Non-linear operator

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

If is the set of real or 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)

where , , , , is a function such that is continuous on for any (for example, is continuous on , , ). If is non-linear in , then is a non-linear Urysohn operator mapping into itself. Under other restrictions on an Urysohn operator acts on other spaces, for instance, or maps one Orlicz space into another .

2)

where is non-linear in and defined for , . Under appropriate restrictions on the operator acts from one function space into another and is called a non-linear Hammerstein operator.

3)

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 into itself.

4)

is a non-linear differential operator of order in divergence form acting on the Sobolev space under suitable restrictions on the non-linear function . Here is the multi-index , , and is a bounded domain in .

5)

is non-linear integro-differential operator acting under appropriate restrictions on the function in the space of continuously-differentiable functions.

To non-linear operators acting from one topological vector space into another one , many concepts and operations of mathematical analysis of real-valued functions of a real variable can be transferred. Thus, a non-linear operator , , is called bounded if is a bounded set in for any bounded set ; a non-linear operator is continuous at a point if the inverse image of a neighbourhood of the point contains for some neighbourhood of . As for functions, a non-linear operator that is continuous at every point of a compact set is bounded on this set. In contrast to linear operators, if a non-linear operator acting on a normed space is bounded on some ball, it does not follow that 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 to one can distinguish certain important classes.

1) Semi-linear operators , linear in each argument. The space of all -linear operators is isomorphic to the space , where is the space of all linear operators from to . If and are normed spaces, then and are isometric. If is symmetric in all arguments, then is denoted by and is called a homogeneous operator of degree .

2) In spaces endowed with a partial order, isotone operators and antitone operators are characterized by the conditions and .

3) In a Hilbert space , monotone operators are defined by the condition for any .

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 acting from an open set of a normed vector space into a normed vector space is called Fréchet differentiable at a point if there exists a continuous linear operator such that for any for which ,

where as . In this case the linear part in of the increment is called the Fréchet differential of at and is denoted by , and is called the remainder of the increment. The bounded linear operator is called the Fréchet derivative of at . Apart from Fréchet differentiability one also introduces Gâteaux differentiability. Namely, an operator is called Gâteaux differentiable at a point if the limit

exists, which is called the Gâteaux differential of at . The Gâteaux differential is homogeneous in , that is, . If is linear in and , then the linear operator is called the Gâteaux derivative of . Fréchet differentiability implies Gâteaux differentiability, and then . Gâteaux differentiability does not, in general, imply Fréchet differentiability, but if exists in a neighbourhood of , is continuous in and uniformly continuous in , then is Fréchet differentiable at . For non-linear functionals Fréchet and Gâteaux differentials and derivatives are defined similarly. Here the Gâteaux derivative is called the gradient of the functional and is an operator from to . If for some non-linear functional , then is called a potential operator.

For operators acting on separable topological vector spaces one can in one way or another define differentiation. Let be a collection of bounded sets in a topological vector space . A mapping is called -small if as uniformly in for any . A mapping (where is open) is called -differentiable at if

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

Higher-order derivatives and of an operator are defined in the usual way, as derivatives of derivatives. These are symmetric multi-linear mappings. A differential of order is then a homogeneous form of degree . Other definitions of higher-order derivatives are possible. Suppose, for example, that and are normed vector spaces, is open, and . If for any for which ,

(*)

where , then the multi-linear form is called the derivative of order . The expression (*) is then called the bounded expansion of order of the difference . Under appropriate restrictions the various definitions of higher-order derivatives are equivalent.

If a scalar countably-additive measure is given in , then a non-linear operator can be integrated, by understanding in the sense of the Bochner integral.

For a non-linear operator , as in the case of a linear operator, the values of the parameter for which exists and is continuous on are naturally called regular, and the remaining points belong to the spectrum. In its properties the spectrum of a non-linear operator 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 of an operator , that is, an element such that , can bifurcate into several eigen element branches (as 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)