Oscillation and asymptotic behaviour of a higher-order nonlinear neutral-type functional differential equation with oscillating coefficients

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Volume 2011, Article ID 158060,8pages doi:10.1155/2011/158060

Research Article

Oscillation and Asymptotic Behaviour of

a Higher-Order Nonlinear Neutral-Type Functional

Differential Equation with Oscillating Coefficients

Mustafa Kemal Yildiz,

1

_{Emrah Karaman,}

2

_{and H ¨ulya Durur}

3

1_{Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus,} 03200 Afyon, Turkey

2_{Department of Mathematics, Faculty of Science and Arts, Karab ¨uk University, 78050 Karab ¨uk, Turkey} 3_{Department of Technical Programs, Vocational High School of Ardahan, Ardahan University,}

75000 Ardahan, Turkey

Correspondence should be addressed to Mustafa Kemal Yildiz,myildiz@aku.edu.tr

Received 6 August 2010; Accepted 16 April 2011 Academic Editor: J. C. Butcher

Copyrightq 2011 Mustafa Kemal Yildiz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We will study oscillation of bounded solutions of higher-order nonlinear neutral delay diﬀerential equations of the following type:yt ptfyτtn qthyσt 0, t ≥ t0, t ∈R, where

p ∈ Ct0, ∞,R, limt → ∞pt 0, q ∈ Ct0, ∞,R, τt, σt ∈ Ct0, ∞,R, τt, σt < t, limt → ∞τt, σt ∞, and f, h ∈ CR, R. We obtain suﬃcient conditions for the oscillation of all

solutions of this equation.

1. Introduction

In this paper, we are concerned with the oscillation of the solutions of a certain more general higher-order nonlinear neutral-type functional diﬀerential equation with an oscillating coeﬃcient of the form

yt ptfyτtn qthyσt 0, t ≥ t0, t ∈ R, 1.1

where p ∈ Ct0, ∞, R is oscillatory and limt → ∞pt 0, q ∈ Ct0, ∞, R, τt, σt ∈

Ct0, ∞, R, τt, σt < t, limt → ∞τt ∞, limt → ∞σt ∞, and f, h ∈ CR, R. As it is

customary, a solution yt is said to be oscillatory if yt is not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A diﬀerential equation

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is called oscillatory if all of its solutions oscillate. Otherwise, it is nonoscillatory. In this paper, we restrict our attention to real-valued solutions y.

In1,2, several authors have investigated the linear delay diﬀerential equation

xt qtxσt 0, t ≥ t0, 1.2

where q ∈ Ct0, ∞, R and σt ∈ Ct0, ∞, R. A classical result is that every solution of

1.2 oscillates if lim inf t → ∞ _t σt qsds > 1 e. 1.3

In 3, Zein and Abu-Kaﬀ have investigated the higher-order nonlinear delay

diﬀerential equation

xt ptxτtn ft, xt, xσt st, t ≥ t0, t ∈ R, 1.4

where p ∈ Ct0, ∞, R, limt → ∞pt 0, τt, σt ∈ Ct0, ∞, R, τt, σt < t, limt → ∞τt

∞, limt → ∞σt ∞, f : R× R × R → R is continuous, yft, x, y > 0 for xy > 0, there exists

an oscillatory function r ∈ Cn_R

, R, such that rnt st, limt → ∞rt 0.

In 4, Bolat and Akin have investigated the higher-order nonlinear diﬀerential

equation yt ptyτtn m i1 qitfi yσit  st, 1.5

where pt, qit, τt, st ∈ Ct0, ∞, R for i 1, . . . , m, pt and st are oscillating

functions, qit ≥ 0 for i 1, . . . , m, σit ∈ C1t0, ∞, R, σ_it > 0, σit ≤ t, limt → ∞σit ∞

for i 1, . . . , m, limt → ∞τt ∞, fiu ∈ CR, R is nondecreasing function, ufu > 0 for

u / 0, and i 1, . . . , m. If n is odd, limt → ∞pt 0, limt → ∞rt 0, and

_∞

t0 νn−1qνdν ∞

for i 1, . . . , m, then every bounded solution of 1.5 is either oscillatory or tends to zero

as t → ∞. If n is even, limt → ∞pt 0, and limt → ∞rt 0, there exists a continuously

diﬀerentiable function ϕt lim sup t → ∞ t t0 ϕν m i1 qiνdν ∞, lim sup t → ∞ t t0 ϕν2 ϕνσ_iνσ_in−2ν dν < ∞, 1.6

then every bounded solution of1.5 is either oscillatory or tends to zero as t → ∞.

Recently, many studies have been made on the oscillatory and asymptotic behaviour of solutions of higher-order neutral-type functional diﬀerential equations. Most of the known results which were studied are the cases when fu Iu, where I is the identity function; see, for example,1–15 and references cited there in.

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The purpose of this paper is to study oscillatory behaviour of solutions of1.1. For

the general theory of diﬀerential equations, one can refer to 5,6,12–14. Many references to

some applications of the diﬀerential equations can be found in 2.

In this paper, the function zt is defined by

zt yt ptfyτt. 1.7

2. Some Auxiliary Lemmas

Lemma 2.1 see 5. Let y be a positive and n-times diﬀerentiable function on t0, ∞. If ynt

is of constant sign and not identically zero in any intervalb, ∞, then there exist a t1 ≥ t0and an

integer l, 0 ≤ l ≤ n such that nl is even, if ynt is nonnegative, or nl odd, if ynt is nonpositive, and that, as t ≥ t1, if l > 0, ynt > 0 for k 0, 1, 2, . . . , l − 1, and if l ≤ n − 1, −1k1ynt > 0 for

k l, l 1, . . . , n − 1.

Lemma 2.2 see 5. Let yt be as in Lemma 2.1. In addition limt → ∞yt / 0 and

yn−1tynt ≤ 0 for every t ≥ ty; then for every λ, 0 < λ < 1, the following hold:

yt ≥ λ

n − 1!tn−1yn−1t for all large t. 2.1

3. Main Results

Theorem 3.1. Assume that n is even,

C1 there exists a function H : R → R such that H is continuous and nondecreasing and

satisfies the inequality

−H−uv ≥ Huv ≥ KHuHv, for u, v > 0, 3.1

where K is a positive constant, and

|hu| ≥ |Hu|, Hu

u ≥ γ > 0, Hu > 0, for u / 0, 3.2

C2 limt → ∞pt 0,

C3

_∞

t0 sn−1qsds ∞

and every solution of the first-order delay diﬀerential equation wt qtKγH 1 2 λ n − 1!σn−1t wσt 0 3.3

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Proof. Assume that1.1 has a bounded nonoscillatory solution y. Without loss of generality,

assume that y is eventually positive the proof is similar when y is eventually negative. That is, yt > 0, yτt > 0, and yσt > 0 for t ≥ t1 ≥ t0. Further, suppose that y does not tend

to zero as t → ∞. By 1.1 and 1.7, we have

znt −qthyσt≤ 0, t ≥ t1. 3.4

It follows that zαtα 0, 1, 2, . . . , n − 1 is strictly monotone and eventually of constant sign. Since y is bounded and does not tend to zero as t → ∞, by virtue of C2,

limt → ∞ptfyτt 0. Then we can find a t2≥ t1such that zt yt ptfyτt > 0

eventually and zt is also bounded for suﬃciently large t ≥ t2. Because n is even and n l

odd for znt ≤ 0 and zt > 0 is bounded, byLemma 2.1, since l 1 otherwise, zt is not bounded, there exists a t3≥ t2such that for t ≥ t3

−1k1_zk_{t > 0 k 1, 2, . . . , n − 1.} _3.5

In particular, since zt > 0 for t ≥ t3, z is increasing. Since y is bounded,

limt → ∞ptfyτt 0 by C2. Then, there exists a t4 ≥ t3by1.7,

yt zt − ptfyτt≥ 1

2zt > 0, 3.6

for t ≥ t4. We may find a t5≥ t4such that for t ≥ t5, we have

yσt ≥ 1

2zσt > 0. 3.7

From3.4 and 3.7, we can obtain the result of

znt qth 1 2zσt ≤ 0, 3.8

for t ≥ t5. Since z is defined for t ≥ t2, and zt > 0 with znt ≤ 0 for t ≥ t2and not identically

zero, applying directlyLemma 2.2second part, since z is positive and increasing, it follows

fromLemma 2.2that

yσt ≥ 1

2

λ

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UsingC1 and 3.7, we find for t ≥ t6≥ t5, hyσt≥ Hyσt ≥ H 1 2 λ n − 1!σn−1tzn−1σt ≥ KH 1 2 λ n − 1!σn−1t H zn−1σt ≥ KγH 1 2 λ n − 1!σn−1t zn−1σt. 3.10

It follows from3.4 and the above inequality that zn−1t is an eventually positive solution

of wt qtKγH 1 2 λ n − 1!σn−1t wσt ≤ 0. 3.11

By a well-known resultsee 14, Theorem 3.1, the diﬀerential equation

wt qtKγH 1 2 λ n − 1!σn−1t wσt 0, t ≥ t7≥ t6 3.12

has an eventually positive solution. This contradicts the fact that1.1 is oscillatory, and the

proof is completed.

Thus, fromTheorem 3.1and 11, Theorem 2.3 see also 11, Example 3.1, we can

obtain the following corollary.

Corollary 3.2. If lim inf t → ∞ t σtqsH 1 2 λ n − 1!σtn−1 ds > 1 eKγ, 3.13

then every bounded solution of1.1 is either oscillatory or tends to zero as t → ∞.

Theorem 3.3. Assume that n is odd and (C2), (C3) hold. Then, every bounded solution of1.1 either

oscillates or tends to zero as t → ∞.

Proof. Assume that1.1 has a bounded nonoscillatory solution y. Without loss of generality,

assume that y is eventually positive the proof is similar when y is eventually negative. That is, yt > 0, yτt > 0, and yσt > 0 for t ≥ t1 ≥ t0. Further, we assume that yt does not

tend to zero as t → ∞. By 1.1 and 1.7, we have for t ≥ t1

znt −qthyσt≤ 0. 3.14

That is, znt ≤ 0. It follows that zαt α 0, 1, 2, . . . , n − 1 is strictly monotone and eventually of constant sign. Since limt → ∞pt 0, there exists a t2 ≥ t1, such that for t ≥ t2,

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we have zt > 0. Since y is bounded, by virtue of C2 and 1.7, there is a t3 ≥ t2 such

that z is also bounded, for t ≥ t3. Because n is odd and z is bounded, byLemma 2.1, since

l 0 otherwise, zt is not bounded, there exists t4 ≥ t3, such that for t ≥ t4, we have

−1k_zk_{t > 0 k 1, 2, . . . , n − 1. In particular, since z}_{t < 0 for t ≥ t}

4, z is decreasing.

Since z is bounded, we may write limt → ∞zt L, −∞ < L < ∞. Assume that 0 ≤ L < ∞.

Let L > 0. Then, there exist a constant c > 0 and a t5 with t5 ≥ t4, such that zt > c > 0 for

t ≥ t5. Since y is bounded, limt → ∞ptfyτt 0 by C1. Therefore, there exists a constant

c1 > 0 and a t6 with t6 ≥ t5, such that yt zt − ptfyτt > c1 > 0 for t ≥ t6. So, we

may find t7with t7≥ t6, such that yσt > c1> 0 for t ≥ t7. From3.14, we have

znt ≤ −qthc1 t ≥ t7. 3.15

If we multiply3.15 by tn−1_{and integrate from t}

7to t, then we obtain Ft − Ft7 ≤ −hc1 _t t7 qssn−1_ds, _3.16 where Ft  t γ2 −1γ tn−1zn−γ−1t γdt. 3.17

Since−1kzkt > 0, for k 1, 2, . . . , n − 1 and t ≥ t4, we have Ft > 0 for t ≥ t7. From3.16,

we have −Ft7 ≤ −hc1 t t7 qssn−1ds. 3.18 ByC3, we obtain −Ft7 ≤ −hc1 t t7 qssn−1ds −∞, 3.19

as t → ∞. This is a contradiction. So, L > 0 is impossible. Therefore, L 0 is the only possible case. That is, limt → ∞zt 0. Since y is bounded, by virtue of C2 and 1.7, we obtain

lim

t → ∞yt limt → ∞zt − limt → ∞ptf

yτt 0. 3.20

Now, let us consider the case of yt < 0 for t ≥ t1. By1.1 and 1.7,

znt −qthyσt≥ 0 t ≥ t1. 3.21

That is, znt ≥ 0. It follow that zαt α 0, 1, 2, . . . , n − 1 is strictly monotone and eventually of constant sign. Since limt → ∞pt 0, there exists a t2 ≥ t1, such that for t ≥ t2,

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we have zt < 0. Since yt is bounded, by virtue of C2 and 1.7, there is a t3≥ t2such that

zt is also bounded, for t ≥ t3. Assume that xt −zt. Then, xnt −znt. Therefore,

xt > 0 and xnt ≤ 0 for t ≥ t3. From this, we observe that xt is bounded. Because n is

odd and x is bounded, byLemma 2.1, since l 0 otherwise, x is not bounded, there exists a t4 ≥ t3, such that−1kxkt > 0 for k 1, 2, . . . , n − 1 and t ≥ t4. That is,−1kzkt < 0

for k 1, 2, . . . , n − 1 and t ≥ t4. In particular, for t ≥ t4, we have zt > 0. Therefore, zt

is increasing. So, we can assume that limt → ∞zt L, −∞ < L ≤ 0. As in the proof of

yt > 0, we may prove that L 0. As for the rest, it is similar to the case yt > 0. That is,

limt → ∞yt 0. This contradicts our assumption. Hence, the proof is completed.

Example 3.4. We consider diﬀerence equation of the form

yt 1 tsint y3t − 2 yt − 2 4 1 t2y 3_{t − 3 0,} _3.22

where n 4, τt t−2, pt 1/t sint, qt 1/t2_{, σt t−3, hy y}3_{, and fy y}3_y.

By taking Hu u, lim inf t → ∞ t t−3 1 s2 1 2 1 3! s − 3 23 3 ds > 1 e, 3.23

we check that all the conditions ofTheorem 3.1are satisfied and that every bounded solution of3.22 oscillates or tends to zero at infinity.

Example 3.5. We consider diﬀerence equation of the form

yt cos te−5t2y5t − 5 2yt − 53 t2y2t − 3 0, t ≥ 2, 3.24

where n 3, qt t2_{, σt t − 3, τt t − 5, and pt cos te}−5t2

, fy y5_{− 2y, hy y}2_.

Hence, we have lim t → ∞pt limt → ∞ 1 e5t2cos t 0, _∞ t0 sn−1qsds  _∞ t0 s4ds ∞. 3.25

Since ConditionsC2 and C3 ofTheorem 3.3are satisfied, every bounded solution of3.24

oscillates or tends to zero at infinity.

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