Applied Mathematics

Volume 7, Issue 10 (June 2016)

ISSN Print: 2152-7385   ISSN Online: 2152-7393

Google-based Impact Factor: 0.83  Citations  

New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations

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DOI: 10.4236/am.2016.710089    2,117 Downloads   2,564 Views  
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ABSTRACT

New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous difference equations relies on the existence of a positive definite Liapunov function that has an indefinitely small upper bound and whose variation along a given nonautonomous difference equations is negative definite. In this paper, we consider the case that the Liapunov function is only positive definite and its variation is semi-negative definite. At these weaker conditions, we put forward a new asymptotical stability theorem of nonautonomous difference equations by adding to extra conditions on the variation. After that, in addition to the hypotheses of our new asymptotical stability theorem, we obtain a new uniformly asymptotical stability theorem of nonautonomous difference equations provided that the Liapunov function has an indefinitely small upper bound. Example is given to verify our results in the last.

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Zhang, L. and Zhang, C. (2016) New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations. Applied Mathematics, 7, 1023-1031. doi: 10.4236/am.2016.710089.

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