J. Appl. Math. & Computing Vol. 21(2006), No. 1 - 2, pp. 99 - 118 Website: http://jamc.net

OSCILLATION OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

ELMETWALLY M. ELABBASY, TAHER S. HASSAN* AND SAMIR H. SAKER

ABSTRACT. In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equations. Several new sufficient conditions which ensure that all solutions are oscillatory are given. The obtained results extend and improve several known results in the literature. Some examples are considered to illustrate the main results.

AMS Mathematics Subject Classification: 34K15, 34C10. Key words and phrases : Oscillation, nonoscillation, neutral delay of differential equations.

1. Introduction

In this paper, we shall consider the first order nonlinear neutral delay differential equation

(x(t) - q(t)x(t - a))' + f (t, x(t(t))) =0, t > to,

q,T e C ([t0, œ), R+), a G (0, to) t (t) < t, lim t (t) = œ, (2)

q(tj )

œ as n

i=ij=i

f G C([t0, œ) x R, R), uf(t, u) > 0, q(t) = 1.

In connection with the nonlinear function f (t,u) in (1) we suppose that the following assumption (H) holds :

Received October 10, 2005. Revised January 16, 2006. * Corresponding author. © 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

(H) There are a piecewise continuous functions

p : [t0, x) ^ R+ = [0, x), g e C (R, R+)

and a number e0 > 0 such that

(i) g is nondecreasing on R+

(ii) g (-u) = g (u), Km g (u) = °

(iii) Jg (e~u) du < x, 0

(iv) j^j- |/(t, u) — p (t) u\ < p (t) g (u) for t > to and 0 < \u\ < £o?

(v) For each tp e C ([t0, x), R) with lim (t) > 0,

Jf (t, p (t (t))) dt = x, Jf (t, -p (t (t))) dt = -x.

A solution x (t) of equation (1) is said to be oscillatory if it has arbitrarily large zeros on [t0, x). Otherwise it is nonoscillatory and the equation (1) is called oscillatory if every solution of this equation is oscillatory. When q (t) = 0, Eq.(1) reduces to the equation

xX (t) + f (t,x (t (t))) =0,

which has been studied recently by Tang and Shen [32]. The oscillatory behavior of various other neutral delay differential equations have been investigated by many authors. For contributions we refer the reader to the papers, [1 — 3, 6 — 12, 14 — 27, 29 — 36], and references cited therein. Also, in a recent papers Elabbasy and Saker [6], Kubiaczyk and Saker [17] and their references obtained an infinite integral conditions for oscillation of the linear neutral delay differential equation

(x (t) — q (t) x (t — a))' + p (t) x (t — t) = 0,

Our aim in this paper is to extend the results in [7,18] and provide some new finite and infinite integral sufficient conditions for the oscillation of all solutions of (1). Some examples that are illustrating our main results are given. The following notations will be used throughout this paper, 5 (t) = max {t (t) : t0 < s < t} and ¿-1 (t) = min (s > t0 : 6 (s) = t} . Clearly, 5 and 5-1 are nondecreasing and satisfy (A) 5(t) <t and 6-1 (t) >t, (b) 6(6-1 (t)) = t and 6-1 (5 (t)) < t. Let 5-k (t) be defined on [t0, x) by

Also, we use the sequence {pk} of functions defined as follows:

S-1(t)

pi (t) = J p (s) ds, t > t0,

<s-1(t)

pk+1 (t) = J p (s)pk (s) ds, t > t0, k =1, 2,... t

In what follows, when we write a functional inequality we will mean that it holds for all sufficiently large values of t.

2. Main results

To prove our main results we shall need the following Lemmas.

Lemma 1. Assume that (2), (3) and (4) hold, let x (t) be an eventually positive solution of (1) and set

z (t) = x (t) - q (t) x (t - a) . (6)

Then z (t) is eventually nonincreasing positive function.

Proof. From (1), (4), we have z (t) = — f (t, x (t (t))) < 0 eventually. We prove that z (t) is a positive function. If not, then there exist T > t0 and a < 0 such that z (t) <a for t > T. Then from (6), we have

x (t) <a + q (t) x (t — a),

which implies that

x (t + a) <a + q (t + a) x (t) . Now we choose k such that tk = t* + ka > T. Then x (tk+1) <a + q (tk+1) x (tk). Applying this inequality by induction it gives

x (tn ) < a

Now define qn and dn by

i+ E n q (tn-j ) i=k+2j=0

+ n q (ti) x (tk).

and let

n n-i n

qn = 1 + E n q (tn-j), dn = n q(ti), i=k+2j=0 i=k+1

I fc+1 i 1

s: = -j- = s« - П ) q '"q ^ 00 as n 00'

n \ i=1 j=1 q ( j )

by condition (3). Using the last inequality, one can see that

x (tk)

■(tn) <

adn ^ —сю as n ^ <x,

and this contradicts the assumption that x (t) > 0. Then z (t) must be positive function. The proof is complete. □

Note that the proof of Lemma 1 is similar to that of Lemma 1 in [4] and we state it here for the sake of completeness.

Lemma 2. Assume that (2), (3), (4) and (H) hold, let x (t) be an eventually positive solution of 1. Then x (t) and z (t) are convergent to zero monotonically as t ^ x.

Proof. By Lemma 1 z (t) is a nonincreasing positive function and satisfy the equation

z' (t) = -f (t,x (t (t))) .

Choose a ti > t0 such that x (t) > 0, z (t) > 0 for t > t\.

It follows from equations (2), (4) and (H) that there exists t2 > t1 such that t (t) > t1 and z (t) < 0 for t> t2. Hence

lim x (t) > lim z (t) = a > 0 exists.

t—t—

If a > 0, then from (1), we have

z (t) - z (to) = -J f (t,x(T (s))) ds.

It follows from the assumption (H) (v) that lim z (t) = —x, which contradicts

that z(t) being positive. Then lim z (t) = 0. Since q (t) = 1, we have also

lim x (t) = 0. Then the proof is complete. □

Lemma 3. Assume that (2), (3), (4) and (H) hold. If x (t) is a nonoscillatory solution of equation (1), there exist A > 0, e > 0 and T G (0, to) such that for t>T

\x(t)\<Aexpi-^Jp(s)ds\+e. (7)

Proof. We shall assume x (t) to be eventually positive [if x(t) is eventually negative the proof is similar]. By Lemma 2, there exists ti > 0 such that

0 <x^r (t)^ <£o for t > t1.

From (H), we find that for t > t1

f (t, x(t (t))) > p (t)[1 - g (x (t (t)))] x (t (t)),

and lim x (t) = 0. By assumption (H), there exists T > ti such that for t > T t—

f (t, x(t (t))) > \p (t) x (r (t)) > \p (t) x (t) ,

and it follows from (1) that for t > T

(x (t) — q(t)x(t — a))' + ip (t) x (t) < 0,

z {t)+l-p{t)z{t)< 0,

where z (t) = x (t) - q (t) x (t - a) . This yields for t > T

z (t) < A exp

Jp(s)ds

\x (t)| < A exp Jp (s) ds^ + e, where A = x (T) — q (T) x (T — a). The proof is complete. □

Lemma 4. Assume that (2), (3), (4) and (H) hold. If equation (1) has a nonoscillatory solution, then eventually t

Jp (s) ds < 2 and pk (t) < 2k, k =1, 2,... (8)

Proof. Let us suppose that x(t) is a nonoscillatory solution of equation (1) which we shall assume to be eventually positive [if x(t) is eventually negative the proof is similar]. By Lemma 2, there exists T > 0 such that

x (t (t)) > x (t) > 0 for t > T,

(x (t) -q(t)x(t- r)j + (t) x (r (t)) < 0,

Z (t) + ^p(t)z(T(t)) <0 for t >T. (9)

Integrating both sides from t (t) to t yields that

z(t)~ z (r (t)) + i Jp (s) z (r (s)) ds < 0 for t > T.

By the decreasing nature of z (t) for large t and the increasing nature of t (t), there exists T1 > T such that

Then, we have

z(t) - z (t (t)) + ^z (t (t)) Jp(s)ds< 0 for t >T\.

j„ w ds < 2

Also, Integrating both sides of equation (9) from t to 5-1 (t) yields

s-1(t)

z (S-1 (t)) - z (t) + ^ J p (s) z (t (s)) ds < 0 for t >T. t

By the decreasing nature of z (t) for large t and the increasing nature of t (t), there exists T1 > T such that

/s-1(t) \

J p (s) ds 1 z (t (5-1 (t))) < 0 for t > T1.

z(5-i(t))-z(t)+l-

z(5-i(t))-z(t) + ^

/s-1(t)

J p (s) ds I z (t) < 0 for t > T1.

Then, we have

s-1(t)

pi (t)= J p (s) ds < 2. t

By iteration we deduce from this that

pk (t) < 2k,

which shows that (8) holds for t > Ti. The proof of Lemma 4 is complete. □

Lemma 5. Assume that (2), (3), (4) and (H) hold, and that

lim inf / p(s)ds > 0. (10)

Z (T (t))

If x (t) is a nonosdilatory solution of equation (1), then-——, which is well

defined for large t, is bounded.

Proof. Let us suppose that x(t) is a nonoscillatory solution of equation (1) which we shall assume to be eventually positive [if x(t) is eventually negative the proof is similar]. By the same argument as in the proof of Lemma 3, there exists T > 0, such that

x (t (t)) > x (t) > 0 for t > T, (x (t) -q(t)x(t- a))' + (t) x (r (t)) < 0,

-2' W + \p{t)z{r{t)) < 0 for t >T.

The rest of the proof is similar to that of Lemma 5 in [24] respectively, and hence is omitted. □

Theorem 1. In addition to the assumptions (2), (3), (4) and (H) assume that

lim inf / P(s)ds > -, (11)

t—W J e

lim sup / P (s) ds > 1, (12)

where P (t) = (1 — e) p (t) . Then every solution of Eq. (1) oscillates.

Proof. Let us suppose that x(t) is a nonoscillatory solution of equation (1) which we shall assume to be eventually positive [if x(t) is eventually negative the proof is similar]. By Lemma 2, there exists ti > 0 such that

0 <x(t (t)) <e0 for t > t1.

From (H), we find that for t > t1

f (t, x(t (t))) > p (t)[1 - g (x (t (t)))] x (t (t)),

and lim x (t) = 0. By assumption (H), there exists T > ti such that for t > T t—

f (t,x(T (t))) > (1 - e)p (t) x (t (t)), and it follows from (1) that for t > T

(x (t) - q (t) x (t - a))' + (1 - e) p (t) x (t (t)) < 0,

z' (t)+P (t) z (t (t)) < 0. (13)

But, then by Corollary 3.2.2 [11] the delay differential equation

z' (t) + P (t) z (t (t))=0, (14)

has an eventually positive solution as well. It is also well known that (11) or (12) implies (14) has no eventually positive solution (see, [11] Theorem 3.4.3). This contradiction completes the proof. □

Remark 1. It is clear that every solution of (1) oscillates if (14) has no eventually positive solutions.

It is clear that there is a gap between (11) and (12) for the oscillation of all solutions of (1). The problem how to fill this gap for the equation ( 1) when the limit

lim / P (s) ds, J

does not exist needs to be considered. This problem has been cleared for the linear Eq. (14). Let the numbers k and l be defined by

k = lim inf / P(s)ds, l = lim sup / P(s)ds, .

t—J t—J

T(t) T(t)

0 < k < / < 1,

and A is the smallest root of the equation A = ekx. Then Eq. (14) will be oscillatory if either of the following conditions is satisfied:

(Ci) (C2) (C3) (C4) (C5) (Ce)

ln A +1

l> 1 -

1 — k — a/1 -2 k-k2 2 '

1 + ln A 1 - k - a/1-2 k-k2

/ > 2k + - - 1, A

ln A - 1 + a/5 - 2A + 2k\ Â '

e - 1 1

[18] [34] [14] [19] [30] [7]

Remark 2. Theorem 1 implies that Eq. (1) will also be oscillatory if either of the conditions (C1) — (Ce) is satisfied.

In the following theorems we present new infinite integral sufficient conditions for the oscillation of all solutions of (1)

Theorem 2. Assume that (2), (3), (4), (10) and (H) hold, and suppose that there exists a positive integer n such that

Jp (t)ln(pn (t) + 1) dt = to.

Then every solution of (1) oscillates.

Proof. Assume that (1) has a nonoscillatory solution x (t) which will be assumed to be eventually positive (if x (t) is eventually negative the proof is similar). By Lemma 2 and assumption (H), there exists tj > t0 such that

0 <x(t) < x (S (t)) < x (t (t)) <£q, g (x (t (t))) < 1, t > t*0, (16)

where e0 is given by assumption (H). (16) and (H) yield that for t > tj,

f (t, x (t (t))) > p (t)[1 - g (x (t (t)))] x (t (t)) > p (t)[1 - g (x (t (t)))] z (S (t)) ,

and it follows from (1) that

JW+p (t) i7(¡T[1"9 {x (T m] - t -(18)

By Lemmas 1 - 5, there exist T > t2, A > 0, e > 0 and M > 0 such that for t > T,

x (t (t)) < A exp

/ T(t) 1

p (s) ds

jp (s) ds < Jp (s) ds < 2, pk (t) < 2k, k =1, 2,...,

z(S(tj) <Z(r(t)) <M

Let tk = 5 k (T), k =1, 2, ...Clearly tk ^Mas k ^m. Set

A = t >T.

■■mi

= exp j X (s) ds, t > t1,

and from (18) we have for t > t1

A (t) > p (t) exp J A (s) ds — p (t) g (x (r (t)))

It follows from (19) - (22) that for t > t1,

z(t) '

A (t) > p (t) exp J X(s)ds - Mp(t) g Aexp -^Jp(s)ds

( T(t)

> p (t) exp J X (s) ds - Mp (t) g i Ai exp i -1Jp (s) ds J + e J ,

s(t) V V T ) ) (23)

where A1 = eA. By the inequality ec > c for c > 0, we have for t > t1

A (t) >p (t) ^ A (s) ds - Mp (t) g exp \ (s) ds j + e j •

:(t) = yP(s)ds, t>T,

( Ao (t) = A (t),

t > T,

Ak (t) = p (t) J Ak-i (s) ds, t > tk,k = 1, 2,...,n,

( Go (t) = 0,

t > T,

Gk (t)=p (t) J Gk-i (s) ds

m (27)

+ Mp(t) g (Ai exp (—a (t)) + e), t > tk, k = 1, 2,...,n,

Clearly (10) implies that a (t) is nondecreasing on [T, to) and a (t) ^ to as t —> to. By iteration we deduce from (24) that

A (t) > Ak (t) — Gk (t), t > tk, k = 1, 2, ...n — 1,

and so by (23)

^ (t) > p (t) exp J \n-l (s) ds

From (27), one can easily obtain Gk+i (t) — Gk (t)

— I Gn-i (s) ds

\ s(t)

— Gi (t), t > tn.

p (t) j [Gk (s) — Gk-i (s)] ds, t > tk+i ,k =1, 2,...,n — 1. (30)

By (20), (25) and (27), for t > t2 we have

J Gi (s) ds = M J p (s) g (Ai exp (—a (s)) + e) ds

= 2M J g (Aie-u + e) du

x(S(t))

< 2M J g (A1e-u + e) du.

a(t) —1

Thus, from (30), we get

G2 (t) - G1 (t) = P (t) j G1 (s) ds

< 2Mp (t) J g (A1e-u + e) du, t > t2,

a(t)-1

G3 (t) - G2 (t) = p (t) j [G2 (s) - G1 (s)] ds

t a(s)

< 2Mp (t) JP (s) J g (A1e-u + e) duds

S(t) a(s)-1 a(t) v

= 4Mp (t) J j g (A1 e-u + e) dudv

a(S(t))v-1 a(t) v

< 4Mp (t) J jg (A1e-u + e) dudv

a(t)-1v-1 a(t)

< 4Mp (t) J g (A1e-u + e) du, t > t3.

a(t)-2

By induction, one can prove in general that for k = 2, 3, ...,n - 1,

Gk (t) - Gk-1 (t) ^k-1

(t) - Gfc-1 (t) < (2)fc-1 (k - 2)!Mp (t) J g (A1e-u + e) du, t > tk,

a(t)-(k-1)

and so

n— 1

Gn-1 (t) = ]T[Gk (t) - Gk—1 (t)]

< G1 (t) + Mp (t)J2 (2e)k—1 (k - 2)! (32)

a(t)-(k-1)

By (20), (21) and (26), we obtain

J g (Aie-u + e) du, t > tn-i.

■■(Ht))

Ai (t) = p (t) J A (s) ds = p (t)ln

< p (t)ln m, t > ti,

A2 (t)=p (t) J Ai (s) ds < p (t)lnM j p (s) ds .

S(t) S(t)

< 2p (t)ln M, t > t2,

An-i (t) < 2n-2p (t)ln M, t > tn-i.

D (t) = p (t)exp

+Gi (t), t > tn.

t t j An-i (s) ds 1 — exp ^y Gn-i (s) ds

One can easily see that

0 < 1 — e-c < c, c > 0.

From, (20), (32), (33), (33) and (34) we have / t \ t

D (t) < p (t) exp

V(t) ) S(t)

< Gi (t)+p (t) exp

s(t) < Gi

J An-i (s) ds I J Gn-i (s) ds + Gi (t) / s(t)

2n-2 ln M J p (s) ds

/n-1 „

Gi (s)+Mp(s)^(2)k-i (k — 2)! / g(Aie-u + e) du

a(s)-(k-1)

) J g (A1 e-u + e) du

1 ln M

,0-i t x^ (2)k-1 (k - 2)! jp (s) J g (A1e-M + e) duds (35)

k=2 S(t) a(s)-(k-1)

n-1 a(tf)

< G1 (t)+M1p (t)^(2)k-1 (k - 1)! / g(A1e-u + e)du, t > tn,

1__1 J

where M1 = 2Mexp ((2)n 1 ln m) . Let T* > tn be such that a (T*) > n + ln A1. It follows from (36) and (H) that

Jd (t) dt < JG1 (t) dt + M1^ (2)k-1 (k - 1)\Jp (t) J g (A1e-u) dudt

T* T* k=1 T* a(t)-k

< 2M J g (A1 e-u) du

a (T *)

n-1 0 v

+2M1Y, (2)k-1 (k - 1)! / / g (A1e-^ dud«

k=1 a(T*)v-k

0 n-1 0

< 2M g (A1 e-u) du + 2M^ (2)k-1 k! g (A^-") du

^ 7-_1 ^

*(T *)

*(T *)-(k+1)

Jd (t) dt < 2M Jg (e-u) du + 2M1 ^ (2)k-1 kjg (e-u) du < m.

T"* n k=1 n

p (t) exp

V(t) 11

= p (t) exp

J Xn-1 (s) ds I exp - J Gn-1 (s) ds I - G1 (t) I / \ <5(t)

I Xn-1 (s) ds I - D (t), t > tn,

it follows from (29) that

A (t) > p (t) exp

J An-i (s) ds

— D (t), t > tr,

A (t) > p (t) exp

pn (t)

■pn (t) J An-1 (s) ds

— D (t), t > tn.

ln (y +1)

One can easily show that e"/x > x H--for all x > 0 and 7 > 0, and so

for t > tn,

pn (t) A (t) — p (t) j An-i (s) ds > p (t)ln(pn (t) + 1) — pn (t) D (t).

For N > S-n (T*), we have

Jpn (t) A (t) dt — jp (t) j An-i (s) dsdt T * T * S(t)

> jp (t)ln(pn (t) + 1) dt — jpn (t) D (t) dt.

S1 (t) = S (t), Sk+1 (t) = S(Sk (t)) , k = 1, 2,,...,n. Then by interchanging the order of integration, we have

S-1(t)

T* S(t)

Jp (t) J An-i (s) dsdt > J An-i (t) j p (s) dsdt

S{N ) t

= J p (t) pi (t) j An-2 (s) dsdt T* S(t)

S2(N) 5-1(t)

> J An-2 (t) J p (s) pi (s) dsdt

s2(N) t

p (t) p2 (t)/ Xn-3 (s) dsdt

t- S(t)

> j X (t) pn (t) dt.

From this and (39) we have

J pn (t) X (t) dt > jp (t)ln(pn (t) + 1) dt - Jpn (t) D (t) dt, (40)

which together with (20) yields

2n J X (t) dt > jp (t)ln(pn (t) + 1) dt - 2n jD (t) dt,

gn(N) T* T*

x (5n (N))

In view of (15) and (36) , we have

> 2-n Jp (t)ln(pn (t) + 1) dt - jD (t) dt. (41)

x (5n (N)) , N

m v ,KnT'J = oo. 42

On the other hand, (21) implies that

x (5n (N)) x (51 (N)) x (52 (N)) x (5n (N))

x (N) x (N) 'x (51 (N'x (5n-1 (N))

This contradicts (42) and completes the proof. □

Remark 3. From Lemma 3 we have

lim inf pk (t) < (2)k 1 lim inf / p (s) ds < (2)k 1 lim inf / p (s) ds.

t—>o t—>o J t—>o J

t T(t)

As a result, by Theorem 2 we have

Corollary 1. Assume that (2) , (3), (4), and (H) hold, and that there exists a positive integer n such that

lim inf pn (t) > 0.

t—>o

Then every solution of (1) oscillates.

Example 1. Consider the neutral delay differential equation

x(t) - Q + sint^ x(t - tt)^ + f(t,x(r(t))) =0, t > 3, (43) where

t (t) = t — 1 and / (t, u) = [exp 3 (sint — 1) + |w|]3 u.

p (t) = exp (sint) — 0.1 and g (u) = e \u\3 . It is easy to see that assumption (H) holds. Clearly

lim inf / p(s)ds < -, t—œ J e

t-1 oo

Jp (t)ln(pi (t) + 1) dt

oo /t+1 \

>j exp(sin t - 1)ln [J exp(sin s) ds) dt

By Jensen's inequality,

A+1 \ oo t+1

Jp (t)ln [ Jp (s) ds +1 ) dt > J exp (sin t — 1) J sin sdsdt

2sin2-1 f ( 1\ - / exp (smt) sin I i — 1 dt.

On the other hand, it is easy to see that

J exp (sin s)cos sds

is bounded and

exp (sin t) sin tdt > 0.

p (t)ln(p1 (t) + 1) dt = x.

By Theorem 2 every solution of (43) oscillates.

Note that most of the results that has been given in [3,11] required the condition that 0 < q (t) < 1, which cannot be applied on (43) since q (t) = (f +sint) > 1.

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Elmetwally M. Elabbasy received his BS (1973) from Mansoura University and MSC (1977) from Ain Shams University Egypt. He got his Ph.D. (1980) from Wales, UK. His subject interest is oscillation theory of differential and difference equations and their applications and also the nonlinear dynamical systems. Now he is teaching in Department of Mathematics, Mansoura University.

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

e-mail: emelabbasy@mans.edu.eg

Taher S. Hassan received his BS (1997) and MSC (2003) from Mansoura University, Egypt. His subject interest is oscillation theory of functional differential equations. Now he is teaching in Department of Mathematics, Mansoura University.

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

e-mail: tshassan@mans.edu.eg

Samir H. Saker received his BS (1993) and MSC (1997) from Mansoura University, Egypt and he got his Ph. D. from Adam Mickiewiz University, Poland (2002). His research interest focus on the oscillation theory of differential, difference equations and their applications and oscillation of dynamic equations on time scales which unify the oscillation of differential and difference equations. Now he is teaching in Department of Department of Mathematics, Mansoura University.

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

e-mail: shsaker@mans.edu.eg