Non-linear operator

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)