Actually in discrete case, existence is obvious and trivial so it doesn't mentioned. consider case: x_n=f(x_{n-1}), starting from x_0, existence of solution comes from existence of value for f(x), it means that if x in definition domain of function, then solution exist. In differential case, existence isn't trivial and that why there are different theorem for it.

Rick, that is not what I am looking for. The theorem is only useful for a first-order equation. I'm looking for a theorem which is valid for a non-linear, nth order, non-autonomous, ordinary difference equation. I wouldn't ask if I had not checked wikipedia and a bunch of texbooks first...

Y. Azizi, I did not get your answer because the redaction is a little bit confusing. Could you clarify what you mean that by the existence of x in the domain of f(x)?

it means that when you define function: f:I-->J which I and J are some specific sets, then only condition on existence comes from x be in x. example: f(x)=arcsin(x), then x can goes beyond [-1,1],  and function becomes non-defined! for difference equation this is only important point in existence problem. Picard theorem deals with ode not difference equation.

in above example: start with x=1-->f(1)=pi/2--->f(pi/2)=?, therefore in this case solution exist only for finite number of iteration! (of course in application usually people define function such that J be a subset of I, otherwise this problem arise).

I hope you see the point. if you have difference equation, you can solve it recurrently, unless above problem arise.  In ode this condition, isn't enough, and non-global solution can arise easily, example: x'=-x^2, while everything seems good but it has singularity in finite time.

Thank you! Now I understand your point. Where can I find a formal theorem stating what you just explained?

it's about functions and maps (more formal name of difference equation is map), you can find it on text related to function definition in analysis

@ Alejandro González

The existence of any solution of the equation of the form   x_{n+1} = f(n; x_n)  for n=0,1,2,. . .  with arbitrary  given initial  x_0 \in  X    is elementary, whenever  the ranges (values' spaces) of  all  functions  [ f(n; x), x \in X],  numbered by    n=0,1,2, ....,  are contained in  X  (in symbols:   f(n; X) \subset X, for every  n ), and if no particular features of the sequence  (x_0, x_1, x_2, ... )  are required.

However, already in a case when e.g. a bounded solutions are sought, the problem  becomes highly dependent on properties of ALL functions   f(n; . ).  Even if  the equation describes an autonomous linear system evolving in  a finite dimensional  real space, it could happen that zero is the only bounded solution.  A simple illustration gives the equation   

f(n; x)  =  2 x , for  x\in R, for every  n.

This is due to the formula  x_n = 2^n \cdot x_0, for all n.

And how about  the following example:    f(n; x) =  c_n \cdot x?  Then, obviously, a bounded non-zero solution EXISTS if, and only if, the sequence of products

c_0 \cdot  c_1 \cdot  c_2  ... \cdot  c_N ,   N=0,1,2, . .

.is bounded.

It should be clear, that there are much more complicated cases, for instance equations of at least second order, neither of the forward, nor of the backward type, like this

x_n =  f(n; x_{n-1}, x_n, x_{n+1} ), for all n

Then, even the conditions sufficient for existence of    the solution   x   of a single equation of the form     x =  f( x_{n-1}, x , x_{n+1} )  might be a problem; and this problem for non-homogeneous (in time  n ) systems should be somehow formulated and  solved (if you want to have a theorem on existence of the global solution).

For some examples of existence theorems, let me suggest the  book by Saber Elaydi:

An Introduction to Difference Equations, Third Edition, edited by Springer, available at

http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/An-introduction-to-difference-equation-by-Elaydi.pdf

found through TheGoogle:)

Regards