Equations of convolution type with monotone coefficients

Abstract

This paper studies convolution type linear difference equations with coefficients satisfying some monotonicity properties. Methods from renewal theory are employed to obtain easily verified conditions for asymptotic stability of the zero solution, in terms of the coefficient sequence. Explicit bounds and rates of convergence are also considered, and an application to norms of matrix inverses is included.

Keywords:

• difference equations
• convolution
• asymptotic stability
• renewal sequences
• exponential convergence
• explicit bounds
• matrix inequalities

AMS Subject Classification::

• 39A10
• 39A11
• 60K05
• 15A09
• 15A57

Acknowledgements

We are very thankful to the referees for comments and insights that substantially improved this manuscript.