Applied Mathematics Letters 26 (2013) 179–183

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

New results for oscillatory behavior of even-order half-linear delaydifferential equationsChenghui Zhang a,∗, Ravi P. Agarwal b, Martin Bohner c, Tongxing Li aa School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, PR Chinab Department of Mathematics, Texas A&M University–Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USAc Department of Mathematics and Statistics, Missouri S&T, Rolla, MO 65409-0020, USA

a r t i c l e i n f o

Article history:Received 7 August 2012Received in revised form 8 August 2012Accepted 8 August 2012

Keywords:Oscillatory solutionDelay differential equationHalf-linear differential equationEven order

a b s t r a c t

Some new oscillation results are presented that improve those reported in [C. Zhang, T. Li,B. Sun, and E. Thandapani, On the oscillation of higher-order half-linear delay differentialequations, Appl. Math. Lett. 24 (2011) 1618–1621].

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the natural sciences, technology, and population dynamics, differential equations find many application fields; see[1]. During the past few years, there has been increasing research activity concerning the oscillation of various classes ofdifferential equations. We refer the reader to [2–14] and the references cited therein. Zhang et al. [14] studied the higher-order differential equation

r(t)(x(n−1)(t))α′

+ q(t)xβ(τ (t)) = 0, (1.1)where(H) α, β are the ratios of odd positive integers,β ≤ α, r ∈ C1

[t0, ∞), r ′(t) ≥ 0, r(t) > 0, q, τ ∈ C[t0, ∞), q(t) ≥ 0, τ (t) <t , and limt→∞ τ(t) = ∞.

In the following, we describe one of the results obtained in [14] for the convenience of the reader.

Theorem 1.1 (See [14, Theorem 2.1]). Let n ≥ 2 and (H) hold. Assume that∞

t0

dtr1/α(t)

< ∞. (1.2)

Further, assume that the differential equation

y′(t) + q(t)

λ0τn−1(t)

(n − 1)!r1/α(τ (t))

β

yβ/α(τ (t)) = 0 (1.3)

∗ Corresponding author.E-mail addresses: zchui@sdu.edu.cn (C. Zhang), agarwal@tamuk.edu (R.P. Agarwal), bohner@mst.edu (M. Bohner), litongx2007@163.com (T. Li).

0893-9659/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.aml.2012.08.004

180 C. Zhang et al. / Applied Mathematics Letters 26 (2013) 179–183

is oscillatory for some constant λ0 ∈ (0, 1). If

lim supt→∞

t

t0

Mβ−αq(s)

λ1

(n − 2)!τ n−2(s)

β

δα(s) −αα+1

(α + 1)α+1

1δ(s)r1/α(s)

ds = ∞ (1.4)

holds for some constant λ1 ∈ (0, 1) and for every constant M > 0, where

δ(t) :=

∞

t

dsr1/α(s)

, (1.5)

then every solution of (1.1) is oscillatory or converges to zero as t → ∞.

We note that the conclusion of Theorem 1.1 is that (1.1) may have a non-oscillatory solution which satisfieslimt→∞ x(t) = 0. For instance, we consider the following equation:

(etx′′′(t))′ + λetx

t − arcsin

√1010

= 0 for λ > 0. (1.6)

By some straightforward computations, we have that (1.6) with λ := λ1 = e−arcsin

√1010

2 /16 has a non-oscillatory solution

x(t) = e−t2 ; (1.6) with λ := λ2 = 2

√10earcsin

√1010 has an oscillatory solution x(t) = et sin t . The natural question now is:

Can one obtain that all solutions of (1.6) with λ := λ2 = 2√10earcsin

√1010 are oscillatory? The purpose of this work is to give

an affirmative answer to this question.By a solution of (1.1) we mean a function x ∈ Cn−1

[Tx, ∞), Tx ≥ t0, which has the property r(x(n−1))α ∈ C1[Tx, ∞) and

satisfies (1.1) on [Tx, ∞). We consider only those solutions x of (1.1) which satisfy sup{|x(t)| : t ≥ T } > 0 for all T ≥ Tx. Weassume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [Tx, ∞);otherwise, it is called non-oscillatory. Eq. (1.1) is called oscillatory if all of its solutions are oscillatory.

In the sequel, all functional inequalities considered in this note are assumed to hold eventually, that is, they are satisfiedfor all t large enough.

2. Oscillation criteria

In this section, we will present some new oscillation criteria for (1.1). We begin with the following lemma.

Lemma 2.1 (See [12]). Let f ∈ Cn([t0, ∞), R+). If f (n)(t) is eventually of one sign for all large t, then there exist a tx ≥ t0 andan integer l, 0 ≤ l ≤ n, with n + l even for f (n)(t) ≥ 0, or n + l odd for f (n)(t) ≤ 0 such that

l > 0 yields f (k)(t) > 0 for t ≥ tx, k = 0, 1, . . . , l − 1, andl ≤ n − 1 yields (−1)l+kf (k)(t) > 0 for t ≥ tx, k = l, l + 1, . . . , n − 1.

Lemma 2.2 (See [4, Lemma 2.2.3]). If f is as in Lemma 2.1, f (n)(t)f (n−1)(t) ≤ 0 for t ≥ tx, and limt→∞ f (t) = 0, then for everyconstant λ ∈ (0, 1), there exists tλ ∈ [tx, ∞) such that

f (t) ≥λ

(n − 1)!tn−1

|f (n−1)(t)|

holds on [tλ, ∞).

Theorem 2.1. Let n ≥ 4 be even, (H) and (1.2) hold. Assume that differential equation (1.3) is oscillatory for some constantλ0 ∈ (0, 1). If (1.4) and

lim supt→∞

t

t0

Mβ−αq(s)Rα(s) −

αα+1

(α + 1)α+1

(R′(s))α+1

R(s)Rα∗(s)

ds = ∞ (2.1)

hold for some constant λ1 ∈ (0, 1) and for every constant M > 0, where δ is as in (1.5) and

R(t) :=

∞

t (η − t)n−3δ(η)dη(n − 3)!

, R∗(t) :=

∞

t (η − t)n−4δ(η) dη(n − 4)!

, (2.2)

then (1.1) is oscillatory.

C. Zhang et al. / Applied Mathematics Letters 26 (2013) 179–183 181

Proof. Assume that (1.1) has a non-oscillatory solution x. Without loss of generality we may assume that x is eventuallypositive. It follows from (1.1) and Lemma 2.1 that there exist three possible cases for t ≥ t1 large enough:

(1) x(t) > 0, x′(t) > 0, x(n−1)(t) > 0, x(n)(t) ≤ 0, (r(t)(x(n−1)(t))α)′ ≤ 0,(2) x(t) > 0, x′(t) > 0, x(n−2)(t) > 0, x(n−1)(t) < 0, (r(t)(x(n−1)(t))α)′ ≤ 0,

and

(3) x(t) > 0, x(j)(t) < 0, x(j+1)(t) > 0 for every odd integer j ∈ {1, 2, . . . , n − 3},

and x(n−1)(t) < 0, (r(t)(x(n−1)(t))α)′ ≤ 0.The proof of the case where (1) or (2) holds is the same as that of [14, Theorem 2.1]. Assume (3) holds. Noting that

r(x(n−1))α is non-increasing, we have

r1/α(s)x(n−1)(s) ≤ r1/α(t)x(n−1)(t), s ≥ t ≥ t1.

Dividing the above inequality by r1/α(s) and integrating the resulting inequality from t to l, we obtain

x(n−2)(l) ≤ x(n−2)(t) + r1/α(t)x(n−1)(t) l

tr−1/α(s) ds.

Letting l → ∞, we get

0 ≤ x(n−2)(t) + r1/α(t)x(n−1)(t)δ(t),

which yields

x(n−2)(t) ≥ −δ(t)r1/α(t)x(n−1)(t). (2.3)

Integrating (2.3) from t to ∞ provides

−x(n−3)(t) ≥ −r1/α(t)x(n−1)(t)

∞

tδ(s) ds.

Similarly, integrating the above inequality from t to ∞ a total of (n − 4) times, we find

− x′(t) ≥−r1/α(t)x(n−1)(t)

(n − 4)!

∞

t(η − t)n−4δ(η) dη. (2.4)

Integrating (2.4) from t to ∞ implies that

x(t) ≥−r1/α(t)x(n−1)(t)

(n − 3)!

∞

t(η − t)n−3δ(η) dη. (2.5)

Define the function w by

w(t) :=r(t)(x(n−1)(t))α

(x(t))α, t ≥ t1. (2.6)

Then w(t) < 0 for t ≥ t1. Differentiating (2.6), we have

w′(t) =(r(t)(x(n−1)(t))α)′

(x(t))α− α

r(t)(x(n−1)(t))αx′(t)(x(t))α+1

.

It follows from (1.1) and (2.4) that

w′(t) ≤ −q(t)xβ(τ (t))(x(t))α

− α

∞

t (η − t)n−4δ(η)dη(n − 4)!

w(α+1)/α(t).

Recalling x′ < 0 and τ(t) < t , then there exists a constantM > 0 such that

w′(t) ≤ −q(t)xα(τ (t))(x(t))α

xβ−α(τ (t)) − α

∞

t (η − t)n−4δ(η) dη(n − 4)!

w(α+1)/α(t)

≤ −Mβ−αq(t) − α

∞

t (η − t)n−4δ(η)dη(n − 4)!

w(α+1)/α(t).

182 C. Zhang et al. / Applied Mathematics Letters 26 (2013) 179–183

Multiplying the above inequality by Rα(t) and integrating the resulting inequality from t1 to t , we have

Rα(t)w(t) − Rα(t1)w(t1) − α

t

t1R′(s)Rα−1(s)w(s)ds

+

t

t1Mβ−αq(s)Rα(s)ds + α

t

t1w(α+1)/α(s)R∗(s)Rα(s)ds ≤ 0.

Set B := −R′(s)Rα−1(s), A := Rα(s)R∗(s), and v := −w(s). Using the inequality

Av(α+1)/α− Bv ≥ −

αα

(α + 1)α+1

Bα+1

Aα, A > 0,

we have t

t1

Mβ−αq(s)Rα(s) −

αα+1

(α + 1)α+1

(R′(s))α+1

R(s)Rα∗(s)

ds ≤ Rα(t1)w(t1) + 1

due to (2.5), which contradicts (2.1). Therefore, every solution of (1.1) is oscillatory. �

From [11, Theorem 2.1.1] and Theorem 2.1, we have the following.

Corollary 2.1. Let α = β, n ≥ 4 be even, and (H) and (1.2) hold. If

lim inft→∞

t

τ(t)q(s)

(τ n−1(s))α

r(τ (s))ds >

((n − 1)!)α

e, (2.7)

lim supt→∞

t

t0

q(s)Rα(s) −

αα+1

(α + 1)α+1

(R′(s))α+1

R(s)Rα∗(s)

ds = ∞, (2.8)

and

lim supt→∞

t

t0

q(s)

λ1

(n − 2)!τ n−2(s)

β

δα(s) −αα+1

(α + 1)α+1

1δ(s)r1/α(s)

ds = ∞ (2.9)

holds for some constant λ1 ∈ (0, 1), where δ, R, and R∗ are as in Theorem 2.1, then (1.1) is oscillatory.

As an application of Corollary 2.1, we consider (1.6) with λ = 2√10earcsin

√1010 . In this equation, we have

n = 4, α = β = 1, r(t) = et , q(t) = 2√10earcsin

√1010 +t , τ (t) = t − arcsin

√1010

.

Then, we obtain

δ(t) = e−t , R(t) = R∗(t) =

∞

t (η − t)n−3δ(η) dη(n − 3)!

= e−t .

One can easily see that all assumptions of Corollary 2.1 are satisfied. Hence (1.6) with λ = 2√10earcsin

√1010 is oscillatory. This

gives an answer to the question in Section 1.On the basis of [10, Theorem 2] and Theorem 2.1, we have the following.

Corollary 2.2. Let α > β, n ≥ 4 be even, and (H) and (1.2) hold. Assume that∞

t0q(t)

τ n−1(t)

r1/α(τ (t))

β

dt = ∞. (2.10)

If (1.4) and (2.1) hold for some constant λ1 ∈ (0, 1) and for every constant M > 0, where δ, R, and R∗ are as in Theorem 2.1,then (1.1) is oscillatory.

3. Conclusions

In this work, we suggest some new oscillation criteria for an even-order delay differential equation (1.1) with a non-canonical form (1.2). We stress that the study of oscillatory properties of Eq. (1.1) in the case (1.2) brings in additionaldifficulties. To ensure that all solutions of Eq. (1.1) oscillate, our results include three assumptions. The conclusion obtainedimproves that given in [14].

C. Zhang et al. / Applied Mathematics Letters 26 (2013) 179–183 183

Acknowledgments

The authors wish to express their sincere gratitude to Professor Alan Tucker for his valuable comments and suggestions.This research was supported by NNSF of PR China (Grant Nos. 61034007, 51277116, and 50977054).

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