OSCILLATION OF CERTAIN NONLINEAR SECOND-ORDER ...

U.P.B. Sci. Bull., Series A, Vol. 73, Iss. 1, 2011

ISSN 1223-7027

OSCILLATION OF CERTAIN NONLINEAR SECOND-ORDER DAMPED DELAY DYNAMIC EQUATIONS ON TIME SCALES Da-Xue Chen1 , Guang-Hui Liu2 , Yu-Hua Long3

The paper present some oscillation criteria for the nonlinear second-order damped delay dynamic equation ³ ´∆ σ σ r(t)|x∆ (t)|β−1 x∆ (t) + p(t)|x∆ (t)|β−1 x∆ (t) + q(t)f (x(τ (t))) = 0 on an arbitrary time scale T, where β > 0 is a constant, sup T = ∞, σ(t) := σ inf{s ∈ T : s > t} is the forward jump operator on T, and x∆ := x∆ ◦ σ. Our results improve and extend some known results in which β > 0 is a quotient of odd positive integers. Examples are given to illustrate our main results. Keywords: oscillation, nonlinear second-order damped delay dynamic equation, time scale MSC2000: 39A 10.

1. Introduction The purpose of this paper is to give several oscillation criteria for the nonlinear second-order damped delay dynamic equation ³ ´∆ σ σ r(t)|x∆ (t)|β−1 x∆ (t) + p(t)|x∆ (t)|β−1 x∆ (t) + q(t)f (x(τ (t))) = 0 (1) on an arbitrary time scale T, where β > 0 is a constant, sup T = ∞, σ(t) := σ inf{s ∈ T : s > t} is the forward jump operator on T, x∆ := x∆ ◦ σ, r, p and q are positive rd-continuous functions on the time scale interval [t0 , ∞), τ : T → T satisfies τ (t) ≤ t for t ∈ T, and limt→∞ τ (t) = ∞. The function f ∈ C(R, R) is assumed to satisfy uf (u) > 0 and |f (u)| ≥ K|uβ |, for u 6= 0 and for some K > 0. Foundation item: The research was financed by the Science and Technology Program of Hunan Province of P. R. China (Grant No. 2010FJ6021). 1

Professor, College of Science, Hunan Institute of Engineering, Xiangtan 411104, P. R. China, e-mail: [email protected] 2 Lecturer, College of Science, Hunan Institute of Engineering, Xiangtan 411104, P. R. China 3 Lecturer, College of Science, Hunan Institute of Engineering, Xiangtan 411104, P. R. China 71

72

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

Recall that a solution of (1) is a nontrivial real function x such that 1 1 x ∈ Crd [tx , ∞) and r(t)|x∆ (t)|β−1 x∆ (t) ∈ Crd [tx , ∞) for a certain tx ≥ t0 and satisfying (1) for t ≥ tx . Our attention is restricted to those solutions of (1) which exist on the half-line [tx , ∞) and satisfy sup{|x(t)| : t > t∗ } > 0 for any t∗ ≥ tx . A solution x of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. Recently, much interest has focused on obtaining sufficient conditions for the oscillation and nonoscillation of solutions of different classes of dynamic equations on time scales, and we refer the reader to [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In particular, much work has been done on second-order damped dynamic equations. For example, Guseinov and Kaymak¸calan [24] considered the linear damped dynamic equation x∆∆ (t) + p(t)x∆ (t) + q(t)x(t) = 0,

(2)

where p and q are positive rd-continuous functions, and established some sufficient conditions for nonoscillation. They proved that if Z ∞ Z ∞ p(t)∆t < ∞ and tq(t)∆t < ∞, t0

t0

then (2) is nonoscillatory. Erbe et al. [25] considered (2) and the nonlinear damped dynamic equation σ x∆∆ (t) + p(t)x∆ (t) + q(t)(f ◦ xσ )(t) = 0, (3) where p and q are positive rd-continuous functions and f ∈ C(R, R) is assumed to satisfy xf (x) > 0 for x 6= 0 and |f (x)| ≥ K|x| for some K > 0,

(4)

and established some sufficient conditions for oscillation by reducing the equations to the self-adjoint form and employing the generalized Riccati transformation technique. Erbe and Peterson [26] considered (3) and obtained an oscillation criterion when p is nonnegative rd-continuous function and f (x) f 0 (x) ≥ ≥ λ > 0 for |x| ≥ L > 0. (5) x No explicit sign assumptions are made with respect to the coefficient q and the oscillation criterion is obtained by comparing the oscillation of (3) with the self-adjoint equation when

R∞

(ep (t, t0 )x∆ (t))∆ + λep (t, t0 )q(t)xσ (t) = 0,

e−p (t, t0 )∆t = ∞. t0 Bohner et al. [27] considered (3) when f 0 (x) > 0 and xf (x) > 0 for x 6= 0

(6)

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 73

and established some new oscillation criteria in which no explicit sign assumptions on p and q are required. The results are obtained by reducing the equation to the nonlinear equation (ep (t, t0 )x∆ (t))∆ + ep (t, t0 )q(t)(f ◦ xσ )(t) = 0. Saker et al. [28] gave some oscillation criteria for the second-order nonlinear damped dynamic equation σ

(r(t)x∆ (t))∆ + p(t)x∆ (t) + q(t)(f ◦ xσ )(t) = 0, where r, p and q are positive rd-continuous functions and f ∈ C(R, R) satisfies (4) or f 0 (x) ≥ K for x 6= 0 and some K > 0. The results are essentially new and complement the nonoscillation conditions for (2) for the linear case that has been established in [24]. Very recently, Erbe et al. [29] presented several oscillation criteria for the second-order nonlinear damped delay dynamic equation ³ ´∆ σ r(t)(x∆ (t))β + p(t)(x∆ (t))β + q(t)f (x(τ (t))) = 0, where β is a quotient of odd positive integers. However, all the results in [24, 25, 26, 27, 28] cannot be applied to (1) when β 6= 1. Also, the results in [29] cannot be applied to (1) when β is not equal to ³a quotient´ of odd positive integers. Furthermore, in the case when 1 f (x) = x 19 + 1+x 2 , the conditions (5) and (6) do not hold and the results in 2

2

−2)(x −5) [26, 27] cannot be applied, since f 0 (x) = (x 9(1+x changes sign four times 2 )2 (see Saker et al. [28]). Therefore, it is of great interest to study (1) when β > 0 is a constant. The main goal of this paper is to establish some new criteria for the oscillation of (1) when β > 0 is a constant. Our results are essentially new and extend and improve the results in [24, 25, 26, 27, 28, 29]. This paper is organized as follows: In the next section, we present some preliminaries on time scales and several lemmas which enable us to prove our main results. In Section 3, we establish several new oscillation criteria for (1). In the last section, we illustrate our results with some examples to which the oscillation criteria in [24, 25, 26, 27, 28, 29] fail to apply. In what follows, for convenience, when we write a functional inequality without specifying its domain of validity we assume that it holds for all sufficiently large t.

2. Preliminaries on time scales and lemmas For completeness, we recall the following concepts related to the notion of time scales. More details can be found in [8, 9]. A time scale T is an arbitrary nonempty closed subset of the real numbers R. We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. Some examples of time scales are

74

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

as follows: the real numbers R, the integers Z, the positive integers N, the nonnegative integers N0 , [0, 1] ∪ [2, 3], [0, 1] ∪ N, hZ := {hk : k ∈ Z, h > 0} and q Z := {q k : k ∈ Z, q > 1} ∪ {0}. But the rational numbers Q, the complex numbers C and the open interval (0, 1) are no time scales. Many other interesting time scales exist, and they give rise to plenty of applications (see [8]). For t ∈ T, the forward jump operator and the backward jump operator are defined by: σ(t) := inf{s ∈ T : s > t},

ρ(t) := sup{s ∈ T : s < t},

where inf ø = sup T (i.e., σ(t) = t if T has a maximum t) and sup ø = inf T (i.e., ρ(t) = t if T has a minimum t), here ø denotes the empty set. Let t ∈ 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 < sup T 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. The graininess function µ : T → [0, ∞) is defined by µ(t) := σ(t) − t. We also need below the set Tκ : If T has a left-scattered maximum m, then Tκ = T − {m}. Otherwise, Tκ = T. Let f : T → R, then we define the function f σ : Tκ → R by f σ (t) := f (σ(t)) for all t ∈ Tκ , i.e., f σ := f ◦ σ. For a, b ∈ T with a < b, we define the interval [a, b] in T by [a, b] := {t ∈ T : a ≤ t ≤ b}. Open intervals and half-open intervals, etc. are defined accordingly. Fix t ∈ Tκ and let f : T → R. Define f ∆ (t) to be the number (provided it exists) with the property that given any ε > 0, there is a neighbourhood U of t such that |[f (σ(t)) − f (s)] − f ∆ (t)[σ(t) − s]| ≤ ε|σ(t) − s| for all s ∈ U. In this case, we say that f ∆ (t) is the (delta) derivative of f at t and that f is (delta) differentiable at t. Assume that f : T → R and let t ∈ Tκ . If f is (delta) differentiable at t, then f (σ(t)) = f (t) + µ(t)f ∆ (t). A function f : T → R is said to be right-dense continuous (rd-continuous) provided it is continuous at each right-dense point in T and its left-sided limits exist (finite) at all left-dense points in T. The set of all such rd-continuous functions is denoted by Crd (T) = Crd (T, R).

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 75

The set of functions f : T → R that are (delta) differentiable and whose (delta) derivative is rd-continuous is denoted by 1 1 Crd (T) = Crd (T, R).

We will make use of the following product and quotient rules for the (delta) derivatives of the product f g and the quotient f /g of two (delta) differentiable functions f and g: (f g)∆ = f ∆ g + f σ g ∆ = f g ∆ + f ∆ g σ and

³ f ´∆

=

(7)

f ∆g − f g∆ , gg σ

(8) g where g σ = g ◦ σ and gg σ 6= 0. For a, b ∈ T and a (delta) differentiable function f , the Cauchy (delta) integral of f ∆ is defined by Z b f ∆ (t)∆t = f (b) − f (a). a

The integration by parts formula reads Z b Z b ∆ f (t)g (t)∆t = f (b)g(b) − f (a)g(a) − f ∆ (t)g σ (t)∆t a

or

Z

(9)

a

b

Z σ

b

∆

f (t)g (t)∆t = f (b)g(b) − f (a)g(a) − a

f ∆ (t)g(t)∆t.

a

The infinite integral is defined as Z t Z ∞ f (s)∆s. f (s)∆s = lim t→∞

a

a

Next we present some lemmas which we will need in the proof of our main results. Throughout this paper, we let Z ∞³ ´1/β 1 E(t) := e p(t) ∆s, (t, t0 ), g(t) := r σ (t) τ (t) E(s)r(s) and

Z

τ (t)

α(t, u) := u

µ

1 r(s)

Lemma 2.1. Suppose that Z ∞³ t0

¶1/β

ÁZ ∆s

σ(t)

µ

u

´1/β 1 ∆t = ∞ E(t)r(t)

1 r(s)

¶1/β ∆s.

(10)

76

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

holds and that x is an eventually positive solution of (1). Then x∆ (t) > 0 and ³ ´∆ r(t)(x∆ (t))β < 0. Proof. Take t1 ≥ t0 such that x(τ (t)) > 0,

t ∈ [t1 , ∞).

(11)

Therefore, from (1) we have for t ∈ [t1 , ∞) ³ ´∆ σ σ r(t)|x∆ (t)|β−1 x∆ (t) + p(t)|x∆ (t)|β−1 x∆ (t) = −q(t)f (x(τ (t))) < 0, (12) which implies ³ ´∆ ³ ´σ p(t) ∆ β−1 ∆ ∆ β−1 ∆ r(t)|x (t)| x (t) E(t) + r(t)|x (t)| x (t) E(t) σ 0 for t ∈ [t1 , ∞). If not, then there exists t2 ≥ t1 such that x∆ (t2 ) ≤ 0. Take t3 > t2 . Since (13) implies that E(t)r(t)|x∆ (t)|β−1 x∆ (t) is strictly decreasing on [t1 , ∞), we get for t ≥ t3 E(t)r(t)|x∆ (t)|β−1 x∆ (t) ≤ c1 := E(t3 )r(t3 )|x∆ (t3 )|β−1 x∆ (t3 ) < E(t2 )r(t2 )|x∆ (t2 )|β−1 x∆ (t2 ) ≤ 0. Therefore, for t ≥ t3 we conclude that x∆ (t) < 0, −E(t)r(t)(−x∆ (t))β ≤ c1 and ³ ´1/β 1 x∆ (t) ≤ −(−c1 )1/β . E(t)r(t) Integrating from t3 to t, we find Z t³ ´1/β 1 1/β x(t) ≤ x(t3 ) − (−c1 ) ∆s, t ∈ [t3 , ∞). E(s)r(s) t3 Thus, from (10) we obtain that x(t) is eventually negative, which implies a contradiction. Hence, we have x∆ (t) > 0 for t ∈ [t1 , ∞) and thus, from (12) we get ´∆ ³ σ ∆ β = −p(t)(x∆ (t))β − q(t)f (x(τ (t))) < 0, t ∈ [t1 , ∞). r(t)(x (t)) The proof is complete.

¤

Lemma 2.2. Suppose that ¸1/β Z ∞· Z v 1 β ∆v = ∞ E(u)q(u)g (u)∆u E(v)r(v) t0 t0

(14)

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 77

holds and that x is an eventually positive solution of (1). Then x∆ (t) > 0 and ³ ´∆ r(t)(x∆ (t))β < 0. Proof. We proceed as in the proof of Lemma 2.1 to get that (11)–(13) hold. Now, we claim x∆ (t) > 0 for t ∈ [t1 , ∞). If not, then we proceed as in the proof of Lemma 2.1 to obtain that there exists t3 ≥ t1 such that x∆ (t) < 0 for t ∈ [t3 , ∞). Take t4 ≥ t3 such that τ (t) ≥ t3 for t ∈ [t4 , ∞). Using the fact that −E(t)r(t)(−x∆ (t))β is strictly decreasing on [t1 , ∞), we have −x(τ (t)) ≤ x(∞) − x(τ (t)) Z ∞ [E(s)r(s)(−x∆ (s))β ]1/β =− ∆s [E(s)r(s)]1/β τ (t) ∆

≤ −[E(τ (t))r(τ (t))(−x (τ (t))) ] Z ∞ ∆ β 1/β ≤ −[E(t3 )r(t3 )(−x (t3 )) ]

τ (t)

= c2 g(t),

Z

∞

1 ∆s 1/β τ (t) [E(s)r(s)] 1 ∆s [E(s)r(s)]1/β

β 1/β

t ∈ [t4 , ∞),

where x(∞) := limt→∞ x(t) ≥ 0 and c2 := −[E(t3 )r(t3 )(−x∆ (t3 ))β ]1/β < 0. Thus, from (12) we obtain ³ ´∆ ∆ β − E(t)r(t)(−x (t)) = −E(t)q(t)f (x(τ (t))) ≤ −KE(t)q(t)xβ (τ (t)) ≤ −K(−c2 )β E(t)q(t)g β (t),

t ∈ [t4 , ∞).

Integrating from t4 to t, we get − E(t)r(t)(−x∆ (t))β ∆

Z β

β

t

≤ −E(t4 )r(t4 )(−x (t4 )) − K(−c2 ) Z ≤ −K(−c2 )β

E(u)q(u)g β (u)∆u

t4 t

E(u)q(u)g β (u)∆u,

t ∈ [t4 , ∞).

t4

From the last inequality, we have ¸1/β · Z t 1 1 β ∆ β , E(u)q(u)g (u)∆u x (t) ≤ K c2 E(t)r(t) t4

t ∈ [t4 , ∞).

Integrating from t4 to t, we find ¸1/β Z v Z t· 1 1 β β ∆v, E(u)q(u)g (u)∆u x(t) ≤ x(t4 ) + K c2 t4 E(v)r(v) t4

t ∈ [t4 , ∞).

Therefore by (14), we obtain limt→∞ x(t) = −∞, which contradicts the fact that x is an eventually positive solution of (1). Hence, we have x∆ (t) > 0 for

78

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

t ∈ [t1 , ∞) and thus, from (12) we get ³ ´∆ σ ∆ β r(t)(x (t)) = −p(t)(x∆ (t))β − q(t)f (x(τ (t))) < 0, The proof is complete.

¤

Lemma 2.3. Assume that Z ∞³ 1

Z

v

E(u)q(u)∆u

E(v)r(v)

t0

t ∈ [t1 , ∞).

´1/β

∆v = ∞

(15)

t0

holds and that x is an eventually positive solution of (1). Then either ³ ´∆ ∆ ∆ β x (t) > 0 and r(t)(x (t)) 0; (ii) x∆ (t) ≤ 0. Case (i). Let x∆ (t) > 0. In this case, it follows from (1) that ³ ´∆ σ r(t)(x∆ (t))β = −p(t)(x∆ (t))β − q(t)f (x(τ (t))) < 0. Case (ii). Let x∆ (t) ≤ 0. In this case, we get limt→∞ x(t) := l ≥ 0 and x(τ (t)) ≥ l. Therefore, from (1) there exists t1 ≥ t0 such that ³ ´∆ − E(t)r(t)(−x∆ (t))β = −E(t)q(t)f (x(τ (t))) ≤ −KE(t)q(t)xβ (τ (t)) ≤ −Klβ E(t)q(t),

t ∈ [t1 , ∞).

Integrating from t1 to t, we have ∆

Z

β

∆

β

−E(t)r(t)(−x (t)) ≤ −E(t1 )r(t1 )(−x (t1 )) − Kl

t

E(u)q(u)∆u t1

Z ≤ −Kl

β

t

β

E(u)q(u)∆u,

t ∈ [t1 , ∞).

t1

Thus, we obtain ³

1 x (t) ≤ −K l E(t)r(t) ∆

1 β

Z

t

E(u)q(u)∆u

´1/β

,

t ∈ [t1 , ∞).

t4

Integrating from t1 to t, we get Z v Z t³ ´1/β 1 1 β x(t) ≤ x(t1 ) − K l E(u)q(u)∆u ∆v, E(v)r(v) t1 t1

t ∈ [t1 , ∞).

If l > 0, then by (15) we obtain limt→∞ x(t) = −∞, which contradicts the fact that x is an eventually positive solution of (1). Hence, we have l = 0, i.e., limt→∞ x(t) = 0. The proof is complete. ¤

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 79

Lemma 2.4. Assume that there exists T ≥ t0 such that ³ ´∆ ∆ ∆ β x(t) > 0, x (t) > 0 and r(t)(x (t)) < 0 f or t ∈ [T, ∞). Then x(τ (t)) ≥ α(t, T )xσ (t) for t ∈ [T1 , ∞), where T1 > T satisfies that τ (t) > T for t ∈ [T1 , ∞). Proof. Since r(t)(x∆ (t))β is strictly decreasing on [T, ∞), for t ∈ [T1 , ∞) we have Z σ(t) [r(s)(x∆ (s))β ]1/β σ x (t) − x(τ (t)) = ∆s r1/β (s) τ (t) Z σ(t) 1 ∆ β 1/β ≤ [r(τ (t))(x (τ (t))) ] ∆s 1/β (s) τ (t) r and xσ (t) [r(τ (t))(x∆ (τ (t)))β ]1/β ≤1+ x(τ (t)) x(τ (t))

Z

σ(t) τ (t)

1 r1/β (s)

∆s.

(16)

Also, for t ∈ [T1 , ∞) we get Z

τ (t)

[r(s)(x∆ (s))β ]1/β x(τ (t)) > x(τ (t)) − x(T ) = ∆s r1/β (s) T Z τ (t) ∆ β 1/β ≥ [r(τ (t))(x (τ (t))) ] T

1 r1/β (s)

∆s

and

Z ´−1 [r(τ (t))(x∆ (τ (t)))β ]1/β ³ τ (t) 1 < ∆s . x(τ (t)) r1/β (s) T Therefore, (16) and (17) imply Z σ(t) ³ Z τ (t) ´−1 xσ (t) 1 1 ≤1+ ∆s ∆s 1/β (s) x(τ (t)) r1/β (s) τ (t) r T Z Z σ(t) ³ τ (t) ´−1 1 1 = ∆s ∆s r1/β (s) r1/β (s) T T

(17)

for t ∈ [T1 , ∞). Hence, we obtain x(τ (t)) ≥ α(t, T )xσ (t) for t ≥ T1 . The proof is complete. ¤ Lemma 2.5. (Bohner and Peterson [8], p. 32, Theorem 1.87) Let f : R → R be continuously differentiable and suppose g : T → R is delta differentiable. Then f ◦ g : T → R is delta differentiable and satisfies ½Z 1 ¾ ∆ 0 ∆ (f ◦ g) (t) = f (g(t) + hµ(t)g (t))dh g ∆ (t). 0

80

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

Lemma 2.6. (Hardy et al. [7]) If X and Y are nonnegative, then λXY λ−1 − X λ ≤ (λ − 1)Y λ

when λ > 1,

where the equality holds if and only if X = Y. 3. Main results In this section, we state and prove our main results. Theorem 3.1. Suppose that one of the conditions (10) or (14) holds. Furthermore, suppose that there exists a positive ∆-differentiable function ϕ such that for all sufficiently large T ≥ t0 , ¸ Z t· r(s)(Q+ (s))β+1 β lim sup Kα (s, T )ϕ(s)q(s) − ∆s = ∞, (18) (β + 1)β+1 ϕβ (s) t→∞ T1 where T1 > T satisfies that τ (t) > T for t ∈ [T1 , ∞) and Q+ (t) := max{0, Q(t)}, here Q(t) := ϕ∆ (t) − ϕ(t)p(t) . Then every solution of (1) is oscillatory. rσ (t) Proof. Suppose that x is a nonoscillatory solution of (1). Without loss of generality, we may assume that x is an eventually positive solution of (1). Then, by Lemmas 2.1 and 2.2 there exists T ≥ t0 such that ³ ´∆ x(τ (t)) > 0, x∆ (t) > 0 and r(t)(x∆ (t))β < 0 for t ∈ [T, ∞). Define the function w by the generalized Riccati substitution w(t) = ϕ(t)

r(t)(x∆ (t))β . xβ (t)

(19)

It is easy to see that w(t) > 0 for t ∈ [T, ∞). By the product and quotient rules (7) and (8) for the ∆-derivative of two ∆-differentiable functions, we have ³ r(t)(x∆ (t))β ´σ ³ r(t)(x∆ (t))β ´∆ w∆ (t) = ϕ∆ (t) + ϕ(t) xβ (t) xβ (t) ³ r(t)(x∆ (t))β ´σ [r(t)(x∆ (t))β ]∆ = ϕ∆ (t) + ϕ(t) xβ (t) (xβ (t))σ r(t)(x∆ (t))β (xβ (t))∆ , t ∈ [T, ∞). − ϕ(t) xβ (t)(xβ (t))σ Hence, from (1), (19) and the definition of Q we obtain for t ∈ [T, ∞) ³ x(τ (t)) ´β Q(t) r(t)(x∆ (t))β (xβ (t))∆ w∆ (t) ≤ σ wσ (t)−Kϕ(t)q(t) −ϕ(t) . (20) ϕ (t) xσ (t) xβ (t)(xβ (t))σ ³ ´∆ ∆ β Since t ≤ σ(t) and r(t)(x (t)) is strictly decreasing on t ∈ [T, ∞), we get ³ ´σ r(t)(x∆ (t))β ≥ r(t)(x∆ (t))β for t ∈ [T, ∞). Thus, from (19) and (20), we

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 81

have w∆ (t) ≤

³ x(τ (t)) ´β Q(t) σ wσ (t) (xβ (t))∆ w (t)−Kϕ(t)q(t) −ϕ(t) , ϕσ (t) xσ (t) ϕσ (t) xβ (t)

t ∈ [T, ∞).

Taking T1 > T such that τ (t) > T for t ∈ [T1 , ∞), it follows from Lemma 2.4 that w∆ (t) ≤

Q(t) σ wσ (t) (xβ (t))∆ β w (t) − Kα (t, T )ϕ(t)q(t) − ϕ(t) , ϕσ (t) ϕσ (t) xβ (t)

t ∈ [T1 , ∞).

By Lemma 2.5, for t ∈ [T1 , ∞) we obtain ½Z 1 ¾ β ∆ ∆ β−1 (x (t)) = β [x(t) + hµ(t)x (t)] dh x∆ (t) 0 ½Z 1 ¾ σ β−1 =β [(1 − h)x(t) + hx (t)] dh x∆ (t) 0 ½ β(x(t))β−1 x∆ (t), β > 1, ≥ β(xσ (t))β−1 x∆ (t), 0 < β ≤ 1. Therefore, for t ∈ [T1 , ∞), if 0 < β ≤ 1, we get Q(t) σ wσ (t) x∆ (t) ³ xσ (t) ´β β w (t) ≤ σ w (t) − Kα (t, T )ϕ(t)q(t) − βϕ(t) σ , (21) ϕ (t) ϕ (t) xσ (t) x(t) ∆

whereas if β > 1, we get w∆ (t) ≤

Q(t) σ wσ (t) x∆ (t) xσ (t) β w (t) − Kα (t, T )ϕ(t)q(t) − βϕ(t) . ϕσ (t) ϕσ (t) xσ (t) x(t)

(22)

Using the fact that x(t) is strictly increasing and r(t)(x∆ (t))β is strictly decreasing, we have ³ rσ (t) ´1/β xσ (t) ≥ x(t) and x∆ (t) ≥ (x∆ (t))σ , t ∈ [T1 , ∞). (23) r(t) From (21)–(23), we obtain for t ∈ [T1 , ∞) w∆ (t) ≤

Q(t) σ wσ (t) ³ rσ (t) ´1/β (x∆ (t))σ β w (t) − Kα (t, T )ϕ(t)q(t) − βϕ(t) . ϕσ (t) ϕσ (t) r(t) xσ (t)

In view of (19), we get for t ∈ [T1 , ∞) w∆ (t) ≤

βϕ(t)(wσ (t))λ Q+ (t) σ β w (t) − Kα (t, T )ϕ(t)q(t) − , ϕσ (t) (ϕσ (t))λ r1/β (t)

where λ := 1 + β1 . Taking X=

(βϕ(t))1/λ wσ (t) ϕσ (t)r1/(β+1) (t)

and Y =

(βr(t))1/λ (Q+ (t))β , (β + 1)β ϕβ/λ (t)

(24)

82

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

by Lemma 2.6 and (24) we have w∆ (t) ≤

r(t)(Q+ (t))β+1 − Kαβ (t, T )ϕ(t)q(t), (β + 1)β+1 ϕβ (t)

t ∈ [T1 , ∞).

Integrating from T1 to t, we obtain for t ∈ [T1 , ∞) ¸ Z t· r(s)(Q+ (s))β+1 β ∆s ≤ w(T1 ) − w(t) < w(T1 ), Kα (s, T )ϕ(s)q(s) − (β + 1)β+1 ϕβ (s) T1 which implies a contradiction to (18). The proof is complete. ¤ The following theorem gives a Philos-type oscillation criterion for (1). Theorem 3.2. Suppose that one of the conditions (10) or (14) holds. Fur1 ([t0 , ∞), R) and thermore, suppose that there exist a positive function ϕ ∈ Crd a function H ∈ Crd (D, R), where D := {(t, s) ∈ T × T : t ≥ s ≥ t0 }, such that H(t, t) = 0 f or t ≥ t0 ,

H(t, s) > 0 f or (t, s) ∈ D0 ,

where D0 := {(t, s) ∈ T × T : t > s ≥ t0 }, and H has a nonpositive rdcontinuous delta partial derivative H ∆s (t, s) on D0 with respect to the second variable and satisfies, for all sufficiently large T ≥ t0 , Z t−1 · 1 lim sup KH(t, s)αβ (s, T )ϕ(s)q(s) t→∞ H(t, T1 ) T1 ¸ r(s)(h+ (t, s)ϕσ (s))β+1 − ∆s = ∞, (25) (β + 1)β+1 (H(t, s)ϕ(s))β where T1 is defined as in Theorem 3.1 and h+ (t, s) := max{0, H ∆s (t, s) + (s) H(t, s) Qϕσ+(s) }, here Q+ is defined as in Theorem 3.1. Then all solutions of (1) are oscillatory. Proof. Suppose that x is a nonoscillatory solution of (1). Without loss of generality, we may assume that x is an eventually positive solution of (1). We proceed as in the proof of Theorem 3.1 to get that (24) holds. Multiplying (24) by H(t, s) and integrating from T1 to t − 1, we find Z t−1 H(t, s)Kαβ (s, T )ϕ(s)q(s)∆s T1

Z

Z

t−1

≤−

∆

t−1

H(t, s)w (s)∆s + Z

T1 t−1

−

σ

H(t, s) T1

H(t, s) T1 λ

βϕ(s)(w (s)) ∆s, (ϕσ (s))λ r1/β (s)

Q+ (s) σ w (s)∆s ϕσ (s)

t ∈ [T1 + 1, ∞).

(26)

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 83

Applying the integration by parts formula (9), we get Z t−1 − H(t, s)w∆ (s)∆s T1

is=t−1 Z = − H(t, s)w(s) + h

s=T1

Z

t−1

< H(t, T1 )w(T1 ) +

t−1

H ∆s (t, s)wσ (s)∆s

T1

H ∆s (t, s)wσ (s)∆s,

t ∈ [T1 + 1, ∞).

(27)

T1

Substituting (27) in (26), we obtain for t ∈ [T1 + 1, ∞) Z t−1 H(t, s)Kαβ (s, T )ϕ(s)q(s)∆s T1

< H(t, T1 )w(T1 ) ¸ ¾ Z t−1 ½· βϕ(s)(wσ (s))λ Q+ (s) σ ∆s + w (s) − H(t, s) σ ∆s H (t, s) + H(t, s) σ ϕ (s) (ϕ (s))λ r1/β (s) T1 ¸ Z t−1 · βϕ(s)(wσ (s))λ σ ≤ H(t, T1 )w(T1 ) + ∆s. (28) h+ (t, s)w (s) − H(t, s) σ (ϕ (s))λ r1/β (s) T1 Therefore by using Lemma 2.6 in (28) with X=

(H(t, s)βϕ(s))1/λ wσ (s) ϕσ (s)r1/(β+1) (s)

and Y =

r1/λ (s)(h+ (t, s)ϕσ (s))β , λβ (H(t, s)βϕ(s))β/λ

we have for t ∈ [T1 + 1, ∞) Z t−1 H(t, s)Kαβ (s, T )ϕ(s)q(s)∆s T1

Z

t−1

< H(t, T1 )w(T1 ) + T1

r(s)(h+ (t, s)ϕσ (s))β+1 ∆s. (β + 1)β+1 (H(t, s)ϕ(s))β

Therefore, we obtain for t ∈ [T1 + 1, ∞) ¸ Z t−1 · 1 r(s)(h+ (t, s)ϕσ (s))β+1 β H(t, s)Kα (s, T )ϕ(s)q(s) − ∆s H(t, T1 ) T1 (β + 1)β+1 (H(t, s)ϕ(s))β < w(T1 ), which implies a contradiction to (25). Thus, this completes the proof. ¤ Remark 3.1. From Theorems 3.1 and 3.2, we can obtain many different sufficient conditions for the oscillation of (1) with different choices of the functions ϕ and H. For example, let ϕ(t) = t, then Theorem 3.1 yields the following result.

84

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

Corollary 3.1. Suppose that one of the conditions (10) or (14) holds and for all sufficiently large T , ¸ Z t· r(s)(V+ (s))β+1 β lim sup Ksα (s, T )q(s) − ∆s = ∞, (29) (β + 1)β+1 sβ t→∞ T1 where T1 is defined as in Theorem 3.1 and V+ (t) := max{0, 1 − every solution of (1) is oscillatory.

tp(t) }. rσ (t)

Then

Let ϕ(t) = 1, then from Theorem 3.1 we obtain the following result. Corollary 3.2. Suppose that one of the conditions (10) or (14) holds and for all sufficiently large T , Z ∞ αβ (t, T )q(t)∆t = ∞, (30) T1

where T1 is defined as in Theorem 3.1. Then every solution of (1) is oscillatory. Let ϕ(t) = 1 and H(t, s) = (t − s)m , (t, s) ∈ D, where m ≥ 1 is a constant, then H ∆s (t, s) ≤ −m(t − σ(s))m−1 ≤ 0 for (t, s) ∈ D0 (see Saker [6]). Therefore, from Theorem 3.2 we obtain the following Kamenev-type oscillation criterion for (1). Corollary 3.3. Suppose that one of the conditions (10) or (14) holds and for all sufficiently large T , Z t 1 lim sup (t − s)m αβ (s, T )q(s)∆s = ∞, (31) m (t − T ) t→∞ 1 T1 where T1 is defined as in Theorem 3.1 and m ≥ 1 is a constant. Then all solutions of (1) are oscillatory. Also, by Lemma 2.3, we obtain some other oscillation criteria for (1) as in Theorems 3.1 and 3.2 and Corollaries 3.1–3.3 as follows. Corollary 3.4. Assume that (15) and (18) hold. Then every solution of (1) is oscillatory or tends to zero as t → ∞. Corollary 3.5. Assume that (15) and (25) hold. Then every solution of (1) is oscillatory or tends to zero as t → ∞. Corollary 3.6. Assume that (15) and (29) hold. Then every solution of (1) is oscillatory or tends to zero as t → ∞.

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 85

Corollary 3.7. Assume that (15) and (30) hold. Then every solution of (1) is oscillatory or tends to zero as t → ∞. Corollary 3.8. Assume that (15) and (31) hold. Then every solution of (1) is oscillatory or tends to zero as t → ∞. 4. Examples In this section, we give some examples to illustrate our main results. Example 4.1. Consider the second-order nonlinear damped delay dynamic equation ³ tβ−1 ´∆ (σ(t))β−1 σ σ |x∆ (t)|β−1 x∆ (t) + |x∆ (t)|β−1 x∆ (t) σ b(t) tb (t) 1 + β |x(τ (t))|β−1 x(τ (t)) = 0, (32) α (t, t∗ )t2 where β > 0 is a constant, t∗ satisfies that t0 > t∗ > 0 and τ (t) > t∗ for β−1 β−1 t ∈ [t0 , ∞), and b(t) := e 1 (t, t0 ). In (32), r(t) = tb(t) , p(t) = (σ(t)) , q(t) = σ tb (t) t 1 β−1 and f (u) = |u| u. αβ (t,t∗ )t2 We will apply Corollary 3.1. We have Z ∞ Z ∞³ ´1/β 1 1 ∆t = 1 ∆t = ∞, 1− E(t)r(t) β t0 t t0 which implies (10) holds. Furthermore, we see that V+ (t) := max{0, 1− rtp(t) σ (t) } = 0 and ¸ Z t· r(s)(V+ (s))β+1 β lim sup ∆s Ksα (s, T )q(s) − (β + 1)β+1 sβ t→∞ T1 ¶β Z t µ Z t α(s, T ) 1 1 = lim sup ∆s = lim sup K K ∆s = ∞, ∗ α(s, t ) s s t→∞ t→∞ T1 T1 since limt→∞ oscillatory.

α(t,T ) α(t,t∗ )

= 1. Therefore by Corollary 3.1, every solution of (32) is

Example 4.2. Consider the second-order nonlinear damped delay dynamic equation ³ (tσ(t))β ´∆ (σ(t))β σ σ |x∆ (t)|β−1 x∆ (t) + σ |x∆ (t)|β−1 x∆ (t) b(t) b (t) tβ |x(τ (t))|β−1 x(τ (t)) = 0, (33) + β α (t, t∗ )

86

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

where 0 < β ≤ 1 is a constant, t∗ satisfies that t0 > t∗ > 0 and τ (t) > t∗ for t ∈ [t0 , ∞), b(t) := e 1 β (t, t0 ) and we suppose (σ(σ(t))) Z ∞ 1 ∆t = ∞ (34) 1 1− β t0 t σ(t) for those time scales T. This holds for many time scales, for example when T = q0N0 := {q0k : k ∈ N0 , q0 > 1}. β β tβ β−1 , p(t) = (σ(t)) , q(t) = αβ (t,t In (33), r(t) = (tσ(t)) u. ∗ ) and f (u) = |u| b(t) bσ (t) It is clear that Z ∞³ Z ∞ Z ∞³ ´1/β 1 1 1 ´∆ 1 < ∞, ∆t = ∆t = − ∆t = E(t)r(t) t t0 t0 t0 tσ(t) t0 which implies (10) does not hold. Now we prove that (14) holds. We have ¸1/β Z ∞· Z v 1 β E(u)q(u)g (u)∆u ∆v E(v)r(v) t0 t0 ¸1/β Z v³ Z ∞· ug(u) ´β 1 ∆u ∆v ≥ (vσ(v))β t0 α(u, t∗ ) t0 ¸1/β Z ∞· v − t0 ≥ ∆v, (35) (vσ(v))β t0 since for t ∈ [t0 , ∞), E(t) := e

1 (σ(σ(t)))β

(t, t0 ) ≥ 1 and

Z ∞ ´1/β 1 1 ∆s = g(t) := ∆s τ (t) sσ(s) τ (t) E(s)r(s) Z ∞³ 1 ´∆ 1 1 α(t, t∗ ) − = ∆s = ≥ ≥ . s τ (t) t t τ (t) Z

∞

³

Take 0 < c < 1 such that t − t0 > ct for t ≥ tc > t0 , then from (35) and (34) we obtain ¸1/β Z ∞· Z v Z ∞ 1 1 1 β E(u)q(u)g (u)∆u ∆v ≥ c β ∆v = ∞, 1 1− β E(v)r(v) t0 t0 tc v σ(v) which implies (14) holds. To apply Corollary 3.2, it remains to prove that (30) holds. We get ¶β Z ∞ Z ∞ Z ∞µ α(t, T ) β β t ∆t = tβ ∆t = ∞, α (t, T )q(t)∆t = ∗ α(t, t ) T1 T1 T1 α(t,T ) since limt→∞ α(t,t ∗ ) = 1. Thus, for those time scales where (34) holds, every solution of (33) is oscillatory by Corollary 3.2.

Oscillation of certain nonlinear second-order damped delay dynamic equations on time scales 87

REFERENCES [1] M. Huang, Oscillation criteria for second order nonlinear dynamic equations with impulses, Comput. Math. Appl., 59(2010), No. 1, 31-41. [2] D. -X. Chen, Oscillation of second-order Emden–Fowler neutral delay dynamic equations on time scales, Math. Comput. Modelling, 51(2010), No. 9-10, 1221-1229. [3] D. -X. Chen, Oscillation and asymptotic behavior for nth-order nonlinear neutral delay dynamic equations on time scales, Acta Appl. Math., 109(2010), No. 3, 703-719. [4] D. -X. Chen and J. -C. Liu, Asymptotic behavior and oscillation of solutions of thirdorder nonlinear neutral delay dynamic equations on time scales, Can. Appl. Math. Q., 16(2008), No. 1, 19-43. [5] A. Zafer, On oscillation and nonoscillation of second-order dynamic equations, Appl. Math. Lett., 22(2009), No. 1, 136-141. [6] S. H. Saker, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, J. Comput. Appl. Math., 187(2006), No. 2, 123-141. [7] G. H. Hardy, J. E. Littlewood and G. P´ olya, Inequalities (second ed.), Cambridge University Press, Cambridge, 1988. [8] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨auser, Boston, 2001. [9] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003. [10] Y. Xu and Z. Xu, Oscillation criteria for two-dimensional dynamic systems on time scales, J. Comput. Appl. Math., 225(2009), No. 1, 9-19. [11] T. S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345(2008), No. 1, 176-185. [12] Z. Han, S. Sun and B. Shi, Oscillation criteria for a class of second-order Emden– Fowler delay dynamic equations on time scales, J. Math. Anal. Appl., 334(2007), No. 2, 847-858. [13] L. Erbe, A. Peterson and S. H. Saker, Hille and Nehari type criteria for third-order dynamic equations, J. Math. Anal. Appl., 329(2007), No. 1, 112-131. [14] H. -W. Wu, R. -K. Zhuang and R. M. Mathsen, Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations, Appl. Math. Comput., 178(2006), No. 2, 321-331. [15] L. Erbe, A. Peterson and S. H. Saker, Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. Comput. Appl. Math., 181(2005), No. 1, 92-102. [16] B. G. Zhang and Z. Shanliang, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comput. Math. Appl., 49(2005), No. 4, 599-609. [17] Y. S ¸ ahiner, Oscillation of second-order delay differential equations on time scales, Nonlinear Anal., 63(2005), No. 5-7, 1073-1080. [18] L. Ou, Atkinson’s super-linear oscillation theorem for matrix dynamic equations on a time scale, J. Math. Anal. Appl., 299(2004), No. 2, 615-629. [19] M. Bohner and S. H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comput. Modelling, 40(2004), No. 3-4, 249-260. [20] R. P. Agarwal, D. O’Regan and S. H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl., 300(2004) 203-217.

88

Da-Xue Chen, Guang-Hui Liu, Yu-Hua Long

[21] O. Doˇsl´y and S. Hilger, A necessary and sufficient condition for oscillation of the Sturm– Liouville dynamic equation on time scales, J. Comput. Appl. Math., 141(2002), No. 1-2, 147-158. [22] R. P. Agarwal, M. Bohner, D. O’Regan and A. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math., 141(2002), No. 1-2, 1-26. [23] R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor & Francis, London, 2003. [24] G. Sh. Guseinov and B. Kaymak¸calan, On a disconjugacy criterion for second-order dynamic equations on time scales, J. Comput. Appl. Math., 141(2002), No. 1-2, 187196. [25] L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London Math. Soc., 67(2003), No. 3, 701-714. [26] L. Erbe and A. Peterson, An oscillation result for a nonlinear dynamic equations on a time scale, Can. Appl. Math. Q., 11(2003), No. 2, 143-158. [27] M. Bohner, L. Erbe and A. Peterson, Oscillation for nonlinear second order dynamic equations on a time scale, J. Math. Anal. Appl., 301(2005), No. 2, 491-507. [28] S. H. Saker, R. P. Agarwal and D. O’Regan, Oscillation of second-order damped dynamic equations on time scales, J. Math. Anal. Appl., 330(2007), No. 2, 1317-1337. [29] L. Erbe, T. S. Hassan and A. Peterson, Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comput., 203(2008), No. 1, 343-357. [30] T. S. Hassan, L. Erbe and A. Peterson, Oscillation of second order superlinear dynamic equations with damping on time scales, Comput. Math. Appl., 59(2010), No. 1, 550-558.