Annali di Matematica Pura ed Applicata

, Volume 109, Issue 1, pp 235–245

# 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
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## References

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