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The great importance of the notion of  "accretive mapping"  consists in the fact that it allows one to treat many partial differential equations and functional differential equations from mathematical physics (such as the heat and wave equations) as suitable ordinary differential equations associated with accretive generators of suitable semi-groups in appropriate functional (Sobolev) spaces. This method, known as the semi-group approach, has significantly clarified and unified the study of many classes of partial and functional differential equations and has solved problems that had been left open by the previous classical methods.
 
The great importance of the notion of  "accretive mapping"  consists in the fact that it allows one to treat many partial differential equations and functional differential equations from mathematical physics (such as the heat and wave equations) as suitable ordinary differential equations associated with accretive generators of suitable semi-groups in appropriate functional (Sobolev) spaces. This method, known as the semi-group approach, has significantly clarified and unified the study of many classes of partial and functional differential equations and has solved problems that had been left open by the previous classical methods.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201001.png" /> be a general [[Banach space|Banach space]] with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201002.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201003.png" /> is a bounded [[Linear operator|linear operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201004.png" /> into itself, then the exponential formula below holds:
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Let $X$ be a general [[Banach space|Banach space]] with norm $|.|$. If $A$ is a bounded [[Linear operator|linear operator]] from $X$ into itself, then the exponential formula below holds:
  
<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/a/a120/a120100/a1201005.png" /></td> </tr></table>
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\begin{equation*} S ( t ) = e ^ { - t A } = \sum _ { m = 0 } ^ { \infty } \frac { ( - t A ) ^ { m } } { m ! }, \end{equation*}
  
as the series is convergent. Moreover, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201006.png" /> is the unique strong solution to the [[Cauchy problem|Cauchy problem]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201008.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a1201009.png" /> is unbounded, then the series above is not convergent, so the exponential formula makes no sense. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010010.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010011.png" />-accretive (see below and [[M-accretive-operator|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010012.png" />-accretive operator]]), then the so-called Crandall–Liggett exponential formula (1971) can be defined. Namely:
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as the series is convergent. Moreover, the function $y ( t ) = e ^ { - t A } x = S ( t ) x$ is the unique strong solution to the [[Cauchy problem|Cauchy problem]] $y ^ { \prime } ( t ) = - A y ( t )$, $y ( 0 ) = x$. If $A$ is unbounded, then the series above is not convergent, so the exponential formula makes no sense. However, if $A$ is $m$-accretive (see below and [[M-accretive-operator|$m$-accretive operator]]), then the so-called Crandall–Liggett exponential formula (1971) can be defined. Namely:
  
<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/a/a120/a120100/a12010013.png" /></td> </tr></table>
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\begin{equation*} e ^ { - t A  }x = \operatorname { lim } _ { n \rightarrow \infty } \left( I + \frac { t } { n } A \right) ^ { - n } x = S ( t ) x , \forall x \in X, \end{equation*}
  
as the limit above exists. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010014.png" /> linear and unbounded, it is due to E. Hille and K. Yosida (who started these investigations in 1948). The one-parameter family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010015.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010016.png" /> is said to be the [[Semi-group|semi-group]] generated by the (possible non-linear and multi-valued) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010017.png" />-accretive mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010018.png" />. The main difference in this unbounded case is that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010019.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010020.png" /> is not differentiable. This is why the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010021.png" /> is said to be a mild (or generalized) solution to the Cauchy problem above.
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as the limit above exists. For $A$ linear and unbounded, it is due to E. Hille and K. Yosida (who started these investigations in 1948). The one-parameter family of operators $S ( t )$ defined by $S ( t ) x = e ^ { - t A } x $ is said to be the [[Semi-group|semi-group]] generated by the (possible non-linear and multi-valued) $m$-accretive mapping $- A$. The main difference in this unbounded case is that for $x \notin D ( A )$, the function $t \rightarrow S ( t ) x$ is not differentiable. This is why the function $y ( t ) = e ^ { - t A } x = S ( t ) x$ is said to be a mild (or generalized) solution to the Cauchy problem above.
  
Roughly speaking, accretive mappings acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010022.png" /> are generalizations of non-decreasing real-valued functions. More precisely, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010023.png" /> is said to be accretive if
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Roughly speaking, accretive mappings acting in $X$ are generalizations of non-decreasing real-valued functions. More precisely, a mapping $A : D ( A ) \subset X \rightarrow 2 ^ { X }$ is said to be accretive if
  
<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/a/a120/a120100/a12010024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \| x _ { 1 } - x _ { 2 } \| \leq \| x _ { 1 } - x _ { 2 } + \lambda ( y _ { 1 } - y _ { 2 } ) \| , \end{equation}
  
<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/a/a120/a120100/a12010025.png" /></td> </tr></table>
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\begin{equation*} \forall x _ { i } \in D ( A ) , y _ { i } \in A x _ { i } , i = 1,2 , \lambda \geq 0. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010027.png" /> stand for the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010028.png" /> and the family of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010029.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010030.png" /> is a real [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010031.png" /> with inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010032.png" />, then (a1) is equivalent to
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Here, $D ( A )$ and $2 ^ { X }$ stand for the domain of $A$ and the family of all subsets of $X$, respectively. If $X$ is a real [[Hilbert space|Hilbert space]] $H$ with inner product $(.)$, then (a1) is equivalent to
  
<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/a/a120/a120100/a12010033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \langle y _ { 1 } - y _ { 2 } , x _ { 1 } - x _ { 2 } \rangle \geq 0, \end{equation}
  
<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/a/a120/a120100/a12010034.png" /></td> </tr></table>
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\begin{equation*} \forall x _ { i } \in D ( A ) , y _ { i } \in A x _ { i } , i = 1,2 , \lambda \geq 0. \end{equation*}
  
It is now clear that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010035.png" /> (the set of all real numbers) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010036.png" /> a single-valued function, accretivity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010037.png" /> is equivalent to
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It is now clear that for $X = \bf R$ (the set of all real numbers) and $A$ a single-valued function, accretivity of $A$ is equivalent to
  
<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/a/a120/a120100/a12010038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} \langle A x _ { 1 } - A x _ { 2 } , x _ { 1 } - x _ { 2 } \rangle \geq 0 \end{equation}
  
<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/a/a120/a120100/a12010039.png" /></td> </tr></table>
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\begin{equation*} \forall x _ { i } \in D ( A ), \end{equation*}
  
i.e. to the classical definition  "x1&lt;x2 implies Ax1≤Ax2"  for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010040.png" /> to be non-decreasing. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010041.png" /> is said to be dissipative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010042.png" /> is accretive. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010043.png" /> is said to be maximal accretive if it is accretive and if it has no accretive extensions. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010044.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010046.png" />-accretive (or hyper-maximal accretive) if it is accretive and if the following range condition holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010047.png" />, or, equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010048.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010050.png" /> denotes the identity operator (cf. also [[M-accretive-operator|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010051.png" />-accretive operator]]).
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i.e. to the classical definition  "x1&lt;x2 implies Ax1≤Ax2"  for $A$ to be non-decreasing. The mapping $A$ is said to be dissipative if $- A$ is accretive. $A$ is said to be maximal accretive if it is accretive and if it has no accretive extensions. $A$ is said to be $m$-accretive (or hyper-maximal accretive) if it is accretive and if the following range condition holds: $R ( I + A ) = X$, or, equivalently, $R ( I + \lambda A = X$, for all $\lambda &gt; 0$, where $I$ denotes the identity operator (cf. also [[M-accretive-operator|$m$-accretive operator]]).
  
In a [[Normed space|normed space]],  "m-accretive"  implies  "maximal accretive" . The converse assertion need not be true. The first counterexample was constructed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010052.png" /> by B.D. Calvert (1970). Moreover, A. Cernes (1974) has proven that even if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010054.png" /> (the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010055.png" />) are uniformly convex (cf. [[Banach space|Banach space]]), but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010056.png" /> is not a Hilbert space, then there are maximal accretive mappings which are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010057.png" />-accretive. However, it was proved by G. Minty (1962) that in Hilbert spaces, the notions of  "m-accretive"  and  "maximal accretive"  are equivalent. Note that in Hilbert spaces,  "accretive"  is also known as  "monotone" .
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In a [[Normed space|normed space]],  "m-accretive"  implies  "maximal accretive" . The converse assertion need not be true. The first counterexample was constructed in $l ^ { p }$ by B.D. Calvert (1970). Moreover, A. Cernes (1974) has proven that even if both $X$ and $X ^ { * }$ (the dual of $X$) are uniformly convex (cf. [[Banach space|Banach space]]), but $X$ is not a Hilbert space, then there are maximal accretive mappings which are not $m$-accretive. However, it was proved by G. Minty (1962) that in Hilbert spaces, the notions of  "m-accretive"  and  "maximal accretive"  are equivalent. Note that in Hilbert spaces,  "accretive"  is also known as  "monotone" .
  
The theory of accretive-type operators is also known as Minty–Browder theory. It has started with some pioneering work of M.M. Vainberg, E.M. Zarantonello and R.I. Kachurovski in the 1960s. As a significant example, consider the [[Laplace operator|Laplace operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010059.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010061.png" /> is a bounded domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010062.png" /> with sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010063.png" />. In view of the Green formula,
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The theory of accretive-type operators is also known as Minty–Browder theory. It has started with some pioneering work of M.M. Vainberg, E.M. Zarantonello and R.I. Kachurovski in the 1960s. As a significant example, consider the [[Laplace operator|Laplace operator]] $A = - \Delta$ in $L ^ { 2 } ( \Omega )$ with $D ( \Delta ) = H _ { o } ^ { 1 } \cap H ^ { 2 } ( \Omega )$, where $\Omega$ is a bounded domain of ${\bf R} ^ { n }$ with sufficiently smooth boundary $\partial \Omega$. In view of the Green formula,
  
<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/a/a120/a120100/a12010064.png" /></td> </tr></table>
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\begin{equation*} \int _ { \Omega } u \Delta u d x = \int _ { \partial \Omega } u \frac { \partial u } { \partial \eta } d \sigma - \int _ { \Omega } | \operatorname { grad } u | ^ { 2 } d x, \end{equation*}
  
<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/a/a120/a120100/a12010065.png" /></td> </tr></table>
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\begin{equation*} u \in D ( \Delta ), \end{equation*}
  
it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010066.png" /> is monotone. Moreover, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010067.png" />, the elliptic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010068.png" /> has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010069.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010070.png" /> is maximal monotone. H. Brézis has proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010071.png" /> is actually the subdifferential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010072.png" /> of a lower semi-continuous convex functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010073.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010074.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010075.png" /> which (according to a more general result of R.T. Rockafellar, 1966), is maximal monotone (accretive). It follows from the definition (a1) that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010076.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010077.png" />-accretive, then for every positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010079.png" /> is invertible and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010080.png" /> is non-expansive (i.e., Lipschitz continuous of Lipschitz constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010081.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010082.png" />. The crucial importance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120100/a12010083.png" />-accretive operators has already been pointed out above.
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it follows that $- \Delta$ is monotone. Moreover, for each $f \in L ^ { 2 } ( \Omega )$, the elliptic equation $u - \Delta u = f$ has a unique solution $u = u _ { f } \in D ( \Delta )$, so $- \Delta$ is maximal monotone. H. Brézis has proved that $- \Delta$ is actually the subdifferential $\partial \phi$ of a lower semi-continuous convex functional $\phi$ from $L ^ { 2 } ( \Omega )$ into $ \mathbf{R}$ which (according to a more general result of R.T. Rockafellar, 1966), is maximal monotone (accretive). It follows from the definition (a1) that if $A$ is $m$-accretive, then for every positive $\lambda$, $( I + \lambda A )$ is invertible and the operator $J _ { \lambda } = ( I + \lambda A ) ^ { - 1 }$ is non-expansive (i.e., Lipschitz continuous of Lipschitz constant $1$) on $X$. The crucial importance of $m$-accretive operators has already been pointed out above.
  
 
There is an extensive literature on this topic.
 
There is an extensive literature on this topic.
Line 46: Line 54:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Noordhoff  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Brezis,  "Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert" , North-Holland  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Motreanu,  N.H. Pavel,  "Tangency, flow-invariance for differential equations and optimization problems" , M. Dekker  (1999)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.H. Pavel,  "Nonlinear evolution operators and semigroups" , ''Lecture Notes Math.'' , '''1260''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to PDE" , Springer  (1983)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Noordhoff  (1975)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H. Brezis,  "Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert" , North-Holland  (1973)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Motreanu,  N.H. Pavel,  "Tangency, flow-invariance for differential equations and optimization problems" , M. Dekker  (1999)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N.H. Pavel,  "Nonlinear evolution operators and semigroups" , ''Lecture Notes Math.'' , '''1260''' , Springer  (1987)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Pazy,  "Semigroups of linear operators and applications to PDE" , Springer  (1983)</td></tr></table>

Latest revision as of 17:03, 1 July 2020

The great importance of the notion of "accretive mapping" consists in the fact that it allows one to treat many partial differential equations and functional differential equations from mathematical physics (such as the heat and wave equations) as suitable ordinary differential equations associated with accretive generators of suitable semi-groups in appropriate functional (Sobolev) spaces. This method, known as the semi-group approach, has significantly clarified and unified the study of many classes of partial and functional differential equations and has solved problems that had been left open by the previous classical methods.

Let $X$ be a general Banach space with norm $|.|$. If $A$ is a bounded linear operator from $X$ into itself, then the exponential formula below holds:

\begin{equation*} S ( t ) = e ^ { - t A } = \sum _ { m = 0 } ^ { \infty } \frac { ( - t A ) ^ { m } } { m ! }, \end{equation*}

as the series is convergent. Moreover, the function $y ( t ) = e ^ { - t A } x = S ( t ) x$ is the unique strong solution to the Cauchy problem $y ^ { \prime } ( t ) = - A y ( t )$, $y ( 0 ) = x$. If $A$ is unbounded, then the series above is not convergent, so the exponential formula makes no sense. However, if $A$ is $m$-accretive (see below and $m$-accretive operator), then the so-called Crandall–Liggett exponential formula (1971) can be defined. Namely:

\begin{equation*} e ^ { - t A }x = \operatorname { lim } _ { n \rightarrow \infty } \left( I + \frac { t } { n } A \right) ^ { - n } x = S ( t ) x , \forall x \in X, \end{equation*}

as the limit above exists. For $A$ linear and unbounded, it is due to E. Hille and K. Yosida (who started these investigations in 1948). The one-parameter family of operators $S ( t )$ defined by $S ( t ) x = e ^ { - t A } x $ is said to be the semi-group generated by the (possible non-linear and multi-valued) $m$-accretive mapping $- A$. The main difference in this unbounded case is that for $x \notin D ( A )$, the function $t \rightarrow S ( t ) x$ is not differentiable. This is why the function $y ( t ) = e ^ { - t A } x = S ( t ) x$ is said to be a mild (or generalized) solution to the Cauchy problem above.

Roughly speaking, accretive mappings acting in $X$ are generalizations of non-decreasing real-valued functions. More precisely, a mapping $A : D ( A ) \subset X \rightarrow 2 ^ { X }$ is said to be accretive if

\begin{equation} \tag{a1} \| x _ { 1 } - x _ { 2 } \| \leq \| x _ { 1 } - x _ { 2 } + \lambda ( y _ { 1 } - y _ { 2 } ) \| , \end{equation}

\begin{equation*} \forall x _ { i } \in D ( A ) , y _ { i } \in A x _ { i } , i = 1,2 , \lambda \geq 0. \end{equation*}

Here, $D ( A )$ and $2 ^ { X }$ stand for the domain of $A$ and the family of all subsets of $X$, respectively. If $X$ is a real Hilbert space $H$ with inner product $(.)$, then (a1) is equivalent to

\begin{equation} \tag{a2} \langle y _ { 1 } - y _ { 2 } , x _ { 1 } - x _ { 2 } \rangle \geq 0, \end{equation}

\begin{equation*} \forall x _ { i } \in D ( A ) , y _ { i } \in A x _ { i } , i = 1,2 , \lambda \geq 0. \end{equation*}

It is now clear that for $X = \bf R$ (the set of all real numbers) and $A$ a single-valued function, accretivity of $A$ is equivalent to

\begin{equation} \tag{a3} \langle A x _ { 1 } - A x _ { 2 } , x _ { 1 } - x _ { 2 } \rangle \geq 0 \end{equation}

\begin{equation*} \forall x _ { i } \in D ( A ), \end{equation*}

i.e. to the classical definition "x1<x2 implies Ax1≤Ax2" for $A$ to be non-decreasing. The mapping $A$ is said to be dissipative if $- A$ is accretive. $A$ is said to be maximal accretive if it is accretive and if it has no accretive extensions. $A$ is said to be $m$-accretive (or hyper-maximal accretive) if it is accretive and if the following range condition holds: $R ( I + A ) = X$, or, equivalently, $R ( I + \lambda A = X$, for all $\lambda > 0$, where $I$ denotes the identity operator (cf. also $m$-accretive operator).

In a normed space, "m-accretive" implies "maximal accretive" . The converse assertion need not be true. The first counterexample was constructed in $l ^ { p }$ by B.D. Calvert (1970). Moreover, A. Cernes (1974) has proven that even if both $X$ and $X ^ { * }$ (the dual of $X$) are uniformly convex (cf. Banach space), but $X$ is not a Hilbert space, then there are maximal accretive mappings which are not $m$-accretive. However, it was proved by G. Minty (1962) that in Hilbert spaces, the notions of "m-accretive" and "maximal accretive" are equivalent. Note that in Hilbert spaces, "accretive" is also known as "monotone" .

The theory of accretive-type operators is also known as Minty–Browder theory. It has started with some pioneering work of M.M. Vainberg, E.M. Zarantonello and R.I. Kachurovski in the 1960s. As a significant example, consider the Laplace operator $A = - \Delta$ in $L ^ { 2 } ( \Omega )$ with $D ( \Delta ) = H _ { o } ^ { 1 } \cap H ^ { 2 } ( \Omega )$, where $\Omega$ is a bounded domain of ${\bf R} ^ { n }$ with sufficiently smooth boundary $\partial \Omega$. In view of the Green formula,

\begin{equation*} \int _ { \Omega } u \Delta u d x = \int _ { \partial \Omega } u \frac { \partial u } { \partial \eta } d \sigma - \int _ { \Omega } | \operatorname { grad } u | ^ { 2 } d x, \end{equation*}

\begin{equation*} u \in D ( \Delta ), \end{equation*}

it follows that $- \Delta$ is monotone. Moreover, for each $f \in L ^ { 2 } ( \Omega )$, the elliptic equation $u - \Delta u = f$ has a unique solution $u = u _ { f } \in D ( \Delta )$, so $- \Delta$ is maximal monotone. H. Brézis has proved that $- \Delta$ is actually the subdifferential $\partial \phi$ of a lower semi-continuous convex functional $\phi$ from $L ^ { 2 } ( \Omega )$ into $ \mathbf{R}$ which (according to a more general result of R.T. Rockafellar, 1966), is maximal monotone (accretive). It follows from the definition (a1) that if $A$ is $m$-accretive, then for every positive $\lambda$, $( I + \lambda A )$ is invertible and the operator $J _ { \lambda } = ( I + \lambda A ) ^ { - 1 }$ is non-expansive (i.e., Lipschitz continuous of Lipschitz constant $1$) on $X$. The crucial importance of $m$-accretive operators has already been pointed out above.

There is an extensive literature on this topic.

Finally, there is a second notion which also goes by the name "dissipative" (the Coddington–Levinson–Taro Yoshizawa dissipative differential systems). However, the notion of dissipative operators as defined above and that of dissipative systems are different.

References

[a1] V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Noordhoff (1975)
[a2] H. Brezis, "Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert" , North-Holland (1973)
[a3] D. Motreanu, N.H. Pavel, "Tangency, flow-invariance for differential equations and optimization problems" , M. Dekker (1999)
[a4] N.H. Pavel, "Nonlinear evolution operators and semigroups" , Lecture Notes Math. , 1260 , Springer (1987)
[a5] A. Pazy, "Semigroups of linear operators and applications to PDE" , Springer (1983)
How to Cite This Entry:
Accretive mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Accretive_mapping&oldid=18808
This article was adapted from an original article by N.H. Pavel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article