Studiying dynamical systems, I found in most textbooks proof's of the existence and uniqueness theorems for differential equations, however, I was not able to find an analogous theorem for difference equations. Could someone state the theorem briefly or point me to proper sources?
PS: I'm trained in economics, so I would appreciate if the source was not to technical or heavy on the mathemathics. I have a background on calculus, matrix algebra, some elementary concepts of topology (closed/open, convex, compact sets, etc) but not much else.
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?
@ 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
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