Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1094505
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] K. Gopalsamy, Oscillations in integrodifferential equations of arbitrary order, J. Math. Anal. Appl. 126 (1987), 100-109. MR 900531 (88g:45012)
  • [2] A. G. Kartsatos, Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, Lecture Notes in Pure and Appl. Math., vol. 28, Dekker, New York, 1977, pp. 17-72. MR 0594954 (58:28853)
  • [3] G S. Ladde, V. Lashmikanthan, and B. G. Zhang, Oscillation theory of differential equations with deviating arguments, Marcel Dekker, New York, 1987. MR 1017244 (90h:34118)
  • [4] J. L. Levin, Boundedness and oscillation of some Volterra and delay equations, J. Differential Equations 5 (1969), 369-398. MR 0236642 (38:4937)
  • [5] A. D. Myskis, Linear differential equations with deviated arguments, 2nd ed., Nauka, Moscow, 1972. (Russian) MR 0352648 (50:5135)
  • [6] V. V. Nemeickii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Univ. Press, Princeton, NJ, 1960. MR 0121520 (22:12258)
  • [7] C. A. Swanson, Comparison and oscillation theory of linear differential equations, Academic Press, New York, 1968. MR 0463570 (57:3515)
  • [8] E. H. Yang, Asymptotic behaviour of certain second order integrodifferential equations, J. Math. Anal. Appl. 106 (1985), 132-139. MR 780324 (86e:45012)
  • [9] -, On asymptotic behaviour of certain integrodifferential equations, Proc. Amer. Math. Soc. 90 (1984), 271-276. MR 727248 (86g:45026)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34K15, 34K25, 45J05

Retrieve articles in all journals with MSC: 34K15, 34K25, 45J05

Additional Information

Keywords: Integrodifferential equations, asymptotic behavior, oscillation
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society