On asymptotic summations of almost diagonal difference systems and their applications

In this paper, we derive some new results of asymptotic summations of left and right almost diagonal difference systems. In particular, we study a special case where the diagonal elements are constants with magnitude 1. Conditions on the perturbation matrices are given under which the system is left and right almost diagonal. An application of this result in a discrete adiabatic oscillator equation is discussed. We also introduce a result for the case when the diagonal matrix is approaching the identity matrix II. An example is given to illustrate that the new result can be applied to some cases that do not satisfy the “Growth Condition”.