Existence of solutions to recurrence relations
There are many open problems in this area. Here is one from the book by Kulenovic and Ladas.
A solution to the recurrence relation is a sequence of real numbers such that for all .
The existence problem for the recurrence relation is this: for which values of is there a solution of the recurrence relation with ?
Kulenovic, M.R.S. & Ladas, G. (2002) Dynamics of second order rational difference equations. Chapman & Hall/CRC.
Camouzis, E., DeVault, R. & Lada, G. (2001) On the recursive sequence . J. Diff Equations Applied, vol 7.