Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

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Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

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Submitted:

29 August 2024

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29 August 2024

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Abstract

This paperfocusesontheoscillatorypropertiesofthesemi-canonicalnonlineardelaydifferential equation ofthird-order (δ2(t)(δ1(t)ζ′(t))′)′ + η(t)ζα(τ(t))= 0, t ≥ t0, under theconditions ∞t01δ2(t)dt = ∞ andZ ∞t01δ1(t)dt < ∞. Using the new canonical transform method, we transformed the studied equation into a canonical type equation and then we find the conditions for the oscillation of the studied equation from the canonical equations. The obtained results are new and complement to the existing results mentioned in the literature. Examples are provided to illustrate the importance and novelty of the main results.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

The aim of this paper is to extend and complement the results in [20,21] for the third-order delay differential equations of the form

L ζ (t) + η (t) ζ^{α} (τ (t)) = 0, t \geq t_{0} > 0,

(E)

where

L

is the differential operator defined by

L ζ (t) = {(δ_{2} (t) {(δ_{1} (t) ζ^{'} (t))}^{'})}^{'} .

(1)

In the sequel, we assume the following hypotheses:

$(H_{1})$ $δ_{1} \in C^{2} ([t_{0}, \infty),$ $δ_{2} \in C^{1} ([t_{0}, \infty),$ $a_{1} (t) > 0$ and $a_{2} (t) > 0$ for all $t \geq t_{0};$
$(H_{2})$ $η (t) \in C ([t_{0}, \infty))$ and $η (t) > 0$ for all $t \geq t_{0};$
$(H_{3})$ $τ (t) \in C^{1} ([t_{0}, \infty)),$ $τ (t) < t,$ ${lim}_{t \to \infty} τ (t) = \infty$ and $τ (t) > 0$ for all $t \geq t_{0};$
$(H_{4})$ $α$ is a ratio of odd positive integers;
$(H_{5})$ the operator $L$ is in semi-canonical form, that is,

$\int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t = \infty a n d Ω_{1} (t_{0}) = \int_{t_{0}}^{\infty} \frac{1}{δ_{1} (t)} d t < \infty .$

Definition 1.

The solution of (E) is defined to be a nontrivial function

ζ (t) \in C ([t_{ζ}, \infty)),

t_{ζ} \geq t_{0}

with the properties

δ_{1} ζ^{'} \in C^{1} ([t_{ζ}, \infty)),

δ_{2} {(δ_{1} ζ^{'})}^{'} \in C^{1} ([t_{ζ}, \infty)),

that satisfies (E) on

[t_{ζ}, \infty) .

We consider only those solutions

ζ (t)

of (E) satisfying

sup {| ζ (t) | : t \geq T} > 0

for all

T \geq t_{ζ}

and assume that (E) possesses such a solution.

Definition 2.

A solution of (E) is called oscillatory if it has infinitely many zeros on

[t_{ζ}, \infty);

otherwise it is called nonoscillatory. Equation (E) itself is called oscillatory if all its solution are oscillatory.

Due to many practical importance of third-order functional differential equations as well as a number of mathematical problems involved [13], the area of qualitative theory of such equations received a great attention in the last several decades. Further this is very useful in predicting the similar behavior of solutions of third-order partial differential equations [9,10].

Oscillatory phenomena play a significant role in understanding the inherent vibrational patterns within dynamic systems, so investigating the oscillatory behavior of solutions to differential equations provides valuable insights into the stability and periodicity of the systems under considerations. In recent years, there has been great interest in investigating the oscillatory behavior of solutions of (E) or its particular cases or its generalizations, see, for example [1,2,3,4,5,6,7,8,11,12,15,16,19,21,22] and the references are cited therein.

From the review of literature, we see that most of the oscillation results for the equation (E) under the conditions

\int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t = \int_{t_{0}}^{\infty} \frac{1}{δ_{1} (t)} d t = \infty

\int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t < \infty a n d \int_{t_{0}}^{\infty} \frac{1}{δ_{1} (t)} d t = \infty

\int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t < \infty a n d \int_{t_{0}}^{\infty} \frac{1}{δ_{1} (t)} d t < \infty,

see, for example [3,4,6,9-13,17-21,24] and the references are cited therein.

In [5,20,21] the authors studied the oscillatory properties of (E) or its generalizations under the condition

\int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t = \infty a n d \int_{t_{0}}^{\infty} \frac{1}{δ_{1} (t)} d t < \infty .

However in [20,21], the authors transformed the equation (E) into canonical form under the condition

Ω_{21} (t_{0}) = \int_{t_{0}}^{\infty} \frac{Ω_{1} (t)}{a_{2} (t)} d t = \infty,

(2)

which greatly simplified the examination of (E) by reducing the set of nonoscillatory solutions to two instead of three.

In view of the above observation, we see that if condition (2) fails to hold then (E) cannot be called as a canonical type equation. Therefore the aim of this paper is to fill this gap, that is, if

Ω_{21} (t_{0}) < \infty

then we present new transform that transforms (E) into canonical form. Thus, the results obtained here are new and complement to that of in [5,20,21]. Examples are provided to show the novelty and significance of our main results.

2. Main Results

For simplicity, we employ the following notation:

Ω_{1} (t) = \int_{t}^{\infty} \frac{1}{δ_{1} (s)} d s, Ω_{21} (t) = \int_{t}^{\infty} \frac{Ω_{1} (s)}{δ_{2} (s)} d s

β_{1} (t) = \frac{δ_{1} (t) Ω_{1}^{2} (t)}{Ω_{21} (t)}, β_{2} (t) = \frac{δ_{2} (t) Ω_{21}^{2} (t)}{Ω_{1} (t)}, z (t) = \frac{ζ (t)}{Ω_{1} (t)} .

We begin with a theorem that is adopted from [12] but presented with a different proof.

Theorem 1.

Ω_{21} (t_{0}) < \infty,

(3)

then the semi-canonical operator

L ζ

has the following unique canonical representation

L ζ (t) = \frac{1}{Ω_{21} (t)} {(\frac{δ_{2} (t) Ω_{21}^{2} (t)}{Ω_{1} (t)} {(\frac{δ_{1} (t) Ω_{1}^{2} (t)}{Ω_{21} (t)} {(\frac{ζ (t)}{Ω_{1} (t)})}^{'})}^{'})}^{'} .

(4)

Proof.

By a direct calculation, we have

\frac{δ_{2} (t) Ω_{21}^{2} (t)}{Ω_{1} (t)} {(\frac{δ_{1} (t) Ω_{1}^{2} (t)}{Ω_{21} (t)} {(\frac{ζ (t)}{Ω_{1} (t)})}^{'})}^{'} = Ω_{21} (t) δ_{2} (t) {(δ_{1} (t) ζ^{'} (t))}^{'} + Ω_{1} (t) δ_{1} (t) ζ^{'} (t) + ζ (t),

\begin{matrix} \frac{1}{Ω_{21} (t)} {(\frac{δ_{2} (t) Ω_{21}^{2} (t)}{Ω_{1} (t)} {(\frac{δ_{1} (t) Ω_{1}^{2} (t)}{Ω_{21} (t)} {(\frac{ζ (t)}{Ω_{1} (t)})}^{'})}^{'})}^{'} = \frac{1}{Ω_{21} (t)} [Ω_{21} (t) δ_{2} (t) {(δ_{1} (t) ζ^{'} (t))}^{'} \\ + Ω_{1} (t) δ_{1} (t) ζ^{'} (t) + ζ (t)]^{'} \end{matrix}

\begin{matrix} = & \frac{1}{Ω_{21} (t)} [Ω_{21} (t) {(δ_{2} (t) {(δ_{1} (t) ζ^{'} (t))}^{'})}^{'} - Ω_{1} (t) {(δ_{1} (t) ζ^{'} (t))}^{'} \\ + & Ω_{1} (t) {(δ_{1} (t) ζ^{'} (t))}^{'} - ζ^{'} (t) + ζ^{'} (t)] = {(δ_{2} (t) {(δ_{1} (t) ζ^{'} (t))}^{'})}^{'} . \end{matrix}

Next, we show that (4) is in canonical form, that is,

\int_{t_{0}}^{\infty} \frac{Ω_{1} (t)}{δ_{2} (t) Ω_{21}^{2} (t)} d t = \infty = \int_{t_{0}}^{\infty} \frac{Ω_{21} (t)}{δ_{1} (t) Ω_{1}^{2} (t)} d t .

Now,

\int_{t_{0}}^{\infty} \frac{Ω_{1} (t)}{δ_{2} (t) Ω_{21}^{2} (t)} d t = \int_{t_{0}}^{\infty} d (\frac{1}{Ω_{21} (t)}) = lim_{t \to \infty} \frac{1}{Ω_{21} (t)} - \frac{1}{Ω_{21} (t_{0})} = \infty .

Further,

\begin{matrix} \int_{t_{0}}^{\infty} \frac{Ω_{21} (t)}{δ_{1} (t) Ω_{1}^{2} (t)} d t & = & \int_{t_{0}}^{\infty} Ω_{21} (t) d (\frac{1}{Ω_{1} (t)}) \\ = & \frac{Ω_{21} (t)}{Ω_{1} (t)} |_{t_{0}}^{\infty} + \int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t \\ > & \int_{t_{0}}^{\infty} \frac{1}{δ_{2} (t)} d t = \infty \end{matrix}

(H_{5}) .

However, Trench proved in [24] that there exists only one canonical representation of

L

(up to multiplicative constants with product 1) and so our canonical form is unique. The proof of the theorem is complete. □

Based on Theorem 2.1, one can write (E) in the canonical form as

{(β_{2} (t) (β_{1} (t) z^{'} (t))}^{'})^{'} + w (t) z^{α} (τ (t)) = 0, t \geq t_{0},

(E_c )

where

w (t) = η (t) Ω_{21} (t) Ω_{1}^{α} (τ (t)),

and the following results are immediate.

Theorem 2.

Let (3) holds. Then the semi-canonical equation (E) possesses a solution

ζ (t)

if and only if the canonical equation (E_c) has the solution

z (t) .

Corollary 1.

Let (3) holds. The semi-canonical equation (E) has an eventually positive solution if and only if the canonical equation (E_c) has an eventually positive solution.

Corollary 2.3 simplifies significantly the investigation of (E) since for (E_c), we deal with only two classes of an eventually positive (nonoscillatory) solutions (see, Lemma 2 [14]), namely, either

z (t) > 0, L_{1} z (t) < 0, L_{2} z (t) > 0, L_{3} z (t) < 0,

and in this case, we say

z \in D_{0}

z (t) > 0, L_{1} z (t) > 0, L_{2} z (t) > 0, L_{3} z (t) < 0,

and in this case we denote that

z \in D_{2},

where

L_{0} z (t) = z (t), L_{i} z (t) = β_{i} (t) {(L_{i - 1} z (t))}^{'}, i = 1, 2, L_{3} z (t) = {(L_{2} z (t))}^{'} .

Theorem 3.

Let (3) holds. Assume that there exists a function

σ \in C^{1} ([t_{0}, \infty))

such that

σ^{'} (t) > 0, σ (t) > t, g (t) = τ (σ (σ (t))) < t .

(5)

If both the first-order delay differential equations

u^{'} (t) + Q_{1} (t) u^{α} (τ (t)) = 0

(6)

and

u^{'} (t) + Q_{2} (t) u^{α} (g (t)) = 0,

(7)

where

\begin{matrix} Q_{1} (t) & = & w (t) {(\int_{t_{1}}^{τ (t)} \frac{1}{β_{1} (s)} \int_{t_{1}}^{s} \frac{1}{β_{2} (s_{1})} d s_{1} d s)}^{α} \\ Q_{2} (t) & = & \frac{1}{β_{1} (t)} \int_{t}^{σ (t)} \frac{1}{β_{2} (s)} \int_{s}^{σ (s)} w (s_{1}) d s_{1} d s \end{matrix}

for all

t_{1} \geq t_{0},

are oscillatory, then equation (E) is oscillatory.

Proof.

Let

ζ (t)

be an eventually positive solution of (E). Then by Corollary 2.3,

z (t)

is also a positive solution of (E_c) and either

z (t) \in D_{0}

z (t) \in D_{2}

for all

t \geq t_{1} \geq t_{0} .

First assume that

z (t) \in D_{2} .

Then using the fact that

L_{2} z (t) > 0

and decreasing, we have

L_{1} z (t) \geq \int_{t_{1}}^{t} \frac{L_{2} z (s)}{β_{2} (s)} d s \geq L_{2} z (t) \int_{t_{1}}^{t} \frac{1}{β_{2} (s)} d s,

z^{'} (t) \geq L_{2} z (t) (\frac{1}{β_{1} (t)} \int_{t_{1}}^{t} \frac{1}{β_{2} (s)} d s) .

Integrating again from

t_{1}

t,

we get

z (t) \geq L_{2} z (t) \int_{t_{1}}^{t} \frac{1}{β_{1} (s)} (\int_{t_{1}}^{s} \frac{1}{β_{2} (s_{1})} d s_{1}) d s .

(8)

Let

u (t) = L_{2} z (t) .

Then combining (8) with (E_c), we see that

u^{'} (t) + w (t) {(\int_{t_{1}}^{τ (t)} \frac{1}{β_{1} (s)} \int_{t_{1}}^{s} \frac{1}{β_{2} (s_{1})} d s_{1} d s)}^{α} u^{α} (τ (t)) \leq 0

for

t \geq t_{1} .

Integrating the latter inequality from t to

\infty,

we have

u (t) \geq \int_{t}^{\infty} Q_{1} (s) u^{α} (τ (s)) d s

for

t \geq t_{1} .

The function

u (t)

is clearly decreasing on

[t_{1}, \infty)

and hence by Theorem 1 in [18], we conclude that there exists a positive solution

u (t)

of (6) with

{lim}_{t \to \infty} u (t) = 0,

which contradicts the fact that (6) is oscillatory.

Next, assume that

z (t) \in D_{0} .

Integrating (E_c) from t to

σ (t)

gives

L_{2} z (t) \geq \int_{t}^{σ (t)} w (s) z^{α} (τ (s)) d s \geq z^{α} (τ (σ (t))) \int_{t}^{σ (t)} w (s) d s .

Then

{(β_{1} (t) z^{'} (t))}^{'} \geq \frac{z^{α} (τ (σ (t)))}{β_{2} (t)} \int_{t}^{σ (t)} w (s) d s .

Integrating last inequality from t to

σ (t),

we get

- z^{'} (t) \geq \frac{z^{α} (g (t))}{β_{1} (t)} \int_{t}^{σ (t)} \frac{1}{β_{2} (s)} \int_{s}^{σ (s)} w (s_{1}) d s_{1} d s .

Finally, integrating from t to ∞ yields

z (t) \geq \int_{t}^{\infty} \frac{z^{α} (g (s))}{β_{1} (s)} \int_{s}^{σ (s)} \frac{1}{β_{2} (s_{1})} \int_{s_{1}}^{σ (s_{1})} w (s_{2}) d s_{2} d s_{1} d s .

Set the right hand side of the last inequality by

u (t),

we have

z (t) \geq u (t) > 0 .

Then, it is easy to see that

0 = u^{'} (t) + Q_{2} (t) z^{α} (g (t))

0 \geq u^{'} (t) + Q_{2} (t) u^{α} (g (t)) .

Since

u (t)

is a positive bounded solution of the last inequality, then by Corollary 1 of [18], we see that the corresponding differential equation (7) has also a positive solution. This is a contradiction to our assumption, and we conclude that (E) oscillates. The proof of the theorem is complete. □

Employing criteria for oscillation of (6) and (7), we immediately obtain explicit criteria for oscillation of (E) for different value of

α .

Corollary 2.

Let (3) holds. Assume that there exists a function

σ \in C^{1} ([t_{0}, \infty))

such that (5) holds. If

α = 1,

lim_{t \to \infty} inf \int_{τ (t)}^{t} Q_{1} (s) d s > \frac{1}{e}

(9)

and

lim_{t \to \infty} inf \int_{g (t)}^{t} Q_{2} (s) d s > \frac{1}{e},

(10)

then (E) is oscillatory.

Proof.

Application of Theorem 2.1.1 of [9] with (9) and (10) imply that the equations (6) and (7) are oscillatory. Now, the proof follows from Theorem 2.4. This ends the proof. □

Corollary 3.

Let (3) holds. Assume that there exists a function

σ \in C^{1} ([t_{0}, \infty))

such that (5) holds. If

0 < α < 1

\int_{t_{*}}^{\infty} Q_{1} (t) d t = \infty

(11)

and

\int_{t_{*}}^{\infty} Q_{2} (t) d t = \infty

(12)

for all

t_{*} \geq t_{0},

then equation (E) is oscillatory.

Proof.

Application of Theorem 3.9.3 of [9] with (11) and (12) imply that the equations (6) and (7) are oscillatory. Now, the proof follows from Theorem 2.4. This ends the proof. □

Corollary 4.

Let (3) holds. Assume that there exists a function

σ \in C^{1} ([t_{0}, \infty))

such that (5) holds. Further assume that

α > 1,

τ (t) = θ_{1} t,

σ (t) = θ_{2} (t),

θ_{1} \in (0, 1),

θ_{2} > 1

and

g (t) = θ_{3} (t)

with

θ_{3} = θ_{1} θ_{2}^{2} < 1 .

If there exist

λ_{1} > - ln α / ln θ_{1}, λ_{2} > - ln α / ln θ_{3}

(13)

such that

lim_{t \to \infty} inf [Q_{1} (t) exp (- t^{λ_{1}})] > 0

(14)

and

lim_{t \to \infty} inf [Q_{2} (t) exp (- t^{λ_{2}})] > 0

(15)

hold, then (E) is oscillatory.

Corollary 5.

Let (3) holds. Assume that there exists a function

σ \in C^{1} ([t_{0}, \infty))

such that (5) holds. Further assume that

α > 1,

τ (t) = t^{θ_{1}},

σ (t) = t^{θ_{2}},

θ_{1} \in (0, 1),

θ_{2} > 1

and

g (t) = t^{θ_{3}}

with

θ_{3} = θ_{1} θ_{2}^{2} < 1 .

If there exist

λ_{1} > - ln α / ln θ_{1}, λ_{2} > - ln α / ln θ_{3}

(16)

such that

lim_{t \to \infty} inf [Q_{1} (t) exp (- {(ln t)}^{λ_{1}})] > 0

(17)

and

lim_{t \to \infty} inf [Q_{2} (t) exp (- {(ln t)}^{λ_{2}})] > 0

(18)

hold, then (E) is oscillatory.

The proofs of Corollary 2.7 and Corollary 2.8 follow by using Theorem 4 and Theorem 5 of [23] with Theorem 2.4, respectively.

Next, we present another criteria for the oscillation of (E) when

α = 1

and

α < 1 .

Theorem 4.

Let (3) and

α = 1

hold. If

lim_{t \to \infty} sup \int_{τ (t)}^{t} (\frac{1}{β_{1} (s)} \int_{s}^{t} \frac{1}{β_{2} (s_{1})} \int_{s_{1}}^{t} w (s_{2}) d s_{2} d s_{1}) d s > 1

(19)

and

lim_{t \to \infty} inf \int_{τ (t)}^{t} Q_{1} (s) d s > \frac{1}{e},

(20)

then equation (E) is oscillatory.

Proof.

Let

ζ (t)

be an eventually positive solution of (E). Then proceeding as in the proof of Theorem 2.4, we see that

z (t)

is a positive solution of (E_c) and belongs to either

D_{0}

D_{2}

for all

t \geq t_{1} \geq t_{0} .

z (t) \in D_{2},

then as in the Proof of Corollary 2.5, we obtain a contradiction with (20) and so

z (t) \in D_{0}

Integrating (E_c) from s to t gives

\begin{matrix} {(β_{1} (s) z^{'} (s))}^{'} & \geq & \frac{1}{β_{2} (s)} \int_{s}^{t} w (s_{1}) z^{α} (τ (s_{1})) d s_{1} \\ \geq & \frac{z^{α} (τ (t))}{β_{2} (s)} \int_{s}^{t} w (s_{1}) d s_{1} . \end{matrix}

Again integrating the inequality twice from s to

t,

we have

z (s) \geq z^{α} (τ (t)) \int_{s}^{t} (\frac{1}{β_{1} (s_{1})} \int_{s_{1}}^{t} \frac{1}{β_{2} (s_{2})} \int_{s_{2}}^{t} w (s_{3}) d s_{3} d s_{2}) d s_{1} .

Letting

s = τ (t)

and

α = 1,

we obtain a contradiction with (19). Hence we conclude that (E) is oscillatory. This completes the proof. □

Theorem 5.

Let (3) and

0 < α < 1

hold. If

lim_{t \to \infty} sup \int_{τ (t)}^{t} (\frac{1}{β_{1} (s)} \int_{s}^{t} \frac{1}{β_{2} (s_{1})} \int_{s_{1}}^{t} w (s_{2}) d s_{2} d s_{1}) d s = \infty

(21)

and

lim_{t \to \infty} \int_{t_{1}}^{\infty} Q_{1} (s) d s = \infty

(22)

for all

t_{1} \geq t_{0},

then equation (E) is oscillatory.

Proof.

Let

ζ (t)

be an eventually positive solution of (E). Then proceeding as in the proof of Theorem 2.4, we see that

z (t)

is a positive solution of (E_c) that belongs to either

D_{0}

D_{2}

for all

t \geq t_{1} \geq t_{0} .

First assume that

z (t) \in D_{2} .

In view of condition (22) and by Corollary 2.6, we get that

D_{2}

is empty and so

z (t) \in D_{0} .

Thus proceeding as in the proof of Theorem 2.9, we obtain

z^{1 - α} (τ (t)) \geq \int_{τ (t)}^{t} (\frac{1}{β_{1} (s)} \int_{s}^{t} \frac{1}{β_{2} (s_{1})} \int_{s_{1}}^{t} w (s_{2}) d s_{2} d s_{1}) d s .

(23)

Since

z (t)

is decreasing and

α < 1,

we see that

z^{1 - α} (τ (t)) \leq M

for all

t \geq t_{1} \geq t_{0},

and using this in (23), one gets a contradiction with (21). The proof of the theorem is complete. □

Remark 1.

Here, we use the canonical equation (E_c) to get oscillation criteria of (E). Therefore, it is clear that one may use the results in [1,2,3,6,11,15,16] to get several oscillatory and asymptotic behaviors of (E_c) which in turn imply that of (E). The details are left to the reader.

3. Examples

In this section, we present some examples to show the importance of the main results.

Example 1.

Consider the semi-canonical third-order linear delay differential equation

{(t {(t^{2} ζ^{'} (t))}^{'})}^{'} + η_{0} ζ (λ t) = 0, t \geq 1,

(24)

where

η_{0} > 0

and

λ \in (0, 1) .

Here

δ_{1} (t) = t^{2},

δ_{2} (t) = t,

η (t) = η_{0},

τ (t) = λ t

and

α = 1 .

By a simple calculation, we have

Ω_{1} (t) = \frac{1}{t},

Ω_{21} (t) = \frac{1}{t},

β_{1} (t) = t,

β_{2} (t) = 1

and

w (t) = \frac{η_{0}}{λ t^{2}} .

The transformed equation is

{(t z^{'} (t))}^{''} + \frac{η_{0}}{λ^{2} t} z (λ t) = 0, t \geq 1,

which is in canonical form. The condition (3) is clearly satisfied. Choose

σ (t) = μ t

with

1 < μ < \frac{1}{\sqrt{λ}}

and

g (t) = λ μ^{2} t < t,

we see that

Q_{1} (t) = η_{0} (\frac{1}{t} - \frac{1}{λ t^{2}} - \frac{ln λ}{λ t^{2}} - \frac{ln t}{λ t^{2}}),

Q_{2} (t) = \frac{η_{0}}{λ t} (1 - \frac{λ}{μ}) ln μ .

The condition (9) becomes

η_{0} ln (\frac{1}{λ}) > \frac{1}{e}

and the condition (10) becomes

\frac{η_{0}}{λ} (1 - \frac{1}{μ}) ln μ ln (\frac{1}{λ μ^{2}}) > \frac{1}{e} .

Therefore, by Corollary 2.5, the equation (24) is oscillatory if

η_{0} > max \{\frac{1}{e ln (\frac{1}{λ})}, \frac{λ μ}{(μ - 1) ln μ ln (\frac{1}{λ μ^{2}})}\} .

Example 2.

Consider the third-order semi-canonical sub-linear delay differential equation

{(t^{3} ζ^{'} (t))}^{''} + η_{0} t ζ^{\frac{1}{3}} (λ t) = 0, t \geq 1,

(25)

where

η_{0} > 1

and

λ \in (0, 1) .

Here

δ_{2} (t) = 1,

δ_{1} (t) = t^{3},

η (t) = η_{0} t,

τ (t) = λ t

and

α = \frac{1}{3} .

By a simple calculation, we have

Ω_{1} (t) = \frac{1}{2 t^{2}},

Ω_{21} (t) = \frac{1}{2 t},

β_{1} (t) = β_{2} (t) = \frac{1}{2},

w (t) = \frac{η_{1}}{t^{\frac{2}{3}}}

where

η_{1} = η_{0} {(\frac{2}{λ})}^{\frac{2}{3}} .

The condition (3) clearly holds and the transformed equation is

z^{'''} (t) + \frac{η}{t^{\frac{2}{3}}} z^{\frac{1}{3}} (λ t) = 0, t \geq 1,

which is in canonical form. With a further calculation, we see that

Q_{1} (t) = 2^{\frac{1}{3}} η_{0} .

Choose

σ (t) = μ t

with

1 < μ < \frac{1}{λ},

we see that

g (t) = λ μ^{2} t < t

and so condition (5) holds. Also

Q_{2} (t) = \frac{9}{4} η_{1} {(μ - 1)}^{\frac{5}{3}} t^{\frac{4}{3}} .

The conditions (11) and (12) are clearly hold. Therefore, by Corollary 2.6, the equation (25) is oscillatory for all

η_{0} > 0 .

Example 3.

Consider the third-order semi-canonical super-linear delay differential equation

{(t^{4} ζ^{'} (t))}^{''} + η_{0} t^{12} e^{t^{2}} ζ^{3} (\frac{t}{4}) = 0, t \geq 1,

(26)

where

η_{0} > 0 .

Here

δ_{1} (t) = t^{4},

δ_{2} (t) = 1,

η (t) = η_{0} t^{12},

τ (t) = \frac{t}{4}

and

α = 3 .

By a simple computation, we see that

Ω_{1} (t) = \frac{1}{3 t^{3}} a n d Ω_{21} (t) = \frac{1}{6 t^{2}}

and hence condition (3) is satisfied. The transformed equation is

{(\frac{1}{t} z^{''} (t))}^{'} + η_{1} t e^{t^{2}} z^{3} (\frac{t}{4}) = 0, t \geq 1,

where

η_{1} = \frac{η_{0} {(4)}^{9}}{9} .

Choose

σ (t) = \frac{3}{2} t,

we have

g (t) = \frac{9}{16} t

and so condition (5) hold. With a further calculation, we see that

Q_{1} (t) = \frac{η_{0}}{9 (6^{3})} t^{10} e^{t^{2}}, Q_{2} (t) = \frac{η_{1}}{36} (4 e^{\frac{81}{16} t^{2}} - 13 e^{\frac{9}{4} t^{2}} + 9 e^{t^{2}}) .

Let

λ_{1} = 1,

then

1 > \frac{ln 3}{ln 4}

and let

λ_{2} = 2,

then

2 > \frac{ln 3}{ln (\frac{16}{9})},

so conditions in (13) holds. The condition (14) becomes

lim_{t \to \infty} inf [\frac{η_{0}}{9 (6^{3})} t^{10} (e^{t^{2} - t})] > 0,

that is, condition (14) holds. The condition (15) becomes

lim_{t \to \infty} inf [\frac{η_{1}}{36} (4 e^{\frac{81}{16} t^{2}} - 13 e^{\frac{9}{4} t^{2}} + 9 e^{t^{2}}) e^{- t^{2}}] > lim_{t \to \infty} \frac{η_{1}}{36} (3 e^{\frac{5}{4} t^{2}} + 9 - 16 e^{- t^{2}}) > 0,

that is, condition (15) hold. Therefore, by Corollary 2.7, the equation (26) is oscillatory.

4. Conclusions

In this paper, by using a new canonical transform, we changed the shape of the equation (E) into a canonical type equation. This significantly simplifies the examination of the equation (E). Therefore, the results obtained here are new and complement to the existing results.

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Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Abstract

1. Introduction

2. Main Results

3. Examples

4. Conclusions

References

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