# Oscillation and Asymptotic Behaviour of a Higher-Order Nonlinear Neutral-Type Functional Differential Equation with Oscillating Coefficients

Authors: Mustafa Kemal Yildiz; Emrah Karaman; Hülya Durur
Journal: Journal of Applied Mathematics
 (2011)

---

## Abstract

We will study oscillation of bounded solutions of higher-order nonlinear neutral delay differential equations of the following type:[y(t)+p(t)f(y(τ(t)))](n)+q(t)h(y(σ(t)))=0, t≥t0, t∈ℝ, where
p∈C([t0,∞),ℝ),  limt→∞p(t)=0, q∈C([t0,∞),ℝ+), τ(t), σ(t)∈C([t0,∞),ℝ), τ(t), σ(t)<t, limt→∞τ(t), σ(t)=∞, and
f,h∈C(ℝ,ℝ). We obtain sufficient conditions for the oscillation of all solutions of this equation.


---

## Body

## 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(1.1)[y(t)+p(t)f(y(τ(t)))](n)+q(t)h(y(σ(t)))=0,t≥t0,t∈R,
wherep∈C([t0,∞),ℝ) is oscillatory and limt→∞p(t)=0, q∈C([t0,∞),ℝ+), τ(t), σ(t)∈C([t0,∞),ℝ), τ(t), σ(t)<t, limt→∞τ(t)=∞, limt→∞σ(t)=∞, and f,h∈C(ℝ,ℝ). As it is customary, a solution y(t) is said to be oscillatory if y(t) is not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A differential equation 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.In [1, 2], several authors have investigated the linear delay differential equation
(1.2)x′(t)+q(t)x(σ(t))=0,t≥t0,
where q∈C([t0,∞),ℝ+) and σ(t)∈C([t0,∞),ℝ). A classical result is that every solution of (1.2) oscillates if
(1.3)lim⁡inf⁡t→∞∫σ(t)tq(s)ds>1e.In [3], Zein and Abu-Kaff have investigated the higher-order nonlinear delay differential equation
(1.4)[x(t)+p(t)x(τ(t))](n)+f(t,x(t),x(σ(t)))=s(t),t≥t0,t∈R,
where p∈C([t0,∞),ℝ), limt→∞p(t)=0, τ(t), σ(t)∈C([t0,∞),ℝ), τ(t), σ(t)<t, limt→∞τ(t)=∞, limt→∞σ(t)=∞, f:ℝ+×ℝ×ℝ→ℝ is continuous, yf(t,x,y)>0 for xy>0, there exists an oscillatory function r∈Cn(ℝ+,ℝ), such that r(n)(t)=s(t), limt→∞r(t)=0.In [4], Bolat and Akin have investigated the higher-order nonlinear differential equation
(1.5)[y(t)+p(t)y(τ(t))](n)+∑i=1mqi(t)fi(y(σi(t)))=s(t),
where p(t), qi(t), τ(t), s(t)∈C([t0,∞),ℝ) for i=1,…,m, p(t) and s(t) are oscillating functions, qi(t)≥0 for i=1,…,m, σi(t)∈C1([t0,∞),ℝ), σi′(t)>0, σi(t)≤t, limt→∞σi(t)=∞ for i=1,…,m, limt→∞τ(t)=∞, fi(u)∈C((ℝ,ℝ)) is nondecreasing function, uf(u)>0 for u≠0, and i=1,…,m. If n is odd, limt→∞p(t)=0, limt→∞r(t)=0, and ∫t0∞νn-1q(ν)dν=∞ 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→∞p(t)=0, and limt→∞r(t)=0, there exists a continuously differentiable function φ(t)(1.6)limsup⁡t→∞∫t0tφ(ν)∑i=1mqi(ν)dν=∞,limsup⁡t→∞∫t0t[[φ′(ν)]2φ(ν)σi′(ν)σin-2(ν)]dν<∞,
then every bounded solution of (1.5) 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 whenf(u)=I(u), where I is the identity function; see, for example, [1–15] and references cited there in.The purpose of this paper is to study oscillatory behaviour of solutions of (1.1). 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 functionz(t) is defined by
(1.7)z(t)=y(t)+p(t)f(y(τ(t))).


## 2. Some Auxiliary Lemmas

Lemma 2.1 (see [5]).
Lety be a positive and n-times differentiable function on [t0,+∞). If y(n)(t) is of constant sign and not identically zero in any interval [b,+∞), then there exist a t1≥t0 and an integer l, 0≤l≤n such that n+l is even, if y(n)(t) is nonnegative, or n+l odd, if y(n)(t) is nonpositive, and that, as t≥t1, if l>0, y(n)(t)>0 for k=0,1,2,…,l-1, and if l≤n-1, (-1)k+1y(n)(t)>0 for k=l,l+1,…,n-1.Lemma 2.2 (see [5]).
Lety(t) be as in Lemma 2.1. In addition limt→∞y(t)≠0 and y(n-1)(t)y(n)(t)≤0 for every t≥ty; then for every λ, 0<λ<1, the following hold:
(2.1)y(t)≥λ(n-1)!tn-1y(n-1)(t)for all larget.


## 3. Main Results

Theorem 3.1.
Assume thatn is even,(C1)
there exists a functionH:ℝ→ℝ such that H is continuous and nondecreasing and satisfies the inequality
(3.1)-H(-uv)≥H(uv)≥KH(u)H(v),foru,v>0,
where K is a positive constant, and
(3.2)|h(u)|≥|H(u)|,H(u)u≥γ>0,H(u)>0,foru≠0,(C2)
limt→∞p(t)=0,(C3)
∫t0∞sn-1q(s)ds=∞
and every solution of the first-order delay differential equation(3.3)w′(t)+q(t)KγH(12λ(n-1)!σn-1(t))w(σ(t))=0
is oscillatory. Then every bounded solution of (1.1) is either oscillates or tends to zero as t→∞.Proof.
Assume that (1.1) 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, y(t)>0, y(τ(t))>0, and y(σ(t))>0 for t≥t1≥t0. Further, suppose that y does not tend to zero as t→∞. By (1.1) and (1.7), we have
(3.4)z(n)(t)=-q(t)h(y(σ(t)))≤0,t≥t1.
It follows that z(α)(t)(α=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→∞p(t)f(y(τ(t)))=0. Then we can find a t2≥t1 such that z(t)=y(t)+p(t)f(y(τ(t)))>0 eventually and z(t) is also bounded for sufficiently large t≥t2. Because n is even and (n+l) odd for z(n)(t)≤0 and z(t)>0 is bounded, by Lemma 2.1, since l=1 (otherwise, z(t) is not bounded), there exists a t3≥t2 such that for t≥t3(3.5)(-1)k+1z(k)(t)>0(k=1,2,…,n-1).
In particular, since z′(t)>0 for t≥t3, z is increasing. Since y is bounded, limt→∞p(t)f(y(τ(t)))=0 by (C2). Then, there exists a t4≥t3 by (1.7),
(3.6)y(t)=z(t)-p(t)f(y(τ(t)))≥12z(t)>0,
for t≥t4. We may find a t5≥t4 such that for t≥t5, we have
(3.7)y(σ(t))≥12z(σ(t))>0.
From (3.4) and (3.7), we can obtain the result of
(3.8)z(n)(t)+q(t)h(12z(σ(t)))≤0,
for t≥t5. Since z is defined for t≥t2, and z(t)>0 with z(n)(t)≤0 for t≥t2 and not identically zero, applying directly Lemma 2.2 (second part, since z is positive and increasing), it follows from Lemma 2.2 that
(3.9)y(σ(t))≥12λ(n-1)!σn-1(t)y(n-1)(σ(t)).
Using (C1) and (3.7), we find for t≥t6≥t5,
(3.10)h(y(σ(t)))≥H(y(σ(t)))≥H(12λ(n-1)!σn-1(t)z(n-1)(σ(t)))≥KH(12λ(n-1)!σn-1(t))H(z(n-1)(σ(t)))≥KγH(12λ(n-1)!σn-1(t))z(n-1)(σ(t)).
It follows from (3.4) and the above inequality that z(n-1)(t) is an eventually positive solution of
(3.11)w′(t)+q(t)KγH(12λ(n-1)!σn-1(t))w(σ(t))≤0.
By a well-known result (see [14, Theorem  3.1]), the differential equation
(3.12)w′(t)+q(t)KγH(12λ(n-1)!σn-1(t))w(σ(t))=0,t≥t7≥t6
has an eventually positive solution. This contradicts the fact that (1.1) is oscillatory, and the proof is completed.Thus, from Theorem3.1 and [11, Theorem  2.3] (see also [11, Example  3.1]), we can obtain the following corollary.Corollary 3.2.
If(3.13)lim⁡inf⁡t→∞∫σ(t)tq(s)H(12λ(n-1)!σ(t)n-1)ds>1eKγ,
then every bounded solution of (1.1) is either oscillatory or tends to zero as t→∞.Theorem 3.3.
Assume thatn is odd and (C2), (C3) hold. Then, every bounded solution of (1.1) either oscillates or tends to zero as t→∞.Proof.
Assume that (1.1) 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, y(t)>0, y(τ(t))>0, and y(σ(t))>0 for t≥t1≥t0. Further, we assume that y(t) does not tend to zero as t→∞. By (1.1) and (1.7), we have for t≥t1(3.14)z(n)(t)=-q(t)h(y(σ(t)))≤0.
That is, z(n)(t)≤0. It follows that z(α)(t)(α=0,1,2,…,n-1) is strictly monotone and eventually of constant sign. Since limt→∞p(t)=0, there exists a t2≥t1, such that for t≥t2, we have z(t)>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, by Lemma 2.1, since l=0 (otherwise, z(t) is not bounded), there exists t4≥t3, such that for t≥t4, we have (-1)kz(k)(t)>0(k=1,2,…,n-1). In particular, since z′(t)<0 for t≥t4, z is decreasing. Since z is bounded, we may write limt→∞z(t)=L, (-∞<L<∞). Assume that 0≤L<∞. Let L>0. Then, there exist a constant c>0 and a t5 with t5≥t4, such that z(t)>c>0 for t≥t5. Since y is bounded, limt→∞p(t)f(y(τ(t)))=0 by (C1). Therefore, there exists a constant c1>0 and a t6 with t6≥t5, such that y(t)=z(t)-p(t)f(y(τ(t)))>c1>0 for t≥t6. So, we may find t7 with t7≥t6, such that y(σ(t))>c1>0 for t≥t7. From (3.14), we have
(3.15)z(n)(t)≤-q(t)h(c1)(t≥t7).
If we multiply (3.15) by tn-1 and integrate from t7 to t, then we obtain
(3.16)F(t)-F(t7)≤-h(c1)∫t7tq(s)sn-1ds,
where
(3.17)F(t)=∫γ=2t(-1)γtn-1z(n-γ-1)(t+γ)dt.
Since (-1)kz(k)(t)>0, for k=1,2,…,n-1 and t≥t4, we have F(t)>0 for t≥t7. From (3.16), we have
(3.18)-F(t7)≤-h(c1)∫t7tq(s)sn-1ds.
By (C3), we obtain
(3.19)-F(t7)≤-h(c1)∫t7tq(s)sn-1ds=-∞,
as t→∞. This is a contradiction. So, L>0 is impossible. Therefore, L=0 is the only possible case. That is, limt→∞z(t)=0. Since y is bounded, by virtue of (C2) and (1.7), we obtain
(3.20)lim⁡t→∞y(t)=lim⁡t→∞z(t)-lim⁡t→∞p(t)f(y(τ(t)))=0.
Now, let us consider the case of y(t)<0 for t≥t1. By (1.1) and (1.7),
(3.21)z(n)(t)=-q(t)h(y(σ(t)))≥0(t≥t1).
That is, z(n)(t)≥0. It follow that z(α)(t)(α=0,1,2,…,n-1) is strictly monotone and eventually of constant sign. Since limt→∞p(t)=0, there exists a t2≥t1, such that for t≥t2, we have z(t)<0. Since y(t) is bounded, by virtue of (C2) and (1.7), there is a t3≥t2 such that z(t) is also bounded, for t≥t3. Assume that x(t)=-z(t). Then, x(n)(t)=-z(n)(t). Therefore, x(t)>0 and x(n)(t)≤0 for t≥t3. From this, we observe that x(t) is bounded. Because n is odd and x is bounded, by Lemma 2.1, since l=0 (otherwise, x is not bounded), there exists a t4≥t3, such that (-1)kx(k)(t)>0 for k=1,2,…,n-1 and t≥t4. That is, (-1)kz(k)(t)<0 for k=1,2,…,n-1 and t≥t4. In particular, for t≥t4, we have z′(t)>0. Therefore, z(t) is increasing. So, we can assume that limt→∞z(t)=L,(-∞<L≤0). As in the proof of y(t)>0, we may prove that L=0. As for the rest, it is similar to the case y(t)>0. That is, limt→∞y(t)=0. This contradicts our assumption. Hence, the proof is completed.Example 3.4.
We consider difference equation of the form(3.22)[y(t)+1tsin⁡(t)(y3(t-2)+y(t-2))](4)+1t2y3(t-3)=0,
where n=4, τ(t)=t-2, p(t)=(1/t)sin(t), q(t)=1/t2, σ(t)=t-3, h(y)=y3, and f(y)=y3+y. By taking H(u)=u,
(3.23)lim⁡inf⁡t→∞∫t-3t1s21213!(s-323)3ds>1e,
we check that all the conditions of Theorem 3.1 are satisfied and that every bounded solution of (3.22) oscillates or tends to zero at infinity.Example 3.5.
We consider difference equation of the form(3.24)[y(t)+cos⁡te-5t2[y5(t-5)+2y(t-5)]](3)+t2y2(t-3)=0,t≥2,
where n=3, q(t)=t2, σ(t)=t-3, τ(t)=t-5, and p(t)=coste-5t2, f(y)=y5-2y, h(y)=y2. Hence, we have
(3.25)lim⁡t→∞p(t)=lim⁡t→∞1e5t2cos⁡⁡t=0,∫t0∞sn-1q(s)ds=∫t0∞s4ds=∞.
Since Conditions (C2) and (C3) of Theorem 3.3 are satisfied, every bounded solution of (3.24) oscillates or tends to zero at infinity.



---

*Source: 158060-2011-06-29.xml*