View source for Non-linear operator

<!--
n0671501.png
$#A+1 = 188 n = 0
$#C+1 = 188 : ~/encyclopedia/old_files/data/N067/N.0607150 Non\AAhlinear operator
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A mapping  $  A $
of a space (as a rule, a vector space)  $  X $
into a vector space  $  Y $
over a common field of scalars that does not have the property of linearity, that is, such that generally speaking

$$ 
A ( \alpha _ {1} x _ {1} + \alpha _ {2} x _ {2} )  \neq \ 
\alpha _ {1} A x _ {1} + \alpha _ {2} A x _ {2} .
$$

If  $  Y $
is the set  $  \mathbf R $
of real or  $  \mathbf C $
of complex numbers, then a non-linear operator is called a [[Non-linear functional|non-linear functional]]. The simplest example of a non-linear operator (non-linear functional) is a real-valued function of a real argument other than a linear function. One of the important sources of the origin of non-linear operators are problems in mathematical physics. If in a local mathematical description of a process small quantities not only of the first but also of higher orders are taken into account, then there arise equations with non-linear operators. Certain problems in mathematical economics, auto-regulation, control theory, etc., also lead to non-linear operator equations.

Examples of non-linear operators.

1)

$$ 
Ax  =  \int\limits _ { a } ^ { b }  K ( t , s , x ( s) )  ds,
$$

where  $  K ( t , s , u ) $, 
$  a \leq  t $, 
$  s \leq  b $, 
$  - \infty < u < \infty $, 
is a function such that  $  g ( t) = \int _ {a}  ^ {b} K ( t , s , x ( s) )  ds $
is continuous on  $  [ a , b ] $
for any  $  x ( s) \in C ( a , b ) $(
for example,  $  K ( t , s , u ) $
is continuous on  $  a \leq  t $, 
$  s \leq  b $, 
$  - \infty < u < \infty $). 
If  $  K ( t , s , u ) $
is non-linear in  $  u $, 
then  $  A $
is a non-linear Urysohn operator mapping  $  C [ a , b ] $
into itself. Under other restrictions on  $  K ( t , s , u ) $
an Urysohn operator acts on other spaces, for instance,  $  L _ {2} [ a , b ] $
or maps one Orlicz space  $  L _ {M _ {1}  } [ a , b ] $
into another  $  L _ {M _ {2}  } [ a , b ] $.

2)

$$ 
Bx  =  \int\limits _ { a } ^ { b }  K ( t , s ) g ( s , x ( s) )  ds ,
$$

where  $  g ( t , u ) $
is non-linear in  $  u $
and defined for  $  a \leq  t \leq  b $, 
$  - \infty < u < \infty $. 
Under appropriate restrictions on  $  g ( t , u ) $
the operator  $  B $
acts from one function space into another and is called a non-linear Hammerstein operator.

3)

$$ 
F ( x)  =  f ( t , x ( t) )
$$

is a superposition operator, also called a Nemytskii operator, and, under suitable restrictions on the non-linearity in the second argument of the function, it transforms the space of measurable functions  $  x ( t) $
into itself.

4)

$$ 
D ( x)  =  \sum _ {| k | \leq  m }
D  ^ {k} ( a _ {k} ( t , x , Dx \dots D  ^ {k} x ))
$$

is a non-linear differential operator of order  $  2m $
in divergence form acting on the Sobolev space  $  W _  \rho   ^ {2m} ( G) $
under suitable restrictions on the non-linear function  $  a _ {k} ( t , u _ {0} \dots u _ {m} ) $. 
Here  $  k $
is the multi-index  $  ( k _ {1} \dots k _ {n} ) $, 
$  | k | = k _ {1} + \dots + k _ {n} $, 
$  D  ^ {k} = {\partial  ^ {| k | } } / {\partial  t _ {1} ^ {k _ {1} } \dots \partial  t _ {n} ^ {k _ {n} } } $
and  $  G $
is a bounded domain in  $  \mathbf R  ^ {n} $.

5)

$$ 
J ( x)  =  \int\limits _ { a } ^ { b } 
K ( t , s , x ( s) , x  ^  \prime  ( s) )  ds
$$

is non-linear integro-differential operator acting under appropriate restrictions on the function  $  K ( t , s , u _ {0} , u _ {1} ) $
in the space  $  C  ^ {1} [ a , b ] $
of continuously-differentiable functions.

To non-linear operators acting from one topological vector space  $  X $
into another one  $  Y $, 
many concepts and operations of mathematical analysis of real-valued functions of a real variable can be transferred. Thus, a non-linear operator  $  A :  M \rightarrow Y $, 
$  M \subset  X $, 
is called bounded if  $  A ( B \cap M ) $
is a bounded set in  $  Y $
for any bounded set  $  B \subset  X $; 
a non-linear operator  $  A $
is continuous at a point  $  x \in M $
if the inverse image  $  A  ^ {-} 1 ( U _ {Ax} ) $
of a neighbourhood  $  U _ {Ax} $
of the point  $  Ax $
contains  $  M \cap U _ {x} $
for some neighbourhood  $  U _ {x} $
of  $  x $. 
As for functions, a non-linear operator that is continuous at every point of a compact set  $  M $
is bounded on this set. In contrast to linear operators, if a non-linear operator  $  A $
acting on a normed space is bounded on some ball, it does not follow that  $  A $
is continuous on this ball. However, in certain cases continuity (boundedness) of a non-linear operator on a ball implies continuity (boundedness) of the operator in its whole domain of definition.

Among the non-linear operators acting from  $  X $
to  $  Y $
one can distinguish certain important classes.

1) Semi-linear operators  $  A :  X \times \dots \times X \rightarrow Y $, 
linear in each argument. The space  $  L _ {n} ( X , Y ) = ( I) $
of all  $  n $-
linear operators is isomorphic to the space  $  L \{ X [ \dots L ( X , Y ) , .  .  . ] \} = ( II) $, 
where  $  L ( X , Y ) $
is the space of all linear operators from  $  X $
to  $  Y $. 
If  $  X $
and  $  Y $
are normed spaces, then  $  ( I) $
and  $  ( II) $
are isometric. If  $  A $
is symmetric in all arguments, then  $  \widetilde{A}  ( x \dots x ) $
is denoted by  $  \widetilde{A}  x  ^ {n} $
and is called a homogeneous operator of degree  $  n $.

2) In spaces endowed with a partial order, isotone operators  $  A $
and antitone operators  $  \widetilde{A}  $
are characterized by the conditions  $  x \leq  y \Rightarrow Ax \leq  Ay $
and  $  x \leq  y \Rightarrow \widetilde{A}  x \geq  \widetilde{A}  y $.

3) In a Hilbert space  $  H $, 
monotone operators  $  M $
are defined by the condition  $  \langle  Mx - My , x - y \rangle \geq  0 $
for any  $  x , y \in H $.

4) Compact operators transform bounded subsets in the domain of definition into pre-compact sets; among them are the completely-continuous operators, which are simultaneously compact and continuous.

For non-linear operators the concepts of a differential and a derivative are non-trivial and useful. An operator  $  A $
acting from an open set  $  G $
of a normed vector space  $  X $
into a normed vector space  $  Y $
is called Fréchet differentiable at a point  $  x \in G $
if there exists a continuous linear operator  $  A  ^  \prime  ( x) :  X \rightarrow Y $
such that for any  $  h \in X $
for which  $  x + h \in G $,

$$ 
A ( x+ h ) - A ( x)  =  A  ^  \prime  ( x) h + \omega ,
$$

where  $  \omega / \| h \| \rightarrow 0 $
as  $  h \rightarrow 0 $. 
In this case the linear part  $  A  ^  \prime  ( x) h $
in  $  h $
of the increment  $  A ( x+ h) - A ( x) $
is called the Fréchet differential of  $  A $
at  $  x $
and is denoted by  $  dA ( x , h ) $, 
and  $  \omega = \omega ( A , x , h ) $
is called the remainder of the increment. The bounded linear operator  $  A  ^  \prime  ( x) $
is called the Fréchet derivative of  $  A $
at  $  x $. 
Apart from Fréchet differentiability one also introduces Gâteaux differentiability. Namely, an operator  $  A $
is called Gâteaux differentiable at a point  $  x $
if the limit

$$ 
\lim\limits _ {t \rightarrow 0 } \ 

\frac{A ( x+ th ) - A ( x) }{t}
  =  DA ( x , h )
$$

exists, which is called the Gâteaux differential of  $  A $
at  $  x $. 
The Gâteaux differential is homogeneous in  $  h $, 
that is,  $  DA ( x , \lambda h ) = \lambda DA ( x , h ) $. 
If  $  DA ( x , h ) $
is linear in  $  h $
and  $  DA ( x , h ) = A _ {0}  ^  \prime  ( x) h $, 
then the linear operator  $  A _ {0}  ^  \prime  ( x) $
is called the Gâteaux derivative of  $  A $. 
Fréchet differentiability implies Gâteaux differentiability, and then  $  A _ {0}  ^  \prime  ( x) = A  ^  \prime  ( x) $. 
Gâteaux differentiability does not, in general, imply Fréchet differentiability, but if  $  DA ( x , h ) $
exists in a neighbourhood of  $  x $, 
is continuous in  $  h $
and uniformly continuous in  $  x $, 
then  $  A $
is Fréchet differentiable at  $  x $. 
For non-linear functionals  $  f :  G \rightarrow \mathbf R $
Fréchet and Gâteaux differentials and derivatives are defined similarly. Here the Gâteaux derivative  $  f _ {0} ^ { \prime } $
is called the gradient of the functional  $  f $
and is an operator from  $  G $
to  $  X  ^ {*} $. 
If  $  Ax =  \mathop{\rm grad}  f ( x) $
for some non-linear functional  $  f $, 
then  $  A $
is called a potential operator.

For operators acting on separable topological vector spaces one can in one way or another define differentiation. Let  $  \mathfrak M $
be a collection of bounded sets in a topological vector space  $  X $. 
A mapping  $  \omega :  G \times X \rightarrow Y $
is called  $  \mathfrak M $-
small if  $  \omega ( x , th ) / t \rightarrow 0 $
as  $  t \rightarrow 0 $
uniformly in  $  h \in \mathfrak M $
for any  $  M \in \mathfrak M $. 
A mapping  $  A :  G \rightarrow Y $(
where  $  G \subset  X $
is open) is called  $  \mathfrak M $-
differentiable at  $  x \in G $
if

$$ 
A ( x+ h ) - A x  =  A  ^  \prime  ( x) h + \omega ( A , x , h ) ,
$$

where  $  \omega $
is an  $  \mathfrak M $-
small mapping. Most frequently  $  \mathfrak M $
is taken to be the collection of all bounded, all compact or all finite sets of  $  X $. 
For non-linear operators on normed spaces the first case leads to Fréchet differentiability and the third to Gâteaux differentiability.

Higher-order derivatives  $  A  ^ {(} n) ( x) $
and  $  A _ {0}  ^ {(} n) ( x) $
of an operator  $  A $
are defined in the usual way, as derivatives of derivatives. These are symmetric multi-linear mappings. A differential of order  $  n $
is then a homogeneous form  $  A  ^ {(} n) ( x) h  ^ {n} $
of degree  $  n $. 
Other definitions of higher-order derivatives are possible. Suppose, for example, that  $  X $
and  $  Y $
are normed vector spaces,  $  G \subset  X $
is open, and  $  x \in G $. 
If for any  $  h $
for which  $  x + h \in G $,

$$ \tag{* }
A ( x+ h) - A( x)  = \ 
a _ {0} ( x) + a _ {1} ( x) h + \dots + a _ {n} ( x) h  ^ {n} +
\omega ,
$$

where  $  \omega = o ( \| h \|  ^ {n} ) $, 
then the multi-linear form  $  k! a _ {k} ( x) $
is called the derivative of order  $  k $. 
The expression (*) is then called the bounded expansion of order  $  n $
of the difference  $  A( x+ h) - A ( x) $. 
Under appropriate restrictions the various definitions of higher-order derivatives are equivalent.

If a scalar countably-additive measure is given in  $  X $, 
then a non-linear operator can be integrated, by understanding  $  \int A ( x)  dx $
in the sense of the [[Bochner integral|Bochner integral]].

For a non-linear operator  $  A :  M \rightarrow Y $, 
as in the case of a linear operator, the values of the parameter  $  \lambda $
for which  $  ( I - \lambda A )  ^ {-} 1 $
exists and is continuous on  $  A ( M) $
are naturally called regular, and the remaining points  $  \lambda $
belong to the spectrum. In its properties the spectrum of a non-linear operator  $  A $
can differ vastly from spectra of linear operators. Thus, the spectrum of a completely-continuous non-linear operator can have continuous parts; an eigen element  $  x _ {0} $
of an operator  $  A $, 
that is, an element  $  x _ {0} $
such that  $  x _ {0} = \lambda A x _ {0} $, 
can bifurcate into several eigen element branches (as  $  \lambda $
varies), cf. [[Bifurcation|Bifurcation]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elemente der Funktionalanalysis" , Akademie Verlag  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Kantorovich,   G.P. Akilov,   "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,   "Variational methods for the study of nonlinear operators" , Holden-Day  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Krasnosel'skii,   P.P. Zabreiko,   "Geometric methods of non-linear analysis" , Springer  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Gajewski,   K. Gröger,   K. Zacharias,   "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag  (1974)</TD></TR></table>

{{SourceattributionEOM|Non-linear operator}}
[[Category:Mathematics]]