Proceedings of the American Mathematical Society

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

 

 

Uniform asymptotic stability via Liapunov-Razumikhin technique


Author: James H. Liu
Journal: Proc. Amer. Math. Soc. 123 (1995), 2465-2471
MSC: Primary 45J05; Secondary 34K20, 45M10
MathSciNet review: 1257116
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Abstract | References | Similar Articles | Additional Information

Abstract: The Liapunov-Razumikhin technique is applied to obtain the uniform asymptotic stability for linear integrodifferential equations in Hilbert spaces,

$\displaystyle x'(t) = A\left[ {x(t) + \int_\char93 ^t {F(t - s)x(s)\,ds} } \right],\quad t \geq {t_0} \geq 0(\char93 = 0\;{\text{or}} - \infty ),$

which occur in viscoelasticity and in heat conduction for materials with memory.

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

  • [1] T. A. Burton, Stability and periodic solutions of ordinary and functional-differential equations, Mathematics in Science and Engineering, vol. 178, Academic Press, Inc., Orlando, FL, 1985. MR 837654
  • [2] R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21 (1990), no. 2, 374–393. MR 1038898, 10.1137/0521021
  • [3] Ronald Grimmer and He Tao Liu, Liapunov-Razumikhin methods for integrodifferential equations in Hilbert space, Delay and differential equations (Ames, IA, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 9–24. MR 1170140
  • [4] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319
  • [5] Ronald Grimmer and George Seifert, Stability properties of Volterra integrodifferential equations, J. Differential Equations 19 (1975), no. 1, 142–166. MR 0388002
  • [6] George Seifert, Liapunov-Razumikhin conditions for stability and boundedness of functional differential equations of Volterra type, J. Differential Equations 14 (1973), 424–430. MR 0492745
  • [7] George Seifert, Liapunov-Razumikhin conditions for asymptotic stability in functional differential equations of Volterra type, J. Differential Equations 16 (1974), 289–297. MR 0460837

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1257116-8
Keywords: Uniform asymptotic stability, Liapunov-Razumikhin technique
Article copyright: © Copyright 1995 American Mathematical Society