Semi-group of non-linear operators

A one-parameter family of operators $S ( t )$, $0 \leq t < \infty$, defined and acting on a closed subset $G$ of a Banach space $X$, with the following properties:

1) $S ( t + \tau ) x = S ( t ) ( S ( \tau ) x )$ for $x \in C$, $t , \tau > 0$;

2) $S ( 0 ) x = x$ for any $x \in C$;

3) for any $x \in C$, the function $S ( t ) x$( with values in $X$) is continuous with respect to $t$ on $[ 0 , \infty )$.

A semi-group $S ( t )$ is of type $\omega$ if

$$\| S ( t ) x - S ( t ) y \| \leq e ^ {\omega t } \| x - y \| ,\ \ x , y \in C ,\ t> 0 .$$

A semi-group of type $0$ 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) $A _ {0}$ of the semi-group $S ( t )$:

$$A _ {0} x = \lim\limits _ {h \rightarrow 0 } \frac{S ( h ) x - x }{h}$$

for those elements $x \in C$ for which the limit exists. If $S ( t )$ is a contraction semi-group, $A _ {0}$ is a dissipative operator. Recall that an operator $A$ on a Banach space $X$ is dissipative if $\| x - y - \lambda ( Ax - Ay ) \| \geq \| x - y \|$ for $x , y \in \overline{ {D ( A) }}\;$, $\lambda > 0$. A dissipative operator may be multi-valued, in which case $Ax$ in the definition stands for any of its values at $x$. A dissipative operator is said to be $m$- dissipative if $\mathop{\rm Range} ( I - \lambda A ) = X$ for $\lambda > 0$. If $S ( t )$ is of type $\omega$, then $A _ {0} - \omega I$ is dissipative.

The fundamental theorem on the generation of semi-groups: If $A - \omega I$ is a dissipative operator and $\mathop{\rm Range} ( I - \lambda A )$ contains $D ( A )$ for sufficiently small $\lambda > 0$, then there exists a semi-group $S _ {A} ( t )$ of type $\omega$ on $\overline{ {D ( A ) }}\;$ such that

$$S _ {A} ( t ) x = \lim\limits _ {n \rightarrow \infty } \ \left ( I - \frac{t}{n} A \right ) ^ {-} n x ,$$

where $x \in \overline{ {D ( A ) }}\;$ and the convergence is uniform on any finite $t$- interval. (The existence of $S _ {A} ( t )$ can also be proved if one replaces the condition $\mathop{\rm Range} ( I - \lambda A ) \supset \overline{ {D ( A ) }}\;$ by the weaker condition

$$\lim\limits _ {\overline{ {\lambda \downarrow 0 }}\; } \ \lambda ^ {-} 1 d ( \mathop{\rm Range} ( I - \lambda A ) , x ) = 0 ,$$

where $d$ is the distance between sets.)

For any operator $A$ one has a corresponding Cauchy problem

$$\tag{* } \frac{du}{dt} ( t) \in Au ( t ) ,\ t > 0 ,\ u ( 0) = x .$$

If the problem (*) has a strong solution, i.e. if there exists a function $u ( t )$ which is continuous on $[ 0 , \infty )$, absolutely continuous on any compact subset of $( 0, \infty )$, takes values in $D ( A )$ for almost all $t > 0$, has a strong derivative for almost all $t > 0$, and satisfies the relation (*), then $u ( t ) = S _ {A} ( t ) x$. Any function $S _ {A} ( t ) x$ is a unique integral solution of the problem (*).

Under the assumptions of the fundamental theorem, if $X$ is a reflexive space and $A$ is closed (cf. Closed operator), then the function $u ( t ) = S _ {A} ( t ) x$ yields a strong solution of the Cauchy problem (*) for $x \in D ( A)$, with $( du / dt) ( t) \in A ^ {0} u ( t )$ almost everywhere, where $A ^ {0} z$ is the set of elements of minimal norm in $Az$. In that case the generating operator $A _ {0}$ of the semi-group $S _ {A} ( t )$ is densely defined: $\overline{ {D ( A _ {0} ) }}\; = \overline{ {D ( A ) }}\;$. If, moreover, $X$ and $X ^ \prime$ are uniformly convex, then the operator $A ^ {0}$ is single-valued and for all $t \geq 0$ there exists a right derivative $d ^ {+} u / dt = A ^ {0} u ( t )$; this function is continuous from the right on $[ 0, \infty )$, and continuous at all points with the possible exception of a countable set; in this case $D ( A _ {0} ) = D ( A )$ and $A _ {0} = A ^ {0}$.

If $X$ is reflexive (or $X = Y ^ \prime$, where $Y$ is separable) and $A$ is a single-valued operator and has the property that $x _ {n} \rightarrow x$ in $X$ and $Ax _ {n} \rightarrow y$ in the weak topology $\sigma ( X , X ^ \prime )$( respectively, in $\sigma ( X , Y )$) imply $y = Ax$, then $u ( t) \in D ( A )$, $t \geq 0$, and $u ( t )$ 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 $\overline{ {D ( A) }}\; = X$ and the functions $u ( t ) = S _ {A} ( t ) x$ do not even have weak derivatives on $X$ at any $x \in X$, $t \geq 0$.

Let $A$ be a continuous operator, defined on all of $X$, such that $A - \omega I$ is dissipative. Then $\mathop{\rm Range} ( I - \lambda A ) = X$ for $\lambda > 0$, $\lambda \omega < 1$, and for any $x \in X$ the problem (*) has a unique continuously-differentiable solution on $[ 0 , \infty )$, given by $u ( t ) = S _ {A} ( t ) x$. If $A$ is continuous on its closed domain $D ( A )$, then it will be the generating operator of a semi-group of type $\omega$ on $D ( A)$ if only and only if $A - \omega I$ is dissipative and $\lim\limits _ {\lambda \rightarrow 0 } \lambda ^ {-} 1 d ( x + \lambda Ax , D ( A ) ) = 0$ for $x \in D ( A)$.

In a Hilbert space $H$, a contraction semi-group on a set $C$ may be extended to a contraction semi-group on a closed convex subset $\widetilde{C}$ of $H$. Moreover, the generating operator $A _ {0}$ of the extended semi-group is defined on a set dense in $\widetilde{C}$. There exists a unique $m$- dissipative operator such that $\overline{ {D ( A) }}\; = C$ and $A _ {0} = A ^ {0}$. If $A$ is $m$- dissipative, then $\overline{ {D ( A) }}\;$ is convex and there exists a unique contraction semi-group $S ( t ) = S _ {A} ( t)$ on $\overline{ {D ( A) }}\;$ such that $A _ {0} = A ^ {0}$.

Let $\phi$ be a convex semi-continuous functional defined on a real Hilbert space $H$ and let $\partial \phi$ be its subdifferential; then the operator $Ax = - \partial \phi ( x)$( for all $x$ such that $\partial \phi ( x)$ is non-empty) is dissipative. The semi-group $S _ {A} ( t )$ possesses properties similar to those of a linear analytic semi-group. In particular, $S _ {A} ( t) x \in D ( A)$( $t> 0$) for any $x \in \overline{ {D ( A) }}\;$, and $u ( t ) = S _ {A} ( t) x$ is a strong solution of the Cauchy problem (*), with

$$\left \| \frac{d ^ {+} u }{dt} ( t) \right \| = \ \| A ^ {0} u ( t ) \| \leq \frac{2}{t} \ \| x - v \| + 2 \| A ^ {0} v \|$$

for all $t > 0$, $v \in D ( A)$. If $\phi$ attains its minimum, then $u ( t )$ converges weakly to some minimum point as $t \rightarrow \infty$.

Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let $X$, $X _ {n}$, $n = 1 , 2 \dots$ be Banach spaces; let $A$, $A _ {n}$ be operators defined and single-valued on $X$, $X _ {n}$, respectively, satisfying the assumptions of the fundamental theorem for the same type $\omega$; let $p _ {n} : X \rightarrow X _ {n}$ be linear operators, $\| p _ {n} \| _ {X \rightarrow X _ {n} } \leq \textrm{ const }$. Then convergence of the resolvents (cf. Resolvent) ( $\lambda > 0$, $\lambda \omega < 1$)

$$\| ( I - \lambda A _ {n} ) ^ {-} 1 p _ {n} x - p _ {n} ( I - \lambda A ) ^ {-} 1 x \| _ {X _ {n} } \rightarrow 0$$

for $x \in \overline{ {D ( A) }}\;$ implies convergence of the semi-groups

$$\| S _ {A _ {n} } ( t) p _ {n} x - p _ {n} S _ {A} ( t) x \| _ {X _ {n} } \rightarrow 0 ,\ x \in \overline{ {D ( A) }}\; ,$$

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 $A$, $B$ and $A + B$ are single-valued $m$- dissipative operators on a Hilbert space and the closed convex set $C \subset \overline{ {D ( A) }}\; \cap \overline{ {D ( B) }}\;$ is invariant under $( I - \lambda A ) ^ {-} 1$ and $( I - \lambda B ) ^ {-} 1$, then, for any $x \in C \cap \overline{ {D ( A) \cap D ( B) }}\;$,

$$\tag{** } S _ {A + B } ( t) x = \lim\limits _ {n \rightarrow \infty } \ \left [ S _ {A} \left ( \frac{t}{n} \right ) S _ {B} \left ( \frac{t}{n} \right ) \right ] ^ {n} x .$$

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

$$S _ {A + B } ( t) x = \lim\limits \ \left [ \left ( I - \frac{t}{n} B \right ) ^ {-} 1 \left ( I - \frac{t}{n} A \right ) ^ {-} 1 \right ] ^ {n} x ,$$

$$x \in \overline{ {D ( A) \cap D ( B) }}\; .$$

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 $X$ and the boundary conditions are indicated, while $D ( A)$ is not described. In all examples, $\Omega$ is a bounded domain in $\mathbf R ^ {n}$ with smooth boundary; $\beta , \gamma$ are multi-valued maximal monotone mappings $\mathbf R \rightarrow \mathbf R$, $\beta ( 0) \ni 0$, $\gamma ( 0) \ni 0$; and $\psi : \mathbf R \rightarrow \mathbf R$ is a continuous strictly-increasing function, $\psi ( 0) = 0$.

Example 1.

$X = L _ {p} ( \Omega )$, $1 \leq p \leq \infty$, $Au = \Delta u - \beta ( u)$, $- \partial u / \partial n \in \gamma ( u)$ on $\Gamma$.

Example 2.

$X = L _ {1} ( \Omega )$, $Au = \Delta \psi ( u)$, $- \partial u / \partial n \in \gamma ( u)$ on $\Gamma$.

Example 3.

$X = W _ {2} ^ {-} 1 ( \Omega )$, $Au = \Delta \psi ( u)$, $u = 0$ on $\Gamma$.

Example 4.

$X = C ( \overline \Omega \; )$ or $X = L _ \infty ( \Omega )$, $Au = \psi ( \Delta u )$, $u = 0$ on $\Gamma$.

Example 5.

$X = L _ {1} ( \mathbf R ^ {n} )$, $Au = \mathop{\rm div} f ( u)$, where $f \in C ^ {1} ( \mathbf R )$ with values in $\mathbf R ^ {n}$, $f ( 0) = 0$.

Example 6.

$X = L _ \infty ( \mathbf R )$, $Au = f ( u _ {x} )$, where $f : \mathbf R \rightarrow \mathbf R$ 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

$$e ^ {it ( A+ B) } = {s - \lim\limits } _ {n \rightarrow \infty } \ ( e ^ {itA/n } e ^ {itB/n } ) ^ {n} ,$$

which holds, e.g., when $A, B$ are self-adjoint operators on a separable Hilbert space so that $A+ B$, defined on $D ( A) \cap D ( B)$, 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