s − Q∗ (s)∆s + (γ + 1) r(T ) r(s) T r(s) T Z t γ σ γ+1 s w (s) γ ∆s. − γ r(s) T
≤
Let 0 < ≤ l be given, then using the definition of l, we can assume, without loss of generality, that T is sufficiently large so that s ≥ l − , s ≥ T. σ(s) It follows that σ(s) ≤ Ls,
s≥T
where
L :=
1 > 0. l−
We then get that tγ+1 w(t) r(t)
Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s r(T ) T r(s) γ σ γ+1 Z t γ σ s w (s) γ γ s w (s) }∆s. + {(γ + 1)L −γ r(s) r(s) T
Let u(s) :=
sγ wσ (s) , r(s)
then uλ (s) = where λ =
γ+1 γ .
sγ wσ (s) r(s)
λ .
It follows that tγ+1 w(t) r(t)
Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s r(T ) T r(s) Z t + {(γ + 1)Lγ u(s) − γuλ (s)}∆s. T
NONLINEAR DAMPED DYNAMIC EQUATION
15
Again, using the inequality Bu − Auλ ≤
γγ B γ+1 , γ+1 (γ + 1) Aγ
where A, B are constants, we get Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s r(T ) T r(s) Z t [(γ + 1)Lγ ]γ+1 γγ + ∆s γ+1 γγ T (γ + 1) Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s + Lγ(γ+1) (t − T ). r(T ) T r(s)
tγ+1 w(t) r(t)
It follows from this that tγ w(t) ≤ r(t)
T γ+1 w(T ) r(T )
t
−
t
1 t
Z
1 t
Z
T
sγ+1 T Q∗ (s)∆s + Lγ(γ+1) (1 − ). r(s) t
∆
Since w (t) ≤ 0, we get tγ wσ (t) ≤ r(t)
T γ+1 w(T ) r(T )
t
−
t
T
sγ+1 T Q∗ (s)∆s + Lγ(γ+1) (1 − ). r(s) t
Taking the lim sup of both sides as t → ∞ we obtain R∗ ≤ −q∗ + Lγ(γ+1) = −q∗ +
1 . (l − )γ(γ+1)
Since > 0 is arbitrary, we get that R∗ ≤ −q∗ +
1 . lγ(γ+1)
Using this and the inequality (2.31) we get 1+ γ1
p∗ ≤ r∗ − l γ r∗
≤ r∗ ≤ R∗ ≤ −q∗ +
1 . lγ(γ+1)
Therefore p∗ + q ∗ ≤
1 lγ(γ+1)
which contradicts (2.24).
Remark 2.2. Note that condition (1.8) is not assumed to hold, in contrast to the results of [6] and [7]. 3. Examples In this section, we give some examples to illustrate our main results. Example 3.1. Consider the nonlinear delay dynamic equation γ−1 γ ∆ (σ(t))γ−1 ∆σ γ 1 t ∆ x (t) + γ xγ (τ (t)) = 0, x (t) + (3.1) a(t) taσ (t) α (t, t0 ) t2
16
L. ERBE, T. S. HASSAN, AND A. PETERSON
where γ is the quotient of odd positive integers and a(t) = e 1t (t, t0 ). Here p(t) = (σ(t))γ−1 taσ (t) ,
1 q(t) = αγ (t,t 2 and r(t) = 0 )t condition (2.1) holds since Z ∞ ∆t t0
1 γ
tγ−1 a(t) ,
then, it is clear that P (t) = 0 and the
∞
Z
∆t
=
R (t)
t
t0
1− γ1
= ∞,
by Example 5.60 in [5]. Also Z t r(s)(P+ (s))γ+1 γ lim sup ∆s Ksα (s, T ) q(s) − (γ + 1)γ+1 sγ t→∞ T Z t ∆s = K lim sup = ∞, t→∞ t0 s R∞ α(s,T ) = 1. Then by Corollary 2.2, every = ∞ implies limt→∞ α(t,t since t0 ∆t 1 0) r γ (t)
solution of (3.1) is oscillatory. Example 3.2. Consider the nonlinear delay dynamic equation ∆ (σ(t))γ−1 ∆σ γ tγ (tσ(t))γ ∆ γ x (t) + x (t) + γ xγ (τ (t)) = 0, (3.2) a(t) t α (t, t0 ) where 0 < γ ≤ 1 is the quotient of odd positive integers, a(t) = e p(t) (t, t0 ) and we r σ (t) assume Z ∞ ∆t (3.3) = ∞, for 0 < γ ≤ 1, 1− γ1 t0 t σ(t) for those time scales [t0 , ∞)T , t0 > 0. This holds for many time scales, for example when T =q N0 = {t : t = q k , k ∈ N0 , q > 1}. It is clear r(t) satisfies ∆ Z ∞ Z ∞ Z ∞ −1 1 ∆t ∆t = ≤ ∆t < ∞. 1 t t0 t0 tσ(t) t0 R γ (t) To see that (2.3) holds note that γ1 Z ∞ Z t 1 γ Q(s)β (s)∆s ∆t r(t) t0 t0
Z
∞
≥ t0
Z
∞
≥ t0
1 (tσ(t))γ
Z t
t − t0 (tσ(t))γ
γ1
t0
sβ(s) α (s, t0 )
γ
γ1 ∆s ∆t
∆t,
since Z
∞
β(t) = τ (t)
∆s 1 γ
Z
∞
=
R (s)
τ (t)
1 = sσ(s)
Z
∞
τ (t)
−1 s
∆ ∆s =
1 1 α(t, t0 ) ≥ ≥ . τ (t) t t
We can find 0 < k < 1 such that t − t0 > kt, for t ≥ tk > t0 . Therefore, we get γ1 Z t Z ∞ Z ∞ 1 1 ∆t (3.3) Q(s)β γ (s)∆s ∆t > k γ = ∞. 1− γ1 r(t) t0 t0 tK t σ(t) To apply Corollary 2.1, it remains to prove that condition (2.15) holds, then Z ∞ Z ∞ αγ (t, T ) q(t)∆t = tγ ∆t = ∞, T
t0
NONLINEAR DAMPED DYNAMIC EQUATION
R∞
∆t
t0
r γ (t)
17
α(s,T ) = ∞ implies limt→∞ α(t,t = 1. We 0) R∞ conclude that if [t0 , ∞)T , t0 > 0 is a time scale where t0 1−∆t = ∞, then, by 1
where we use, as in Example 3.1,
1
t
γ
σ(t)
Corollary 2.1, every solution of (3.2) is oscillatory. Example 3.3. Consider the nonlinear dynamic equation γ−1 γ ∆ (γ − 1) (σ(t))γ−1 ∆σ γ η t x (t) + γ xγ (τ (t)) = 0, x∆ (t) + (3.4) a(t) taσ (t) α (t, t0 ) t2 where γ ≥ 1 is the quotient of odd positive integers, η is a positive constant and γ−1 γ−1 η a(t) = e p(t) and q(t) = αγ (t,t (t, t0 ). Here r(t) = ta(t) , p(t) = (γ−1)(σ(t)) 2. taσ (t) 0 )t σ r (t)
Note that Z
∞
t0
r∆ (t)
= ≥
Z
∆t
∞
∆t
=
1 γ
R (t)
t0
t
1− γ1
= ∞,
1 (tγ−1 )∆ a(t) − (γ − 1)tγ−2 a(t) σ a(t)a (t) 1 (γ − 1)tγ−2 − (γ − 1)tγ−2 = 0, σ a (t)
and, as in Example 3.1, p∗
=
lim inf t→∞
tγ r(t)
∞
Z
Kαγ (s, T )q (s) ∆s
σ(t)
Z
∞
= ηK lim inf ta(t) t→∞
σ(t)
Z
∞
≥ ηK lim inf t t→∞
σ(t) ∞
Z ≥ ηK lim inf t t→∞
if η η>
γγ > lγ 2 −1 K(γ+1) . γ+1 γγ . lγ 2 −1 K(γ+1)γ+1
σ(t)
αγ (s, T ) 1 ∆s αγ (s, t0 ) s2
∆s s2 γγ ∆s = ηlK > γ 2 , sσ(s) l (γ + 1)γ+1
Then, by Theorem 2.3, we get that (3.4) is oscillatory if
Additional examples may be readily given. We leave this to interested reader. 4. Applications In this section, we apply the oscillation criteria to different types of time scales, Rb Rb for example if T = R then σ(t) = t, µ(t) = 0, f ∆ (t) = f 0 (t), a f (t)∆t = a f (t)dt, and (1.1) becomes the nonlinear damped delay differential equation (4.1)
(r(t)(x0 (t))γ )0 + p(t)(x0 (t))γ + q(t)f (x(τ (t))) = 0.
then we have from Theorems 2.1-2.3 and Corollaries 2.1-2.6 the following oscillation criteria for equation (4.1). Theorem 4.1. Assume one of the conditions Z ∞ dt (4.2) = ∞, 1 t0 R γ (t)
18
L. ERBE, T. S. HASSAN, AND A. PETERSON
or Z
∞
(4.3) t0
1 R(t)
Z
t γ
γ1
Q(s)β (s)ds
dt = ∞,
t0
holds. Furthermore, suppose that there exists a positive differentiable function δ(t) such that, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 (4.4) lim sup Kαγ (s, T ) δ(s)q(s) − ds = ∞. (γ + 1)γ+1 δ γ (s) t→∞ T Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Corollary 4.1. Assume one of the conditions (4.2) or (4.3) holds and, for all sufficiently large T, Z ∞ αγ (t, T ) q(t)dt = ∞. (4.5) T
Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Corollary 4.2. Assume one of the conditions (4.2) or (4.3) holds and, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 γ ds = ∞. (4.6) lim sup Ksα (s, T ) q(s) − (γ + 1)γ+1 sγ t→∞ T Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Theorem 4.2. Assume one of the conditions (4.2) or (4.3) holds. Furthermore, suppose that there exist functions H, h ∈ C (D, R), where D ≡ {(t, s) : t ≥ s ≥ t0 } such that (4.7)
H (t, t) = 0,
t ≥ t0 , H (t, s) > 0,
t > s ≥ t0 ,
γ ∂H (t, s) P (s) h (t, s) + H (t, s) σ =− σ (H (t, s)) γ+1 , ∂s δ (s) δ (s) and, for all sufficiently large T, (4.9) # γ+1 Z t" (h− (t, s)) r (s) 1 γ Kα (s, T ) δ (s) q (s) H (t, s) − ds = ∞, lim sup (γ + 1)γ+1 δ γ (s) t→∞ H (t, T ) T
(4.8)
Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Corollary 4.3. Assume that (4.4) holds and γ1 Z ∞ Z t 1 (4.10) Q(s)ds dt = ∞, R(t) t0 t0 Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.4. Assume that (4.5) and (4.10) hold. Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.5. Assume that (4.6) and (4.10) hold. Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.6. Assume that (4.7), (4.8), (4.9) and (4.10) hold. Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero.
NONLINEAR DAMPED DYNAMIC EQUATION
19
Theorem 4.3. Assume (4.2) holds and r(t) is differentiable function with r0 (t) ≥ 0 and Z ∞ αγ (t)q (t) dt < ∞ t0
hold. Furthermore, assume that p∗ >
γγ , (γ + 1)γ+1
or p∗ + q∗ > 1, where, for all sufficiently large T, Z Ktγ ∞ γ p∗ := lim inf α (s)q (s) ds, t→∞ r(t) t
q∗ := lim inf t→∞
K t
Z
t
T
sγ+1 γ α (s)q (s) ds. r(s)
Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Rb Pb−1 If T = Z, then σ(t) = t + 1, µ(t) = 1, f ∆ (t) = ∆f (t), a f (t)∆t = t=a f (t), and (1.1) becomes the nonlinear damped delay difference equation (4.11)
∆(r(t)(∆x(t))γ ) + p(t)(∆x(t + 1))γ + q(t)f (x(τ (t))) = 0.
then we have from Theorems 2.1-2.3 and Corollaries 2.1-2.6 the following oscillation criteria for equation (4.11). Theorem 4.4. Assume one of the conditions ∞ X 1 (4.12) = ∞, 1 γ t=t0 R (t) or " # γ1 ∞ t−1 X 1 X γ (4.13) Q(s)β (s) = ∞, R(t) s=t t=t 0
0
holds. Furthermore, suppose that there exists a sequence δ(t) such that, for all sufficiently large N, t−1 X r(s)(P+ (s))γ+1 = ∞. (4.14) lim sup Kαγ (s, N ) δ(s)q(s) − (γ + 1)γ+1 δ γ (s) t→∞ s=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Corollary 4.7. Assume one of the conditions (4.12) or (4.13) holds and, for all sufficiently large N, ∞ X (4.15) αγ (t, N ) q(t) = ∞. t=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Corollary 4.8. Assume one of the conditions (4.2) or (4.3) holds and, for all sufficiently large N, t−1 X r(s)(P+ (s))γ+1 γ (4.16) lim sup Ksα (s, N ) q(s) − = ∞. (γ + 1)γ+1 sγ t→∞ s=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T .
20
L. ERBE, T. S. HASSAN, AND A. PETERSON
Theorem 4.5. Assume one of the conditions (4.12) or (4.13) holds. Furthermore, suppose that there exist two sequences H, h on D, where D ≡ {(t, s) : t ≥ s ≥ t0 } such that (4.17) (4.18)
H (t, t) = 0,
t ≥ t0 , H (t, s) > 0,
t > s ≥ t0 ,
γ h (t, s) P (s) γ+1 = − (H (t, s)) , δ σ (s) δ σ (s)
∆s H (t, s) + H (t, s)
and, for all sufficiently large N, (4.19) " # γ+1 t−1 X (h (t, s)) r (s) 1 − = ∞, Kαγ (s, N ) δ (s) q (s) H (t, s) − lim sup (γ + 1)γ+1 δ γ (s) t→∞ H (t, N ) s=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Corollary 4.9. Assume that (4.14) holds and " # γ1 ∞ t−1 X 1 X (4.20) = ∞, Q(s) R(t) s=t t=t 0
0
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.10. Assume that (4.15) and (4.20) hold. Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.11. Assume that (4.16) and (4.20) hold. Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.12. Assume that (4.17), (4.18), (4.19) and (4.20) hold. Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Theorem 4.6. Assume (4.12) holds, ∆r(t) ≥ 0 and ∞ X
αγ (t)q (t) < ∞
t=t0
hold. Furthermore, assume that p∗ >
γγ , (γ + 1)γ+1
or p∗ + q∗ > 1, where, for all sufficiently large N, p∗ := lim inf t→∞
∞ Ktγ X γ α (s)q (s) , r(t) t=t+1
q∗ := lim inf t→∞
t−1 K X sγ+1 γ α (s)q (s) . t r(s) t=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Similarly, we can state oscillation criteria for many other time scales, e.g., T =hZ, h > 0, T = {t : t = q k , k ∈ N0 , q > 1}, T = N20 := {n2 : n ∈ N0 }, or T = {Hn : n ∈ where Hn is the so-called n-th harmonic number defined by PN} n H0 = 0, Hn = k=1 k1 , n ∈ N0 .
NONLINEAR DAMPED DYNAMIC EQUATION
21
References [1] R. P. Agarwal, M. Bohner and S. H. Saker, Oscillation of second order delay dynamic equation, Canadian Appl. Math. Quart., 13 (2005) 1–17. [2] R. P. Agarwal, D. O’Regan and S. H. Saker, Philos- type oscillation criteria for second order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104. [3] R. P. Agarwal, S. L. Shien and C. C. Yeh, Oscillation criteria for second-order retarded differential equations, Math. Comp. Modelling 26 (1997), 1–11. [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨ auser, Boston, 2001. [5] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston, 2003. [6] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dynam. Sys. Th.(Submitted). [7] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, J. Comp. Appl. Math. (Submitted). [8] L. Erbe, A. Peterson and S. H. Saker, Hille-Kneser-type criteria for second-order dynamic equations on time scales, Adv. Diff. Eq. 2006 (2006) 1-18. [9] L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London Math. Soc. 76 (2003) 701–714. [10] L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations on time scales, J. Math. Anal. Appl. 333 (2007) 505–522. [11] W.B. Fite, Concerning the zeros of solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1917) 341–352. [12] S. Hilger, Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [13] V. Kac and P. Cheung, Quantum Calculus, Universitext, 2002. [14] J. V. Manojlovic, Oscillation criteria for second-order half-linear differential equations, Math. Comp. Mod. 30 (1999) 109-119. [15] Y. Sahiner, Oscillation of second-order delay dynamic equations on time scales, Nonlinear Analysis: Th. Meth. Appl. 63 (2005) 1073–1080. [16] S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comp. Appl. Math. 177 (2005) 375–387. [17] E. Thandapani, K. Ravi, and J. Graef, Oscillation and comparison theorems for half-linear second order difference equations, Comp. Math. Appl. 42 (2001), 953–960. [18] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951) 368–380. [19] B. G. Zhang and S. Zhu, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comp. Math. Appl. 49 (2005) 599–609. Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 685880130, U.S.A. E-mail address: [email protected] URL: http://www.math.unl.edu/~lerbe2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt Current address: Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, U.S.A. E-mail address: [email protected]; [email protected] URL: http://www.mans.edu.eg/pcvs/30140/ Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 685880130, U.S.A. E-mail address: [email protected] URL: http://www.math.unl.edu/~apeterson1
Abstract. We present new oscillation criteria for the second order nonlinear damped delay dynamic equation (r(t)(x∆ (t))γ )∆ + p(t)(x∆σ (t))γ + q(t)f (x(τ (t))) = 0. Our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation. Our results are illustrated with examples.
1. Introduction In this paper, we are concerned with oscillation behavior of the second order nonlinear damped delay dynamic equation (1.1)
(r(t)(x∆ (t))γ )∆ + p(t)(x∆σ (t))γ + q(t)f (x(τ (t))) = 0,
on an arbitrary time scale T, where γ is a quotient of odd positive integers, r, p and q are positive rd−continuous functions on T, and the so-called delay function τ : T → T satisfies τ (t) ≤ t for t ∈ T and lim t→∞ τ (t) = ∞. The function f ∈ C (R, R) is assume to satisfy uf (u) > 0 and fu(u) γ ≥ K, for u 6= 0 and for some K > 0. Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that sup T = ∞, and define the time scale interval [t0 , ∞)T by [t0 , ∞)T := [t0 , ∞) ∩ T. By a solution of (1.1) we mean a 1 [Tx , ∞), Tx ≥ t0 which has the property that nontrivial real–valued function x ∈ Crd γ ∆ 1 r(t) x (t) ∈ Crd [Tx , ∞) and satisfies equation (1.1) on [Tx , ∞), where Crd is the space of rd−continuous functions. The solutions vanishing 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. Recently there has been a large number of papers devoted to second order nonlinear dynamic equations on time scales. For example. Agarwal et al [1] considered the second order delay dynamic equation (1.2)
x∆∆ (t) + q(t)x(τ (t)) = 0,
2000 Mathematics Subject Classification. 34K11, 39A10, 39A99. Key words and phrases. Oscillation, delay nonlinear dynamic equations, time scales. 1
2
L. ERBE, T. S. HASSAN, AND A. PETERSON
and established some sufficient conditions for oscillation of (1.2). Zhang et al [19] study the oscillation of the second order nonlinear delay dynamic equation (1.3)
x∆∆ (t) + q(t)f (x(t − τ )) = 0,
and the second order nonlinear dynamic equation (1.4)
x∆∆ (t) + q(t)f (x(σ(t))) = 0,
and established the equivalence of the oscillation of (1.3) and (1.4), from which they obtained some oscillation criteria and comparison theorems for (1.3). Sahiner [15] considered the second order nonlinear delay dynamic equation (1.5)
x∆∆ (t) + q(t)f (x(τ (t))) = 0,
and obtained some sufficient conditions for oscillation of (1.5) by means of the Riccati transformation technique. Erbe et al [10] extended Sahiner’s result to the second order nonlinear delay dynamic equation (r(t)x∆ (t))∆ + q(t)f (x(τ (t))) = 0, Erbe et al [8] considered the pair of second order dynamic equations (1.6)
(r(t)(x∆ (t))γ )∆ + q(t)xγ (t) = 0, (r(t)(x∆ (t))γ )∆ + q(t)xγ (σ(t)) = 0,
and established some necessary and sufficient conditions for nonoscillation of HilleKneser type. Saker [16] examined oscillation for half-linear dynamic equations (1.6), where γ > 1 is an odd positive integer and Agarwal et al [2] studied oscillation for the same equation (1.6), where γ > 1 is the quotient of odd positive integers. These results can not be applied when 0 < γ ≤ 1. Very recently Erbe et al [6] and [7] considered the half-linear delay dynamic equation (r(t)(x∆ (t))γ )∆ + q(t)xγ (τ (t)) = 0, for the cases γ ≥ 1 and 0 < γ ≤ 1 respectively. In addition, it was assumed that r∆ (t) ≥ 0,
(1.7) and Z
∞
τ γ (t)q(t)∆t = ∞.
(1.8) t0
Both of the two cases Z
∞
(1.9)
∆t 1
t0
= ∞,
r γ (t)
and Z
∞
(1.10)
∆t 1
t0
< ∞,
r γ (t)
were considered. Therefore it is of great interest to study the equation (1.1) without necessarily assuming the conditions (1.7)-(1.10). We will still assume γ > 0 is the quotient of odd positive integers and, hence our results will improve and extend many known results on half-linear oscillation.
NONLINEAR DAMPED DYNAMIC EQUATION
Note that in the special case when T = R, then Z b Z σ(t) = t, µ(t) = 0, g ∆ (t) = g 0 (t), g(t)∆t = a
3
b
g(t)dt,
a
and (1.1) becomes the second order nonlinear damped delay differential equation 0
(r(t)(x0 (t))γ ) + p(t)(x0 (t))γ + q(t)f (x(τ (t))) = 0.
(1.11)
The oscillation of equation (1.11) when r (t) = 1, p(t) = 0 and f (x(τ (t))) = xγ (τ (t)), was studied by Agarwal et al [3] and proved that if γ Z ∞ τ (s) lim sup tγ q(s) ds > 1, s t→∞ t then every solution of (1.11) oscillates. When T = Z, then Z b b−1 X σ(t) = t + 1, µ(t) = 1, g ∆ (t) = ∆g(t), g(t)∆t = g(t), a
t=a
and (1.1) becomes the second order nonlinear damped delay difference equation (1.12)
∆(r(t)(∆x(t))γ ) + p(t)(∆x(t))γ + q(t)f (x(τ (t))) = 0.
The oscillation of equation (1.12) when r (t) = 1, p(t) = 0 and f (x(τ (t))) = x(t), was studied by Thandapani et al [17] where γ > 0, p(t) is a positive sequence, and it was shown that every solution of (1.12) is oscillatory, if ∞ X
q(n) = ∞.
n=n0
We will see that our results not only unify some of the known oscillation results for differential and difference equations but can be applied to other cases to determine the oscillatory behavior. Note that, if T =hZ, h > 0, then σ(t) = t + h, µ(t) = h, y ∆ (t) = ∆h y(t) = b−a−h h
b
Z
g(t)∆t = a
y(t + h) − y(t) , h
X
g(a + kh)h,
k=0
and (1.1) becomes the second order nonlinear damped delay difference equation (1.13)
∆h (r(t)(∆h x(t))γ ) + p(t)(∆h x(t))γ + q(t)f (x(τ (t))) = 0.
If T =q N0 = {t : t = q k , k ∈ N0 , q > 1}, then σ(t) = qt, µ(t) = (q − 1)t, x∆ (t) = ∆q x(t) = (x(q t) − x(t))/(q − 1) t, Z ∞ ∞ X g(t)∆t = g(q k )µ(q k ), t0
k=n0
n0
where t0 = q , and (1.1) becomes the second order q−nonlinear damped delay difference equation (1.14)
∆q (r(t)(∆q x(t))γ ) + p(t)(∆q x(t))γ + q(t)f (x(τ (t))) = 0.
4
L. ERBE, T. S. HASSAN, AND A. PETERSON
If T = N20 := {n2 : n ∈ N0 }, then
√ y(( t + 1)2 ) − y(t) √ σ(t) = ( t + 1) , µ(t) = 1 + 2 t, ∆N y(t) = , 1+2 t and (1.1) becomes the second order nonlinear damped delay difference equation √
(1.15)
√
2
∆N (r(t)(∆N x(t))γ ) + p(t)(∆N x(t))γ + q(t)f (x(τ (t))) = 0.
If T = {Hn : n ∈ N0 } where Hn denotes the set of numbers defined by H0 = 0, Hn =
n X 1 , n ∈ N, k
k=1
then 1 , y ∆ (t) = ∆Hn y(Hn ) = (n + 1)∆y(Hn ), n+1 and (1.1) becomes the second order nonlinear damped delay difference equation σ(Hn ) = Hn+1 , µ(Hn ) =
(1.16) ∆Hn (r(Hn )(∆Hn x(Hn ))γ )+p(Hn )(∆Hn x(Hn ))γ +q(Hn )f (x(τ (Hn ))) = 0. If
√ T = T2 = { n : n ∈ N0 },
then
√ p p x( t2 + 1) − x(t) ∆ 2 2 √ σ(t) = t + 1, µ(t) = t + 1 − t, x (t) = ∆2 x(t) = , t2 + 1 − t and (1.1) becomes the second order nonlinear delay damped difference equation (1.17)
∆2 (r(t)(∆2 x(t))γ ) + p(t)(∆2 x(t))γ + q(t)f (x(τ (t))) = 0.
We recall that for a discrete time scale, we have Z b X g(t)µ(t). g(t)∆t = a
t∈[a,b)T
We will utilize in the following a Riccati transformation technique to establish oscillation criteria for (1.1), where γ is the quotient of odd positive integers. These results improve and generalize the results that have been established in [2], [6], [7], and [16] and our results are essentially new for equations (1.11)–(1.17). Also, interesting examples that illustrate the importance of our results are included in section 3 below. 2. Main Results Throughout this paper, we let and d− (t) := max {0, −d(t)} ,
d+ (t) := max{0, d(t)} R a ∆s u
θ(a, b; u) := R b u
1
r γ (s) ∆s 1 rγ
,
α(t, u) := θ(τ (t), σ(t); u),
(s)
1 P (t) := σ (δ ∆ (t)rσ (t) − δ(t)p(t)), r (t)
a(t) := e
p(t) r σ (t)
(t, t0 ),
NONLINEAR DAMPED DYNAMIC EQUATION
5
and ∞
∆s
τ (t)
R γ (s)
Z R(t) := a(t)r(t),
Q(t) := a(t)q(t),
β(t) :=
1
.
The first two Lemmas give some sufficient conditions in order that a positive solution of (1.1) is eventually increasing. Lemma 2.1. Assume that Z
∞
(2.1)
∆t
= ∞,
1
t0
R γ (t)
holds and (1.1) has a positive solution x on [t0 , ∞)T . Then there exists a T ∈ [t0 , ∞)T , sufficiently large, so that γ x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T . Proof. Pick t1 ∈ [t0 , ∞)T such that x(τ (t)) > 0 on [t1 , ∞)T . From (1.1), we have γ (2.2) (r(t) x∆ (t) )∆ + p(t)(x∆σ (t))γ = −q(t)f (x(τ (t))) < 0. For t ∈ [t1 , ∞)T . we can write the left hand side of (2.2) in the form γ γ p(t) (r(t) x∆ (t) )∆ a(t) + σ (r(t) x∆ (t) )σ a(t) < 0, r (t) which implies
γ ∆ R(t) x∆ (t) < 0.
Now, we claim that x∆ (t) > 0 on [t1 , ∞)T . If not, then there exists t2 ≥ t1 such γ that x∆ (t2 ) < 0. Using the fact that R(t) x∆ (t) is decreasing, we obtain, for t ∈ [t2 , ∞)T R(t)(x∆ (t))γ < c := R(t2 )(x∆ (t2 ))γ < 0. Integrating from t2 to t, we find that x(t) < x(t2 ) + c
1 γ
Z
t
∆s 1
t2
,
R γ (s)
for t ≥ t2 . Condition (2.1) implies that x(t) is eventually negative, which is a ∆ contradiction. γ ∆Therefore, x (t) > 0 on [t1 , ∞)T and hence, from (1.1), we get ∆ (r(t) x (t) ) < 0 on [t1 , ∞)T . The proof is complete. Lemma 2.2. Assume that γ1 Z ∞ Z t 1 γ Q(s)β (s)∆s ∆t = ∞, (2.3) R(t) t0 t0 holds and (1.1) has a positive solution x on [t0 , ∞)T . Then there exists a T ∈ [t0 , ∞)T , sufficiently large, so that γ x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T .
6
L. ERBE, T. S. HASSAN, AND A. PETERSON
Proof. As in the proof of Lemma 2.1, assume there is a t2 ≥ t0 such that x∆ (t) < 0 on γ [t2 , ∞)T . Pick t3 ≥ t2 so that τ (t) ≥ t2 , for t ≥ t3 . Using the fact that R(t) x∆ (t) is decreasing, we obtain γ 1 Z ∞ (R(s) x∆ (s) ) γ ∆s −x(τ (t)) < x (∞) − x(τ (t)) = 1 τ (t) R γ (s) Z γ 1 ∞ ∆s ≤ (R(τ (t)) x∆ (τ (t)) ) γ 1 τ (t) R γ (s) Z ∞ γ 1 ∆s ≤ (R(t2 ) x∆ (t2 ) ) γ = Lβ(t), 1 τ (t) R γ (s) γ 1 where L := (R(t2 ) x∆ (t2 ) ) γ < 0. From (1.1), we get, for t ≥ t3 γ (R(t) x∆ (t) )∆ = −Q(t)f (x(τ (t))) ≤ −KQ(t)xγ (τ (t)) ≤ KLγ Q(t)β γ (t), Hence, for t ≥ t3 , we have R(t)(x∆ (t))γ
≤ R(t3 )(x∆ (t3 ))γ + KLγ
Z
t
Q(u)β γ (u)∆u
t3
≤ KLγ
Z
t
Q(u)β γ (u)∆u.
t3
It follows from this last inequality that γ1 Z s Z t 1 1 γ γ Q(u)β (u)∆u ∆s. x(t) − x(t3 ) ≤ K L t3 R(s) t3 Hence by (2.3), we have limt→∞ x(t) = −∞, which contradicts the fact that x is ∆ a positive solution γ ∆of (1.1). Thus x (t) > 0 on [t1 , ∞)T and hence, from (1.1), we ∆ get (r(t) x (t) ) < 0 on [t1 , ∞)T . Lemma 2.3. Assume that γ1 Z t Z ∞ 1 Q(s)∆s ∆t = ∞, (2.4) R(t) t0 t0 holds and (1.1) has a positive solution x on [t0 , ∞)T . Then either there exists a T ∈ [t0 , ∞)T , sufficiently large, so that γ x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T , or limt→∞ x(t) = 0. Proof. As in the proof of Lemma 2.1, assume there is a t2 ≥ t0 such that x∆ (t) < 0 on [t2 , ∞)T . Thus, we get limt→∞ x(t) = l ≥ 0. If we assume l > 0, then x(τ (t)) ≥ l, for t ≥ t3 . From (1.1), we get γ (R(t) x∆ (t) )∆ = −Q(t)xγ (τ (t)) ≤
−lγ Q(t),
NONLINEAR DAMPED DYNAMIC EQUATION
7
Hence, for t ≥ t3 , we have ∆
γ
R(t)(x (t))
∆
γ
≤ R(t3 )(x (t3 )) − l
γ
Z
t
Q(u)∆u t3
≤
−lγ
Z
t
Q(u)∆u. t3
It follows from this last inequality that γ1 Z t Z s 1 x(t) − x(t3 ) ≤ −l Q(u)∆u ∆s. t3 R(s) t3 Hence by (2.4), we have limt→∞ x(t) = −∞, which contradicts the fact that x is a positive solution of (1.1). Thus, l = 0. This completes the proof. Lemma 2.4. Assume that there exists T ≥ t0 , sufficiently large, such that γ x(t) > 0, x∆ (t) > 0, and (r(t) x∆ (t) )∆ < 0, on [T, ∞)T . Then x(τ (t)) > α (t, T ) xσ (t) , for t ≥ T1 ≥ T. γ Proof. Since r(t) x∆ (t) is strictly decreasing on [T, ∞). We can choose T1 ≥ T so that τ (t) ≥ T, for t ≥ T1 . Then, we have that γ 1 Z σ(t) (r(s) x∆ (s) ) γ xσ (t) − x(τ (t)) = ∆s 1 τ (t) r γ (s) Z γ 1 σ(t) ∆s , ≤ (r(τ (t)) x∆ (τ (t)) ) γ 1 τ (t) r γ (s) and so γ 1 Z (r(τ (t)) x∆ (τ (t)) ) γ σ(t) ∆s xσ (t) (2.5) ≤1+ . 1 x(τ (t)) x(τ (t)) τ (t) r γ (s) Also, we see that γ 1 Z τ (t) (r(s) x∆ (s) ) γ x(τ (t)) > x(τ (t)) − x (T ) = ∆s 1 T r γ (s) Z γ 1 τ (t) ∆s ≥ (r(τ (t)) x∆ (τ (t)) ) γ , 1 T r γ (s) and hence !−1 γ 1 Z τ (t) (r(τ (t)) x∆ (τ (t)) ) γ ∆s < . (2.6) 1 x(τ (t)) T r γ (s) Therefore, (2.5) and (2.6) imply Z σ(t) ∆s xσ (t) < 1 x(τ (t)) T r γ (s)
Z
τ (t)
∆s 1
T
!−1 ,
r γ (s)
and hence we get the desired inequality x(τ (t)) > α (t, T ) xσ (t) , This completes the proof.
for t ≥ T1 ≥ T.
8
L. ERBE, T. S. HASSAN, AND A. PETERSON
¿From Lemmas 2.1-2.2, we get the following oscillation criteria for equation (1.1). Theorem 2.1. Assume one of the conditions (2.1) or (2.3) holds. Furthermore, suppose that there exists a positive ∆−differentiable function δ(t) such that, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 γ (2.7) lim sup Kα (s, T ) δ(s)q(s) − ∆s = ∞. (γ + 1)γ+1 δ γ (s) t→∞ T Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume (1.1) has a nonoscillatory solution on [t0 , ∞)T . Then, without loss of generality, there is a T ∈ [t0 , ∞)T , sufficiently large, so that x(t) satisfies the conclusions of Lemmas 2.1-2.2 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . In particular, we have γ x(τ (t)) > 0, x∆ (t) > 0, (r(t) x∆ (t) )∆ < 0, for t ≥ T. Consider the generalized Riccati substitution ∆ γ x (t) . (2.8) w(t) = δ(t)r(t) x(t) By the product rule and then the quotient rule σ ∆ r(t)(x∆ (t))γ r(t)(x∆ (t))γ + δ(t) w∆ (t) = δ ∆ (t) xγ (t) xγ (t) σ (r(t)(x∆ (t))γ )∆ r(t)(x∆ (t))γ r(t)(x∆ (t))γ (xγ (t))∆ + δ(t) = δ ∆ (t) − δ(t) . xγ (t) xγσ (t) xγ (t)xγσ (t) ¿From (1.1) and the definition of w(t) and P (t), we have γ x(τ (t)) r(t)(x∆ (t))γ (xγ (t))∆ P (t) σ ∆ − δ(t) . w (t) ≤ σ w (t) − Kδ(t)q(t) δ (t) xσ (t) xγ (t)xγσ (t) Using the fact that r(t)(x∆ (t))γ is strictly decreasing and the definition of w(t), we obtain γ P (t) x(τ (t)) δ(t)wσ (t) (xγ (t))∆ w∆ (t) ≤ σ wσ (t) − Kδ(t)q(t) − . σ δ (t) x (t) δ σ (t) xγ (t) ¿From Lemma 2.4, we get (2.9)
w∆ (t) ≤
P (t) σ δ(t)wσ (t) (xγ (t))∆ γ w (t) − Kα (t, T )δ(t)q(t) − . δ σ (t) δ σ (t) xγ (t)
By the P¨ otzsche chain rule ([4, Theorem 1.90]), we obtain Z 1 γ−1 ∆ (xγ (t)) = γ x(t) + hµ(t)x∆ (t) dh x∆ (t) 0 1
Z = γ ≥
γ−1
[(1 − h) x(t) + hxσ (t)]
dh x∆ (t)
0 γ−1
γ (x(t))
x∆ (t),
γ>1 .
γ−1
γ (xσ (t))
x∆ (t),
0 1, we have that (2.11)
w∆ (t) ≤
γδ(t)wσ (t) x∆ (t) xσ (t) P (t) σ γ w (t) − Kα (t, T ) δ(t)q(t) − . δ σ (t) δ σ (t) xσ (t) x(t)
Using the fact that x(t) is strictly increasing and r(t)(x∆ (t))γ is strictly decreasing, we get that σ γ1 r (t) σ ∆ (x∆ (t))σ . (2.12) x (t) ≥ x(t), x (t) ≥ r(t) ¿From (2.10), (2.11) and (2.12), we obtain (2.13) where λ :=
w∆ (t) ≤ γ+1 γ .
P+ (t) σ γδ(t)(wσ (t))λ . w (t) − Kαγ (t, T ) δ(t)q(t) − 1 σ δ (t) (δ σ (t))λ r γ (t)
Define A > 0 and B > 0 by λ
A :=
γδ(t)(wσ (t))λ 1
(δ σ (t))λ r γ (t)
1
,
B
λ−1
:=
1
(r γ (t)) λ P+ (t) 1
1
λγ λ (δ(t)) λ
.
Then, using the inequality λAB λ−1 − Aλ ≤ (λ − 1)B λ ,
(2.14) we get that
γδ(t)(wσ (t))λ P+ (t) σ w (t) − 1 δ σ (t) (δ σ (t))λ r γ (t)
= λAB λ−1 − Aλ ≤ (λ − 1)B λ r(t)(P+ (t))γ+1 = . (γ + 1)γ+1 δ γ (t)
¿From this last inequality and (2.13) we get w∆ (t) ≤
r(t)(P+ (t))γ+1 − Kδ(t)αγ (t, T ) q(t) (γ + 1)γ+1 δ γ (t)
Integrating both sides from T to t we get Z t r(s)(P+ (s))γ+1 γ ∆s ≤ w(T ) − w(t) ≤ w(T ), Kα (s, T ) δ(s)q(s) − (γ + 1)γ+1 δ γ (s) T which leads to a contradiction to (2.7).
By choosing δ(t) = 1 and δ(t) = t, t ≥ t0 in Theorem 2.1 we have the following oscillation results. Corollary 2.1. Assume one of the conditions (2.1) or (2.3) holds and, for all sufficiently large T, Z ∞ αγ (t, T ) q(t)∆t = ∞. (2.15) T
Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T .
10
L. ERBE, T. S. HASSAN, AND A. PETERSON
Corollary 2.2. Assume one of the conditions (2.1) or (2.3) holds and, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 γ (2.16) lim sup ∆s = ∞, Ksα (s, T ) q(s) − (γ + 1)γ+1 sγ t→∞ T where P (t) = 1 − [t0 , ∞)T
tp(t) r σ (t) .
Then every solution of equation (1.1) is oscillatory on
We are now ready to state and prove Philos-type oscillation criteria for the equation (1.1). Theorem 2.2. Assume one of the conditions (2.1) or (2.3) holds. Furthermore, suppose that there exist functions H, h ∈ Crd (D, R), where D ≡ {(t, s) : t ≥ s ≥ t0 } such that (2.17)
H (t, t) = 0,
t ≥ t0 , H (t, s) > 0,
t > s ≥ t0 ,
and H has a nonpositive continuous ∆-partial derivative H ∆s (t, s) with respect to the second variable and satisfies H ∆s (t, s) + H (t, s)
(2.18)
γ h (t, s) P (s) =− σ (H (t, s)) γ+1 , σ δ (s) δ (s)
and, for all sufficiently large T, (2.19) # γ+1 Z t" 1 (h (t, s)) r (s) − ∆s = ∞, lim sup Kαγ (s, T ) δ (s) q (s) H (t, s) − (γ + 1)γ+1 δ γ (s) t→∞ H (t, T ) T where δ(t) is a positive ∆−differentiable function. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume (1.1) has a nonoscillatory solution on [t0 , ∞)T . Then, without loss of generality, there is a T ∈ [t0 , ∞)T , sufficiently large, so that x(t) satisfies the conclusions of Lemmas 2.1-2.2 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . In particular, we have γ x(τ (t)) > 0, x∆ (t) > 0, (r(t) x∆ (t) )∆ < 0, for t ≥ T. We define w (t) also, as in Theorem 2.1. From (2.13) with P+ (t) replaced by P (t), we have Kαγ (t, T )δ(t)q(t) ≤ −w∆ (t) +
(2.20)
P (t) σ γδ(t) w (t) − 1 (wσ (t))λ . σ λ δ σ (t) γ r (t)(δ (t))
Multiplying both sides of (2.20), with t replaced by s, by H (t, s), integrating with respect to s from T to t, t ≥ T, Z t Z t γ H (t, s) Kα (s, T )δ(s)q(s)∆s ≤ − H (t, s) w∆ (s) ∆s T
Z
t
+
H (t, s) T
T
P (s) σ w (s) ∆s − δ σ (s)
Z
t
H (t, s) T
γδ(s) 1 γ
r (s)(δ σ (s))λ
(wσ (s))λ ∆s.
NONLINEAR DAMPED DYNAMIC EQUATION
11
Integrating by parts and using (2.17) and (2.18), we obtain Z t H (t, s) Kαγ (s, T )δ(s)q(s)∆s T Z t ≤ H (t, T ) w (T ) + H ∆s (t, s) wσ (s) ∆s T
Z
t
+
H (t, s) T
P (s) σ w (s) ∆s − δ σ (s)
Z
t
H (t, s) T
γδ(s) 1 γ
r (s)(δ σ (s))λ
(wσ (s))λ ∆s
≤ H (t, T ) w (T ) Z t" + T
# 1 γδ(s) h (t, s) σ σ λ (H (t, s)) λ w (s) − H (t, s) 1 (w (s)) ∆s − σ δ (s) r γ (s)(δ σ (s))λ ≤ H (t, T ) w (T )
Z t" (2.21)
+ T
# 1 h− (t, s) γδ(s) (wσ (s))λ ∆s. (H (t, s)) λ wσ (s) − H (t, s) 1 δ σ (s) r γ (s)(δ σ (s))λ
Again, define A > 0 and B > 0 by Aλ :=
γH (t, s) δ(s)(wσ (s))λ 1
r γ (s)(δ σ (s))λ
1
,
B λ−1 :=
h− (t, s) r γ+1 (s) 1
,
λ (γδ(s)) λ
and using the inequality (2.14), we obtain 1 γH (t, s) δ(s)(wσ (s))λ h− (t, s) σ λ (H (t, s)) w (s) − = λAB λ−1 − Aλ 1 δ σ (s) r γ (s)(δ σ (s))λ
hγ+1 (t, s) r (s) − . (γ + 1)γ+1 δ γ (t) ¿From this last inequality and (2.21), we have # γ+1 Z t" (h− (t, s)) r (s) γ Kα (s, T ) δ (s) q (s) H (t, s) − ∆s ≤ H (t, T ) w (T ) , (γ + 1)γ+1 δ γ (s) T ≤ (λ − 1)B λ =
and this implies that # γ+1 Z t" (h− (t, s)) r (s) 1 γ Kα (s, T ) δ (s) q (s) H (t, s) − ∆s ≤ w (T ) , H (t, T ) T (γ + 1)γ+1 δ γ (s) which contradicts assumption (2.19). This completes the proof.
Also, by Lemma 2.3, we obtain another oscillation criterion for the equation (1.1) as in Theorems 2.1 and 2.2 and Corollaries 2.1 and 2.2 as follows. Corollary 2.3. Assume that (2.4) and (2.7) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 2.4. Assume that (2.4) and (2.15) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 2.5. Assume that (2.4) and (2.16) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero.
12
L. ERBE, T. S. HASSAN, AND A. PETERSON
Corollary 2.6. Assume that (2.4), (2.17), (2.18) and (2.19) hold. Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T or tends to zero. Remark 2.1. Note that conditions (1.7) and (1.8) are not assumed to hold and condition (1.9) is also not necessary in order that (1.1) be oscillatory, in contrast to the results of [6] and [7]. We introduce the following notation, for all sufficiently large T, Z ∞ Z 1 t sγ+1 tγ Q∗ (s)∆s, Q∗ (s)∆s, q∗ := lim inf p∗ := lim inf t→∞ t T r(s) t→∞ r(t) σ(t) tγ wσ (t) tγ wσ (t) , R∗ := lim sup , t→∞ r (t) r(t) t→∞ t . Note that where Q∗ (t) = Kαγ (t, T )q (t) , and assume that l := lim inf t→∞ σ(t) 0 ≤ l ≤ 1. In order for the definition of p∗ to make sense we assume that Z ∞ (2.22) Q∗ (s)∆s < ∞. r∗ := lim inf
t0
Theorem 2.3. Assume (2.1) holds and r(t) is a (delta) differentiable function with r∆ (t) ≥ 0 and (2.22) holds. Furthermore, assume l > 0 and (2.23)
p∗ >
lγ 2 (γ
γγ , + 1)γ+1
or (2.24)
p∗ + q ∗ >
1 lγ(γ+1)
.
Then every solution of equation (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume (1.1) has a nonoscillatory solution on [t0 , ∞)T . Then, without loss of generality, there is a T ∈ [t0 , ∞)T , sufficiently large, so that x(t) satisfies the conclusions of Lemmas 2.1-2.2 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . In particular, we have γ x(τ (t)) > 0, x∆ (t) > 0, (r(t) x∆ (t) )∆ < 0, for t ≥ T. We define w (t) also, as in Theorem 2.1 by putting δ(t) = 1. Note in this case P+ (t) = 0. From (2.13), we have γ+1 γ (2.25) w∆ (t) ≤ −Q∗ (t) − 1 (wσ (t)) γ ≤ 0, for t ∈ [T, ∞)T . γ r (t) γ First, we assume (2.23) holds. It follows from (2.8) and r(t) x∆ (t) is strictly decreasing that !−γ ∆ γ Z t x (t) ∆s w(t) = r(t) < , for t ∈ [T, ∞)T . 1 x(t) t0 r γ (s) Rt Since (2.1) implies t0 ∆s = ∞, we have that limt→∞ w(t) = 0. Integrating (2.25) 1 r γ (s)
from σ(t) to ∞ and using limt→∞ w(t) = 0, we have 1 Z ∞ Z ∞ (wσ (s)) γ wσ (s) (2.26) wσ (t) ≥ Q∗ (s)∆s + γ ∆s. 1 σ(t) σ(t) r γ (s)
NONLINEAR DAMPED DYNAMIC EQUATION
13
It follows from (2.26) that tγ wσ (t) tγ ≥ r(t) r(t)
(2.27)
∞
Z
Q∗ (s)∆s + γ σ(t)
tγ r(t)
1
∞
(wσ (s)) γ wσ (s)
σ(t)
r γ (s)
Z
1
∆s.
Let > 0, then by the definition of p∗ and r∗ we can pick t1 ∈ [T, ∞)T , sufficiently large, so that Z ∞ tγ tγ wσ (t) (2.28) ≥ r∗ − , Q∗ (s)∆s ≥ p∗ − , and r(t) σ(t) r(t) for t ∈ [t1 , ∞)T . From (2.27) and (2.28) and using the fact r∆ (t) ≥ 0, we get that 1 Z ∞ tγ wσ (t) s (wσ (s)) γ sγ wσ (s) tγ ∆s ≥ (p∗ − ) + γ 1 r(t) r(t) σ(t) sγ+1 r γ (s) γ Z ∞ γr(s) 1+ γ1 t ∆s ≥ (p∗ − ) + (r∗ − ) r(t) σ(t) sγ+1 Z ∞ γ 1+ 1 ≥ (p∗ − ) + (r∗ − ) γ tγ (2.29) ∆s. γ+1 σ(t) s Using the P¨ otzsche chain rule ([4, Theorem 1.90]), we get ∆ Z 1 −1 1 dh = γ γ+1 sγ [s + hµ(s)] 0 Z 1 γ dh ≤ sγ+1 0 γ (2.30) = . sγ+1 Then from (2.29) and (2.30), we have γ tγ wσ (t) t 1+ γ1 ≥ (p∗ − ) + (r∗ − ) . r(t) σ(t) Taking the lim inf of both sides as t → ∞ we get that 1+ γ1 γ
r∗ ≥ p∗ − + (r∗ − )
l .
Since > 0 is arbitrary, we get 1+ γ1 γ
p∗ ≤ r∗ − r∗
(2.31)
l .
Using the inequality Bu − Au
γ+1 γ
≤
γγ B γ+1 γ+1 (γ + 1) Aγ
with B = 1 and A = lγ we get that p∗ ≤
γγ , l (γ + 1)γ+1 γ2
which contradicts (2.23). Next, we assume (2.24) holds. Multiplying both sides of γ+1 (2.25) by tr(t) , and integrating from T to t (t ≥ T ) we get Z
t
T
sγ+1 ∆ w (s)∆s ≤ − r(s)
Z
t
T
sγ+1 Q∗ (s)∆s − γ r(s)
Z t T
sγ wσ (s) r(s)
γ+1 γ ∆s.
14
L. ERBE, T. S. HASSAN, AND A. PETERSON
Using integration by parts, we obtain Z t γ+1 Z t γ+1 ∆ tγ+1 w(t) s T γ+1 w(T ) s wσ (s)∆s − ≤ + Q∗ (s)∆s r(t) r(T ) r(s) T r(s) T Z t γ σ γ+1 s w (s) γ ∆s. −γ r(s) T By the quotient rule and applying the P¨otzsche chain rule, γ+1 ∆ s (sγ+1 )∆ sγ+1 r∆ (s) = − σ r(s) r (s) r(s)rσ (s) (γ + 1)σ γ (s) ≤ rσ (s) (γ + 1)σ γ (s) (2.32) ≤ . r(s) Hence tγ+1 w(t) r(t)
γ Z t γ+1 Z t σ (s)wσ (s) T γ+1 w(T ) s ∆s − Q∗ (s)∆s + (γ + 1) r(T ) r(s) T r(s) T Z t γ σ γ+1 s w (s) γ ∆s. − γ r(s) T
≤
Let 0 < ≤ l be given, then using the definition of l, we can assume, without loss of generality, that T is sufficiently large so that s ≥ l − , s ≥ T. σ(s) It follows that σ(s) ≤ Ls,
s≥T
where
L :=
1 > 0. l−
We then get that tγ+1 w(t) r(t)
Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s r(T ) T r(s) γ σ γ+1 Z t γ σ s w (s) γ γ s w (s) }∆s. + {(γ + 1)L −γ r(s) r(s) T
Let u(s) :=
sγ wσ (s) , r(s)
then uλ (s) = where λ =
γ+1 γ .
sγ wσ (s) r(s)
λ .
It follows that tγ+1 w(t) r(t)
Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s r(T ) T r(s) Z t + {(γ + 1)Lγ u(s) − γuλ (s)}∆s. T
NONLINEAR DAMPED DYNAMIC EQUATION
15
Again, using the inequality Bu − Auλ ≤
γγ B γ+1 , γ+1 (γ + 1) Aγ
where A, B are constants, we get Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s r(T ) T r(s) Z t [(γ + 1)Lγ ]γ+1 γγ + ∆s γ+1 γγ T (γ + 1) Z t γ+1 T γ+1 w(T ) s ≤ − Q∗ (s)∆s + Lγ(γ+1) (t − T ). r(T ) T r(s)
tγ+1 w(t) r(t)
It follows from this that tγ w(t) ≤ r(t)
T γ+1 w(T ) r(T )
t
−
t
1 t
Z
1 t
Z
T
sγ+1 T Q∗ (s)∆s + Lγ(γ+1) (1 − ). r(s) t
∆
Since w (t) ≤ 0, we get tγ wσ (t) ≤ r(t)
T γ+1 w(T ) r(T )
t
−
t
T
sγ+1 T Q∗ (s)∆s + Lγ(γ+1) (1 − ). r(s) t
Taking the lim sup of both sides as t → ∞ we obtain R∗ ≤ −q∗ + Lγ(γ+1) = −q∗ +
1 . (l − )γ(γ+1)
Since > 0 is arbitrary, we get that R∗ ≤ −q∗ +
1 . lγ(γ+1)
Using this and the inequality (2.31) we get 1+ γ1
p∗ ≤ r∗ − l γ r∗
≤ r∗ ≤ R∗ ≤ −q∗ +
1 . lγ(γ+1)
Therefore p∗ + q ∗ ≤
1 lγ(γ+1)
which contradicts (2.24).
Remark 2.2. Note that condition (1.8) is not assumed to hold, in contrast to the results of [6] and [7]. 3. Examples In this section, we give some examples to illustrate our main results. Example 3.1. Consider the nonlinear delay dynamic equation γ−1 γ ∆ (σ(t))γ−1 ∆σ γ 1 t ∆ x (t) + γ xγ (τ (t)) = 0, x (t) + (3.1) a(t) taσ (t) α (t, t0 ) t2
16
L. ERBE, T. S. HASSAN, AND A. PETERSON
where γ is the quotient of odd positive integers and a(t) = e 1t (t, t0 ). Here p(t) = (σ(t))γ−1 taσ (t) ,
1 q(t) = αγ (t,t 2 and r(t) = 0 )t condition (2.1) holds since Z ∞ ∆t t0
1 γ
tγ−1 a(t) ,
then, it is clear that P (t) = 0 and the
∞
Z
∆t
=
R (t)
t
t0
1− γ1
= ∞,
by Example 5.60 in [5]. Also Z t r(s)(P+ (s))γ+1 γ lim sup ∆s Ksα (s, T ) q(s) − (γ + 1)γ+1 sγ t→∞ T Z t ∆s = K lim sup = ∞, t→∞ t0 s R∞ α(s,T ) = 1. Then by Corollary 2.2, every = ∞ implies limt→∞ α(t,t since t0 ∆t 1 0) r γ (t)
solution of (3.1) is oscillatory. Example 3.2. Consider the nonlinear delay dynamic equation ∆ (σ(t))γ−1 ∆σ γ tγ (tσ(t))γ ∆ γ x (t) + x (t) + γ xγ (τ (t)) = 0, (3.2) a(t) t α (t, t0 ) where 0 < γ ≤ 1 is the quotient of odd positive integers, a(t) = e p(t) (t, t0 ) and we r σ (t) assume Z ∞ ∆t (3.3) = ∞, for 0 < γ ≤ 1, 1− γ1 t0 t σ(t) for those time scales [t0 , ∞)T , t0 > 0. This holds for many time scales, for example when T =q N0 = {t : t = q k , k ∈ N0 , q > 1}. It is clear r(t) satisfies ∆ Z ∞ Z ∞ Z ∞ −1 1 ∆t ∆t = ≤ ∆t < ∞. 1 t t0 t0 tσ(t) t0 R γ (t) To see that (2.3) holds note that γ1 Z ∞ Z t 1 γ Q(s)β (s)∆s ∆t r(t) t0 t0
Z
∞
≥ t0
Z
∞
≥ t0
1 (tσ(t))γ
Z t
t − t0 (tσ(t))γ
γ1
t0
sβ(s) α (s, t0 )
γ
γ1 ∆s ∆t
∆t,
since Z
∞
β(t) = τ (t)
∆s 1 γ
Z
∞
=
R (s)
τ (t)
1 = sσ(s)
Z
∞
τ (t)
−1 s
∆ ∆s =
1 1 α(t, t0 ) ≥ ≥ . τ (t) t t
We can find 0 < k < 1 such that t − t0 > kt, for t ≥ tk > t0 . Therefore, we get γ1 Z t Z ∞ Z ∞ 1 1 ∆t (3.3) Q(s)β γ (s)∆s ∆t > k γ = ∞. 1− γ1 r(t) t0 t0 tK t σ(t) To apply Corollary 2.1, it remains to prove that condition (2.15) holds, then Z ∞ Z ∞ αγ (t, T ) q(t)∆t = tγ ∆t = ∞, T
t0
NONLINEAR DAMPED DYNAMIC EQUATION
R∞
∆t
t0
r γ (t)
17
α(s,T ) = ∞ implies limt→∞ α(t,t = 1. We 0) R∞ conclude that if [t0 , ∞)T , t0 > 0 is a time scale where t0 1−∆t = ∞, then, by 1
where we use, as in Example 3.1,
1
t
γ
σ(t)
Corollary 2.1, every solution of (3.2) is oscillatory. Example 3.3. Consider the nonlinear dynamic equation γ−1 γ ∆ (γ − 1) (σ(t))γ−1 ∆σ γ η t x (t) + γ xγ (τ (t)) = 0, x∆ (t) + (3.4) a(t) taσ (t) α (t, t0 ) t2 where γ ≥ 1 is the quotient of odd positive integers, η is a positive constant and γ−1 γ−1 η a(t) = e p(t) and q(t) = αγ (t,t (t, t0 ). Here r(t) = ta(t) , p(t) = (γ−1)(σ(t)) 2. taσ (t) 0 )t σ r (t)
Note that Z
∞
t0
r∆ (t)
= ≥
Z
∆t
∞
∆t
=
1 γ
R (t)
t0
t
1− γ1
= ∞,
1 (tγ−1 )∆ a(t) − (γ − 1)tγ−2 a(t) σ a(t)a (t) 1 (γ − 1)tγ−2 − (γ − 1)tγ−2 = 0, σ a (t)
and, as in Example 3.1, p∗
=
lim inf t→∞
tγ r(t)
∞
Z
Kαγ (s, T )q (s) ∆s
σ(t)
Z
∞
= ηK lim inf ta(t) t→∞
σ(t)
Z
∞
≥ ηK lim inf t t→∞
σ(t) ∞
Z ≥ ηK lim inf t t→∞
if η η>
γγ > lγ 2 −1 K(γ+1) . γ+1 γγ . lγ 2 −1 K(γ+1)γ+1
σ(t)
αγ (s, T ) 1 ∆s αγ (s, t0 ) s2
∆s s2 γγ ∆s = ηlK > γ 2 , sσ(s) l (γ + 1)γ+1
Then, by Theorem 2.3, we get that (3.4) is oscillatory if
Additional examples may be readily given. We leave this to interested reader. 4. Applications In this section, we apply the oscillation criteria to different types of time scales, Rb Rb for example if T = R then σ(t) = t, µ(t) = 0, f ∆ (t) = f 0 (t), a f (t)∆t = a f (t)dt, and (1.1) becomes the nonlinear damped delay differential equation (4.1)
(r(t)(x0 (t))γ )0 + p(t)(x0 (t))γ + q(t)f (x(τ (t))) = 0.
then we have from Theorems 2.1-2.3 and Corollaries 2.1-2.6 the following oscillation criteria for equation (4.1). Theorem 4.1. Assume one of the conditions Z ∞ dt (4.2) = ∞, 1 t0 R γ (t)
18
L. ERBE, T. S. HASSAN, AND A. PETERSON
or Z
∞
(4.3) t0
1 R(t)
Z
t γ
γ1
Q(s)β (s)ds
dt = ∞,
t0
holds. Furthermore, suppose that there exists a positive differentiable function δ(t) such that, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 (4.4) lim sup Kαγ (s, T ) δ(s)q(s) − ds = ∞. (γ + 1)γ+1 δ γ (s) t→∞ T Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Corollary 4.1. Assume one of the conditions (4.2) or (4.3) holds and, for all sufficiently large T, Z ∞ αγ (t, T ) q(t)dt = ∞. (4.5) T
Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Corollary 4.2. Assume one of the conditions (4.2) or (4.3) holds and, for all sufficiently large T, Z t r(s)(P+ (s))γ+1 γ ds = ∞. (4.6) lim sup Ksα (s, T ) q(s) − (γ + 1)γ+1 sγ t→∞ T Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Theorem 4.2. Assume one of the conditions (4.2) or (4.3) holds. Furthermore, suppose that there exist functions H, h ∈ C (D, R), where D ≡ {(t, s) : t ≥ s ≥ t0 } such that (4.7)
H (t, t) = 0,
t ≥ t0 , H (t, s) > 0,
t > s ≥ t0 ,
γ ∂H (t, s) P (s) h (t, s) + H (t, s) σ =− σ (H (t, s)) γ+1 , ∂s δ (s) δ (s) and, for all sufficiently large T, (4.9) # γ+1 Z t" (h− (t, s)) r (s) 1 γ Kα (s, T ) δ (s) q (s) H (t, s) − ds = ∞, lim sup (γ + 1)γ+1 δ γ (s) t→∞ H (t, T ) T
(4.8)
Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Corollary 4.3. Assume that (4.4) holds and γ1 Z ∞ Z t 1 (4.10) Q(s)ds dt = ∞, R(t) t0 t0 Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.4. Assume that (4.5) and (4.10) hold. Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.5. Assume that (4.6) and (4.10) hold. Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.6. Assume that (4.7), (4.8), (4.9) and (4.10) hold. Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T or tends to zero.
NONLINEAR DAMPED DYNAMIC EQUATION
19
Theorem 4.3. Assume (4.2) holds and r(t) is differentiable function with r0 (t) ≥ 0 and Z ∞ αγ (t)q (t) dt < ∞ t0
hold. Furthermore, assume that p∗ >
γγ , (γ + 1)γ+1
or p∗ + q∗ > 1, where, for all sufficiently large T, Z Ktγ ∞ γ p∗ := lim inf α (s)q (s) ds, t→∞ r(t) t
q∗ := lim inf t→∞
K t
Z
t
T
sγ+1 γ α (s)q (s) ds. r(s)
Then every solution of equation (4.1) is oscillatory on [t0 , ∞)T . Rb Pb−1 If T = Z, then σ(t) = t + 1, µ(t) = 1, f ∆ (t) = ∆f (t), a f (t)∆t = t=a f (t), and (1.1) becomes the nonlinear damped delay difference equation (4.11)
∆(r(t)(∆x(t))γ ) + p(t)(∆x(t + 1))γ + q(t)f (x(τ (t))) = 0.
then we have from Theorems 2.1-2.3 and Corollaries 2.1-2.6 the following oscillation criteria for equation (4.11). Theorem 4.4. Assume one of the conditions ∞ X 1 (4.12) = ∞, 1 γ t=t0 R (t) or " # γ1 ∞ t−1 X 1 X γ (4.13) Q(s)β (s) = ∞, R(t) s=t t=t 0
0
holds. Furthermore, suppose that there exists a sequence δ(t) such that, for all sufficiently large N, t−1 X r(s)(P+ (s))γ+1 = ∞. (4.14) lim sup Kαγ (s, N ) δ(s)q(s) − (γ + 1)γ+1 δ γ (s) t→∞ s=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Corollary 4.7. Assume one of the conditions (4.12) or (4.13) holds and, for all sufficiently large N, ∞ X (4.15) αγ (t, N ) q(t) = ∞. t=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Corollary 4.8. Assume one of the conditions (4.2) or (4.3) holds and, for all sufficiently large N, t−1 X r(s)(P+ (s))γ+1 γ (4.16) lim sup Ksα (s, N ) q(s) − = ∞. (γ + 1)γ+1 sγ t→∞ s=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T .
20
L. ERBE, T. S. HASSAN, AND A. PETERSON
Theorem 4.5. Assume one of the conditions (4.12) or (4.13) holds. Furthermore, suppose that there exist two sequences H, h on D, where D ≡ {(t, s) : t ≥ s ≥ t0 } such that (4.17) (4.18)
H (t, t) = 0,
t ≥ t0 , H (t, s) > 0,
t > s ≥ t0 ,
γ h (t, s) P (s) γ+1 = − (H (t, s)) , δ σ (s) δ σ (s)
∆s H (t, s) + H (t, s)
and, for all sufficiently large N, (4.19) " # γ+1 t−1 X (h (t, s)) r (s) 1 − = ∞, Kαγ (s, N ) δ (s) q (s) H (t, s) − lim sup (γ + 1)γ+1 δ γ (s) t→∞ H (t, N ) s=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Corollary 4.9. Assume that (4.14) holds and " # γ1 ∞ t−1 X 1 X (4.20) = ∞, Q(s) R(t) s=t t=t 0
0
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.10. Assume that (4.15) and (4.20) hold. Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.11. Assume that (4.16) and (4.20) hold. Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Corollary 4.12. Assume that (4.17), (4.18), (4.19) and (4.20) hold. Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T or tends to zero. Theorem 4.6. Assume (4.12) holds, ∆r(t) ≥ 0 and ∞ X
αγ (t)q (t) < ∞
t=t0
hold. Furthermore, assume that p∗ >
γγ , (γ + 1)γ+1
or p∗ + q∗ > 1, where, for all sufficiently large N, p∗ := lim inf t→∞
∞ Ktγ X γ α (s)q (s) , r(t) t=t+1
q∗ := lim inf t→∞
t−1 K X sγ+1 γ α (s)q (s) . t r(s) t=N
Then every solution of equation (4.11) is oscillatory on [t0 , ∞)T . Similarly, we can state oscillation criteria for many other time scales, e.g., T =hZ, h > 0, T = {t : t = q k , k ∈ N0 , q > 1}, T = N20 := {n2 : n ∈ N0 }, or T = {Hn : n ∈ where Hn is the so-called n-th harmonic number defined by PN} n H0 = 0, Hn = k=1 k1 , n ∈ N0 .
NONLINEAR DAMPED DYNAMIC EQUATION
21
References [1] R. P. Agarwal, M. Bohner and S. H. Saker, Oscillation of second order delay dynamic equation, Canadian Appl. Math. Quart., 13 (2005) 1–17. [2] R. P. Agarwal, D. O’Regan and S. H. Saker, Philos- type oscillation criteria for second order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104. [3] R. P. Agarwal, S. L. Shien and C. C. Yeh, Oscillation criteria for second-order retarded differential equations, Math. Comp. Modelling 26 (1997), 1–11. [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨ auser, Boston, 2001. [5] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston, 2003. [6] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dynam. Sys. Th.(Submitted). [7] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, J. Comp. Appl. Math. (Submitted). [8] L. Erbe, A. Peterson and S. H. Saker, Hille-Kneser-type criteria for second-order dynamic equations on time scales, Adv. Diff. Eq. 2006 (2006) 1-18. [9] L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London Math. Soc. 76 (2003) 701–714. [10] L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations on time scales, J. Math. Anal. Appl. 333 (2007) 505–522. [11] W.B. Fite, Concerning the zeros of solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1917) 341–352. [12] S. Hilger, Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [13] V. Kac and P. Cheung, Quantum Calculus, Universitext, 2002. [14] J. V. Manojlovic, Oscillation criteria for second-order half-linear differential equations, Math. Comp. Mod. 30 (1999) 109-119. [15] Y. Sahiner, Oscillation of second-order delay dynamic equations on time scales, Nonlinear Analysis: Th. Meth. Appl. 63 (2005) 1073–1080. [16] S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comp. Appl. Math. 177 (2005) 375–387. [17] E. Thandapani, K. Ravi, and J. Graef, Oscillation and comparison theorems for half-linear second order difference equations, Comp. Math. Appl. 42 (2001), 953–960. [18] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951) 368–380. [19] B. G. Zhang and S. Zhu, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comp. Math. Appl. 49 (2005) 599–609. Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 685880130, U.S.A. E-mail address: [email protected] URL: http://www.math.unl.edu/~lerbe2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt Current address: Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, U.S.A. E-mail address: [email protected]; [email protected] URL: http://www.mans.edu.eg/pcvs/30140/ Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 685880130, U.S.A. E-mail address: [email protected] URL: http://www.math.unl.edu/~apeterson1
