Asymptotic behaviour and oscillation of classes of integrodifferential equations

Abstract:

Under some conditions on the integrodifferential equations \[ \ddot y\left ( t \right ) + \int _0^t {k\left ( {t - s} \right )y\left ( s \right )ds + \varphi \left ( t \right )} \int _0^t {K\left ( {t - s} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _0^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ]} ,\quad t \geq 0,\], \[ \ddot y\left ( t \right ) + \int _1^t {k\left ( {\frac {t}{s}} \right )y\left ( s \right )} \frac {1}{s}ds + \varphi \left ( t \right )\int _1^t {K\left ( {\frac {t}{s}} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _1^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ],\quad t \geq 1,} \], the explicit asymptote of solutions is proved to be $y\left ( t \right ) = A\sin \left ( {\omega t + \delta } \right )$ as $t \to \infty$. 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.