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ArticleOriginal scientific text

Title

Stabilization of solutions to a differential-delay equation in a Banach space

Authors 1, 2

Affiliations

  1. Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
  2. Mathematical Institute, Academy of Sciences of The Czech Republic, 11567 Praha 1, Czech Republic

Abstract

A parameter dependent nonlinear differential-delay equation in a Banach space is investigated. It is shown that if at the critical value of the parameter the problem satisfies a condition of linearized stability then the problem exhibits a stability which is uniform with respect to the whole range of the parameter values. The general theorem is applied to a diffusion system with applications in biology.

Keywords

ENG
abstract differential-delay equation, dependence on parameter, uniform stability

Bibliography

  1. J. K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977.
  2. J. S. Jung, J. Y. Park and H. J. Kang, Asymptotic behavior of solutions of nonlinear functional differential equations, Internat. J. Math. Math. Sci. 17 (1994), 703-712.
  3. J. J. Koliha and I. Straškraba, Stability in nonlinear evolution problems by means of fixed point theorems, Comment. Math. Univ. Carolin. 38 (1) (1997), to appear.
  4. S. Murakami, Stable equilibrium point of some diffusive functional differential equations, Nonlinear Anal. 25 (1995), 1037-1043.
  5. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  6. C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395-418.
  7. C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable, J. Math. Anal. Appl. 56 (1976), 397-409.
  8. T. Wang, Stability in abstract functional differential equations. Part I. General theorems, J. Math. Anal. Appl. 186 (1994), 534-558.
  9. T. Wang, Stability in abstract functional differential equations, Part II. Applications, J. Math. Anal. Appl. 186 (1994), 835-861.
  10. G. F. Webb, Asymptotic stability for abstract nonlinear functional differential equations, Proc. Amer. Math. Soc. 54 (1976), 225-230.
Annales Polonici Mathematici
1996-1997/65/3
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Pages:
271-281
Main language of publication
English
Received
1996-06-19
Accepted
1996-10-17
Published
1997
Exact and natural sciences