The fourth order strongly noncanonical operators

Blanka Baculikova; Jozef Dzurina

doi:10.1515/math-2018-0135

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The fourth order strongly noncanonical operators

Blanka Baculikova und Jozef Dzurina

Veröffentlicht/Copyright: 31. Dezember 2018

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Abstract

It is shown that the strongly noncanonical fourth order operator

Ly=r3(t)r2(t)r1(t)y′(t)′′′

can be written in essentially unique canonical form as

Ly=q4(t)q3(t)q2(t)q1(t)q0(t)y(t)′′′′.

The canonical representation essentially simplifies examination of the fourth order strongly noncanonical equations

r3(t)r2(t)r1(t)y′(t)′′′+p(t)y(τ(t))=0.

Keywords: canonical operator; fourth order differential equations

MSC 2010: 34K11; 34C10

1 Introduction

In the paper, we consider the fourth order delay differential equation

r3(t)r2(t)r1(t)y′(t)′′′+p(t)y(τ(t))=0,(E)

where r_i ∈ C^(4–i)(t₀, ∞), r_i(t) > 0, i = 1, …, 3, p > 0, τ(t) ≤ t, τ′(t) > 0 and τ(t) → ∞ as t → ∞.

Fourth-order differential equations naturally appear in models concerning physical, biological, and chemical phenomena, such as, for instance, problems of elasticity, deformation of structures, or soil settlement, for example, see [1]. In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the past decades, there has been a constant interest in obtaining sufficient conditions for oscillatory properties of different class of fourth order differential equations with deviating argument, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

As far as the oscillation theory of fourth-order differential equations is concerned, the problem of investigating ways of factoring disconjugated operators

Ly≡r3(t)r2(t)r1(t)y′(t)′′′,(1.1)

which are crucial in studying the perturbed differential equations such as (E), has been of special interest. Motivated by the famous work of George Polya, Trench [2] showed that if operator 𝓛y is strongly noncanonical, that is,

∫∞1ri(s)ds<∞,i=1,2,3,(1.2)

then it can be written in an essentially unique canonical form as

Ly=q4(t)q3(t)q2(t)q1(t)q0(t)y(t)′′′′,

so that q_i ∈ C^(4–i)(t₀, ∞), q_i(t) > 0, i = 0, …, 4 and

∫∞1qi(s)ds=∞,i=1,2,3.

However, the computation using the Lemmas 1 and 2 from [2] leading to canonical representation is very complicated and does not provide closed formulas for q_i(t). This brings us to the question whether is it possible to establish a closed form formulas for q_i. The aim of this paper is to positively answer to this question, showing simultaneously the advantage of the result in the investigation of oscillatory properties of strongly noncanonical equations.

2 Main results

Throughout the paper we assume that (1.2) hold and so we can employ the notation

πi(t)=∫t∞1ri(s)ds,πij(t)=∫t∞1ri(s)πj(s)ds

and

πijk(t)=∫t∞1ri(s)πjk(s)ds,

where i, j, k ∈ {1, 2, 3} are mutually different.

We start with the following auxiliary results which are elementary but very useful.

Lemma 1

Let(1.2)hold. Then

πij(t)+πji(t)=πi(t)πj(t).

Proof

Since

πi(t)πj(t)′=−πj(t)ri(t)−πi(t)rj(t),

an integration of this equality from t to ∞, yields

πi(t)πj(t)=∫t∞1ri(s)πj(s)ds+∫t∞1rj(s)πi(s)ds=πij(t)+πji(t).

□

To simplify our notation, we denote

Ω(t)=π12(t)π23(t)−π2(t)π123(t).

The following result provides an alternative formula for Ω(t).

Lemma 2

Let(1.2)hold. Then

π12(t)π23(t)−π2(t)π123(t)=π21(t)π32(t)−π2(t)π321(t).(2.1)

Proof

Proof of this lemma is similar to that of Lemma 1 and so it can be omitted.□

Now, we are prepared to introduce the main result.

Theorem 1

The strongly noncanonical operator 𝓛 yhas the following unique canonical representation

Ly=1π321(t)r3π3212Ωr2Ω2π321π123r1π1232Ωyπ123′′′′(t).(2.2)

Proof

Straightforward evaluation of 𝓛 y, with 𝓛 as defined by (2.2) yields

L1y=r1π1232Ωyπ123′=r1y′π123+yπ23Ω.(2.3)

Employing (2.1), we see that

L2y=Ω2r2π321π123(L1y)′=r2r1y′′Ω−r1y′π321−π1π32+yπ32π321.

It follows from Lemma 2 that π₁₂(t) + π₂₁(t) = π₁(t)π₂(t) and so

(L2y)′=r2r1y′′′Ω−r1y′1r3π12−y1r3π2π321+r2r1y′′Ωπ21+r1y′π1π32π21−π321π21+yπ32π21r3π3212

On the other hand, by Lemma 1 and 2

r1y′(π1π32π21−π321π21−π12π321)=r1y′(π1π32π21−π321π1π2)=r1y′π1Ω.

Consequently,

(L2y)′=r2r1y′′′Ωπ321+r2r1y′′Ωπ21+r1y′π1Ω+yΩr3π3212.

Then

L3y=r3π3212ΩL2y′s=r3r2r1y′′′π321+r2r1y′′π21+r1y′π1+y.

Finally

1π321(t)(L3y)′=r3(t)r2(t)r1(t)y′(t)′′′=Ly,

which means that the operators (1.1) and (2.2) are equivalent.

Now we shall show that operator (2.2) is canonical. Direct computation shows that

∫t0t1q1(s)ds=∫t0tΩ(s)r1(s)π1232(s)ds=∫t0tπ12(s)π123(s)′ds=π12(t)π123(t)−c1.

Applying twice the L’Hospital rule, we get

limt→∞π12(t)π123(t)=limt→∞1π3(t)=∞

and so

∫t0∞1q1(s)ds=∞.

Similarly

∫t0t1q3(s)ds=∫t0tΩ(s)r3(s)π3212(s)ds=π32(t)π321(t)−c3→∞ as t→∞.

To evaluate the last one integral it is useful to see that

π321(t)π23(t)π32(t)Ω(t)+π3(t)π32(t)′=π3(t)π21(t)−π321(t)Ω(t)′ =π321(t)π123(t)r2(t)Ω2(t).

Therefore,

∫t0t1q2(s)ds=∫t0tπ321(s)π123(s)r2(s)Ω2(s)ds=∫t0tπ321(s)π23(s)π32(s)Ω(s)+π3(s)π32(s)′ds=π321(t)π23(t)π32(t)Ω(t)+π3(t)π32(t)−c2.

Moreover,

limt→∞π3(t)π32(t)=limt→∞1π2(t)=∞

and we conclude that the operator (2.2) is canonical. By Trench’s result [2] there exists the only one canonical representation of 𝓛 (up to multiplicative constants with product 1) and so our canonical form is unique.□

We support our results with couple of illustrative examples.

Example 1

Let us consider the following operator

Ly=tγtβtαy′(t)′′′,α>1,β>1,γ>1.(2.4)

By Theorem 1, this operator can be rewritten in canonical form as

Ly=1t3−α−β−γt2−αt2−βt2−γy(t)t3−α−β−γ′′′′.

Example 2

The operator

Ly=eγteβteαty′(t)′′′,α>0,β>0,γ>0.

can be represented in canonical form as

Ly=e(α+β+γ)te−αte−βte−γte(α+β+γ)ty(t)′′′′.

3 Applications to differential equations

Theorem 1 can be applied to study oscillatory properties of differential equations. We are going to outline one such application based on comparison principle.

Corollary 1

Noncanonical differential equation(E)can be written in canonical form as

1π321(t)r3π3212Ωr2Ω2π321π123r1π1232Ωyπ123′′′′(t)+p(t)y(τ(t))=0.

Setting z(t) = y(t)/π₁₂₃(t) we get the following comparison result that reduces oscillation of strongly noncanonical equation to that of canonical equation.

Corollary 2

Noncanonical equation(E)is oscillatory if and only if the canonical equation

r3π3212Ωr2Ω2π321π123r1π1232Ωz′′′′(t)+π321(t)π123(τ(t))p(t)z(τ(t))=0.(E_c)

is oscillatory.

Example 3

Let us consider the fourth order differential equation

tγtβtαy′(t)′′′+p(t)y(τ(t))=0(3.1)

withα > 1, β > 1, γ > 1. By Corollary 2, this equation is oscillatory if and only if the canonical equation

t2−αt2−βt2−γz′(t)′′′+(tτ(t))3−α−β−γp(t)z(τ(t))=0(3.2)

is oscillatory. It is more convenient to study oscillation of(3.12)instead of(3.1).

Now, we are ready to study the properties of (E) with the help of (E_c). Without loss of generality, we can consider only with the positive solutions of (E_c). The following result is a modification of Kiguradze’s lemma [3].

Let us denote

q1(t)=r1(t)π1232(t)Ω(t),q2(t)=r2(t)Ω2(t)π321(t)π123(t),q3(t)=r3(t)π3212(t)Ω(t)

and

P(t)=π321(t)π123(τ(t))p(t).

Then (E_c) can be rewritten as

q3q2q1z′′′′(t)+P(t)z(τ(t))=0.

Lemma 3

Assume that z(t) is an eventually positive solution of(E_c), then either

z(t)∈N1⟺z′(t)>0,q1z′′(t)<0,q2q1z′′′(t)>0

z(t)∈N3⟺z′(t)>0,q1z′′(t)>0,q2q1z′′′(t)>0

Consequently, the set 𝒩 of all positive solutions of (E) has the decomposition

N=N1∪N3.

To obtain oscillation of studied equation (E), we need to eliminate both cases of possible non-oscillatory solutions.

Let us denote

Q1(t)=1q2(t)∫t∞1q3(u)∫u∞P(s)dsdu∫t1τ(t)1q1(u)du

and

Q2(t)=P(t)∫t1t1q1(s1)∫t1s11q2(u)∫t1s1q3(s)dsduds1.

Theorem 2

Let(1.2)hold. Assume that both first-order delay differential equations

x′(t)+Q1(t)x(τ(t))=0(3.3)

and

x′(t)+Q2(t)x(τ(t))=0.(3.4)

are oscillatory. Then(E)is oscillatory.

Proof

Assume that y(t) is an eventually positive solution of (E), say for t ≥ t₁. Then by Corollary 2, z(t) = y(t)/π₁₂₃(t) is a solution of (E_c).

It follows from Lemma 3 that either z(t) ∈ 𝒩₁ or z(t) ∈ 𝒩₃. At first, we admit that z(t) ∈ 𝒩₁. Noting that q₁(t)z′(t) is decreasing, we see that

z(t)≥∫t1t1q1(u)q1(u)z′(u)du≥q1(t)z′(t)∫t1t1q1(u)du.(3.5)

Integrating (E_c) from t to ∞, we have

q3q2q1z′′′(t)≥∫t∞P(s)z(τ(s))ds.(3.6)

Taking into account that z(τ(t)) is increasing, the last inequality yields

q2q1z′′′(t)≥z(τ(t))1q3(t)∫t∞P(s)ds.(3.7)

Integrating once more, we are led to

−q1z′′(t)≥z(τ(t))1q2(t)∫t∞1q3(u)∫u∞P(s)dsdu.(3.8)

Combining the last inequality with (3.5), one gets

−q1z′′(t)≥z′(τ(t))q1(τ(t))q2(t)∫t∞1q3(u)∫u∞P(s)dsdu∫t1τ(t)1q1(u)du=q1(τ(t))z′(τ(t)Q1(t).(3.9)

Thus, the function x(t) = q₁(t)z′(t) is a positive solution of the differential inequality

x′(t)+Q1(t)x(τ(t))≤0.

Hence, by Philos theorem [4], we conclude that the corresponding differential equation (3.3) also has a positive solution, which contradicts the assumptions of the theorem.

Now, we shall assume that z(t) ∈ 𝒩₃. Since q3q2q1z′′′ is decreasing, we are led to

q2q1z′′(t)≥∫t1t1q3(s)q3(s)q2q1z′′′(s)ds≥q3q2q1z′′′(t)∫t1t1q3(s)ds.

Integrating the above inequality, one can verify that

z′(t)≥q3q2q1z′′′(t)1q1(t)∫t1t1q2(u)∫t1s1q3(s)dsdu

Integrating once more, we see that x(t) = q3q2q1z′′′(t) satisfies

z(t)≥x(t)∫t1t1q1(s1)∫t1s11q2(u)∫t1s1q3(s)dsduds1.

Setting the last estimate into (E_c), we see that x(t) is a positive solution of the differential inequality

x′(t)+Q2(t)x(τ(t))≤0,

which in view of Philos theorem in [4] guarantees that the corresponding differential equation (3.4) has also a positive solution. This is a contradiction and the proof is complete now.□

Applying suitable criteria for oscillation of (3.3) and (3.4), we immediately obtain the criteria for oscillation of (E). We use the one which is due to Ladde et al. [5].

Corollary 3

Let(1.2)hold. Assume that fori = 1, 2

lim inft→∞∫τ(t)tQi(s)ds>1e(3.10)

hold. Then(E)is oscillatory.

We support our results by another example.

Example 4

Let us consider the general Euler delay differential equation

tγtβtαy′(t)′′′+atα+β+γ−4y(λt)=0(3.11)

withα > 1, β > 1, γ > 1, a > 0, and 0 < λ < 1. By Corollary 2 and Example 3, this equation is oscillatory if and only if the canonical equation

t2−αt2−βt2−γz′(t)′′′+aλα+β+γ−3tα+β+γ−4z(λt)=0(3.12)

is oscillatory. The straightforward computation yields that

Q1(t)∼aλ2−α−β(α+β+γ−3)(β+γ−2)(γ−1)t−1

and

Q2(t)∼aλ3−α−β−γ(α+β+γ−3)(α+β−2)(α−1)t−1.

By Corollary 4 considered equation is oscillatory, provided that both conditions

aλ2−α−β(α+β+γ−3)(β+γ−2)(γ−1)ln1λ>1e

and

aλ3−α−β−γ(α+β+γ−3)(α+β−2)(α−1)ln1λ>1e

are satisfied.

4 Summary

In this paper we provided canonical representation for strongly noncanonical operator. This canonical transformation is easy and immediate. Moreover, we point out its application in the oscillation theory.

Acknowledgement

The paper has been supported by the grant project KEGA 035TUKE-4/2017.

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Received: 2018-05-22

Accepted: 2018-11-26

Published Online: 2018-12-31

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canonical operator; fourth order differential equations

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