Accretive mapping

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)