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Annali di Matematica Pura ed Applicata

, Volume 109, Issue 1, pp 235–245 | Cite as

Asymptotic behavior of the solutions of an integrodifferential system

  • Deh-phone Kung Hsing
Article

Summary

We consider the system(L):\(y'(t) = \sum\limits_{i = 1}^\infty {A_i y(t - \tau _i )} + \int\limits_{ - \infty }^t {B(t - s)y(s)ds} \), t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each Ai, B(t) are n × n matrices. System(L) generates a semigroup by means of Ttf(s)=y (t+s, f), f(s) ∈ BCl(− ∞, 0]. Under some hypotheses concerning the roots ofdet\([\lambda 1 - \mathop \Sigma \limits_{i = 1}^\infty A_i exp[ - \lambda \tau _i ] - \widehat{B(\lambda )}] = 0\) where\(\widehat{B(\lambda )}\) is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L1[0, ∞),\(\left\| {B(t)} \right\|\varepsilon L_1 [0,\infty ),\mathop \Sigma \limits_{i = 1}^\infty \left\| {A_i } \right\|< \infty \) thendet\(\left[ {\lambda I - \mathop \Sigma \limits_{i = 1}^\infty A_i \exp [ - \lambda \tau _i ] - \widehat{B(\lambda )}} \right] \ne 0\) forRe λ>0 iff for every ɛ>0 there is an Mɛ>0 such that ∥Ttf∥l ⩽ ⩽ Mɛexp [ɛt]∥f∥l for t ⩾ 0. Corollary 3.1.1: suppose\(\mathop \Sigma \limits_{i = 1}^\infty \left\| {A_i } \right\|\exp [a\tau _i ]< \infty \)exp [at]B(t) ∈ ∈ L1[0, ∞) for some a>0 anddet\(\left[ {\lambda I - \mathop \Sigma \limits_{i = 1}^\infty A_i \exp [ - \lambda \tau _i ] - \widehat{B(\lambda )}} \right] \ne 0\) forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable.

Keywords

Asymptotic Behavior Typical Result Integrodifferential System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    E. Hille -R. S. Phillips,Functional analysis and semigroups, rev. ed. Amer. Math. Colloq. Publ. 1, vol. 31, Amer. Math. Soc. Providence, R. I. (1957), MR 19, pag. 664.Google Scholar
  2. [2]
    V. Barbu -S. Grossman,Asymptotic behavior of linear integrodifferential systems, Trans. of the Amer. Math. Soc.,173 (1972), pp. 277–288.CrossRefMathSciNetGoogle Scholar
  3. [3]
    J. K. Hale,Functional differential equations with infinite delays, J. Math. Analysis Appl. (in print).Google Scholar

Copyright information

© Fondazione Annali di Matematica Para ed Applicata 1975

Authors and Affiliations

  • Deh-phone Kung Hsing
    • 1
  1. 1.KingstonU.S.A.

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