View source for Semi-group of non-linear operators

A one-parameter family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841302.png" />, defined and acting on a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841303.png" /> of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841304.png" />, with the following properties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841305.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841307.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841308.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841309.png" />;

3) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413010.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413011.png" /> (with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413012.png" />) is continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413014.png" />.

A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413016.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413017.png" /> if

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A semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413019.png" /> is called a contraction semi-group.

As in the case of semi-groups of linear operators (cf. [[Semi-group of operators|Semi-group of operators]]), one introduces the concept of the generating operator (or infinitesimal generator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413020.png" /> of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413021.png" />:

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for those elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413023.png" /> for which the limit exists. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413024.png" /> is a contraction semi-group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413025.png" /> is a dissipative operator. Recall that an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413026.png" /> on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413027.png" /> is dissipative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413030.png" />. A dissipative operator may be multi-valued, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413031.png" /> in the definition stands for any of its values at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413032.png" />. A dissipative operator is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413034.png" />-dissipative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413037.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413039.png" /> is dissipative.

The fundamental theorem on the generation of semi-groups: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413040.png" /> is a dissipative operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413041.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413042.png" /> for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413043.png" />, then there exists a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413044.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413046.png" /> such that

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413048.png" /> and the convergence is uniform on any finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413049.png" />-interval. (The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413050.png" /> can also be proved if one replaces the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413051.png" /> by the weaker condition

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413053.png" /> is the distance between sets.)

For any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413054.png" /> one has a corresponding Cauchy problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

If the problem (*) has a strong solution, i.e. if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413056.png" /> which is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413057.png" />, absolutely continuous on any compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413058.png" />, takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413059.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413060.png" />, has a strong derivative for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413061.png" />, and satisfies the relation (*), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413062.png" />. Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413063.png" /> is a unique integral solution of the problem (*).

Under the assumptions of the fundamental theorem, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413064.png" /> is a [[Reflexive space|reflexive space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413065.png" /> is closed (cf. [[Closed operator|Closed operator]]), then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413066.png" /> yields a strong solution of the Cauchy problem (*) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413067.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413068.png" /> almost everywhere, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413069.png" /> is the set of elements of minimal norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413070.png" />. In that case the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413071.png" /> of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413072.png" /> is densely defined: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413073.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413075.png" /> are uniformly convex, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413076.png" /> is single-valued and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413077.png" /> there exists a right derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413078.png" />; this function is continuous from the right on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413079.png" />, and continuous at all points with the possible exception of a countable set; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413081.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413082.png" /> is reflexive (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413083.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413084.png" /> is separable) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413085.png" /> is a single-valued operator and has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413088.png" /> in the [[Weak topology|weak topology]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413089.png" /> (respectively, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413090.png" />) imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413091.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413093.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413094.png" /> is a weakly (weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413095.png" />) continuously-differentiable solution of the problem (*). In the non-reflexive case, examples are known where the assumptions of the fundamental theorem hold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413096.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413097.png" /> do not even have weak derivatives on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413098.png" /> at any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130100.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130101.png" /> be a continuous operator, defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130102.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130103.png" /> is dissipative. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130104.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130106.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130107.png" /> the problem (*) has a unique continuously-differentiable solution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130108.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130109.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130110.png" /> is continuous on its closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130111.png" />, then it will be the generating operator of a semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130112.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130113.png" /> if only and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130114.png" /> is dissipative and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130115.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130116.png" />.

In a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130117.png" />, a contraction semi-group on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130118.png" /> may be extended to a contraction semi-group on a closed convex subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130119.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130120.png" />. Moreover, the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130121.png" /> of the extended semi-group is defined on a set dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130122.png" />. There exists a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130123.png" />-dissipative operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130125.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130126.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130127.png" />-dissipative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130128.png" /> is convex and there exists a unique contraction semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130129.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130130.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130131.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130132.png" /> be a convex semi-continuous functional defined on a real Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130133.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130134.png" /> be its [[Subdifferential|subdifferential]]; then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130135.png" /> (for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130136.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130137.png" /> is non-empty) is dissipative. The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130138.png" /> possesses properties similar to those of a linear analytic semi-group. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130139.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130140.png" />) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130141.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130142.png" /> is a strong solution of the Cauchy problem (*), with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130143.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130144.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130145.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130146.png" /> attains its minimum, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130147.png" /> converges weakly to some minimum point as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130148.png" />.

Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130150.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130151.png" /> be Banach spaces; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130153.png" /> be operators defined and single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130155.png" />, respectively, satisfying the assumptions of the fundamental theorem for the same type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130156.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130157.png" /> be linear operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130158.png" />. Then convergence of the resolvents (cf. [[Resolvent|Resolvent]]) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130160.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130161.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130162.png" /> implies convergence of the semi-groups

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130163.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130165.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130166.png" /> are single-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130167.png" />-dissipative operators on a Hilbert space and the closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130168.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130170.png" />, then, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130171.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130172.png" /></td> <td valign="top" style="width:5%;text-align:right;">(**)</td></tr></table>

This formula is also valid in an arbitrary Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130173.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130174.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130175.png" /> is a densely-defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130176.png" />-dissipative linear operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130177.png" /> is a continuous dissipative operator defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130178.png" />. In both cases

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130179.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130180.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130181.png" /> and the boundary conditions are indicated, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130182.png" /> is not described. In all examples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130183.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130184.png" /> with smooth boundary; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130185.png" /> are multi-valued maximal monotone mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130186.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130188.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130189.png" /> is a continuous strictly-increasing function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130190.png" />.

==Example 1.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130191.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130193.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130194.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130195.png" />.

==Example 2.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130198.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130199.png" />.

==Example 3.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130200.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130202.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130203.png" />.

==Example 4.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130204.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130207.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130208.png" />.

==Example 5.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130210.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130211.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130212.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130213.png" />.

==Example 6.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130214.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130215.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130216.png" /> is continuous.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Barbu,   "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Brézis,   "Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Brézis,   A. Pazy,   "Convergence and approximation of semigroups of nonlinear operators in Banach spaces"  ''J. Funct. Anal.'' , '''9''' :  1  (1972)  pp. 63–74</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.H. Martin,   "Differential equations on closed subsets of a Banach space"  ''Trans. Amer. Math. Soc.'' , '''179'''  (1973)  pp. 399–414</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.F. Webb,   "Continuous nonlinear perturbations of linear accretive operators in Banach spaces"  ''J. Funct. Anal.'' , '''10''' :  2  (1972)  pp. 191–203</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.I. [M.I. Khazan] Hazan,   "Differentiability of nonlinear semigroups and the classical solvability of nonlinear boundary value problems for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130217.png" />"  ''Soviet Math. Dokl.'' , '''17''' :  3  (1976)  pp. 839–843  ''Dokl. Akad. Nauk SSSR'' , '''228''' :  4  (1976)  pp. 805–808</TD></TR></table>



====Comments====
See also [[Semi-group of operators|Semi-group of operators]]; [[One-parameter semi-group|One-parameter semi-group]].

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130218.png" /></td> </tr></table>

which holds, e.g., when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130219.png" /> are self-adjoint operators on a separable Hilbert space so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130220.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130221.png" />, is self-adjoint, is known as the Trotter product formula, [[#References|[a5]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Pazy,   "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.H. Martin,   "Nonlinear operators and differential equations in Banach spaces" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Simon,   "Functional integration and quantum physics" , Acad. Press  (1979)  pp. 4–6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Trotter,   "On the product of semigroups of operators"  ''Proc. Amer. Math. Soc.'' , '''10'''  (1959)  pp. 545–551</TD></TR></table>