Difference Equations, Existence and Uniqueness

Existence and Uniqueness

In the world of differential equations, existence is hard to prove. In fact I still don't have all the details right. Also, the theorem is somewhat constrained. The n^th derivative has to be a linear combination of lower derivatives, where continuous functions of x act as coefficients. In the world of difference equations, the n^th difference can be any function f of the lower order differences, such as 3xy²y′+x². And the theorem is easier to prove.

The solution y(x) is unique only if the initial conditions are set. Let y(0) = c₀, y′(0) = c₁, y′′(0) = c₂, and so on up to n-1. Given these initial constants, a deterministic process builds y(1), y(2), y(3), and so on.

Since the differences at 0 are known, up to order n-1, plug them into f and find the value of the n^th difference at 0. Call this c_n.

If w is the n-1^st difference sequence evaluated at 1, write c_n = w-c_n-1 and solve for w. Do this all the way down the line, until y(1) = y′(0)+y(0).

The difference sequences up to order n-1 have been evaluated at x = 1; apply f and find the n^th difference, evaluated at 1. This allows us to compute the lesser differences at x = 2, then the n^th difference at x = 2, then the lesser differences at x = 3, and so on, building the unique solution y(x).

There are other ways to specify the initial conditions. For instance, specifying y(0) through y(n-1) provides enough information to construct the difference values at 0, up through order n-1. This ensures a unique solution, as described above.