Asymptotic behaviour and oscillation of classes of integrodifferential equations
Authors: Nasr A. H. 
Published: June 2010

Under some conditions on the integrodifferential equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove y With two-dots left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline k left-parenthesis t minus s right-parenthesis y left-parenthesis s right-parenthesis d s plus phi left-parenthesis t right-parenthesis integral Subscript 0 Superscript t Baseline upper K left-parenthesis t minus s right-parenthesis ModifyingAbove y With dot left-parenthesis s right-parenthesis d s equals f left-bracket t comma y left-parenthesis t right-parenthesis comma ModifyingAbove y With dot left-parenthesis t right-parenthesis comma integral Subscript 0 Superscript t Baseline g left-parenthesis t comma s comma y left-parenthesis s right-parenthesis comma ModifyingAbove y With dot left-parenthesis s right-parenthesis right-parenthesis d s right-bracket comma t greater-than-or-equal-to 0 comma">

y
¨

(
t
)

+

∫
0
t

k

(

t
−
s

)

y

(
s
)

d
s
+
φ

(
t
)

∫
0
t

K

(

t
−
s

)

y
˙

(
s
)

d
s
=
f

[

t
,
y

(
t
)

,

y
˙

(
t
)

,

∫
0
t

g

(

t
,
s
,
y

(
s
)

,

y
˙

(
s
)

)

d
s

]

,

t
≥
0
,

\ddot y\left ( t \right ) + \int _0^t {k\left ( {t - s} \right )y\left ( s \right )ds + \varphi \left ( t \right )} \int _0^t {K\left ( {t - s} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _0^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ]} ,\quad t \geq 0,

\]
,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove y With two-dots left-parenthesis t right-parenthesis plus integral Subscript 1 Superscript t Baseline k left-parenthesis StartFraction t Over s EndFraction right-parenthesis y left-parenthesis s right-parenthesis StartFraction 1 Over s EndFraction d s plus phi left-parenthesis t right-parenthesis integral Subscript 1 Superscript t Baseline upper K left-parenthesis StartFraction t Over s EndFraction right-parenthesis ModifyingAbove y With dot left-parenthesis s right-parenthesis d s equals f left-bracket t comma y left-parenthesis t right-parenthesis comma ModifyingAbove y With dot left-parenthesis t right-parenthesis comma integral Subscript 1 Superscript t Baseline g left-parenthesis t comma s comma y left-parenthesis s right-parenthesis comma ModifyingAbove y With dot left-parenthesis s right-parenthesis right-parenthesis d s right-bracket comma t greater-than-or-equal-to 1 comma">

y
¨

(
t
)

+

∫
1
t

k

(

t
s

)

y

(
s
)

1
s

d
s
+
φ

(
t
)

∫
1
t

K

(

t
s

)

y
˙

(
s
)

d
s
=
f

[

t
,
y

(
t
)

,

y
˙

(
t
)

,

∫
1
t

g

(

t
,
s
,
y

(
s
)

,

y
˙

(
s
)

)

d
s

]

,

t
≥
1
,

\ddot y\left ( t \right ) + \int _1^t {k\left ( {\frac {t}{s}} \right )y\left ( s \right )} \frac {1}{s}ds + \varphi \left ( t \right )\int _1^t {K\left ( {\frac {t}{s}} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _1^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ],\quad t \geq 1,}

\]
, the explicit asymptote of solutions is proved to be

y

(
t
)

=
A
sin
⁡

(

ω
t
+
δ

)

y\left ( t \right ) = A\sin \left ( {\omega t + \delta } \right )

as

t
→
∞

t \to \infty

. From this asymptote, the oscillatory behavior of the equations, the limit of the amplitudes, and the limit of the distance between consecutive zeros of the solutions are evident. Their definite values are also determined.