Oscillation and nonoscillation of even-order nonlinear delay-differential equations
Authors: Singh Bhagat 
Published: December 2016

The purpose of this paper is to compare the equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis 2 n right-parenthesis Baseline left-parenthesis x right-parenthesis plus p left-parenthesis x right-parenthesis g left-parenthesis y left-parenthesis x right-parenthesis comma y Subscript tau Baseline left-parenthesis x right-parenthesis right-parenthesis equals 0">

y

(

2
n

)

(
x
)

+
p

(
x
)

g

(

y

(
x
)

,

y
τ

(
x
)

)

=
0

{y^{\left ( {2n} \right )}}\left ( x \right ) + p\left ( x \right )g\left ( {y\left ( x \right ),{y_\tau }\left ( x \right )} \right ) = 0

\]
(1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis 2 n right-parenthesis Baseline left-parenthesis x right-parenthesis plus p left-parenthesis x right-parenthesis g left-parenthesis y left-parenthesis x right-parenthesis comma y Subscript tau Baseline left-parenthesis x right-parenthesis right-parenthesis equals f left-parenthesis x right-parenthesis">

y

(

2
n

)

(
x
)

+
p

(
x
)

g

(

y

(
x
)

,

y
τ

(
x
)

)

=
f

(
x
)

{y^{\left ( {2n} \right )}}\left ( x \right ) + p\left ( x \right )g\left ( {y\left ( x \right ),{y_\tau }\left ( x \right )} \right ) = f\left ( x \right )

\]
(2) for their oscillatory and nonoscillatory nature. In Eqs. (1) and (2)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis x right-parenthesis identical-to left-parenthesis d Superscript i Baseline slash d x Superscript i Baseline right-parenthesis y left-parenthesis x right-parenthesis comma i equals 1 comma 2 comma period period period comma 2 n">

y

(
i
)

(
x
)
≡

(

d
i

/

d

x
i

)

y
(
x
)
,
i
=
1
,
2
,
.
.
.
,
2
n

{y^{(i)}}(x) \equiv \\ \left ( {{d^i}/d{x^i}} \right )y(x),i = 1,2,...,2n

;

t
2

>

t
1

{t_2} > {t_1}

;

y
τ

(
x
)
≡
y

(

x
−
τ

(
x
)

)

{y_\tau }(x) \equiv y\left ( {x - \tau \left ( x \right )} \right )

;

d
y

/

d
x

dy/dx

and

d
2

y

/

d

x
2

{d^2}y/d{x^2}

will also be denoted by

y
′

y’

and

y
y

respectively. Throughout this paper it will be assumed that

p

(
x
)

p\left ( x \right )

,

f

(
x
)

f\left ( x \right )

,

τ

(
x
)

\tau \left ( x \right )

are continuous real-valued functions on the real line

(

−
∞
,
∞

)

\left ( { - \infty ,\infty } \right )

;

f

(
x
)

f\left ( x \right )

,

p

(
x
)

p\left ( x \right )

and

τ

(
x
)

\tau \left ( x \right )

, in addition, are nonnegative,

τ

(
x
)

\tau \left ( x \right )

is bounded and

f

(
x
)

f\left ( x \right )

,

p

(
x
)

p\left ( x \right )

eventually become positive to the right of the origin. In regard to the function

g
g

we assume the following: (i)

g
:

R
2

→
R

g:{R^2} \to R

is continuous,

R
R

being the real line, (ii)

g

(

λ
x
,
λ
y

)

=

λ

2
q
+
1

g

(

x
,
y

)

g\left ( {\lambda x,\lambda y} \right ) = {\lambda ^{2q + 1}}g\left ( {x,y} \right )

for all real

λ
≠
0

\lambda \ne 0

and some integer

q
≥
1

q \ge 1

, (iii)

sgn
⁡
g

(

x
,
y

)

=
sgn
⁡
x

\operatorname {sgn} g\left ( {x,y} \right ) = \operatorname {sgn} x

, (iv)

g

(

x
,
y

)

→
∞

g\left ( {x,y} \right ) \to \infty

as

x
,
y
→
∞

x,y \to \infty

;

g
g

is increasing in both arguments monotonically. Eq. (1) is called oscillatory if every nontrivial solution

y

(
t
)

∈

[

t
0

,
∞

)

y\left ( t \right ) \in \left [ {{t_0},\infty } \right )

has arbitrarily large zeros; i.e., for every such solution

y

(
t
)

y\left ( t \right )

, if

y

(

t
1

)

=
0

y\left ( {{t_1}} \right ) = 0

then there exists

t
2

>

t
1

{t_2} > {t_1}

such that

y

(

t
2

)

=
0

y\left ( {{t_2}} \right ) = 0

. Eq. (1) is called nonoscillatory if it has a solution with a last zero or no zero in

[

t
0

,
∞

)

,

t
0

≥
a
>
0

\left [ {{t_0},\infty } \right ), {t_0} \ge a > 0

. A similar definition holds for eq. (2). All solutions of (1) and (2) considered henceforth are continuous and nontrivial, existing on some halfline

[

t
0

,
∞

)

\left [ {{t_0},\infty } \right )

.