Functional equations are those in which a function is sought which is
to satisfy certain relations among its values at all points. For example,
we may look for functions satisfying f(x*y)=f(x)+f(y) and enquire whether
the logarithm function f(x)=log(x) is the only solution. (It's not.)
In some cases the nature of the answer is different when we insist that
the functional equation hold for all real x, or all complex x, or only
those in certain domains, for example.
A special case involves difference equations, that is, equations
comparing f(x) - f(x-1), for example, with some expression involving
x and f(x). In some ways these are discrete analogues of differential
equations; thus we face similar questions of existence and uniqueness
of solutions, global behaviour, and computational stability.
When the focus of a functional equation is on continuity of
functions and a domain is specified, this becomes a question of
topology (in particular this sometimes becomes
questions about the group of homeomorphism or diffeomorphisms of a
set. Thus see the manifolds page, for
Functions whose domains are integers are sequences,
of course; thus a functional equation with this domain is essentially
a recursion problem. These are frequently seen among
sequences of integers.
Functional equations are often studied by considering the orbits of points
in the domain under iterates of some function. This then becomes the
purview of dynamical systems (58FXX) .
Functions of one variable which satisfy a difference equation will
tend to follow patterns set by ordinary differential
equations; naturally functions of two or more variables behave more
like solutions of partial differential equations.
There are only two subfields, which are then further subdivided:
- 39A: Difference equations, For dynamical systems, see 58FXX
- 39A05: General
- 39A10: Difference equations, See also 33Dxx
- 39A11: Stability of difference equations
- 39A12: Discrete version of topics in analysis
- 39A70: Difference operators, See also 47B39
- 39A99: None of the above but in this section
- 39B: Functional equations, see also 30D05
- 39B05: General
- 39B12: Iteration theory, iterative and composite equations, See also 26A18, 30D05, 58F08
- 39B22: Equations for real functions
- 39B32: Equations for complex functions, See also 30D05
- 39B42: Matrix and operator equations
- 39B52: Equations for functions with more general domains and/or ranges
- 39B62: Systems of functional equations
- 39B72: Inequalities involving unknown functions, See also 26A51, 26Dxx
- 39B99: None of the above but in this section
This is among the smaller areas in the Math Reviews database.
Browse all (old) classifications for this area at the AMS.
- Introductory remarks to the calculus of finite differences.
- Linear difference equations with constant coefficients: summary of pointers
- General remarks on solving functional equations, using f( x^2/(4x-2) ) = (x-1)/x f(x) as the example.
- A sample functional equation: solve f(x) + a = f( x + a*sqrt(x) )
- How many homeomorphisms of an interval are there, having order 2, that is, having f(f(x))=x ?
- Using the Intermediate Value Theorem to disallow functions with f(f(f(x)))=x
- What functions have the property that their n-fold iterates are the identity? (e.g. f(f(f(f(f(x)))))=x).
- Need bounded functions with reciprocal symmetry, that is, f(x)+f(1/x)=1.
- Recognizing the reciprocal function from the equation xy f(x+y) [f(x)+f(y)] = 1.
- Solving the functional equation f(ax+b)=cf(x)+d
- Square roots of the exponential function, that is, f(f(x))=exp(x). Note that this is closely related to the construction of a "tower" function, e.g. the solutions to f(x+1)=exp(f(x)).
- Defining k-fold iteration of a function where k is a real parameter: the Abel equation.
- Cesàro's (singular) solution to f(x)= p*f(2x) for x < 1/2, f(x)=(1-p)*f(2x-1) + p for x > 1/2.
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