Scholarly article on topic 'Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order'

Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order Academic research paper on "Mathematics"

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Academic research paper on topic "Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order"

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 281581, 5 pages http://dx.doi.org/10.1155/2013/281581

Research Article

Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

Fatima N. Ahmed, Rokiah Rozita Ahmad,

Ummul Khair Salma Din, and Mohd Salmi Md Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Fatima N. Ahmed; zahra80zahra@yahoo.com Received 10 May 2013; Revised 11 August 2013; Accepted 12 August 2013 Academic Editor: Aref Jeribi

Copyright © 2013 Fatima N. Ahmed 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.

Some new sufficient conditions for oscillation of all solutions of the first-order linear neutral delay differential equations are obtained. Our new results improve many well-known results in the literature. Some examples are inserted to illustrate our results.

1. Introduction

A neutral delay differential equation (NDDE) is a differential equation in which the highest-order derivative of the unknown function is evaluated both at the present state at time t and at the past state at time t - k for some positive constant k.

In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of neutral delay differential equations. Particularly, we mention the papers by Ladas and Sficas [1], Chuanxi and Ladas [2], Ruan [3], Elabbasy and Saker [4], Kulenovic et al. [5], and Karpuz and Ocalan [6] who investigated NDDEs with variable coefficients. To a large extent, this is due to its theoretical interest as well as to its importance in applications. It suffices to note that NDDEs appear in the study of networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) in population dynamics and also in many applications in epidemics and infection diseases. We refer reader to [1-18] for relevant studies on this subject.

In this paper, we consider the linear first-order NDDE of the type

(x (t) - px(t - t))' + qlx (t)

+ q2 (t) x(t-a) = 0; t > t0,

where p,q1,T,a e (0,œ) and q2(t) e C[[i0, œ), R]. When q1 = 0 and q2(t) = q,q is a constant, Jaros [9] established some new oscillation conditions for all solutions of (1), and his technique was based on the study of the characteristic equation

A - Xpe Xr + qe la = 0.

Zhang [19], Ladas and Sficas [1], Grammatikopoulos et al. [10], and Yu et al. [8] considered (1) when q1 = 0, and they obtained some sufficient conditions for oscillation of (1). The purpose of this work is to present some new sufficient conditions under which all solutions of (1) are oscillatory. In order to achieve this object, we are first concerned with NDDE (1) with constant coefficients (when q2(t) = q2, q2 is a constant). That is,

(x (t) - px(t - t))' + q1x (t) + q2x (t-a) = 0, t >0.

Some illustrating examples are given. In some sense, the established results extend and improve some previous investigations such as [1, 8-10,19].

As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is eventually positive or eventually negative. A function x(t) is called eventually positive (or negative) if there exists t0 such

that x(t) > 0 (or x(t) < 0) for all t > t0. Equation (1) is called oscillatory if all its solutions are oscillatory; otherwise, it is called nonoscillatory.

2. Main Results

In this section, we give some new sufficient conditions for the oscillation of all solutions of (1) and (3). This is done by using the following well-known lemmas which are from [11,12].

Lemma 1. Consider the NDDE

(x(t)+px(t-T))' + Y%x(t-ai) = 0, t>t0, (4)

where t > 0,qi > 0, and ai > 0 for all i = l,2,...,n. Let x(t) be a positive solution of (4). Set

That is,

z(t) = x (t) - px(t-r).

If p > -1, then z(t) is a positive and decreasing solution of (4); that is,

z (t) + pz' (t-T) + Y%z(t-°'i) = 0, t>t0. (6) i=i

Lemma 2. Let p and r be positive constants. Let x(t) be an eventually positive solution of the delay differential inequality

x (t) +px(t-r)< 0. Then for t sufficiently large,

x(t-r)<Bx (t),

b = —.

Our main results can now be given as follows. Theorem 3. Consider NDDE (3). Assume that

(i) q2 e [0, m), a>T, 0 < peqiT < 1 and

(ii) r(qlpeqiX + q2eqi°) > (1 - peqirm)2/m, where m is the unique real root of the equation

1 - peqiXm = ln m, 1 <m <

Then all solutions of (3) are oscillatory.

Proof. Assume, for the sake of a contradiction, that (3) has a nonoscillatory solution x(t). Without loss of generality, assume that x(t) > 0 Vt>t0 > 0. Let

x (t) = e-qity (t). So that y(t) is also a positive solution of (3).

(y (t) - Piy (t - t))' + X aiy (t - ri) = 0, (12)

Pi = peqi\ ai = qipeqi\ a2 = q2eqi°,

t, = t, tt = a.

Set for t>tn + 2t

z(t) = y(t)-piy(t-r),

z(t-T) (14)

w(t) =

Thus it follows from Lemma 1 that z(t) is a positive and decreasing solution of

z (t) - p1z' (t-T) + Y aiz (t - Ti) = 0, (15)

and in particular (as a > t implies that t-Ti <t-r, i = 1,2), it follows that

z (t) - p1z' (t-r) + (a1 + a2) z(t-r) < 0. (16)

But we have

z (t-r)< 0. (17)

This implies that

z (t) + (a1 +a2)z(t-r) <0. (18)

Applying Lemma 2 with (18) we get

z(t-r)<Bz(t); B=——4--j. (19)

T2(al + a2)

Then w(t) is bounded.

Dividing (16) by z(t) > 0 and integrating from t -t to t, we get

ln w (t) > (a1 + a2) I w (s) Jt-r

(t d p, I w(s) — (ln z(s-r))ds. Jt-r ds

Let m = limt^TO inf w(t).

Then, it follows from (20) that for e > 0 and sufficiently small,

ln (m + e) > (a, + a2) (m - e) t + p, (m- e) ln (m- e).

As e is arbitrary, so we have

(a, + a2) t <

(1 - p1m) ln m

(1 - p,m) lnm F(m) = ---, 1<m<B.

■I 1 - Pim - lni

-, 1<m<B.

F (m) =

Let m be the unique real root of the equation 1 - p1m = ln m, me

(1-Pim)2

max F (m) = F (m) = -—

(qipe^ + q2e'ia)r<

(1-peqitmf

This contradicts condition (ii) and then completes the proof.

Example 4. Consider the NDDE

x(t)-9x(t-^)) + 1x(t)

+ x\t- 5- ) = 0, t>0.

We note that

P=9, h = 9, I2 = 1, t=2>

5n (29)

Then we have

(i) 0 < peqiT = (1/9)en/18 <1 and a > t,

r(qipeqiT + q2en = 2(l/118 + \<?"1%)

(1-pe*irm)2 (1-(1/9V/i8)2

where m = 2 is the unique real root of the equation

1 - 1en/18m = lnm, m e [l,9e-"/18]. (31)

Then all the hypotheses of Theorem 3 are satisfied, and therefore every solution of (28) oscillates. (Indeed x(t) = sin t is such a solution.)

Theorem 5. Consider the NDDE (1). Assume that

(iii) 0 < peqiT < 1, a = t, and q2(t) e C[[io>œ), (0, œ)] is periodic with period t,

(iv) limt^œ inf Î_re^ir(q2(s) + qip)ds > (1 - pe^mf /m,

where m is defined as in Theorem 3. Then all solutions of (1) are oscillatory.

Proof. Assume, for the sake of contradiction, that (1) has a nonoscillatory solution x(t). Without loss of generality, assume that x(t) > 0 Vt>t0 > 0. Let

x (t) = e-qity (t),

which is oscillation invariant transformation. Then y(t) is a positive solution of the equation

(y (t) - pi y (t - t))' + q(t)y(t-r) = 0, (33)

where p1 = peqiT and q(t) = eqiT(q2(t) + q1p) is periodic with period r. Let

z(t) = y(t)-p1y(t-T). (34)

Then z(t) is decreasing positive solution of the equation

(z (t) - P1Z (t - t))' + q(t)z(t-T) = 0. (35)

z(t-T)

w(t) =

This implies that w(t) > 1, since z(t -t) > z(t).

Dividing both sides of (33) by z(t) and then integrating from t - t to t,we obtain that

ln w (t)

g (s) (y (s-t)- piy (s - 2t) + piy (s - 2t)) y(s)-piy(s-r)

ds. (37)

Hence ln w(t)

q (s) w (s) ds + pi I

it-r Jt-

q(s)y(s-2r) ]t-r\y(s)- piy(s-r)

Since q(t) is periodic with period r, then we obtain (y(t-T)-p1y(t-2r))'

q(t) = q(t-r) = —

y(t-2r)

ds. (38)

Substituting in (38) we find, for all t > t0,

Now, let

ln w(t) = I q (s) w (s)

-pi \ w(s)d ln (y (s - t) - pxy (s - 2t))

ds. (40)

Now, we want to prove that w(t) is bounded.

Applying the assumption (iv), we can find t* e (t - t, t) such that

f q(s)

, f q(s)ds>^, (41) 2 it- 2

where F(m) is similar as in the proof of Theorem 3. Integrating (33) from t* to t we obtain

y(t*)-piy(t* -*)> \ q(s)y(s-r)ds if

> \ q(s)(y(s-r)-p1y(s-2r))ds, if

Using Bonnet's Theorem and in particular (as z'(t - t) < o), we get

y(t*)-Piy(t* -r)

>[y(t-T)- p1y(t-2r)]- \ q(s)ds.

Integrating (33) from t - t to t* ,we get y(t-r)-ply(t-2r)

>\ 4 (s) y(s-r)ds (44)

> \ q(s)(y(s-T)-p1y(s-2r))ds. Jt-T

Using Bonnet's Theorem and in particular (as z'(t - t) < o), we get

y(t-T)-p1y(t-2r)

t' (45)

>[y(t* -r)-p1y(t* -2t)]-\ q(s)ds.

Combining (43) and (45), we conclude

y(t*)-Piy(t* -r)

/Fffi)\2 (46)

>(y(t*-r)-Ply(t*-2T))(yp),

w(?) =

y(f -r)-Ply(t* - 2t) 4

y(t*)-Ply(t* -t) (F(m))2' Then w(t) is bounded.

m = lim inf w (t).

But we have proved that w(t) is bounded; that is, m is finite. From (40), we obtain

lnm>plmlnm+ lim infm \ q(s)ds. (49)

Therefore, we get

C^ 1 _ -ty ^^

lim inf m \ q(s)ds< -—— ln m. (50)

Jt-T m

it (1 - p1m)

lim inf I q(s)ds<±-= '

^™ Jt-T m

This contradicts our assumption (iv) and then completes the proof. □

Example 6. Consider the NDDE

(x(t)-9x(t-n)) +i^8x(t)

+ (1 + cos 2t)x(t-n) = 0, t>0,

(43) where

P=7-,> 4i =

T = o = n,

9 " 18

q2 (t) = 1 + cos 2t.

Then we have

(1) 0 < peqiT = en/18/9 < 1;

(2) q2(t) = 1 + cos 2t e C[[0, ot), (0, ot)] is periodic with period n and satisfies

lim inf I eqiT (q2 (s)+q1p)ds

= lim inf I en/18 ((1 + cos 2s) +——) t^™ Jt-n V 162 J

= en/18 lim inf ( s +—-—+ 1 sin 2s 162 2

= ot >

(1-peqiTm)2 (1-(2/9)enl18)2

where m = 2 is the unique real root of the equation

1 1 k/18 T , „ „ n -n/18

1--e m = ln m; 1<m<9e .

Therefore (52) satisfies all the hypotheses of Theorem 5. Hence every solution of this equation is oscillatory.

Theorem 7. Suppose that condition (iii) holds. If

(v) limt^œ inf jt[r e^^s) + q1p]ds > (1 - pe^T)/e, then every solution of (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 5, we get (49) which implies that

\nm>p1 lnm + m lim inf I q(s)ds. (56) t^rn Jt_r

1 - P 1 - P lim inf I q (s) ds < -— ln m < -—.

Jt-T m e

But this is a contradiction of assumption (v), and then the proof is complete. □

Example 8. Consider the NDDE

x(t)-hx{t-î)),+22x(t)

+ (e + sin 4t)x(t-^2) = 0, t>0.

Here we have

t = a =

q2 (t) = e + sin At.

Note that q2(t) = e + sin 4t is positive and periodic with period n/2, and also

(1) 0 < peqiT = 1/5 <1,

lim inf I eq'T [q2 (s) + qip] ds

t^rn J—

im inf I

(e + sin At) +

ds = œ, (60)

(1-pe^) A

Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.

Funding

This research has been completed with the support of these Grants: ukm-DLP-2011-049, DIP-2012-31, and FRGS/1/ 2012/SG04/ukm/01/1.

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