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 \[ \ddot y\left ( t \right ) + \int _0^t {k\left ( {t - s} \right )y\left ( s \right )ds + \varphi \left ( t \right )} \int _0^t {K\left ( {t - s} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _0^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ]} ,\quad t \geq 0,\], \[ \ddot y\left ( t \right ) + \int _1^t {k\left ( {\frac {t}{s}} \right )y\left ( s \right )} \frac {1}{s}ds + \varphi \left ( t \right )\int _1^t {K\left ( {\frac {t}{s}} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _1^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ],\quad t \geq 1,} \], 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.

- K. Gopalsamy,
*Oscillations in integro-differential equations of arbitrary order*, J. Math. Anal. Appl.**126**(1987), no. 1, 100–109. MR**900531**, DOI https://doi.org/10.1016/0022-247X%2887%2990078-3 - Athanassios G. Kartsatos,
*Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order*, Stability of dynamical systems, theory and applications (Proc. Regional Conf., Mississippi State Univ., Mississippi State, Miss., 1975) Dekker, New York, 1977, pp. 17–72. Lecture Notes in Pure and Appl. Math., Vol. 28. MR**0594954** - G. S. Ladde, V. Lakshmikantham, and B. G. Zhang,
*Oscillation theory of differential equations with deviating arguments*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 110, Marcel Dekker, Inc., New York, 1987. MR**1017244** - J. J. Levin,
*Boundedness and oscillation of some Volterra and delay equations*, J. Differential Equations**5**(1969), 369–398. MR**236642**, DOI https://doi.org/10.1016/0022-0396%2869%2990051-5 - A. D. Myshkis,
*Lineĭ nye differentsial′nye uravneniya s zapazdyvayushchim argumentom*, 2nd ed., Izdat. “Nauka”, Moscow, 1972 (Russian). MR**0352648** - V. V. Nemytskii and V. V. Stepanov,
*Qualitative theory of differential equations*, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR**0121520** - C. A. Swanson,
*Comparison and oscillation theory of linear differential equations*, Academic Press, New York-London, 1968. Mathematics in Science and Engineering, Vol. 48. MR**0463570** - En Hao Yang,
*Asymptotic behaviour of certain second order integro-differential equations*, J. Math. Anal. Appl.**106**(1985), no. 1, 132–139. MR**780324**, DOI https://doi.org/10.1016/0022-247X%2885%2990136-2 - En Hao Yang,
*On asymptotic behaviour of certain integro-differential equations*, Proc. Amer. Math. Soc.**90**(1984), no. 2, 271–276. MR**727248**, DOI https://doi.org/10.1090/S0002-9939-1984-0727248-1

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Keywords:
Integrodifferential equations,
asymptotic behavior,
oscillation

Article copyright:
© Copyright 1992
American Mathematical Society