OSCILLATION OF NONLINEAR SECOND ORDER NEUTRAL ...

Bull. Korean Math. Soc. 45 (2008), No. 2, pp. 299–312

OSCILLATION OF NONLINEAR SECOND ORDER NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES Hassan A. Agwo

Reprinted from the Bulletin of the Korean Mathematical Society Vol. 45, No. 2, May 2008 c °2008 The Korean Mathematical Society

Bull. Korean Math. Soc. 45 (2008), No. 2, pp. 299–312

OSCILLATION OF NONLINEAR SECOND ORDER NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES Hassan A. Agwo Abstract. In this paper, we establish some oscillation criteria for nonautonomous second order neutral delay dynamic equations (x(t) ± r(t)x(τ (t)))44 + H(t, x(h1 (t)), x4 (h2 (t))) = 0 on a time scale T. Oscillatory behavior of such equations is not studied before. This is a first paper concerning these equations. The results are not only can be applied on neutral differential equations when T = R, neutral delay difference equations when T = N and for neutral delay q−difference equations when T =q N for q > 1, but also improved most previous results. Finally, we give some examples to illustrate our main results. These examples are not discussed before and there is no previous theorems determine the oscillatory behavior of such equations.

1. Introduction In the recent years, the theory of time scales has received a lot of attention which was introduced by Stefan Hilger in his Ph. D. thesis in 1988 in order to unify continuous and discrete analysis (see [10]). In fact there has been much activities concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales (or measure chains). We refer the reader to recent papers [1-5, 8, 11, 13-15, 17] and the references cited therein. A book on the subject of time scales, by Bohner and Peterson [7] summarizes and organizes much of time scales calculus, see also the book by Bohner and Peterson [6] for advances in dynamic equations on time scales. In this paper, we are concerned with the oscillation of the second-order nonlinear dynamic equations (A)

(x(t) + r(t)x(τ (t)))44 + H(t, x(h1 (t)), x4 (h2 (t))) = 0

and (B)

(x(t) − r(t)x(τ (t)))44 + H(t, x(h1 (t)), x4 (h2 (t))) = 0

on a time scale T. Since we are interested in asymptotic behavior of solutions, we will suppose that the time scale T under consideration is not bounded above, Received May 20, 2007. 2000 Mathematics Subject Classification. 34K11, 39A10, 39A99. Key words and phrases. oscillation, time scales, neutral delay, dynamic equation. c °2008 The Korean Mathematical Society

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i.e., it is a time scale interval of the form [t0 , ∞)T = [t0 , ∞) ∩ T. Through this paper, we assume that: (H1 ) r ∈ Crd (T, R+ ), nonincreasing, h and g ∈ Crd (T, T), τ (t) < t, h1 (t) < t, h2 (t) < τ (t), lim τ (t) = ∞, lim h1 (t) = ∞, lim h2 (t) = ∞ and t→∞

t→∞

t→∞

0 ≤ r(t) ≤ r < 1, Crd (T, S) denotes the set of all functions f : T → S which are right-dense continuous on T. (H2 ) H(t, u, v) ∈ C(T × R2 , R) for each t ∈ T which are nondecreasing in u and v and H(t, u, v) > 0 for uv > 0. λ λ (H3 ) |H(t, u, v)| ≥ α(t) |u| + β(t) |v| where α(t), β(t) ≥ 0 and 0 ≤ λ = p q < 1 with p, q are odd integers. By a solution of equation (A & B), we mean a nontrivial real valued function 2 x(t) which has the properties (x(t) ± r(t)x(τ (t)) ∈ Crd [tx , ∞), tx > t0 and satisfying equation (A & B) for all t > tx . Our attention is restricted to those solutions of equation (A & B) which exist on some half line [tx , ∞) and satisfy sup{|x(t)| : t > t1 } > 0 for any t1 > tx . A solution x(t) of (A & B) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory. 0 Note that if T = R, we have σ(t) = ρ(t) = t, f ∆ (t) = f (t), and (A), (B) become respectively as second-order neutral delay differential equations (1.1)

00

[x(t) + r(t)x(τ (t))] + H(t, x(h1 (t)), x0 (h2 (t))) = 0

and (1.2)

00

[x(t) − r(t)x(τ (t))] + H(t, x(h1 (t)), x0 (h2 (t))) = 0.

As a special case of equation (1.2), Wong in [16] considered the second order sublinear neutral differential equation (1.3)

00

[x(t) − rx(t − τ )] + a(t)f (x(t − σ)) = 0,

t ≥ t0 ,

where 0 < r < 1, a(t) ∈ C ([0, ∞), [0, ∞)), f ∈ C 1 ((−∞, ∞), (−∞, ∞)) satisfying x f (x) > 0 for x 6= 0, f 0 (x) ≥ 0 and that Z ² Z 0 dx dx 0< , < ∞, ∀² > 0 0 f (x) −² −f (x) and f (uv) ≥ f (u)f (v), if uv ≥ 0 and |v| ≥ M for some large M > 0. It was proved that equation (1.3) is oscillatory if and only if Z ∞ (1.4) a(t)f (t)dt = ∞.

OSCILLATION OF NONLINEAR SECOND ORDER

301

Recently, X. Lin [13] considered also a sublinear neutral differential equation which is a special case of equation (1.2) in the form (1.5)

00

[x(t) − r(t)x(t − τ )] + a(t)f (x(t − σ)) = 0,

t ≥ t0 .

It was assumed that r(t), a(t) ∈ C([0, ∞), [0, ∞)) and 0 ≤ r(t) ≤ r < 1, f is nondecreasing, satisfying x f (x) > 0 for x 6= 0 and σ > τ. Then every solution of equation (1.5) is oscillatory if Z ∞ (1.6) a(t)dt = ∞. Note that condition (1.6) can not be applied for the second order neutral equation 00 γ (1.7) [x(t) − r(t)x(τ (t))] + xλ (t − h) = 0, (t − h)2 where γ > 0 and 0 ≤ λ = pq < 1 with p, q are odd integers, for other results see [9], [12]. If T = Z, we have σ(t) = t + 1, µ(t) = 1, f ∆ = ∆f, and (A & B) become the second-order neutral delay difference equations (1.8)

∆2 [x(t) ± r(t)x(τ (t))] + H(t, x(h1 (t)), 4x(h2 (t))) = 0.

(t) If T =hZ, h > 0, we have σ(t) = t + h, µ(t) = h, f ∆ = ∆h f = f (t+h)−f and h (A & B) become the second-order neutral delay difference equations

(1.9)

∆2h [y(t) ± r(t)x(τ (t))] + H(t, x(h1 (t)), 4h x(h2 (t))) = 0.

If T=q N = {t : t = q n , n ∈ N, q > 1}, we have σ(t) = qt, µ(t) = (q − 1)t, x(qt)−x(t) x∆ q (t) = (q−1)t , and (A & B) become the second order q−neutral delay difference equation (1.10)

∆2q [y(t) ± r(t)x(τ (t))] + H(t, x(h1 (t)), x∆ q (h2 (t))) = 0.

The paper is organized as follows: In Section 2 we present some preliminaries on time scales. In Section 3, we establish some new sufficient conditions for oscillation of (A & B). In Section 4, we present some illustrative examples to show that our results not only new but also improved most previous results. 2. Some preliminaries on time scales A time scale T is an arbitrary nonempty closed subset of the real numbers R. On any time scale T, we defined the forward and backward jump operators by (2.1)

σ(t) := inf{s ∈ T : s > t} and ρ(t) := sup{s ∈ T : s < t}.

A point t ∈ T, t > inf T is said to be left-dense if ρ(t) = t, right-dense if t > sup T and σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) >

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t. The graininess function µ : T → [0, ∞) is defined by µ(t) := σ(t) − t. For the function f : T → R the (delta) derivative is defined by f 4 (t) :=

(2.2)

f (σ(t)) − f (t) , σ(t) − t

f is said to be differentiable if its derivative exists. A useful formula is (2.3)

f σ := f (σ(t)) = f (t) + µ(t)f 4 (t).

If f, g are differentiable, then f g and the quotient differentiable with (2.4)

f g

(where gg σ 6= 0) are

(f g)4 = f 4 g + f σ g 4 = f g 4 + f 4 g σ ,

and µ ¶4 f 4 g − f g4 f := . g gg σ

(2.5)

If f 4 (t) ≥ 0, then f is nondecreasing. A function f : [a, b] → R is said to be right-dense continuous if it right continuous at each right-dense point and there exists a finite left limit at all leftdense points. A function f : T → R is called regressive, if 1 + µ(t)f (t) 6= 0 for all t ∈ T. The set of all functions f : T → R which are regressive and rd-continuous will be denoted by Cr . We define the set R+ of all positively regressive elements of R by R+ = {f ∈ R : 1+µ(t)f (t) 6= 0, t ∈ T}. A function F with F 4 = f is called an antiderivative of f and then we define Z b (2.6) f (t)4t = F (b) − F (a), a

where a, b ∈ T. It is well known that rd-continuous functions possess antiderivatives. A simple consequence of formula (2.3) is Z σ(t) (2.7) f (s)4s = µ(t)f (t), t

and infinite integrals are defined as Z ∞ Z (2.8) f (t)4t = lim a

b→∞

b

f (t)4t.

a

3. Main results In this section, we establish some sufficient conditions for the oscillation of equations (A & B). For the remainder of the paper we assume that δ −1 (t) is the inverse of the function δ(t) exists and satisfies δ −(n+1) (t) = t + nδ.


303

I. Oscillatory behavior of solutions of equation (A) Theorem 3.1. Assume that H1 − H3 hold. Then every solution of equation (A) oscillates, if Z ∞ 1 (3.1) {α(s)(k[1 − r(h1 (s))]h1 (s))λ + λ β(s)}4s = ∞. 2 t5 Proof. Suppose to the contrary that equation (A) has a nonoscillatory solution x(t). We may assume that there exists t1 ≥ t0 such that x(t) > 0, x(τ (t)) > 0 and x(δ(t)) > 0 where δ = min{h1 , h2 } for all t > t1 . Set (3.2)

y(t) = x(t) + r(t)x(τ (t)).

Then, equation (A) takes the form y 44 (t) + H(t, x(h1 (t)), x4 (h2 (t))) = 0. and y(t) > 0. Hence, from equation (3.2) it follows that y(t) > x(t) for all t > t1 . Assume that x4 (δ(t)) < 0 for all t > t1 .Then (H2 ) implies that y 44 > 0 and then either y 4 < 0 or y 4 > 0. If y 4 < 0 then y → −∞ which contradicts y > 0. Also, from nonincreasing of r(t) and x4 (t) < 0 implies that y 4 < 0, which contradicts y 4 > 0. Now, it follows that x4 (t) > 0 and then from (H2 ) implies that, y 44 < 0 for all t > t1 . Thus y ∆ (t) is strictly decreasing. We prove that y ∆ (t) > 0 on the interval [t1 , ∞)T . Assume not. Then there exists t2 ≥ t1 such that y ∆ (t2 ) = C < 0. Then, since y ∆∆ (t) < 0, we have (3.3)

y ∆ (t) ≤ y ∆ (t2 ) = C,

for t ≥ t2 ,

and therefore (3.4)

y ∆ (t) ≤ C

for all t ≥ t2 .

Integrating inequality (3.4) from t2 to t, we obtain Z t (3.5) y(t) = y(t2 ) + y ∆ (s)∆s ≤ y(t2 ) + C(t − t2 ), t2

and consequently y(t) → −∞ as t → ∞ which contradicts y > 0. Hence y ∆ (t) > 0 and consequently, y(t) is strictly increasing, then from (3.2) we have, y(t) − r(t)y(τ (t)) = x(t) − r(t)r(τ (t))x(τ (τ (t))) < x(t), therefore (3.6)

x(t) > (1 − r(t))y(t).

Since y ∆∆ (t) < 0 and y(t) > 0, then Z t y 4 (s)4s y(t) = y(t4 ) + t4

(3.7)

> (t − t4 )y 4 (t) > kty 4 (t) for t >

t1 := t2 , 0 < k < 1. (1 − k)

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Since r(t) is nonincreasing, x > 0 and x4 > 0, it follows that y 4 (t) <

1 1 4 y (τ (t)) > y 4 (t). 2 2

On the other side, if x4 (t) ≤ (x(τ (t)))4 , then x4 (τ (t)) > 21 y 4 (t). Thus, we have x4 (τ (t)) >

(3.9)

1 4 y (t). 2

Since τ (h(t)) < h(t) < t and from nonincreasing of y 4 (t), we get y 4 (τ (h(t))) ≥ y 4 (h(t)) ≥ y 4 (t).

(3.10)

Since x > 0, x4 > 0 and H(t, u, v) nondecreasing in u, v, then, by using (H1 ), (H3 ), (3.6), (3.7) and (3.9), we get 0

= ≥ ≥ ≥ ≥

y ∆∆ (t) + H(t, x(h1 (t)), x4 (h2 (t))) 1 y ∆∆ (t) + H(t, [1 − r(h1 (t))]y(h1 (t)), y 4 (τ −1 (h2 (t)))) 2 1 ∆∆ 4 y (t) + H(t, k[1 − r(h1 (t))]h1 (t)y (h1 (t)), y 4 (t)) 2 1 ∆∆ 4 λ y (t) + α(t)(k[1 − r(h1 (t))]h1 (t)y (t)) + β(t)( y 4 (t))λ 2 1 ∆∆ λ 4 y (t) + [α(t)(k[1 − r(h1 (t))]h1 (t)) + λ β(t)]y (t))λ . 2

Hence, −

y ∆∆ (t) 1 ≥ α(t)(k[1 − r(h1 (t))]h1 (t))λ + λ β(t). y 4 (t))λ 2

Integrating the above inequality from t2 to ∞, we get Z ∞ 1 {α(s)(k[1 − r(h1 (s))]h1 (s))λ + λ β(s)}4s 2 t5


Z

≤ = =

305

∞

y 44 (s) 4s. 4 λ t5 (y (s)) Z y4 (t) 4s − lim t→∞ y 4 (t ) sλ 5 Z 0 4s < ∞. − λ y 4 (t5 ) s

−

which contradicts (3.1), and consequently, equation (A) has no eventually positive solution. Similarly, by using the same technique we can prove that equation (A) has no eventually negative solution. Thus equation (A) is oscillatory. The proof is complete. ¤ II. Oscillatory behavior of solutions of equation (B) Theorem 3.2. Assume that H1 − H3 hold with 0 ≤ r(t) = r < 1. Then every solution of equation (B) oscillates, if Z ∞ (3.11) {(kr)λ α(s)(τ (h1 (s))λ + rλ β(s)}4s = ∞. t5

Proof. Suppose to the contrary that equation (B) has a nonoscillatory solution x(t). We may assume that there exists t1 ≥ t0 such that x(t) > 0, x(τ (t)) > 0 and x(δ(t)) > 0 where δ = min{h1 , h2 } for all t > t1 . Set (3.12)

z(t) = x(t) − rx(τ (t)).

Then, equation (B) takes the form (3.13)

z 44 (t) + H(t, x(h1 (t)), x4 (h2 (t))) = 0,

and from (3.12), we get z(t) < x(t). Assume that x4 (δ(t)) < 0 for all t > t1 . Then x(t) is nonincreasing and consequently (H2 ) implies that, H(t, x(h1 (t)), x4 (h2 (t))) < 0. Thus from (3.13) it follows that z 44 > 0 and then z 4 is strictly increasing. So, we may have either z 4 < 0 or z 4 > 0. Assume that z 4 < 0 then from (3.12) we have x4 (t) < r(x( τ (t)))4 and consequently, x4 (t) < r(x(τ (t)))4 < r2 (x(τ 2 (t)))4 < · · · < rk (x(τ k (t)))4 = rk x4 (t0 ) for sufficiently large k such that t0 = τ k (t). Hence, x(t) < x(t0 ) + rk x4 (t0 )(t − t0 ) which implies that x(t) < 0 which contradicts x(t) > 0. So z 4 < 0 is impossible and then z 4 > 0. Also, since x > 0 and x4 < 0, then 0 ≤ lim x(t) = p < ∞ and consequently, t→∞

from (3.12) it follows that lim z(t) = (1 − r)p < ∞. But, since z 44 > 0 then t→∞

z 4 (t) is an increasing. So, 0 < z(t1 ) < z(t) for all t > t1 . Then z(t) = z(t1 ) + Rt 4 z (s)4s > z(t1 ) + z 4 (t1 )(t − t1 ) and therefore z(t) → ∞ as t → ∞ which t1

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HASSAN A. AGWO

is a contradiction. This shows that x4 < 0 is impossible, and consequently x4 > 0. Now x > 0, x4 > 0. Then (H2 ) implies that H(t, x(h1 (t)), x4 (h2 (t))) > 0 and then z 44 < 0. Thus z ∆ (t) is strictly decreasing. We prove that z ∆ (t) > 0 on the interval [t3 , ∞)T . Assume not. Then there exists t4 ≥ t3 such that z ∆ (t4 ) = C < 0. Then, since z ∆∆ (t) < 0, we have (3.14)

z ∆ (t) ≤ z ∆ (t4 ) = C

for t ≥ t4 ,

and therefore (3.15)

z ∆ (t) ≤ C

for all t ≥ t4 .

Integrating the last inequality from t2 to t, we obtain Z t (3.16) z(t) = z(t2 ) + z ∆ (s)∆s ≤ z(t4 ) + C(t − t4 ), t4

and consequently z(t) → −∞ as t → ∞ which implies that there exists c > 0 and t5 ≥ t4 such that z(t) < −c for t ≥ t5 . Then, we have from (3.2) that (3.17)

x(t) < −c + rx(τ (t)) for t ≥ t5 ,

which implies that x(δ −1 (t5 )) < −c + rx(t5 ). Thus (3.18)

x(δ −(n+1) (t5 )) ≤ −c

n X

ri + rn+1 x(t5 ) ≤ −c + rn+1 x(t5 ),

i=0

and so x(δ −(n+1) (t5 )) < 0 for large n, which contradicts the fact that x(t) > 0 for all t ≥ t1 . Hence z ∆ (t) > 0 and this implies that z(t) is strictly increasing on [t1 , ∞). We prove now that z(t) > 0 for t ≥ t2 where t2 is large enough. Suppose not. Then there exists a t3 ≥ t1 with z(t3 ) < 0. Now, since z(t) is strictly increasing then z(t) > 0 for t ≥ t3 (for if there exists a t4 > t3 with z(t4 ) > 0, then z(t) > 0 for t ≥ t4 , but we are assuming that z(t) > 0 for t large enough is not true). Then from (3.2) that x(t) < rx(τ (t)), for t ≥ t3 . Thus x(τ −1 (t)) ≤ rx(t) and this implies after iteration that x(δ −(n+1) (t)) ≤ rn+1 x(t) → 0 for large n, since 0 ≤ r < 1 and so x(δ −(n+1) (t)) < 0 again, which contradicts the fact that x(t) > 0 for all t ≥ t1 . Then, we have (3.19)

z(t) > 0, z ∆ (t) > 0, z ∆∆ (t) < 0 for t ≥ t1 .

Since z ∆∆ (t) < 0 and z(t) > 0, then Z t z 4 (s)4s z(t) = z(t4 ) + t4

(3.20)

> (t − t4 )z 4 (t) > ktz 4 (t) for t >

t4 := t5 , 0 < k < 1. (1 − k)


307

Since z 4 (t) > 0, x4 (t) > 0, 0 ≤ r < 1, and x(t) > z(t) > ktz 4 (t) then we have z 4 (t) = x4 (t) − r(x(τ (t)))4

(3.21)

x4 (t) = z 4 (t) + r(x(τ (t))4 ≥ z 4 (t).

Now z(t) > 0 implies that x(t) > rx(τ (t)). Since z(t) < x(t), 0 < z 4 (t) is nonincreasing and H(t, u, v) nondecreasing in u and v. Then, using (3.20) and (3.21) we get 0

=

z ∆∆ (t) + H(t, x(h1 (t)), x4 (h2 (t)))

≥

z ∆∆ (t) + H(t, rx(τ (h1 (t))), rx4 (τ (h2 (t))))

≥

z ∆∆ (t) + H(t, rz(τ (h1 (t))), rz 4 (τ (h2 (t)))))

≥

z ∆∆ (t) + α(t)(rz(τ (h1 (t))))λ + β(t)(rz 4 (τ (h2 (t))))λ

≥

z ∆∆ (t) + α(t)(krτ (h1 (t))z 4 (τ (h1 (t))))λ +rλ β(t)(z 4 (t))λ

≥

z ∆∆ (t) + ((kr)λ α(t)(τ (h1 (t))λ + rλ β(t))(z 4 (t))λ .

Then (3.22)

((kr)λ α(t)(τ (h1 (t))λ + rλ β(t)) ≤ −

z 44 (t) . (z 4 (t))λ

Integrating inequality (3.22) from t5 to ∞, we get Z ∞ {(kr)λ α(s)(τ (h1 (s))λ + rλ β(s)}4s t5

Z

≤

∞

z 44 (s) 4s. (z 4 (s))λ

− t5

=

Z

z 4 (t)

− lim t→∞

Z =

z 4 (t5 )

0

− y 4 (t5 )

Hence,

Z

∞

4s sλ

4s < ∞, 0 ≤ λ < 1. sλ

{(kr)λ α(s)(τ (h1 (s))λ + rλ β(s)}4s < ∞,

t5

which contradicts (3.11). So, equation (B) has no eventually positive solution. Similarly, by using the same procedure we can prove that equation (B) has no eventually negative solution. Thus equation (B) is oscillatory. ¤ From Theorems (3.1) and (3.2), we have the following results: Theorem 3.3. Assume that H1 − H3 hold. If Z ∞ (3.23) α(s)(k[1 − r(h1 (s))]h1 (s))λ 4s = ∞. t5

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HASSAN A. AGWO

Then every solution of the equation (x(t) + r(t)x(τ (t)))44 + H(t, x(h1 (t))) = 0

(3.24) oscillates.

Theorem 3.4. Assume that H1 − H3 hold with 0 ≤ r(t) = r < 1 . Then every solution of the equation (x(t) − r(t)x(τ (t)))44 + H(t, x(h1 (t))) = 0

(3.25) oscillates, if

Z

∞

(3.26)

α(s)(τ (h1 (s))λ 4s = ∞.

t5

Theorem 3.5. Assume that H1 − H2 hold. Then every solution of (A) oscillates, if the inequality 1 (3.27) z 4 (t) + [α(t)(k[1 − r(h1 (t))]h1 (t))λ + λ β(t)]z λ (τ (t)) ≤ 0, 0 < k < 1 2 has no eventually positive solution. Theorem 3.6. Assume that H1 − H2 hold. Then every solution of (B) oscillates, if the inequality (3.28)

z 4 (t) + [(kr)λ α(t)(τ (h1 (t))λ + rλ β(t)]z λ (τ (t)) ≤ 0, 0 < k < 1

has no eventually positive solution. Theorems 3.5 and 3.6 reduce the question of oscillation of equation (A) and (B) to that of the absence of eventually positive solutions of the dynamic inequality (3.27) and (3.28) respectively. The following result concerning with oscillation of second-order nonlinear delay dynamic equations on time scales. In [17], it was proved the equivalence of the oscillation of nonlinear dynamic equations (3.29)

x44 (t) + a(t)f (x(t − τ )) = 0

and (3.30)

x44 (t) + a(t)(f ◦ xσ ) = 0.

It was proved ([17], Corollary 2.2) that equation (3.29) is oscillatory if Z ∞ (3.31) a(s)4s = ∞, which is the same result obtained by M. Bohner et. al. (see [5], Theorem 3.2). Also, E. Akin et. al. [4] considered the equation (3.32)

x44 (t) + a(t)(xσ )λ = 0,

and proved that, if (3.31) satisfied then equation (3.32) is oscillatory. This is equivalent to oscillation of (3.33)

x44 (t) + a(t)xλ (t − τ ) = 0.


309

Note that condition (3.31) can not be applied for the second order neutral equation (3.34)

x44 (t) +

where γ > 0 and 0 ≤ λ =

p q

γ xλ (t − h) = 0, (t − h)2

< 1 with p, q are odd integers.

Theorem 3.7. Assume that H1 − H3 hold with r(t) = 0 and β(t) = 0. If Z ∞ (3.35) α(s)(h1 (s))λ 4s = ∞. t5

Then every solution of the equation (3.36)

x44 (t) + a(t)xλ (h1 (t)) = 0

oscillates. 4. Examples In this section, we give some examples to illustrate our main results. In fact, Example 4.1 and Example 4.2 are not discussed before and there is no previous theorems determine the oscillatory behavior of such equations. But Example 4.3 shows that our results improve some previous results. Example 4.1. Consider the following second order neutral delay dynamic equation (4.1) ¯ 4 ¯ 1 1 1 λ ¯x (t − h2 )¯λ ) (x(t) + e− λ (t−τ ) x(t − τ ))44 + (|x(t − h1 )| + −t t e +1 × (sgn x(t − h1 ))(sgn x4 (t − h2 )) = 0, 1

where 0 ≤ λ = pq < 1 , p, q are odd integers, τ , h1 , h2 > 0, r(t) = e− λ (t−τ ) , τ (t) = t − τ , h1 (t) = t − h1 , h2 (t) = t − h2 , H(t, x(h1 (t)), x4 (h2 (t)) = ¯ ¯λ λ (|x(t − h1 )| + ¯x4 (t − h2 )¯ ) (sgn x(t − h1 )) (sgn x4 (t − h2 )) (i.e., α(t) = 1t and β = 1 + e−t ). Then we have Z ∞ 1 {α(s)(k[1 − r(h1 (s))]h1 (s))λ + λ β(s)}4s 2 t5 Z ∞ Z ∞ 1 1 1 λ = k [1 − e− λ (s−τ −h1 ) ](s − h1 )λ 4s + λ (1 + e−s )4s s 2 t5 t5 Z ∞ Z ∞ λ (s − h ) 1 1 ≥ kλ 4s + λ 4s = ∞. s 2 t5 t5 Therefore, by Theorem (3.1), equation (4.1) is oscillatory.

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HASSAN A. AGWO

Example 4.2. Consider the following second order neutral delay dynamic equation (4.2)

¯λ 1 (2 + sin t) (3 + cos t) ¯¯ 4 λ ¯ } (x(t) − x(γ1 t))44 + { |x(γ t)| + x (γ t) 2 3 α α c t 1 t 2 × (sgn x(γ2 t)) (sgn x4 (γ3 t)) = 0,

where c > 1, 0 ≤ λ = pq < 1 , p, q are odd integers, α1 , α2 ∈ [λ, λ + 1) and γ1 , γ2 , γ3 ∈ (0, 1). In equation (B) r(t) = 1c , τ (t) = γ1 t, h1 (t) = γ2 t, h2 (t) = γ2 t, H(t, x(h1 (t)), ¢¯λ (2 + sin t) (3 + cos t) ¯¯¡ 4 λ |x(γ2 t)| + x (γ3 t) ¯ } tα1 tβ × (sgn x(γ2 t))(sgn x4 (γ3 t)).

x4 (h2 (t)) = {

(i.e., α(t) = Z

(2+sin t) tα1

and β =

(3+cos t) ). tα2

Then, we have

∞

{(kr)λ α(s)(τ (h1 (s))λ + rλ β(s)}4s Z ∞ Z ∞ (2 + sin s) (3 + cos s) λ λ λ = (kr) { (γ1 γ2 s) }4s + r 4s α1 s sα 2 t t5 Z 5∞ Z ∞ 1 1 ≥ (kr)λ { α1 (γ1 γ2 s)λ }4s + rλ 4s α2 s s t5 t Z ∞ Z ∞ 5 kγ1 γ2 λ 1 4s 4s = ( ) + ( )λ = ∞ for α1 , α2 ∈ [λ, λ + 1). α1 −λ α2 c s c s t5 t5 t5

Hence, by Theorem (3.2) every solution of equation (4.2) oscillates. Example 4.3. Consider the following specific second order neutral delay dynamic equation (4.3)

1 γ (x(t) − x(t − τ ))00 + xλ (t − h) = 0, t ∈ T, 2 (t − h)2

where T is a time scale, and τ, h > 0, r(t) =

1 , τ (t) = t − τ, h1 (t) = t − h, H(t, x(h1 (t)), x0 (h2 (t)) = x(h(t)) 2

γ (i.e., α(t) = (t−h) 2 , γ > 0 and β(t) = 0). Then X. Lin’s result [13] fail to determine oscillatory behavior of this equation, since Z ∞ a(t)dt < ∞


311

But, according to Theorem 3.3 when T = R we have Z ∞ α(s)(τ (h1 (s))λ ds t5 Z ∞ γ(s − τ − h) = ds 2(s − h)2 t5 Z ∞ τ γ 1 (1 − )ds = ∞. = 2 t5 s − h s−h Hence, every solution of equation (4.3) is oscillatory. Example 4.4. Consider the following specific sublinear second order delay dynamic equation 1 1 3 (4.4) x44 (t) + 1 x (h1 (t)) = 0. t(h1 (t)) 3 Here a(t) =

1

1

t(h1 (t)) 3

. Then ( [4], Corollary 5.3), ([5], Theorem 3.2) and ([17],

Corollary 2.2) results fail to determine oscillatory behavior of this equation, since Z ∞ a(t)dt < ∞. But

Z

∞

Z

∞

4s = ∞. s t5 t5 Hence, by Theorem 3.7, equation (4.4) is oscillatory. α(s)(h1 (s))λ 4s =

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[10] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18–56. [11] J. Jiang and X. Li, Oscillation of second order nonlinear neutral differential equations, Appl. Math. Comput. 135 (2003), no. 2-3, 531–540. [12] G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, 110. Marcel Dekker, Inc., New York, 1987. [13] X. Lin, Oscillation of second-order nonlinear neutral differential equations, J. Math. Anal. Appl. 309 (2005), no. 2, 442–452. ¸ ahiner, On oscillation of second order neutral type delay differential equations, Appl. [14] Y. S Math. Comput. 150 (2004), no. 3, 697–706. [15] 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. [16] J. S. W. Wong, Necessary and sufficient conditions for oscillation of second-order neutral differential equations, J. Math. Anal. Appl. 252 (2000), no. 1, 342–352. [17] 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. Department of Mathematics Ain Shams University aculty of Education, Roxy, Cairo, Egypt E-mail address: [email protected]