# On the decomposition problem of stability for Volterra integrodifferential equations

On the decomposition problem of stability for Volterra integrodifferential equations In this paper, we discuss the problem of stability of Volterra integrodifferential equation $$\dot x = A(t)x + \int_0^t {C(t,s)} x(s)ds$$ with the decomposition $$\begin{gathered} \dot x_i = A_{ii} (t)x_i + \sum\limits_{\mathop {i = 1}\limits_{j \ne i} }^r {A_{ij} (t)x_j + \int_0^t {\left[ {C_{ii} (t,s)x_i (s) + \sum\limits_{\mathop {j = 1}\limits_{j \ne i} }^r {C_{ij} (t,s)x_j (s)} } \right]} ds,} \hfill \\ i = 1,2, \cdot \cdot \cdot ,r, \hfill \\ \end{gathered}$$ wherex ∈R n,x T=(x 1 T , ...,x r/T),x i ∈R ni and ∑ i=1 r n i=n;A(t)=(A ij(t)) in whichA ij(t) (i,j=1, ...,r) is ann i ×n j matrix of functions continuous for 0 ≤t<∞;C(t,s)=(C ij (t,s,)) in whichC ij(t,s)(i,j=1, ...,r) is ann i ×n j matrix of functions continuous for 0≤s≤t<∞. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# On the decomposition problem of stability for Volterra integrodifferential equations

, Volume 8 (1) – Jul 13, 2005
15 pages      /lp/springer-journals/on-the-decomposition-problem-of-stability-for-volterra-6NdrtCHCrl
Publisher
Springer Journals
Copyright © 1992 by Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02006075
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we discuss the problem of stability of Volterra integrodifferential equation $$\dot x = A(t)x + \int_0^t {C(t,s)} x(s)ds$$ with the decomposition $$\begin{gathered} \dot x_i = A_{ii} (t)x_i + \sum\limits_{\mathop {i = 1}\limits_{j \ne i} }^r {A_{ij} (t)x_j + \int_0^t {\left[ {C_{ii} (t,s)x_i (s) + \sum\limits_{\mathop {j = 1}\limits_{j \ne i} }^r {C_{ij} (t,s)x_j (s)} } \right]} ds,} \hfill \\ i = 1,2, \cdot \cdot \cdot ,r, \hfill \\ \end{gathered}$$ wherex ∈R n,x T=(x 1 T , ...,x r/T),x i ∈R ni and ∑ i=1 r n i=n;A(t)=(A ij(t)) in whichA ij(t) (i,j=1, ...,r) is ann i ×n j matrix of functions continuous for 0 ≤t<∞;C(t,s)=(C ij (t,s,)) in whichC ij(t,s)(i,j=1, ...,r) is ann i ×n j matrix of functions continuous for 0≤s≤t<∞.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 13, 2005

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