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Asymptotic behaviour and oscillation of classes of integrodifferential equations


Author: A. H. Nasr
Journal: Proc. Amer. Math. Soc. 116 (1992), 143-148
MSC: Primary 34K15; Secondary 34K25, 45J05
DOI: https://doi.org/10.1090/S0002-9939-1992-1094505-6
MathSciNet review: 1094505
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Abstract: Under some conditions on the integrodifferential equations

$\displaystyle \ddot y\left( t \right) + \int_0^t {k\left( {t - s} \right)y\left... ...\left( s \right),\dot y\left( s \right)} \right)ds} } \right]} ,\quad t \geq 0,$

,

$\displaystyle \ddot y\left( t \right) + \int_1^t {k\left( {\frac{t}{s}} \right)... ...\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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1094505-6
Keywords: Integrodifferential equations, asymptotic behavior, oscillation
Article copyright: © Copyright 1992 American Mathematical Society

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