Semi-group of non-linear operators

A one-parameter family of operators , , defined and acting on a closed subset of a Banach space , with the following properties:

1) for , ;

2) for any ;

3) for any , the function (with values in ) is continuous with respect to on .

A semi-group is of type if

A semi-group of type is called a contraction semi-group.

As in the case of semi-groups of linear operators (cf. Semi-group of operators), one introduces the concept of the generating operator (or infinitesimal generator) of the semi-group :

for those elements for which the limit exists. If is a contraction semi-group, is a dissipative operator. Recall that an operator on a Banach space is dissipative if for , . A dissipative operator may be multi-valued, in which case in the definition stands for any of its values at . A dissipative operator is said to be -dissipative if for . If is of type , then is dissipative.

The fundamental theorem on the generation of semi-groups: If is a dissipative operator and contains for sufficiently small , then there exists a semi-group of type on such that

where and the convergence is uniform on any finite -interval. (The existence of can also be proved if one replaces the condition by the weaker condition

where is the distance between sets.)

For any operator one has a corresponding Cauchy problem

(*)

If the problem (*) has a strong solution, i.e. if there exists a function which is continuous on , absolutely continuous on any compact subset of , takes values in for almost all , has a strong derivative for almost all , and satisfies the relation (*), then . Any function is a unique integral solution of the problem (*).

Under the assumptions of the fundamental theorem, if is a reflexive space and is closed (cf. Closed operator), then the function yields a strong solution of the Cauchy problem (*) for , with almost everywhere, where is the set of elements of minimal norm in . In that case the generating operator of the semi-group is densely defined: . If, moreover, and are uniformly convex, then the operator is single-valued and for all there exists a right derivative ; this function is continuous from the right on , and continuous at all points with the possible exception of a countable set; in this case and .

If is reflexive (or , where is separable) and is a single-valued operator and has the property that in and in the weak topology (respectively, in ) imply , then , , and is a weakly (weak-) continuously-differentiable solution of the problem (*). In the non-reflexive case, examples are known where the assumptions of the fundamental theorem hold with and the functions do not even have weak derivatives on at any , .

Let be a continuous operator, defined on all of , such that is dissipative. Then for , , and for any the problem (*) has a unique continuously-differentiable solution on , given by . If is continuous on its closed domain , then it will be the generating operator of a semi-group of type on if only and only if is dissipative and for .

In a Hilbert space , a contraction semi-group on a set may be extended to a contraction semi-group on a closed convex subset of . Moreover, the generating operator of the extended semi-group is defined on a set dense in . There exists a unique -dissipative operator such that and . If is -dissipative, then is convex and there exists a unique contraction semi-group on such that .

Let be a convex semi-continuous functional defined on a real Hilbert space and let be its subdifferential; then the operator (for all such that is non-empty) is dissipative. The semi-group possesses properties similar to those of a linear analytic semi-group. In particular, () for any , and is a strong solution of the Cauchy problem (*), with

for all , . If attains its minimum, then converges weakly to some minimum point as .

Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let , , be Banach spaces; let , be operators defined and single-valued on , , respectively, satisfying the assumptions of the fundamental theorem for the same type ; let be linear operators, . Then convergence of the resolvents (cf. Resolvent) (, )

for implies convergence of the semi-groups

uniformly on any finite closed interval.

The multiplicative formulas developed by S. Lie in the finite-dimensional linear case can be generalized to the non-linear case. If , and are single-valued -dissipative operators on a Hilbert space and the closed convex set is invariant under and , then, for any ,

(**)

This formula is also valid in an arbitrary Banach space for any , provided is a densely-defined -dissipative linear operator and is a continuous dissipative operator defined on all of . In both cases

Examples of non-linear differential operators satisfying the conditions of the fundamental theorem on the generation of semi-groups are given below. In each case only the space and the boundary conditions are indicated, while is not described. In all examples, is a bounded domain in with smooth boundary; are multi-valued maximal monotone mappings , , ; and is a continuous strictly-increasing function, .

Example 1.

, , , on .

Example 2.

, , on .

Example 3.

, , on .

Example 4.

or , , on .

Example 5.

, , where with values in , .

Example 6.

, , where is continuous.

References

[1]	V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian)
[2]	H. Brézis, "Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert" , North-Holland (1973)
[3]	H. Brézis, A. Pazy, "Convergence and approximation of semigroups of nonlinear operators in Banach spaces" J. Funct. Anal. , 9 : 1 (1972) pp. 63–74
[4]	M.G. Crandall, T.M. Liggett, "Generation of semi-groups of nonlinear transformations on general Banach spaces" Amer. J. Math. , 93 : 2 (1971) pp. 265–298
[5]	Y. Kobayashi, "Difference approximation of Gauchy problems for quasi-dissipative operators and generation of nonlinear semigroups" J. Math. Soc. Japan , 27 : 4 (1975) pp. 640–665
[6]	Y. Konishi, "On the uniform convergence of a finite difference scheme for a nonlinear heat equation" Proc. Japan. Acad. , 48 : 2 (1972) pp. 62–66
[7]	R.H. Martin, "Differential equations on closed subsets of a Banach space" Trans. Amer. Math. Soc. , 179 (1973) pp. 399–414
[8]	G.F. Webb, "Continuous nonlinear perturbations of linear accretive operators in Banach spaces" J. Funct. Anal. , 10 : 2 (1972) pp. 191–203
[9]	M.I. [M.I. Khazan] Hazan, "Nonlinear evolution equations in locally convex spaces" Soviet Math. Dokl. , 14 : 5 (1973) pp. 1608–1614 Dokl. Akad. Nauk SSSR , 212 : 6 (1973) pp. 1309–1312
[10]	M.I. [M.I. Khazan] Hazan, "Differentiability of nonlinear semigroups and the classical solvability of nonlinear boundary value problems for the equation " Soviet Math. Dokl. , 17 : 3 (1976) pp. 839–843 Dokl. Akad. Nauk SSSR , 228 : 4 (1976) pp. 805–808

Comments

See also Semi-group of operators; One-parameter semi-group.

The formula (**) above, especially in the form

which holds, e.g., when are self-adjoint operators on a separable Hilbert space so that , defined on , is self-adjoint, is known as the Trotter product formula, [a5], [a4].

References

[a1]	Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , CWI Monographs , 5 , North-Holland (1987)
[a2]	A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
[a3]	R.H. Martin, "Nonlinear operators and differential equations in Banach spaces" , Wiley (1976)
[a4]	B. Simon, "Functional integration and quantum physics" , Acad. Press (1979) pp. 4–6
[a5]	H. Trotter, "On the product of semigroups of operators" Proc. Amer. Math. Soc. , 10 (1959) pp. 545–551