Hiaa2010,

I am assuming that you are asking about the asymptotic nature of the difference equation

for

under the initial conditions

, and .

Here, is the forward difference operator, i.e., and .

For such problems, you need to compute the fundamental solutions, i.e.,

the solutions of the following initial value problems

: for with , and

: for with , and

: for with , and .

Then, by the variation of constants formula, we can write the solution as follows

for .

Here, we have

for

for

and

for .

Assume that , then for , which yields

.........

for all .

Thus, if (the coefficients of the dominant term ), then as .

Obtaining a complete result for your problem requires a deep inspection since the forcing term is not a nice one.