# Some Properties of Third-Order Differential Equations with Mixed Arguments

Authors: B. Baculíková; J. Džurina
Journal: Journal of Mathematics
 (2013)

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## Abstract

We offer a new comparison the principle for deducing properties of third-order differential equations with mixed arguments,(r(t)[x'(t)]γ)′′+q(t)f(x(τ(t)))+p(t)h(x(σ(t)))=0, from those of the corresponding differential equations, without deviating arguments. The presented technique permits to extend immediately the results known for an equation without deviating arguments to a more general equation with advanced and delay arguments.


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## Body

## 1. Introduction

We consider third-order differential equations with mixed arguments(E)(r(t)[x′(t)]γ)′′+q(t)f(x(τ(t)))+p(t)h(x(σ(t)))=0,
where we assume that r,q,τ,p,σ∈C([t0,∞)), f,h∈C((-∞,∞)), and (H1)
γ is the ratio of two positive odd integers,(H2)
r(t)>0, q(t)>0, and p(t)>0,(H3)
σ(t)≥t, τ(t)≤t, limt→∞τ(t)=∞, and τ′(t)>0,(H4)
xf(x)>0, f′(x)≥0 for x≠0, and -f(-xy)≥f(xy)≥f(x)f(y) for xy>0,(H5)
xh(x)>0, h′(x)≥0 for x≠0, and -h(-xy)≥h(xy)≥h(x)h(y) for xy>0.In this paper, we study the canonical case of(E), that is,
(1)R(t)=∫t0t‍r-1/γ(s)ds→∞ast→∞.There is a permanent interest in studying the properties of third-order differential equations. Various techniques have been presented for such equations (see [1–21]). Comparison theorems, especially,  are a very strong and effective tool in the oscillation theory. Mahfoud [18] has presented a very useful comparison technique for studying the properties of delay differential equations through those of differential equations without delay. However, the corresponding result for advanced differential equation is still missing. In this paper, we fill up this gap in the oscillation theory; and moreover, we present such a comparison result that works also for the differential equation with mixed arguments.


## 2. The Main Results

The following result is crucial for our intended comparison theorem.Lemma 1.
Assume thatz(t)>0, z′(t)>0, and (r(t)[z′(t)]γ)′>0, eventually. Then, for arbitrary k∈(0,1),
(2)z[σ(t)]≥kR(σ(t))R(t)z(t),
eventually.Proof.
It follows from the monotonicity ofw(t)=r(t)[z′(t)]γ that
(3)z[σ(t)]-z(t)=∫tσ(t)‍z′(s)ds=∫tσ(t)‍w1/γ(s)r-1/γ(s)ds≥w1/γ(t)(R(σ(t))-R(t)),
that is,
(4)z[σ(t)]z(t)≥1+w1/γ(t)z(t)[R(σ(t))-R(t)].
On the other hand, since z(t)→∞ as t→∞, then for any k∈(0,1) there exists a t1 large enough, such that
(5)kz(t)≤z(t)-z(t1)=∫t1t‍w1/γ(s)r-1/γ(s)ds≤w1/γ(t)∫t1t‍r-1/γ(s)ds=w1/γ(t)[R(t)-R(t1)]≤w1/γ(t)R(t)
or equivalently
(6)w1/γ(t)z(t)≥kR(t).
Using (6) in (4), we obtain that
(7)z[σ(t)]z(t)≥1+kR(t)[R(σ(t))-R(t)]≥kR(σ(t))R(t).We introduce the structure of nonoscillatory solutions.Lemma 2.
Every nonoscillatory solutionx(t) of (E) satisfies one of the folowing:
(N0)x(t)x′(t)<0,x(t)[r(t)[x′(t)]γ]′>0,x(t)[r(t)[x′(t)]γ]′′<0,(N2)x(t)x′(t)>0,x(t)[r(t)[x′(t)]γ]′>0,x(t)[r(t)[x′(t)]γ]′′<0,
eventually.This structure follows from canonical case of(E), and the proof can be omitted. It is well known that for a particular case of (E), namely, for the equation
(8)x′′′(t)+p(t)x(t)=0,
there always exists a solution satisfying (N0), and this fact leads to the following definition, which is due to Kiguradze and Chanturia [15].Definition 3.
One can say that(E) enjoys property (A) if all its nonoscillatory solutions x(t) satisfy (N0).We are prepared to present the main result.Theorem 4.
Assume that for somek∈(0,1) the differential inequality
(9){(r(t)[z′(t)]γ)′′+q(τ-1(t))τ′(τ-1(t))f(z(t))+p(t)h(kR(σ(t))R(t))h(z(t)){q(τ-1(t))τ′(τ-1(t))}}sgnz(t)≤0
enjoys property (A), then so does (E).Proof.
Assume the contrary, that is, we admit that(E) possesses a positive solution x(t) satisfying (N2). Thus, it follows from Lemma 1 that, for every k∈(0,1),
(10)(r(t)[x′(t)]γ)′′+q(t)f(x(τ(t)))+p(t)h(kR(σ(t))R(t))h(x(t))≤0.
Integrating from t to ∞, we obtain that
(11)(r(t)[x′(t)]γ)′≥∫t∞‍q(s)f(x(τ(s)))ds+∫t∞‍p(s)h(kR(σ(s))R(s))h(x(s))ds=∫τ(t)∞‍q(τ-1(s))τ′(τ-1(s))f(x(s))ds+∫t∞‍p(s)h(kR(σ(s))R(s))h(x(s))ds≥∫t∞‍[q(τ-1(s))τ′(τ-1(s))f(x(s))+p(s)h(kR(σ(s))R(s))h(x(s)){q(τ-1(s))τ′(τ-1(s))}]ds.
Integrating twice from t1 to t, we are led to
(12)x(t)≥∫t1t‍r-1/γ(v)×[∫t1v‍∫u∞‍q(τ-1(s))τ′(τ-1(s))f(x(s))+p(s)h(kR(σ(s))R(s))h(x(s))dsdu{∫t1v‍∫u∞‍q(τ-1(s))τ′(τ-1(s))}]1/γdv.
Let us denote the right-hand side by z(t). Then, x(t)≥z(t), z(t) satisfies (N2), and
(13)0=(r(t)[z′(t)]γ)′′+q(τ-1(t))τ′(τ-1(t))f(x(t))+p(t)h(kR(σ(t))R(t))h(x(t))≥(r(t)[z′(t)]γ)′′+q(τ-1(t))τ′(τ-1(t))f(x(t))+p(t)h(kR(σ(t))R(t))h(z(t)),
which contradicts with the assumptions of the theorem, and we conclude that (E) has property (A).Theorem4 reduces the examination of properties of differential equations with mixed arguments to that of simpler equations without deviating arguments and permits to extend immediately the criteria known for property (A) of equations without deviating arguments to more general equations with both advanced and delay arguments. We provide some applications of our main result.For our further references, we set that(14)Q1(t)=[q(τ-1(s))τ′(τ-1(s))+p(s)(kR(σ(s))R(t))γ],Q(t)=∫t∞‍Q1(s)ds,
where k∈(0,1).Theorem 5.
If, for somek∈(0,1),
(15)liminft→∞1Q(t)∫t∞‍(sr(s))1/γQ1+1/γ(s)ds>1(γ+1)1+1/γ,
then
(E1)(r(t)[x′(t)]γ)′′+q(t)xγ(τ(t))+p(t)xγ(σ(t))=0
enjoys property (A).Proof.
By Theorem4, it is sufficient to show that the differential inequality
(E2){(r(t)[x′(t)]γ)′′+Q1(t)xγ(t)}sgnx(t)≤0
has property (A) for some k∈(0,1). Assuming the contrary, we admit that (E2) does not property (A), that is, it possesses an eventually positive solution x(t) satisfying (N2). We define have the following:
(16)w(t)=[r(t)[x′(t)]γ]′xγ(t)>0.
Differentiating w(t), one gets in view of (E2) that
(17)w′(t)=[r(t)[x′(t)]γ]′′xγ(t)-γ[r(t)[x′(t)]γ]′xγ(t)x′(t)x(t)≤Q1(t)-γw(t)x′(t)x(t).
On the other hand, using the monotonicity of [r(t)[x′(t)]γ]′, we have
(18)r(t)[x′(t)]γ≥∫t0t‍[r(s)[x′(s)]γ]′≥[r(t)[x′(t)]γ]′(t-t0)≥ℓt[r(t)[x′(t)]γ]′,
eventually; let us say that for t≥t1 or equivalently
(19)x′(t)≥(ℓt[r(t)[x′(t)]γ]′)1/γr-1/γ(t),
where ℓ∈(0,1) will be specified latter. Setting the last inequality into (17), we obtain
(20)w′(t)≤-Q1(t)-γℓ1/γw1+1/γ(t)t1/γr1/γ(t).
Integrating the last inequality from t to ∞, we have
(21)w(t)≥Q(t)+γℓ1/γ∫t∞‍w1+1/γ(s)(sr(s))1/γds
or
(22)w(t)Q(t)≥1+γℓ1/γQ(t)∫t∞‍(sr(s))1/γQ1+1/γ(s)(w(s)Q(s))1+1/γds,
eventually; let us say that t≥t1. Since w(t)>Q(t), then
(23)inft≥t1w(t)Q(t)=λ≥1.
Thus,
(24)w(t)Q(t)≥1+γℓ1/γλ1+1/γQ(t)∫t∞‍(sr(s))1/γQ1+1/γ(s)ds.
From (15), we see that there exist some ℓ∈(0,1) and some positive η, such that
(25)ℓ1/γQ(t)∫t∞‍(sr(s))1/γQ1+1/γ(s)ds>η>(γ+1)-(γ+1)/γ.
Combining (24) with (25), we have
(26)w(t)Q(t)≥1+γλ1+1/γη.
Therefore,
(27)λ≥1+γλ1+1/γη>1+γλ1+1/γ(γ+1)-(γ+1)/γ
or equivalently
(28)0>1γ+1+γγ+1(λγ+1)1+1/γ-λγ+1.
This contradicts with the fact that the function
(29)g(α)=1γ+1+γγ+1α1+1/γ-α
is positive for all α>0, and we conclude that x(t) cannot satisfy (N2); consequently, (E1) has property (A).In the following criteria, we utilize existing results known for differential equations without deviating arguments to provide new criteria for differential equations with mixed arguments.Corollary 6.
Assume that for somek∈(0,1) the differential equation
(E3)x′′′(t)+{q(τ-1(t))τ′(τ-1(t))+kp(t)σ(t)t}x(t)=0
enjoys property (A), then so does the equation with mixed arguments
(E4)x′′′(t)+q(t)x(τ(t))+p(t)x(σ(t))=0.Proof.
By Theorem4,  (E4) enjoys property (A), provided that, for some k∈(0,1), differential inequality
(30)[x′′′(t)+{q(τ-1(t))τ′(τ-1(t))+kp(t)σ(t)t}x(t)]sgnx(t)≤0
has property (A). But by Corollary 1 in [16], this is equivalent to property (A) of the corresponding differential equation (E3).Corollary 7.
Assume that at least one of the following conditions:(31)liminft→∞t2∫t∞‍(q(τ-1(s))τ′(τ-1(s))+p(s)σ(s)s)ds>133,liminft→∞t∫t∞‍(sq(τ-1(s))τ′(τ-1(s))+p(s)σ(s))ds>233
holds, then (E3) has property (A).Proof.
It is known (see, e.g., [12, 15]) that
(32)liminft→∞t2∫t∞‍P(s)ds>133
guarantees property (A) of
(33)x′′′(t)+P(t)x(t)=0.
The rest follows from Corollary 6, where constant k∈(0,1) is eliminated due to the sharp inequality of the used criterion.We support our results by several illustrative examples.


## 3. Examples

Example 8.
Consider the third-order nonlinear differential equation(Ex1)(t-1(x′(t))3)′′+at6x3(δt)+bt6x3(λt)=0,
where a,b>0, 0<δ≤1, and λ≥1. Simple computation shows that here (15) reduces to
(34)δ5a+kbλ4>(54)4,
for some k∈(0,1), which is equivalent to
(35)δ5a+bλ4>(54)4
and this by Theorem 4 guarantees that (Ex1) enjoys property (A).Example 9.
Consider the third-order differential equation with mixed arguments(Ex2)x′′′(t)+at3x(δt)+bt3x(λt)=0,
where a,b>0, 0<δ≤1, and λ≥1. Both conditions of Corollary 7 reduce to
(36)δ2a+bλ>233,
which guarantees property (A) of (Ex2).Our results take into account the value of advanced argument, which can be seen in the following example.Example 10.
Consider the third-order nonlinear differential equation(Ex3)x′′′(t)+at3x(δt)+bt4x(t2)=0,
where a,b>0, 0<δ<1. Both conditions of Corollary 7 reduce to
(37)δ2a+b>233,
which yields property (A) of (Ex3).


## 4. Conclusion and Discussion

In the paper, we presented new comparison theorems for studying the properties of third-order differential equations with mixed arguments through those of the corresponding differential equations without deviating arguments. The presented technique permits to extend immediately the results known for an equation without deviating arguments to a more general equation with advanced and delay arguments. The results obtained have been supported by several illustrative examples.



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