Oscillation Criteria for Sublinear Half-Linear Delay ... - Semantic Scholar

International Journal of Difference Equations ISSN 0973-6069, Volume 3, Number 2, pp. 227–245 (2008) http://campus.mst.edu/ijde

Oscillation Criteria for Sublinear Half-Linear Delay Dynamic Equations Lynn Erbe, Taher S. Hassan∗, and Allan Peterson University of Nebraska–Lincoln Department of Mathematics Lincoln, NE 68588-0130, U.S.A. [email protected], [email protected], and [email protected] Samir H. Saker King Saud University Department of Mathematics, College of Science Riyadh 11451, Saudi Arabia [email protected]

Abstract This paper is concerned with oscillation of the second-order half-linear delay dynamic equation (r(t)(x∆ )γ )∆ + p(t)xγ (τ (t)) = 0, on a time scale T where 0 < γ ≤ 1 is the quotient of odd positive integers, p : T → [0, ∞), and τ : T → T are positive rd-continuous functions, r(t) is a positive and (delta) differentiable function, τ (t) ≤ t, and lim τ (t) = ∞. We est→∞ tablish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results in the special cases when T = R and T = N involve and improve some oscillation results for second-order differential and difference equations; and when T = hN, T = q N0 and T = N2 our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.

AMS Subject Classifications: 34K11, 39A10, 39A99L. Keywords: Oscillation, delay half-linear dynamic equations, time scales. Received February 2, 2008; Accepted July 28, 2008 Communicated by Johnny Henderson ∗ Supported by the Egyptian Government while visiting the University of Nebraska–Lincoln

228

1

L. Erbe, T. Hassan, A. Peterson, and S. Saker

Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD Thesis in 1988 in order to unify continuous and discrete analysis, see [12]. A time scale T is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to many applications (see [5]). This theory of these so-called “dynamic equations” not only unifies the corresponding theories for the differential equations and difference equations cases, but it also extends these classical cases to cases “in between”. That is, we are able to treat the so-called q-difference equations when T =q N0 := {q n : n ∈ N0 for q > 1} (which has important applications in quantum theory (see [13])) and can be applied to different types of time scales like T =hN, T = N2 and T = Tn the set of the harmonic numbers. The books on the subject of time scales by Bohner and Peterson [5, 6] summarize and organize much of time scale calculus. In the last few years, there has been increasing interest in obtaining sufficient conditions for the oscillation/nonoscillation of solutions of different classes of dynamic equations on time scales, and we refer the reader to the papers [1, 4, 8, 10, 11, 14] and the references cited therein. In this paper, we are concerned with the oscillatory behavior of the second-order half-linear delay dynamic equation
γ (r(t) x∆ (t) )∆ + p(t)xγ (τ (t)) = 0, (1.1) on an arbitrary time scale T, where 0 < γ ≤ 1 is a quotient of odd positive integers, p is a positive rd-continuous function on T, r(t) is a positive and (delta) differentiable function and the so-called delay function τ : T → T satisfies τ (t) ≤ t for t ∈ T and lim τ (t) = t→∞ ∞. 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 nontrivial
real-valued function γ x ∈ Cr1 [Tx , ∞), Tx ≥ t0 which has the property that r(t) x∆ (t) ∈ Cr1 [Tx , ∞) and satisfies equation (1.1) on [Tx , ∞), where Cr 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 also been a spate of papers on second-order nonlinear dynamic equations on time scales. For a few examples of work since then, Agarwal et al. [1] considered the second-order delay dynamic equations on time scales x∆∆ (t) + p(t)x(τ (t)) = 0,

(1.2)

and established some sufficient conditions for oscillation of (1.2). Erbe et al. [9] considered the pair of second-order dynamic equations (r(t)(x∆ (t))γ )∆ + p(t)xγ (t) = 0,

(1.3)

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229

(r(t)(x∆ (t))γ )∆ + p(t)xγ (σ(t)) = 0, and established some necessary and sufficient conditions for nonoscillation of Hille– Kneser type. Saker [14] examines oscillation for half-linear dynamic equations on time scales (1.3), where γ > 1 is an odd positive integer and Agarwal et al. [3] studies oscillation for the same equation (1.3), where γ > 1 is the quotient of odd positive integers. Hassan [11] improved Agarwal’s and Saker’s results for the equation (1.3), when γ > 0 is the quotient of odd positive integers. Erbe et al. [7] considered the half-linear delay dynamic equations on time scales (r(t)(x∆ (t))γ )∆ + p(t)xγ (τ (t)) = 0, where γ > 1 is the quotient of odd positive integers. We herein utilize a Riccati transformation technique to establish oscillation criteria for (1.1), where 0 < γ ≤ 1 is the quotient of odd positive integers, which complete, improve and generalize the results that have been established by Agarwal et al. [3], Saker [14], Erbe et al. [7] and others. Also, interesting examples that illustrate the importance of our results are included in Section 4.

2

Main Results

Throughout the paper we assume that ∆

r (t) ≥ 0,

Z

∞

and

τ γ (t)p(t)∆t = ∞

(2.1)

t0

is satisfied. Before stating our main results, we begin with the following lemma which will play an important role in the proof of our main results. Lemma 2.1. [7] Assume that (2.1) and Z ∞

∆t 1

t0

=∞

(2.2)

r γ (t)

hold and (1.1) has a positive solution x on [t0 , ∞)T . Then there exists a T ∈ [t0 , ∞)T , sufficiently large, so that (i) x∆ (t) > 0, x∆∆ (t) < 0, x(t) > tx∆ (t), for t ∈ [T, ∞)T ; x(t) (ii) is strictly decreasing on [T, ∞)T . t Motivated by [7, Theorem 2.9], we can prove the following result which is a new oscillation result for equation (1.1).

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Theorem 2.2. Assume that (2.1) and (2.2) hold. Furthermore, assume that there exists a positive ∆-differentiable function δ(t) such that  γ  Z t τ (s) r(s)((δ ∆ (s))+ )γ+1 δ(s)p(s) ∆s = ∞, lim sup − γ σ(s) δ (s)(γ + 1)γ+1 t→∞ t0

(2.3)

where d+ (t) := max{d(t), 0} is the positive part of any function d(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 t1 ∈ [t0 , ∞)T such that x(t) satisfies the conclusions of Lemma 2.1 on [t1 , ∞)T with x(τ (t)) > 0 on [t1 , ∞)T . Let δ(t) be a positive ∆ differentiable function and consider the generalized Riccati substitution  w(t) = δ(t)r(t)

x∆ (t) x(t)

γ .

Then by Lemma 2.1, we see that the function w(t) is positive on [t1 , ∞)T . By the product rule and then the quotient rule (suppressing arguments) w

∆

= = = =

Using the fact that

 σ ∆ r(x∆ )γ r(x∆ )γ δ +δ xγ xγ xγ (r(x∆ )γ )∆ − r(x∆ )γ (xγ )∆ δ∆ σ w + δ δσ xγ xγσ ∆ γ τγ δ σ δx (−px ) δr(x∆ )γ (xγ )∆ w + − δσ xγ (xσ )γ xγ (xσ )γ   γ δ∆ σ xτ r(x∆ )γ (xγ )∆ w − pδ . − δ δσ xσ xγ (xσ )γ ∆



x(t) and r(t)(x∆ (t))γ are decreasing (from Lemma 2.1) we get t

xτ (t) τ (t) ≥ σ x (t) σ(t)

and

r(t)(x∆ (t))γ ≥ rσ (t)(x∆ (t))γσ .

By these last two inequalities we obtain w∆ ≤

 τ γ δ∆ σ rσ (x∆σ )γ (xγ )∆ w − δp − δ . δσ σ xγ (xσ )γ

(2.4)

By the P¨otzsche chain rule (see [5, Theorem 1.90]), and the fact that x∆ (t) > 0, we

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231

obtain 1

Z

γ ∆

(x ) (t) = γ Z0 1 = γ Z0 1 ≥ γ

 γ−1 x(t) + hµ(t)x∆ (t) dh x∆ (t) [(1 − h) x(t) + hxσ (t)]γ−1 dh x∆ (t) (xσ (t))γ−1 dh x∆ (t)

0

= γ(xσ (t))γ−1 x∆ (t).

(2.5)

Using (2.4) and (2.5), we have that  τ γ δ∆ σ rσ (x∆σ )γ x∆ w ≤ σ w − δp . − γδ δ σ xγ xσ ∆

Since

1

∆

x (t) ≥

(rσ (t)) γ (x∆ (t))σ 1 γ

,

and xσ (t) ≥ x(t),

r (t) we get that 1

 τ γ δ∆ δrσ(1+ γ ) w ≤ σ wσ − δp −γ 1 δ σ rγ ∆



x∆σ xσ

γ+1 .

Using the definition of w we finally obtain w∆ ≤

where λ :=

 τ γ δ (δ ∆ )+ σ σ λ w − δp −γ 1 (w ) , σ σ λ δ σ γ (δ ) r

γ+1 . Define A ≥ 0 and B ≥ 0 by γ λ

A :=

1

γδ

σ λ

1

(δ σ )λ r γ

(w ) ,

B

λ−1

:=

r γ+1 λ(γδ)

1 λ

(δ ∆ )+ .

Then, using the inequality λAB λ−1 − Aλ ≤ (λ − 1)B λ , we get that (δ ∆ )+ σ δ σ λ w −γ = λAB λ−1 − Aλ 1 (w ) σ σ λ δ γ (δ ) r ≤ (λ − 1)B λ r((δ ∆ )+ )γ+1 ≤ γ . δ (γ + 1)γ+1

(2.6)

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By this last inequality and (2.6) we get  τ γ r((δ ∆ )+ )γ+1 − δp . δ γ (γ + 1)γ+1 σ

w∆ ≤

Integrating both sides from t1 to t we get Z t  τ γ  r((δ ∆ )+ )γ+1 −w(t1 ) ≤ w(t) − w(t1 ) ≤ ∆s, − δp γ γ+1 σ t1 δ (γ + 1) which leads to a contradiction, since the right-hand side tends to −∞ by (2.3). We introduce the notation Z ∞ tγ p∗ := lim inf P (s)∆s, t→∞ r(t) σ(t) r∗ := lim inf t→∞



tγ wσ (t) , r (t)

1 q∗ := lim inf t→∞ t R := lim sup t→∞

Z T

t

sγ+1 P (s)∆s, r(s)

tγ wσ (t) , r(t)

γ

τ (t) t . Note that 0 ≤ l ≤ 1. In p(t) and assume that l := lim inf t→∞ σ(t) σ(t) order for the definition of p∗ to make sense we assume that γ  Z ∞ Z ∞ τ (s) ∆s < ∞. (2.7) P (s)∆s = p(s) σ(s) t0 t0 where P (t) =

Theorem 2.3. Assume that (2.1), (2.2) and (2.7) hold. Furthermore, assume that l > 0 and γγ p∗ > γ 2 , (2.8) l (γ + 1)γ+1 or 1 p∗ + q∗ > γ(γ+1) . (2.9) l 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 such that x(t) satisfies the conclusions of Lemma 2.1 on [T, ∞)T with x(τ (t)) > 0 on [T, ∞)T . Again we define w(t) as in Theorem 2.2 with δ(t) = 1. We get from (2.6) that −w∆ (t) ≥ P (t) +

γ 1 γ

r (t)

(wσ (t))

γ+1 γ

,

for t ∈ [T, ∞)T .

First, we assume (2.8) holds. It follows from Lemma 2.1 that !−γ  ∆ γ Z t x (t) ∆s w(t) = r(t) < , for t ∈ [T, ∞)T , 1 x(t) t0 r γ (s)

(2.10)

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233

which implies (using (2.2)) that lim w(t) = 0. Integrating (2.10) from σ(t) to ∞ and t→∞

using lim w(t) = 0, we have t→∞

Z

σ

∞

w (t) ≥

1

∞

(wσ (s)) γ wσ (s)

σ(t)

r γ (s)

Z P (s)∆s + γ

σ(t)

1

∆s.

(2.11)

It follows from (2.11) that tγ wσ (t) tγ ≥ r(t) r(t)

∞

tγ P (s)∆s + γ r(t) σ(t)

Z

1

∞

(wσ (s)) γ wσ (s)

σ(t)

r γ (s)

Z

1

∆s.

(2.12)

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) P (s)∆s ≥ p∗ − , and ≥ r∗ − , (2.13) r(t) σ(t) r(t) for t ∈ [t1 , ∞)T . From (2.12) and (2.13) and using the fact r∆ (t) ≥ 0, we get that tγ tγ wσ (t) ≥ (p∗ − ) + γ r(t) r(t)

1

∞

s (wσ (s)) γ sγ wσ (s)

σ(t)

sγ+1 r γ (s) Z ∞ tγ γr(s) ∆s r(t) σ(t) sγ+1 Z ∞ γ γ t ∆s. γ+1 σ(t) s

Z

1

1

≥ (p∗ − ) + (r∗ − )1+ γ

1

≥ (p∗ − ) + (r∗ − )1+ γ

Using the P¨otzsche chain rule [5, Theorem 1.90], we get  ∆ Z 1 −1 1 = γ dh γ+1 sγ 0 [s + hµ(s)] Z 1 γ  ≤ dh sγ+1 0 γ = γ+1 . s



t σ(t)

Taking the lim inf of both sides as t → ∞ we get that 1

r∗ ≥ p∗ −  + (r∗ − )1+ γ lγ .

(2.14)

(2.15)

Then from (2.14) and (2.15), we have 1 tγ wσ (t) ≥ (p∗ − ) + (r∗ − )1+ γ r(t)

∆s

γ .

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Since  > 0 is arbitrary, we get 1+ γ1 γ

p ∗ ≤ r∗ − r∗

l .

(2.16)

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 , lγ 2 (γ + 1)γ+1

which contradicts (2.8). Next, we assume (2.9) holds. Multiplying both sides of (2.10) tγ+1 by , and integrating from T to t (t ≥ T ) we get r(t) Z T

t

sγ+1 ∆ w (s)∆s ≤ − r(s)

Z T

t

sγ+1 P (s)∆s − γ r(s)

Z t T

sγ wσ (s) r(s)

 γ+1 γ ∆s.

Using integration by parts, we obtain Z t  γ+1 ∆ Z t γ+1 tγ+1 w(t) T γ+1 w(T ) s s σ ≤ + w (s)∆s − P (s)∆s r(t) r(T ) r(s) T T r(s) Z t  γ σ  γ+1 s w (s) γ −γ ∆s. r(s) T By the quotient rule and applying the P¨otzsche chain rule, 

sγ+1 r(s)

∆

(sγ+1 )∆ sγ+1 r∆ (s) − rσ (s) r(s)rσ (s) γ (γ + 1)σ (s) ≤ rσ (s) (γ + 1)σ γ (s) ≤ . r(s) =

(2.17)

Hence  γ  Z t γ+1 Z t tγ+1 w(t) T γ+1 w(T ) s σ (s)wσ (s) ≤ − P (s)∆s + (γ + 1) ∆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

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235

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 − , σ(s)

s ≥ T.

It follows that σ(s) ≤ Ks,

s≥T

where

K :=

1 . l−

We then get that Z t γ+1 tγ+1 w(t) T γ+1 w(T ) s ≤ − P (s)∆s r(t) r(T ) T r(s)  γ σ  γ+1 Z t γ σ s w (s) γ γ s w (s) + {(γ + 1)K −γ }∆s. r(s) r(s) T Let u(s) :=

sγ wσ (s) . r(s)

Then 

λ

u (s) = where λ =

sγ wσ (s) r(s)

λ ,

γ+1 . It follows that γ Z t γ+1 tγ+1 w(t) T γ+1 w(T ) s ≤ − P (s)∆s r(t) r(T ) T r(s) Z t + {(γ + 1)K γ u(s) − γuλ (s)}∆s. T

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 ) tγ+1 w(t) s ≤ − P (s)∆s r(t) r(T ) T r(s) Z t γγ [(γ + 1)K γ ]γ+1 + ∆s γ+1 γγ T (γ + 1) Z t γ+1 s T γ+1 w(T ) − P (s)∆s + K γ(γ+1) (t − T ). ≤ r(T ) r(s) T

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It follows from this that T γ+1 w(T ) r(T )

tγ w(t) ≤ r(t)

t

t

1 − t

Z

1 − t

Z

  T sγ+1 γ(γ+1) 1− P (s)∆s + K . r(s) t

T

Since wσ (t) ≤ w(t) we get tγ wσ (t) ≤ r(t)

T γ+1 w(T ) r(T )

t

t

  T sγ+1 γ(γ+1) 1− P (s)∆s + K . r(s) t

T

Taking the lim sup of both sides as t → ∞ we obtain R ≤ −q∗ + K γ(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 (2.16) we get 1+ γ1

p ∗ ≤ r∗ − l γ r∗

≤ r∗ ≤ R ≤ −q∗ +

Therefore p ∗ + q∗ ≤

1 lγ(γ+1)

1 lγ(γ+1)

.

,

which contradicts (2.9). Remark 2.4. We give an example which shows that the inequality (2.8) and hence the inequality (2.9) cannot be weakened. To see this let T = [1, ∞), r(t) = 1, and p(t) :=

1 γ γ+1 , γ+1 γ+1 (γ + 1) t

We have that

t ≥ 1.

∞

γγ , t→∞ (γ + 1)γ+1 t and the second-order half-linear differential equation γ
0 (x0 (t)) + p(t)xγ (t) = 0, γ

Z

p(s)ds =

p∗ = lim inf t

γγ is (γ + 1)γ+1 sharp for the oscillation for all solutions of this equation. Note in the case when γ = 1 1 this constant is . 4 γ

has a nonoscillatory solution x(t) = t γ+1 . This shows that the constant

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237

Theorem 2.5. Assume that (2.1) and (2.2) hold and  γ Z ∞ τ (s) tγ p(s) ∆s > 1. lim sup s t→∞ r(t) t

(2.18)

Then every solution of (1.1) is oscillatory on [t0 , ∞)T . Proof. Assume x is an eventually positive solution of (1.1) on [t0 , ∞)T . Using Lemma 2.1 there is a t1 ∈ [t0 , ∞)T such that x(t) > 0,

x∆ (t) > 0,

x(τ (t)) > 0,

x∆∆ (t) < 0,

x(t) > x∆ (t), t

x(t) on [t1 , ∞)T and is strictly decreasing on [t1 , ∞)T . Then integrating both sides of t the dynamic equation (1.1) from t to T , T ≥ t ≥ t1 , we obtain Z

T

p(s)xγ (τ (s))∆s = r(t)(x∆ (t))γ − r(T )(x∆ (T ))γ .

t

Since x∆ (t) > 0, we get that 1 r(t) Since

Z

T

p(s)xγ (τ (s))∆s ≤ (x∆ (t))γ .

t

x(t) x(t) is strictly decreasing and using x∆ (t) < we obtain t t  γ Z T τ (s) xγ (t) 1 p(s) xγ (s)∆s ≤ γ . r(t) t s t

Since x(t) is increasing we get tγ r(t)

Z

tγ r(t)

Z

T

 p(s)

t

which implies that

∞

τ (s) , s

 p(s)

t

τ (s) s

γ ∆s ≤ 1

γ ∆s ≤ 1,

which gives us the contradiction tγ lim sup t→∞ r(t) This concludes the proof.

Z

∞

 p(s)

t

τ (s) s

γ ∆s ≤ 1.

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In the following, we assume that Z

∞

∆t 1



(3.5)

or p∗ + q∗ > 1, where

∞

tγ p∗ = lim inf t→∞ r(t)

Z

1 q∗ = lim inf t→∞ t

Z



t

and

T

t

(3.6)

τ (s) s

γ p(s)ds,

sτ γ (s) p(s)ds. r(s)

Then every solution of equation (3.1) is oscillatory on [t0 , ∞). Theorem 3.3. Assume that (3.2) and (3.3) hold. Furthermore, assume  γ Z ∞ τ (s) tγ p(s) ds > 1. lim sup s t→∞ r(t) t

(3.7)

Then every solution of (3.1) is oscillatory on [t0 , ∞). Theorem 3.4. Assume that (3.2) and Z ∞

dt 1 γ



(3.12)

or p∗ + q∗ > 1, where

and

(3.13)

γ ∞  tγ X τ (s) p∗ = lim inf p(s), t→∞ r(t) s + 1 s=t+1 t

1 X sγ+1 q∗ = lim inf t→∞ t r(s) s=N



τ (s) s+1

γ p(s),

where N is sufficiently large. Then every solution of equation (3.8) is oscillatory on N. Theorem 3.7. Assume that (3.9) and (3.10) hold. Furthermore, assume that ∞

tγ X lim sup p(s) t→∞ r(t) s=t



Then every solution of (3.8) is oscillatory on N.

τ (s) s

γ >1

(3.14)

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241

Theorem 3.8. Assume that (3.9) and ∞ X t=t0

1 1 γ