A projected discrete Gronwall's inequality with sub-exponential growth

Abstract

Along with the increasing interest in (h, k)-dichotomy, more attentions are paid to sub-exponential growth in research of asymptotic behaviours. In this paper, we generalize a projected discrete Gronwall's inequality given in [J. Differ. Equ. Appl. 10 (2004), 661–689] to a general one, which may include both terms of sub-exponential growth inside the summation and non-monotonic terms outside the summation. We demonstrate our results with concrete non-monotonic functions and sub-exponential functions. We apply our results to estimating bounded solutions of a non-linear difference equation with an (h, k)-dichotomy.

Keywords:

• Gronwall's inequality
• difference equation
• sub-exponential growth
• non-monotonicity
• (h, k)-dichotomy

AMS (2000) Classification::

• 34A40
• 34D09
• 39A11

Acknowledgements

The authors are very grateful to the referees for their carefully checking and patiently suggesting. The article was supported by NSFC #10825104 and MOE research grant.