# Resource on Infinite Systems of Difference Equations

In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's properties) infinite systems of difference (NOT differential) equations (linear, autonomous, first-order). Solving countable systems is sufficient for my purpose.

However, I have almost no idea about how to solve such systems (and what are the solvability conditions). I have a feeling that a possible approach is to seek generalizations of the finite-dimensional method to infinite-dimensional spaces via the spectral theory of linear operators, but I have no idea about the details.

So my question is whether there are any resources that deal systematically with infinite systems of difference equations. I would be grateful either for a resource that can be read also without a knowledge of advanced mathematics (that is something readable for non-mathematicians) or, alternatively, for a more advanced resource with a recommendation where to learn the prerequisities. My mathematical background is unfortunately quite basic (I am a computer science major): possibly relevant fields I have a background in are calculus, linear algebra, some fundamentals of modern analysis, difference/differential equations, and basics of functional analysis (however I have very poor knowledge of spectral theory).

I would appreciate any resources dealing with this issue, as well as any recommendations about which parts of mathematics I am supposed to learn (preferably with some recommended resources). I am willing to invest a considerable amount of time into the study, but I would like to follow some recommended path that leads to the goal. In the case that the theory of infinite systems of differential equations is similar, I would appreciate also resources dealing with them.

EDIT: an example of such system: $$x_{n+1} = A \cdot x_n,$$ where $x_n$ are in $\ell^{\infty}$, $x_0$ is given (for instance, let $x_0 = (1,0,1,0,1\ldots)$), and $A$ is a linear operator that can be viewed as an infinite matrix (for instance) $$A = \left(\begin{array}{ccccc}0 & 2 & 0 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 0 & 1 & 0 & 2 & \ldots \\ 0 & 0 & 2 & 0 & \ldots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right).$$ I have not been very careful, if this particular system makes sense and is well-behaved, but the form of my systems should be clear. $A$ is always a bounded operator on $\ell^{\infty}$ that can be (if needed) viewed as an infinite matrix that is $k$-diagonal for some fixed (relative to the system) constant $k$. However, these are details. I am not asking for a detailed solution, but mainly for a resource that treats such (or similar) systems and that will enable me to work out the details by myself.

ONE MORE EDIT: The matrix from the above example is symmetric by mistake. My matrices are not symmetric nor Hermitian in general (the operator is not necessarily self-adjoint).

• Can you give an example of such a system? Even a toy system would help clarify the concepts a bit... Commented Jul 21, 2013 at 17:40
• @StevenStadnicki Thank you for your comment, I have edited the question with an example.
– 042
Commented Jul 22, 2013 at 6:40
• Can you give an example of a problem that has motivated you to consider systems of this form? Commented Jul 27, 2013 at 17:19