Accurate solution estimates for nonlinear nonautonomous vector difference equations
The paper deals with the vector discrete dynamical system xk+1=Akxk+fk(xk). Thewell-known result by Perron states that this system is asymptotically stable if Ak≡A=const is stable and fk(x)≡f˜(x)=o(‖x‖). Perron's result gives no information about the size of the region of asymptotic stability...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2004-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/S1085337504306184 |
| Summary: | The paper deals with the vector discrete dynamical system
xk+1=Akxk+fk(xk). Thewell-known result by
Perron states that this system is asymptotically stable if
Ak≡A=const
is stable and fk(x)≡f˜(x)=o(‖x‖). Perron's result gives no information about the
size of the region of asymptotic stability and norms of
solutions. In this paper, accurate estimates for the norms of
solutions are derived. They give us stability conditions for
(1.1) and bounds for the region of attraction of the
stationary solution. Our approach is based on the freezing
method for difference equations and on recent estimates for the
powers of a constant matrix. We also discuss applications of our
main result to partial reaction-diffusion difference equations. |
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| ISSN: | 1085-3375 1687-0409 |