Abstract:
The paper considers nonlinear autonomous first-order
difference systems in real finite-dimensional spaces. For these systems, we study
the asymptotic stability of equilibria. The classical sufficient conditions for asymptotic stability of an equilibrium for difference equation generated by a smooth mapping $f$
are as follows. If the spectral radius of the first derivative of the mapping $f$ at the given equilibrium point is strictly less than one, then this equilibrium point is asymptotically stable. In the present paper, new sufficient conditions for asymptotic stability of the equilibrium are given. The obtained conditions are also applicable to some mappings for which the spectral radius mentioned above is equal to one. These conditions are as follows. There exists a punctured neighborhood of the given equilibrium point such that the mapping defining the difference equation is locally contractive around each point of this neighborhood.
We present an example in which the spectral radius mentioned above equals one,
however, all the assumptions of the obtained stability theorem are fulfilled.
It is shown that the known stability sufficient conditions follow from the obtained results.
An important feature of our stability sufficient conditions is that they are applicable to difference equations generated by continuous non-smooth mappings.
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