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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 39: Difference and functional equations

Introduction

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.

History

Applications and related fields

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 example.)

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.

Subfields

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
  • 39A13: Difference equations, scaling (q-differences) [new in 2000]
  • 39A20: Multiplicative and other generalized difference equations, e.g. of Lyness type [new in 2000]
  • 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
  • 39B55: Orthogonal additivity and other conditional equations [new in 2000]
  • 39B62: Systems of functional equations
  • 39B72: Inequalities involving unknown functions, See also 26A51, 26Dxx
  • 39B82: Stability, separation, extension, and related topics [new in 2000]
  • 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.

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

• Here are the AMS and Goettingen resource pages for area 39.

Selected topics at this site

• 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) )
• Schroeder's equation (iterations of functions near a fixed point)
• 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)).
• Solve f o f = exp to find f
• What comes next after addition, multiplication, exponentiation?
• 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.
• Exploiting the parallels between differential equations and difference equations