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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

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

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

leads to a functional equation

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

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,   G.P. Akilov,   "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,   B. Szökefalvi-Nagy,   "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.F. Davidenko,  , ''Mathematical programming and related problems. Computational methods'' , Moscow  (1976)  pp. 187–212  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.V. Kantorovich,   ''Uspekhi Mat. Nauk'' , '''11''' :  6  (1956)  pp. 99–116</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.F. Davidenko,  , ''The theory of cubature formulas and computing'' , Novosibirsk  (1980)  pp. 59–65  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.L. Daletskii,   M.G. Krein,   "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> , ''Encyclopaedia of elementary mathematics'' , '''3. Functions and limits''' , Moscow-Leningrad  (1952)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Ya. Atsel',   ''Uspekhi Mat. Nauk'' , '''11''' :  3  (1956)  pp. 3–68</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  G.M. Fichtenholz,   "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR></table>

====Comments====
While equations of such types as  $  A f = g $, 
where  $  A $
is a linear or non-linear operator and  $  f $, 
$  g $
are elements of some (Banach) spaces of functions, can be considered functional equations, they can hardly be considered as typical for the field of mathematics which goes by the name functional equations. Far more typical are the Cauchy equations mentioned above (possibly the most elementary example) and such equations as the Poincaré equation  $  F ( a z ) = a F ( z) ( 1 - F ( z) ) $, 
the equation which is at the heart of chaos theory (and responsible for scaling universality):  $  G ( x) = \lambda  ^ {-} 1 G ( G ( \lambda x ) ) $, 
the functional equation satisfied by the Riemann [[Zeta-function|zeta-function]], and the Yang–Baxter equation, which asks for an  $  n  ^ {2} \times n  ^ {2} $
matrix  $  R ( u) $, 
i.e. a linear morphism  $  \mathbf C  ^ {n} \otimes \mathbf C  ^ {n} \rightarrow \mathbf C  ^ {n} \otimes \mathbf C  ^ {n} $
depending on  $  u $, 
such that

$$ 
( I \otimes R ( u - v ) ) ( R ( u) \otimes I ) ( I \otimes R ( v) ) =
$$

$$ 
= \ 
( R ( v) \otimes I ) ( I \otimes R ( u) ) ( R ( u - v ) \otimes I )
$$

(as morphisms of  $  \mathbf C  ^ {n} \otimes \mathbf C  ^ {n} \otimes \mathbf C  ^ {n} $
into itself).

The first functional equations go back to Antiquity (cf. the first paper of [[#References|[a1]]]) and other early examples (besides the Cauchy equation) are Jensen's functional equation  $  f ( ( x + y ) /2) = ( f ( x) + f ( y) ) /2 $
and d'Alembert's equation  $  g ( x + y ) + g ( x - y ) = 2 g ( x) g ( y) $.

The [[Gamma-function|gamma-function]]  $  \Gamma ( z) $
satisfies the functional equations  $  \Gamma ( z + 1 ) = z \Gamma ( z) $
and  $  \sqrt \pi \Gamma ( 2 z ) = ( 2  ^ {2z} - 1 ) \Gamma ( z) \Gamma ( z + 1 / 2 ) $, 
of which the first one is also known as the Euler functional equation.

For a first idea of the theory of functional equations and its manifold applications cf. [[#References|[a1]]]–[[#References|[a6]]]. Reference [[#References|[a2]]] contains a very extensive and complete bibliography up to 1964.

Two other classical functional equations are the Schröder functional equation

$$ 
f ( h ( x) )  =  c f ( x)
$$

and the narrowly related Abel functional equation

$$ 
f ( h ( x) )  =  f ( x) + 1 .
$$

Consider again the additional theorem functional equation  $  F ( f ( x) , f ( y) , f ( x + y ) ) = 0 $. 
Suppose that  $  F ( u , v , w ) $
is a polynomial in  $  u $, 
$  v $, 
$  w $. 
It is then a theorem of Weierstrass that if  $  f ( x) $
is a meromorphic solution, then  $  f ( x) $
must be a rational function, a rational function of  $   \mathop{\rm exp} ( c x ) $
or an elliptic function. A generalization of this result is at the basis of the classification by Belavin and Drinfel'd [[#References|[a7]]] of the solutions of the classical Yang–Baxter equations with values in a simple Lie algebra  $  \mathfrak g $:

$$ 
[ X  ^ {12} ( u _ {1} , u _ {2} ) ,\ 
X  ^ {13} ( u _ {1} , u _ {2} ) ] +
[ X  ^ {12} ( u _ {1} , u _ {2} ) ,\ 
X  ^ {23} ( u _ {2} , u _ {3} ) ] +
$$

$$ 
+ 
[ X  ^ {13} ( u _ {1} , u _ {3} ) , X  ^ {23} ( u _ {2} , u _ {3} ) ]  =  0 .
$$

Here  $  X ( u , v ) $
is an element of  $  \mathfrak g \otimes \mathfrak g $
and  $  X  ^ {12} ( u , v ) $, 
$  X  ^ {13} ( u , v ) $, 
$  X  ^ {23} ( u , v ) $
are the images of  $  X ( u , v ) $
under the imbeddings  $  \mathfrak g \otimes \mathfrak g \rightarrow U \mathfrak g \otimes U \mathfrak g \otimes U \mathfrak g $, 
$  a \otimes b \mapsto a \otimes b \otimes 1 $, 
$  a \otimes b \mapsto a \otimes 1 \otimes b $, 
$  a \otimes b \mapsto 1 \otimes a \otimes b $, 
respectively, where  $  U \mathfrak g $
is the universal enveloping algebra of  $  \mathfrak g $. 
The Yang–Baxter equations are important in the theory of classical and quantum completely-integrable systems.

Functional differential equations (and delay differential equations) are a topic rather separate from [[#References|[a1]]]–[[#References|[a6]]] and the above. They include such integro-differential equations as studied by V. Volterra in predator-prey models,

$$ 
\dot{N} _ {1} ( t)  =  \left ( \epsilon _ {1} - \gamma _ {1} N _ {2} ( t) - \int\limits _ { - } r ^ { 0 } 
F _ {1} ( - \theta ) N _ {2} ( t + \theta )  d \theta \right ) N _ {1} ( t) ,
$$

$$ 
\dot{N} _ {2} ( t)  =  \left ( - \epsilon _ {2} + \gamma _ {2} N _ {1} ( t) + \int\limits _ { - } r ^ { 0 }  F _ {2} ( - \theta ) N _ {1} ( t + \theta )  d \theta \right ) N _ {2} ( t) ,
$$

and the equation

$$ 
\dot{x} ( t)  =  - \int\limits _ { t- } r ^ { t }  a ( t - u ) g ( ( u ) )  d u ,
$$

which occurs in the study of circulating fuel reactors. Differential equations with deviating arguments (cf. [[Differential equations, ordinary, with distributed arguments|Differential equations, ordinary, with distributed arguments]]) also belong to this general class. An important class is the class of general functional differential equations of retarded type,

$$ 
\dot{x}  =  f ( t , x _ {t} ),
$$

where  $  f $
is a function  $  \mathbf R \times C \rightarrow \mathbf R  ^ {n} $, 
$  C $
is a suitable space of functions  $  ( - \infty , 0 ] \rightarrow \mathbf R  ^ {n} $, 
and  $  x _ {t} $
stands for the function  $  \theta \mapsto x ( t + \theta ) $, 
$  - \infty < \theta \leq  0 $. 
The abbreviation RFDE ( $  = $
retarded functional differential equation) is often used. More general are neutral functional differential equations (NFDE),

$$ 
\dot{x}  =  f ( t , x _ {t} , \dot{x} _ {t} ),
$$

where  $  \dot{x} _ {t} ( \theta ) = \dot{x} ( t + \theta ) $, 
$  - \infty < \theta \leq  0 $. 
The simplest kinds of neutral functional differential equations are the form

$$ 
\dot{x} ( t)  =  f ( t , x ( t) , x ( t - h ) , \dot{x} ( t - h ) )
$$

for a fixed  $  h > 0 $, 
where now  $  f $
is a function  $  \mathbf R \times ( \mathbf R  ^ {n} )  ^ {4} \rightarrow \mathbf R  ^ {n} $, 
$  x \in \mathbf R  ^ {n} $.

A natural classification of functional differential equations is in terms of retarded, neutral and advanced type. For differential equations with deviating argument

$$ 
x ^ {( m _ {0} ) } ( t)  =  f
( t , x ( t) ;  x  ^ {(} 1) ( t) \dots x ^ {( m _ {0} - 1 ) } ( t) ;
$$

$$ 
x ( t - \tau _ {1} ( t) ) \dots x ^ {( m _ {1} ) } ( t - \tau _ {1} ( t) ); \dots ;
$$

$$ 
{} x( t - \tau _ {l} ( t) ) \dots x ^ {( m _ {l} ) } ( t - \tau _ {l} ( t) ))
$$

this classification is as follows. Let  $  \mu = \max _ {1 \leq  i \leq  l }   m _ {i} $. 
Then the equation is respectively of advanced, neutral or retarded type if, respectively,  $  m _ {0} < \mu $, 
$  m _ {0} = \mu $, 
or  $  m _ {0} > \mu $.

A number of standard references on functional differential equations are [[#References|[a8]]]–[[#References|[a10]]]. Reference [[#References|[a11]]] deals with neutral functional differential equations in a control-theoretic setting and [[#References|[a12]]] treats retarded functional differential equations on differentiable manifolds.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Aczél (ed.) , ''Functional equations: history, applications and theory'' , Reidel  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Aczél,   "Functional equations and their applications" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Aczél,   "A short course on functional equations" , Reidel  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Dhombres,   "Some aspects of functional equations" , Chulalongkorn Univ.  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Kuczma,   "Functional equations in a single variable" , PWN  (1968)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Kuczma,   "An introduction to the theory of functional equations and inequalities" , PWN &amp; Univ. Sląski  (1985)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.A. Belavin,   V.G. Drinfel'd,   "On the solutions of the classical Yang–Baxter equations for simple Lie algebras"  ''Funct. Anal. Appl.'' , '''16'''  (1982)  pp. 159–180  ''Funkts. Anal. Prilozh.'' , '''16''' :  3  (1982)  pp. 1–29</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.K. Hale,   "Theory of functional differential equations" , Springer  (1977)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  L.E. El'sgol'ts,   S.B Norkin,   "Introduction to the theory and application of differential equations with deviating arguments" , Acad. Press  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  V.B. Kolmonovskii,   V.R. Nosov,   "Stability of functional differential equations" , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D. Salamon,   "Control and observation of neutral systems" , Pitman  (1984)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S.E.A. Mohammed,   "Retarded functional differential equations. A global point of view" , Pitman  (1978)</TD></TR></table>