# Non-linear operator

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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. 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.

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.

#### References

 [1] L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1968) (Translated from Russian) [2] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian) [3] M.M. Vainberg, "Variational methods for the study of nonlinear operators" , Holden-Day (1964) (Translated from Russian) [4] M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian) [5] H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974)
How to Cite This Entry:
Non-linear operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_operator&oldid=47996
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article