# Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

Authors: J. Džurina; B. Baculíková
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The aim of this paper is to offer a generalization of the Philos and
Staikos lemma. As a possible application of the lemma in the oscillation
theory, we study the asymptotic properties and oscillation of thenth order
delay differential equations (E)(r(t)[x(n−1)(t)]γ)′+q(t)xγ(τ(t))=0. The results obtained utilize also the comparison theorems.


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## Body

## 1. Introduction

In this paper, we will study the asymptotic and oscillation behavior of the solutions of the higher-order advanced differential equations:(E)(r(t)[x(n-1)(t)]γ)′+q(t)xγ(τ(t))=0.Throughout the paper, we will assumeq,τ,r∈C([t0,∞)), and  (H1)
n≥3, γ is the ratio of two positive odd integers,(H2)
r(t)>0, q(t)>0, τ(t)≤t, limt→∞τ(t)=∞.Whenever, it is assumed(1.1)R(t)=∫t0t‍r-1/γ(s)ds⟶∞ast⟶∞.By a solution of(E) we mean a function x(t)∈Cn-1([Tx,∞)),  Tx≥t0, which has the property r(t)(x(n-1)(t))γ∈C1([Tx,∞)) and satisfies  (E) on [Tx,∞). We consider only those solutions x(t) of (E) which satisfy sup{|x(t)|:t≥T}>0 for all T≥Tx. We assume that (E) possesses such a solution. A solution of (E) is called oscillatory if it has arbitrarily large zeros on [Tx,∞), and otherwise it is called to be nonoscillatory.The problem of the oscillation of higher-order differential equations has been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for studied equations (see, e.g., [1–19]).Philos and Staikos lemma (see [16, 17]) essentially simplifies the examination of nth-order differential equations of the form
(1.2)y(n)(t)+q(t)yγ(τ(t))=0,
since it provides needed relationship between y(t) and y(n-1)(t), and this fact permit us to establish just one condition for asymptotic behavior of (1.2). If we try to apply the Philos and Staikos lemma to (E), the strong condition r′(t)≥0 appears (see, e.g., [2, 18, 20]). In this paper we offer such generalization of the Philos and Staikos lemma, where this restriction is relaxed. Moreover, the obtained lemma yields many applications in the oscillation theory. As an example of it, we offer its disposal in the comparison theory and we establish new oscillation criteria for (E).


## 2. Main Results

The following result is a well-known lemma of Kiguradze, see, for example, [6] or [13].Lemma 2.1.
Letz(t)∈Ck-1([t0,∞)) and r(t)(z(k-1)(t))γ∈C1([t0,∞)) with z(t)>0, (r(t)(z(k-1)(t))γ)′≤0, and not identically zero on a subray of [t0,∞). Then there exist a t1≥t0 and an integer ℓ, 0≤ℓ≤k-1, with k+ℓ odd so that
(2.1)(-1)l+jz(j)(t)>0,j=l,…,k-1,z(i)(t)>0,i=1,…,l-1,whenl>1,
on [t1,∞).Now we are prepared to provide a generalization of the Philos and Staikos lemma.Lemma 2.2.
Letz(t) be as in Lemma 2.1 and numbers t1 and ℓ assigned to z(t) by Lemma 2.1. Then for 0<ℓ<k-1,
(2.2)z(t)≥r1/γ(t)z(k-1)(t)(k-2)!∫t1t‍r-1/γ(s)(s-t1)k-2ds,
for ℓ=k-1;
(2.3)z(t)≥r1/γ(t)z(k-1)(t)(k-2)!∫t1t‍r-1/γ(s)(t-s)k-2ds,
for t≥t1.Proof.
Letℓ be the integer assigned to function z(t) as in Lemma 2.1. Assume that ℓ<k-1, then, for any s,t with t≥s≥t1,
(2.4)-z(k-2)(s)≥∫st‍z(k-1)(u)du=∫st‍r1/γ(u)z(k-1)(u)r-1/γ(u)du≥r1/γ(t)z(k-1)(t)∫st‍r-1/γ(u)du.
Repeated integration in s from s to t yields
(2.5)z(l)(s)≥r1/γ(t)z(k-1)(t)∫st‍r-1/γ(u)(u-s)k-2-l(k-2-l)!du.
On the other hand, if ℓ≥1, then, for every t≥t1,
(2.6)z(l-1)(t)≥∫t1t‍z(l)(s)ds.
Repeated integration from t1 to t leads to
(2.7)z(t)≥1(l-1)!∫t1t‍z(l)(s)(t-s)l-1ds.
Setting (2.5) into (2.7), one gets
(2.8)z(t)≥r1/γ(t)z(k-1)(t)(l-1)!(k-l-2)!∫t1t‍(t-s)l-1∫st‍r-1/γ(u)(u-s)k-2-lduds≥r1/γ(t)z(k-1)(t)(k-3)!∫t1t‍∫st‍r-1/γ(u)(u-s)k-3duds=r1/γ(t)z(k-1)(t)(k-2)!∫t1t‍r-1/γ(u)(u-t1)k-2ds.
We have verified the first part of the lemma. Now assume that ℓ=k-1. It follows from (2.7) that
(2.9)z(t)≥1(k-2)!∫t1t‍z(k-1)(s)(t-s)k-2du≥1(k-2)!∫t1t‍r1/γ(s)z(k-1)(s)r-1/γ(s)(t-s)k-2ds=r1/γ(t)z(k-1)(t)(k-2)!∫t1t‍r-1/γ(s)(t-s)k-2ds.
The proof is complete now.Employing additional conditions, we are able to joint (2.5) and (2.7) to just one estimate.Lemma 2.3.
Letz(t) be as in Lemma 2.1 and limt→∞z(t)≠0. Let r′(t)≥0,  Then for any λ∈(0,1) there exists some tλ≥t1 such that
(2.10)z(t)≥λ(k-1)!tk-1z(k-1)(t),
for t≥tλ.Proof.
Note thatr′(t)≥0 implies that r-1/γ(t) is nonincreasing. Assume that ℓ is the integer associated with z(t) in Lemma 2.1. If 0<ℓ<k-1, then using (2.2), we have
(2.11)z(t)≥z(k-1)(t)(k-2)!∫t1t‍(s-t1)k-2ds=z(k-1)(t)(k-1)!(t-t1)k-1.
It is easy to see that for any λ∈(0,1) there exists a tλ≥t1 such that t-t1≥λ1/(k-1)t for t≥tλ, which in view of (2.11) yields (2.10).
Ifℓ=k-1, then proceeding similarly as above it can be shown that (2.3) implies (2.10).
Ifℓ=0, then it follows from (2.5) that
(2.12)z(s)≥r1/γ(t)z(k-1)(t)∫st‍r-1/γ(u)(u-s)k-2(k-2)!du=z(k-1)(t)(k-1)!(t-s)k-1.
Setting s=(1-λ1/2(k-1))t, we have
(2.13)x((1-λ1/2(k-1))t)≥z(k-1)(t)(k-1)!λ1/2tk-1.
Moreover,
(2.14)lim⁡t→∞x(t)x((1-λ1/2(k-1))t)=1>λ1/2.
Therefore, for all large t,
(2.15)x(t)≥λ1/2x((1-λ1/2(k-1))t)≥λ(k-1)!tk-1z(k-1)(t).
The proof is complete now.In the following result, we compare (2.2) and (2.3) to verify that both estimates are independent; that is, one does not result from the other, although for r(t)≡1 they are equivalent.Remark 2.4.
Letz(t) be as in Lemma 2.1, such that z(t) is increasing. At first we consider r(t)=t3, γ=3, t1=1, and k=3 then (2.2), (2.3), and (2.10) yield
(2.16)z(t)≥z′′(t)(t2-tln⁡⁡t-t),z(t)≥z′′(t)(t2ln⁡⁡t-t2+t),z(t)≥z′′(t)λt22,
respectively.
Now we modifyr(t)=t-3. Then (2.2) and (2.3) reduce to
(2.17)z(t)≥z′′(t)(t23-t2+16t),z(t)≥z′′(t)(t26-12+13t),
respectively, and (2.10) is not applicable.


## 3. Applications

Lemma2.2 can be applied in various techniques for investigations of the higher-order differential equations. We offer one such application in comparison theory.Theorem 3.1.
Assume that both first-order delay differential equations(E1)y′(t)+q(t)((n-2)!)γ(∫t1τ(t)‍r-1/γ(s)(s-t1)n-2ds)γy(τ(t))=0,(E2)y′(t)+q(t)((n-2)!)γ(∫t1τ(t)‍r-1/γ(s)(τ(t)-s)n-2ds)γy(τ(t))=0
are oscillatory. Moreover, for n-odd assume that
(P0)∫t0∞‍un-2r-1/γ(u)(∫u∞‍q(s)ds)1/γdu=∞.
Then  (i)
forn even, (E) is oscillatory;(ii)
forn odd, every nonoscillatory solution x(t) of (E) satisfies limt→∞x(t)=0.Proof.
Assume thatx(t) is a nonoscillatory solution of (E); let us say positive. Then (r(t)[x(n-1)(t)]γ)′<0, and there exist a t1≥t0 and an integer ℓ with n+ℓ odd such that (2.1) holds.
If0<ℓ<n-1, then by Lemma 2.2(3.1)x(t)≥r1/γ(t)x(n-1)(t)(n-2)!∫t1t‍r-1/γ(s)(s-t1)n-2ds.Setting to (E), we get
(3.2)(r(t)[x(n-1)(t)]γ)′+q(t)r(τ(t))[x(n-1)(τ(t))]γ((n-2)!)γ×(∫t1τ(t)‍r-1/γ(s)(τ(s)-t1)n-2ds)γ≤0.
Then y(t)=r(t)[x(n-1)(t)]γ is positive and satisfies the differential inequality:
(3.3)y′(t)+q(t)((n-2)!)γ(∫t1τ(t)‍r-1/γ(s)(s-t1)n-2ds)γy(τ(t))≤0.
By Theorem 1 in [15], the corresponding equation (E1) has also a positive solution. A contradiction.
Ifℓ=n-1, then by Lemma 2.2(3.4)x(t)≥r1/γ(t)x(n-1)(t)(n-2)!∫t1t‍r-1/γ(s)(t-s)k-2ds,
and proceeding as above, we find out that (E2) has a positive solution. A contradiction and the proof are finished for n even.
Assume thatℓ=0; note that it is possible only for n is odd. Since x′(t)<0, then there exists a finite limt→∞x(t)=c≥0. We claim that c=0. If not, then x(τ(t))≥c>0, eventually, let us say, for t≥t2. An integration of (E) from t to ∞ yields
(3.5)r(t)[x(n-1)(t)]γ≥∫t∞‍q(s)xγ(τ(s))ds
or equivalently
(3.6)x(n-1)(t)≥r-1/γ(t)(∫t∞‍q(s)xγ(τ(s))ds)1/γ.
Integrating n-1 times from t to ∞, we get
(3.7)x(t)≥∫t∞‍(u-t)n-2(n-2)!r-1/γ(u)(∫u∞‍q(s)xγ(τ(s))ds)1/γdu,
then
(3.8)x(t2)≥c(n-2)!∫t2∞‍(u-t2)n-2r-1/γ(u)(∫u∞‍q(s)ds)1/γdu,
which contradicts (P0). The proof is complete.Employing any result (e.g., Theorem 2.1.1 in [13]) for the oscillation of (E1) and (E2), we immediately obtain criteria for studied properties of (E).Corollary 3.2.
Let(3.9)liminf⁡t→∞∫τ(t)t‍q(u)(∫t1τ(u)‍r-1/γ(s)(s-t1)n-2ds)γdu>((n-2)!)γe,liminf⁡t→∞∫τ(t)t‍q(u)(∫t1τ(u)‍r-1/γ(s)(τ(s)-t)n-2ds)γdu>((n-2)!)γe.
Moreover, for n-odd assume that (P0) holds. Then  (i)
forn even, (E) is oscillatory;(ii)
forn odd, every nonoscillatory solution x(t) of (E) satisfies limt→∞x(t)=0.If we replace(P0) by stronger condition, we can establish oscillation of (E) even if n is odd.Theorem 3.3.
Letτ′(t)≥0. Assume that both first-order delay differential equations (E1) and (E1) are oscillatory. Moreover, for n-odd assume that
(P1)limsup⁡t→∞∫τ(t)t‍(u-τ(t))n-2r-1/γ(u)(∫u∞‍q(s)ds)1/γdu>(n-2)!.
Then (E) is oscillatory.Proof.
Assume thatx(t) is a positive solution of (E). Then there exists an integer ℓ assigned with x(t) by Lemma 2.1. If 1≤ℓ≤n-1, then proceeding as in the proof of Theorem 3.1 we eliminate x(t). If ℓ=0, then it follows from (3.7) that
(3.10)x(τ(t))≥∫τ(t)∞‍(u-τ(t))n-2(n-2)!r-1/γ(u)(∫u∞‍q(s)xγ(τ(s))ds)1/γdu≥x(τ(t))∫τ(t)t‍(u-τ(t))n-2(n-2)!r-1/γ(u)(∫ut‍q(s)ds)1/γdu,
which contradicts (P1). The proof is complete.Corollary 3.4.
Let (3.9) hold. Moreover, for n-odd assume that (P1) holds. Then (E) is oscillatory.Example 3.5.
We consider the forth-order delay differential equation:(E3)(t(x′′(t))3)′+at6x3(λt)=0,a>0,0<λ<1,t≥1.
It is easy to verify that (P0) holds for (E3); moreover, conditions (3.9) reduce to
(3.11)aλ5ln⁡(1λ)>1e(53)3,(3.12)a27λ2[λ6125ln⁡(1λ)-3λ3(1-λ)50+3(1-λ2)40-(1-λ)324λ3]>1e,
respectively. Thus, by Corollary 3.4, if both (3.11) and (3.12) hold, then all nonoscillatory solutions of (E3) tend to zero. For a=α3(α+1)3(3α+5)λ3α, with α>0, one such solution is x(t)=t-α. On the other hand, condition (P1) takes the form
(3.13)a(ln⁡(1λ)+1-λ)3>5.
Therefore, by Corollary 3.4, (E3) is oscillatory, provided that all (3.11), (3.12), and (3.13) hold.



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*Source: 951898-2012-07-05.xml*