# Functional equation

An equation (linear or non-linear) in which the unknown is an element of some specific (function) or abstract Banach space, that is, an equation of the form

$$\tag{1 } P ( x) = y$$

where $P ( x)$ is some, generally speaking non-linear, operator transforming elements of a $B$- space $X$ into elements of a space $Y$ of the same type. If the functional equation contains another numerical (or, in general, functional) parameter $\lambda$, then instead of (1) one writes

$$P ( x; \lambda ) = y,$$

where $x \in X$, $y \in Y$, $\lambda \in \Lambda$, and $\Lambda$ is the parameter space.

Specific or abstract differential equations, both ordinary and partial, integral equations, integro-differential equations, functional-differential, as well as more complicated equations in mathematical analysis are all equations of the type (1), as are also systems of algebraic equations, both finite and infinite, finite-difference equations, etc.

In the linear case one considers functional equations of the first kind $Ax = y$ and of the second kind $x - \lambda Ax = y$, where $A$ is a linear operator from $X$ into $Y$ and $\lambda$ is a parameter. A functional equation of the second kind can formally be written as an equation of the first kind $Tx = y$( $T = I - \lambda A$). However, it turns out to be expedient to separate the identity operator $I$, since $A$ can have better properties than $T$, which enables one to investigate the equation under consideration more fully.

Functional equations are also considered in other spaces, for example in spaces normed by elements from a partially ordered set.

If the solutions of a functional equation are elements of a space of operators, then such a functional equation is called an operator equation, specific or abstract. Here algebraic operator equations can also be linear or non-linear, differential, integral, etc. For example, consider the normed ring $[ X] = [ X \rightarrow X]$ of linear operators sending a $B$- space $X$ into itself and regard in it an ordinary differential equation on the infinite interval $0 \leq \lambda < \infty$:

$$\tag{2 } \frac{dx }{d \lambda } = - Ax (= - xA),\ \ x ( 0) = x _ {0} ,$$

where $A$, $x _ {0} \in [ X]$, and $x ( \lambda )$ is an abstract function with values in the Banach space $[ X]$. This equation is the simplest abstract linear differential operator equation; it is obtained, for example, by applying the direct method of variation of the parameter to construct operators of the form $P ( A) = e ^ {- A \lambda }$, $0 \leq \lambda < \infty$, and, in particular, projection operators $P ( A)$ with unit norm $([ P ( A)] ^ {2} = P ( A))$. Projection operators of the form $P ( A)$, $P ( AC)$ and $P( CA)$, $C \in [ X]$, are used, for example, to construct by the direct method of variation of the parameter explicit and implicit pseudo-inverse operators and pseudo-solutions of linear functional equations, as well as eigen values (eigen spaces) of the operator $A$. Reducing various problems to equation (2) and others is very convenient when developing approximate methods of solution. Operator equations of the form

$$\frac{dx }{d \lambda } = \ A ( \lambda ) x + F ( \lambda ) \ (= xA ( \lambda ) + F ( \lambda )),\ \ x ( 0) = x _ {0} ,$$

where $A ( \lambda )$ and $F ( \lambda )$ are abstract functions with values in $[ X]$, as well as other linear and non-linear operator equations, are also interesting.

In some problems connected with differential and other equations one has to investigate linear algebraic operator equations such as those of the form $Ax + xB = y$. Here $x$ is the unknown and $A$, $B$, $y$ are given linear operators which may take zero values.

Functional equations in the narrow sense are equations in which the unknown functions are connected with the known functions in one or several variables by using composition of functions. For example, let $\phi _ {i} ( x)$, $i = 1 \dots n$, be given functions and let $\Psi ( x) = f ( x, C _ {1} \dots C _ {n} )$, where $C _ {i}$ are arbitrary constants. Eliminating $C _ {i}$ from the $n + 1$ equations

$$\Psi ( \phi _ \nu ( x)) = \ f ( \phi _ \nu ( x), C _ {1} \dots C _ {n} ),\ \ \nu = 0 \dots n,$$

$$\phi _ {0} ( x) = x,$$

$$\tag{3 } F [ x, \Psi ( x), \Psi ( \phi _ {1} ( x)) \dots \Psi ( \phi _ {n} ( x))] = 0,$$

which has solution $\Psi ( x) = f ( x, C _ {1} \dots C _ {n} )$.

Constructing functional equations is a direct problem in functional calculus, analogous to determining higher-order derivatives in differential calculus.

Eliminating $C _ {i}$ from the $n + 1$ equations of the form

$$\Psi ^ {\nu + 1 } ( x) = \ f ( \Psi ^ \nu ( x), C _ {1} \dots C _ {n} ),\ \ \nu = 0 \dots n,$$

$$\Psi ^ {0} ( x) = x, \Psi ^ {1} ( x) = \Psi ( x), \Psi ^ {2} ( x) = \Psi ( \Psi ^ {1} ( x)) \dots$$

leads to a functional equation of the form

$$\tag{4 } F [ x, \Psi ( x), \Psi ^ {2} ( x) \dots \Psi ^ {n + 1 } ( x)] = 0,$$

having solution $\Psi ( x) = f ( x, C _ {1} \dots C _ {n} )$.

Sometimes functional equations are distinguished by order and class. The order of a functional equation is the order of the unknown function in the equation, and the class of a functional equation is the number of given functions to which the unknown function is applied. Thus, (3) is a functional equation of order one and class $n + 1$. Equation (4) is a functional equation of order $n + 1$ and class one.

The relations (3) and (4) are identities with respect to $x$ but are called equations in so much as the function $\Psi ( x)$ is unknown.

Equations (3) and (4) are functional equations with one unknown variable. One can consider functional equations with several independent variables, functional equations of fractional order, etc., as well as systems of compatible functional equations. Moreover, functional equations or systems of functional equations can contain a greater number of essential, essentially-different variables than does the unknown function with the maximum number of variables.

Systems of functional equations occur, for example, when determining arbitrary functions which enter into the integrals of partial differential equations and satisfy the conditions of the problem. If $n$ arbitrary functions enter into the integral, then by subjecting them to $n$ conditions one obtains $n$ compatible functional equations. In certain cases it is convenient to write systems of functional equations in a shorter form as vector or matrix functional equations.

Functional equations can also be regarded as expressing a property characterizing some class of functions (for example, the functional equation $f ( x) = f (- x)$( respectively, $f(- x) = - f( x)$) characterizes the class of even (respectively, odd) functions; the functional equation $f ( x + 1) = f ( x)$ characterizes the class of functions with period 1, etc.).

Some of the simplest forms of functional equations are, for example, the Cauchy equations

$$\tag{5 } \left . \begin{array}{c} f ( x + y) = f ( x) + f ( y),\ \ f ( x + y) = f ( x) f ( y), \\ f ( xy) = f ( x) + f ( y),\ \ f ( xy) = f ( x) f ( y), \\ \end{array} \right \}$$

the continuous solutions of which are, respectively,

$$f ( x) = Cx,\ \ e ^ {Cx} ,\ \ C \mathop{\rm log} x,\ \ x ^ {C} \ \ ( x > 0)$$

(in the class of discontinuous functions there can be other solutions). The functional equations (5), with the additional requirement of continuity, can be used to define the indicated functions. Generalized Cauchy functional equations have also been considered in three and more unknown functions, etc. One has also considered functional equations in the complex domain. Functional equations $F ( f ( x), f ( y), f ( x + y)) = 0$ and $\phi ( f ( x), f ( y), f ( xy)) = 0$ are called the addition theorem and the multiplication theorem for the function $f ( t)$, respectively. For example, the simplest functional equations in which the unknown function depends on two variables are the equations

$$\phi ( x, y) + \phi ( y, z) = \ \phi ( x, z) \ \textrm{ and } \ \ \phi ( x, y) \phi ( y, z) = \ \phi ( x, z),$$

the solutions to which are, respectively,

$$\phi ( y, z) = \ \Psi ( y) - \Psi ( z) \ \ \textrm{ and } \ \ \phi ( x, y) = \ \frac{\Psi ( y) }{\Psi ( x) } ,$$

where $\Psi$ is an arbitrary function.

How to Cite This Entry:
Functional equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_equation&oldid=47014
This article was adapted from an original article by D.F. Davidenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article