数理解析研究所講究録
1197 巻 2001 年 122-136

122

Generation and Approximation of Semigroups of Lipschitz Operators
Naoki Tanaka (田中直樹)
Department of Mathematics, Faculty of Science
Okayama University, 700-8530 (岡山大学・理学部)

Introduction

After a pioneering work by
the generation theorem of quasi-contractive
semigroups in Banach spaces has been studied intensively and applied to the well-posednessof Cauchy problems for porous medium equations, Hamilton-Jacobi equations and scalar
first order equations. (Some of the main points of theory of quasi-contractive semigroups
have been outlined in a review paper by Crandall [5].) However, Temple [18] showed thatthe theory of quasi-contractive semigroups could not be in general applied to solve genuinely$\mathrm{K}_{\overline{\mathrm{O}}\mathrm{m}}\mathrm{u}\mathrm{r}\mathrm{a}[10]$nonlinear symmetric hyperbolic systems with initial data dense in the whole underlying
Banach space based on the space of integrable functions. It is expected that solution
operators of the Cauchy problem for first order systems of conservation laws are Lipschitz
continuous with respect tonorm. In this case, it is conjectured that such solution
operators form a semigroup of Lipschitz operators. In fact, an attempt has recently been
made by Bressan, Liu and Yang [1] to prove that a family of solution operators of the
Cauchy problem for strictly hyperbolic systems of conservation laws is a semigroup of
Lipschitz operators in the space of integrable functions if its domain is defined by the setof integrable functions whose total variation are sufficiently small.
Throughout this paper $X$ denotes a real Banach space with norm
and $D$ a closed
subset of $X$ . By a semigroup of Lipschitz operators on $D$ we mean a one-parameter family $\{T(t);t\geq 0\}$ of Lipschitz operators from $D$ into itself satisfying the following three
conditions:$L^{1}$$||\cdot||$(S1)$T(\mathrm{O})_{X}=X,$$T(t)\tau(s)_{X}=\tau(t+s)x$ for $x\in D$ and $t,$ $s\geq 0$.

(S2) For each $x\in D,$ $T(\cdot)X:[0, \infty)arrow X$ is continuous.
(S3) For $\tau>0$ there exists$M_{\tau}\geq 1$such that$||T(t)X-T(t)y||\leq M_{\tau}||x-y||$for*This is a joint work with Y. Kobayashi of Niigata University.

$x,$ $y\inD$

and $t\in[0, \tau]$ .

123
An operator$A_{0}$

in $X$ defined by

for $x\in D(A_{0})$$D(A_{0})=$ { $x \in D;\lim_{h\downarrow(}0T(h)X-x)/h$
exists in $X$ }$A_{0}x= \lim_{h\downarrow}\mathrm{o}(\tau(h)x-x)/h$$\{$is called the infinitesimal generator of $\{T(t);t\geq 0\}$ .

We are interested in studying a basic property of semigroups of Lipschitz operators anda characterization of infinitesimal generators of such semigroups which are roughly statedas follows:(i) A nonlinear analogue of Feller’s theorem for semigroups of class(Theorem 1.1):
A semigroup of Lipschitz operators is a quasi-contractive semigroup with respect toa certain metric or metric-like functional.
$(C_{0})$(ii) A characterization of infinitesimal generators of semigroups of Lipschitz operators(Theorem 1.2): A continuous operator from $D$ into $X$ is the infinitesimal generator
of a semigroup of Lipschitz operators on $D$ if and only if it satisfies the subtangential condition and a general type of dissipative condition that there is a metric-likefunctional with respect to which is dissipative.
$A$

$A$

Our discussion is restricted to a special case in which infinitesimal generators are continuous. However this does not mean that the abstract theory obtained here cannot be applied
to any partial differential equations. In fact, in Section 1 we show that the generation
theorem of
semigroups of bounded linear operators can be derived from our theory,
and we also give an application of our results to the Cauchy problem for quasi-linear wave$(C_{0})$equation with damping.
Section 2 contains an approximation of semigroups of Lipschitz operators; namely the
problem of approximation of a semigroup of Lipschitz operators by a sequence of discrete
parameter semigroups (Theorem 2.1). This was discussed by, Chernoff [3] andKurtz [11] for semigroups of linear operators. In the case of nonlinear quasi-contractive
semigroups, a number of results were obtained by Miyadera and Oharu [17], Brezis and Pazy[2], Kurtz [12], and Miyadera and Kobayashi [16]. Although our discussion is restricted$\mathrm{h}_{0}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}[19]$to the special case as mentioned above, the results obtained here are not covered withthe results for quasi-contractive semigroups and are applicable to the existence problemof the global solution of the quasi-linear wave equation of Kirchhoff type by using a finite
difference scheme of Lax-Riedrichs type. As for the related topics, sufficient conditions forthe convergence of Chernoff’s formula were obtained by Marsden [14], and an approximation
theorem of Lax type for semigroups of Lipschitz operators was obtained by Kobayashi.
al. [7], which is an improvement of Chorin et al.

$et$$[4,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5.1]$

124
1. Semigroups of Lipschitz Operators

We begin by stating a nonlinear analogue of Feller’s theorem.
. Let $\{T(t);t\geq 0\}$ be one-parameter family of
Theorem 1.1
Lipschitz operators from $D$ into itself satisfying two conditions (S1) and (S2). Then thefollowing statements are mutually equivalent:$a$$([8,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}4.1])$(i) $\{T(t);t\geq 0\}$ is a semigroup of Lipschitz operators on $D$ .

(ii) There exist $M\geq 1$ and$\omega\in \mathrm{R}$such that$||T(t)X-T(t)y||\leq Me^{\omega t}||x-y||$for$x,$ $y\inD$

and $t\geq 0$ .

and a nonnegative and Lipschitz continuous functional(iii) There exist$X\cross X$ , satisfying the property$\omega\in \mathrm{R}$(V) there exist $M\geq m>0$ such that $m||x-y||\leq V(x, y)\leq M||x-y||$ for$x,$ $y\in

$V$

D$

on

,

such that

(1.1)

for

$V(T(t)x,\tau(t)y)\leq e^{\omega t}V(x,y)$$x,$ $y\in

and $t\geq 0$ .

D$

Remark. In general, the functional on $X\cross X$ in Theorem 1.1 cannot be represented
equivalent to theas $V(x, y)=N(x-y)$ for $(x, y)\in X\cross X$ , by using any norm
be a continuous function satisfying the property. Indeed, let :original normthat there exists $M>1$ such that $1/M\leq f(r)\leq 1$ for $r\in$ R. The unique solutionof the Cauchy problem$V$$N(\cdot)$$||\cdot||$$f$$\mathrm{R}arrow \mathrm{R}$$u(\cdot;x)\in C^{1}([0, \infty);\mathrm{R})$$u’(t)=f(u(t))$ ,is given by $u(t;x)=g^{-1}(t+g(x))$ where$u(\mathrm{O})=X\in \mathrm{R}$$g(r)= \int_{0}^{r}\frac{d\sigma}{f(\sigma)}$for$r\in \mathrm{R}$. A family $\{T(t);t\geq 0\}$defined by $T(t)x=u(t;x)$ is a semigroup on , and (1.1) is satisfied with $\omega=0$ and a
R. The functional
defined by $V(x, y)=|g(x)-g(y)|$ for
functional on
also satisfies condition (V) of Theorem 1.1.
. Then there exists
Now, let us consider the function $f(r)=(1/M+\sqrt{|r|})$ A 1 for
and $t\geq 0$ , because
no real number such that $|T(t)x-T(t)y|\leq e^{\omega t}|x-y|$ forsign $(x-y)(f(X)-f(y))=\sqrt{x}$ for $0=y\leq x\leq(1-1/M)^{2}$ .
$\mathrm{R}$$V$$\mathrm{R}\cross \mathrm{R}$$(x, y)\in \mathrm{R}\cross$$V$$r\in \mathrm{R}$$\omega$$x,y\in \mathrm{R}$A characterization of the continuous infinitesimal generators of semigroups of Lipschitz(see
operators is given by the following theorem which is a generalization ofalso [13] .
$[15,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5]$

$)$

125

. Let be a continuous operator from $D$ into $X$ .
Theorem 1.2
is the infinitesimal generator of a semigroup $\{T(t);t\geq 0\}$ of Lipschitz operators
Then
on $D$ if and only if it satisfies the following two conditions.
$A$$([8,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}4.2])$$A$(A1) $\lim\inf_{h\downarrow 0}d(x+hAx, D)/h=0$ for all $x\in D$ .
and a nonnegative and Lipschitz continuous functional(A2) There exist
satisfying property (V) of Theorem 1.1 such that$\omega\in \mathrm{R}$for$D_{+}V(x,y)(AX, Ay)\leq\omega V(x, y)$$x,$ $y\in

D$

$V$

on

$X\cross X$,

where $D_{+}V$ is a directional derivative defined by

for $(x, y),$$D_{+^{V(y)}}X,( \xi,\eta)=\lim_{h\downarrow}\inf_{0}(V(X+h\xi, y+h\eta)-V(x, y))/h$$(\xi, \eta)\in x_{\mathrm{X}}x$.

In this case, for each $x\in D$ the abstract Cauchy problem$u’(t)=Au(t)$for$t\geq 0_{f}$and$u(\mathrm{O})=X$

has a unique global solution $u\in C^{1}([0, \infty);X)$ given by $u(t)=T(t)x$ for $t\geq 0$ .

We explain a way to find a functional so that (A2) is satisfied, in an abstract fashion.
must be chosen so that the solution operator $T(t)$ is quasi-contractive
A functionalwith respect to , because the quasi-contractivity of $T(t)$ implies the dissipativity condition(A2). For this purpose, let $y\in D$ , and assume that there exists a curve lying in, the Cauchy problem
and $c(1)=y$ and that for eachsuch that
$V$

$V$

$V$

$c$

$x,$

$D$

$\theta\in[0,1]$$c(\mathrm{O})=x$$u’(t;\theta)=Au(t;\theta)$(1.2)

for $t\geq 0$ ,$\{$$u(0;\theta)=C(\theta)$

has a global “smooth” solution$u(t;\theta)$. Differentiating (1.2) in$\dot{u}’(t;\theta)=dA(u(t;\theta))\dot{u}(t;\theta)$$\{$$\dot{u}(0;\theta)=\dot{C}(\theta)$$\theta$

we havefor $t\geq 0$ ,,where $dA(w) \xi=\lim_{h\downarrow 0}(A(w+h\xi)-A(w))/h$ and the limit is taken in some sense.
Moreover, we assume that for each $w\in D$ , the operator $dA(w)$ generates a quasidepending
semigroup on the Banach space $X$ equipped with norm
contractive
Lipschitz continuously on in the sense of [6]; namely there exists a real Banach space $E$
continuously embedded in $X$ such that the set {Au; $u\in D$ } is bounded in $E$ , and there$||\cdot||_{w}$$(C_{0})$$w$

exist $L>0$ and$\beta\geq 0$such thatfor $u\in X$ , and $w,$ $z\in D$ ,for $u\in D(dA(w))$ and $w\in D$$||u||_{w}\leq(1+L||w-z||_{E})||u||_{z}$$\{$$[u, dA(w)u]_{w}\leq\beta||u||_{w}$

126
where
$[u, \xi]_{w}=\mathrm{l}\mathrm{i}\mathrm{m}h\downarrow 0(||u||_{w}-||u-h\xi||_{w})/h$

By using the Lipschitz continuity of$||\cdot||_{w}$for$u,$ $\xi\in X$

with respect to$w$

and $w\in D$ .

, we have$(d/dt)||\dot{u}(t;\theta)||u(t;\theta)\leq[\dot{u}(t;\theta),\dot{u}’(t;\theta)]u(t;\theta)+L||Au(t;\theta)||_{E}||\dot{u}(t;\theta)||_{u(t;\theta)}$$\leq(\beta+L||Au(t;\theta)||E)||\dot{u}(t;\theta)||_{u}(t;\theta)$

Since the set {Au; $u\in D$ } is bounded in$E$.

, we havefor $t\geq 0$ .

$||\dot{u}(t;\theta)||_{u(t\theta};)\leq e^{\omega t}||\dot{c}(\theta)||_{c}(\theta)$

Therefore, the family $\{T(t);t\geq 0\}$ is quasi-contractive with respect to the non-negative
functional defined by$V$$V(x, y)= \inf_{\mathrm{P}^{\mathrm{a}\mathrm{t}}\mathrm{h}}\{\int_{0}^{1}||\dot{c}(\theta)||_{C(}\theta);C(\mathrm{o})=x,$$c(1)=y\}$ ;namelyfor$V(T(t)x, \tau(t)y)\leq e^{\omega t}V(x, y)$$x,$ $y\inD$

and $t\geq 0$ .

We shall give two applications of Theorem 1.2.
Application 1. It will be shown that the generation theorem of contractive
semigroups can be derived from Martin’s result
which is a special case of Theorem 1.2.
If is a densely defined linear operator in $X$ satisfying the Hille-Yosida condition$(C_{0})$$[15,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5]$$A_{0}$(1.3)$||(I-\lambda A\mathrm{o})^{-}1||\leq 1$

for$\lambda>0$,

then the following assertions hold.

(i) If $D$ is defined as the closure of the set $D_{0}=\{x\in D(A_{0}^{2});||A_{0}^{2_{X}}||\leq r\}$ where $r>0$ ,then there exists a H\"older continuous operator from $D$ into $X$ such that $Ax=A_{0}x$
$A$

for

$x\in D_{0}$(ii) The limit.

$S(t)x= \lim_{\lambda\downarrow 0}(I-\lambda A_{0})^{-[t}/\lambda]x$

exists in(iii) The family $\{S(t);t\geq 0\}$ is a contractive
generator is.

$(C_{0})$$X_{f}$

for every $x\in X$ and$t\geq 0$.

semigroup on $X$ whose infinitesimal$A_{0}$We show that Theorem 2.1 is applicable to prove assertion (ii). To this end, let A $>0$
and $x\in D(A_{0}^{2})$ . Since $(I-\lambda A\mathrm{o})^{-1}z=z+\lambda(I-\lambda A\mathrm{o})^{-1}A0z$ for $z\in D(A_{0})$ , we have
( $I-\lambda A_{0)^{-}=}1XX+\lambda(A_{0}x+\lambda(I-\lambda A_{0})^{-1}A_{0}^{2_{X}})$ , and so$\lambda A_{0}x=(I-\lambda A\mathrm{o})^{-}1_{XX}--\lambda^{2}(I-\lambda A_{0})-1A^{2}x0$.

127
We estimate this identity by (1.3). This yields, which implies$||A_{0^{X}}||2-8||x||||A_{0}^{2_{X}}||\leq 0$ . We therefore obtain the generalized Landau
inequality$\lambda||A_{0^{x}}||\leq 2||x||+\lambda^{2}||A^{2_{X}}|0|$(1.4)$||A_{0}X||\leq 2^{\sqrt{2}/2}||A_{0}^{2}X||^{1}||x||^{1}/2$

for $x\in D(A_{0}^{2})$ .

Now, to prove (i) let $r>0$ and let $D$ be the closure of$||A_{0}x-A_{0}y||\leq 4\sqrt{r}||x-y||^{1/2}$$D_{0}$. Then we have by (1.4)

for $x,$ $y\in D_{0}$ .

from $D$ into $X$ , constructed in the way that $Ax= \lim_{narrow\infty}A0x_{n}$ if $x\in D$
andsatisfy $\lim_{narrow\infty}x_{n}=x$ , is H\"older continuous. This means that assertion (i) istrue. To prove (ii), we first show that
The operator$A$$x_{n}\in D_{0}$(1.5) $(I-\lambda A_{0})-1x\in D$ and $A(I-\lambda A_{0})-1x=A_{0}(I-\lambda A_{0})-1x$ for $x\in D$ and(1.6)$\lim_{\lambda\downarrow 0^{A}\mathrm{o}(-}I\lambda A0)^{-1}X=Ax$$\lambda>0$,

for $x\in D$ .

To demonstrate (1.5), let
and $x\in D$ . Then there exists a sequencein$\lim_{narrow\infty}x_{n}=x$that. By the definition of
we have $(I-\lambda A_{0})-1_{X_{n}}\in D(A_{0}^{2})$ and$\lambda>0$$\{x_{n}\}$$D_{0}$

such$D_{0}$$||A_{0}^{2}(I-\lambda A_{\mathit{0}})^{-1}X_{n}||=||(I-\lambda A0)-1A^{2}X_{n}|0|\leq||A_{0^{X}}^{2}n||\leq r$

for $n\geq 1$ , which implies that $(I-\lambda A_{0})-1_{X_{n}}\in D_{0}$ for $n\geq 1$ . Since$(I-\lambda A_{0})-1x$ , we have $(I-\lambda A_{0})-1x\in D$ and $A(I- \lambda A_{0})-1x=\lim_{narrow\infty}A_{\mathrm{o}(A)^{-1}X_{n}}I-\lambda 0=$$A_{0}(I-\lambda A_{0})-1x$ . Assertion (1.6) is derived from the continuity offrom $D$ into $X,$ $(1.5)$
andfor $x\in D$ .
The subtangential condition (A1) follows from (1.5) and (1.6), because $d(x+hA_{X}, D)/h\leq$$||x+hAx-(I-hA_{0})^{-}1|x|/h=||A_{0}(I-hA_{0})^{-1}x-AX||arrow 0$ as. Since$\lim_{narrow\infty}(I-\lambda A_{0})-1_{X_{n}}=$$A$$\lim_{\lambda\downarrow 0}(I-\lambda A_{0})-1x=x$$h\downarrow \mathrm{O}$$(||x+hAx-(y+hAy)||-||x-y||)/h$$\leq(||(I-hA\mathrm{o})^{-1_{X}}-(I-hA_{0})^{-}1y||-||x-y||)/h$$+||x+hAx-(I-hA_{0})^{-}1|x|+||y+hAy-(I-hA_{0})-1y||$ ,it follows from (1.3) that (A2) is satisfied with $V(x, y)=||x-y||$ and $\omega=0$ . Therefore, we
deduce from Theorem 1.2 that is the infinitesimal generator of a contractive semigroup$\{T(t);t\geq 0\}$ on $D$ .
$A$

Now, we turn to the proof of assertion (ii). Let $\lambda>0$ and $x\in D$ . For simplicity, set
for $i=1,2,$ . Then we have by (1.5)$x_{i}^{\lambda}=(I-\lambda A_{0})-ix$$\ldots$$(x_{i}^{\lambda}-x^{\lambda})i-1/\lambda=A_{X_{i}^{\lambda}}$for $i=1,2,$$\ldots$. Since $(d/dt)T(t)X=AT(t)x$ for $t\geq 0$ , we have,$(T(i\lambda)_{X}-\tau((i-1)\lambda)_{X)}/\lambda=AT(i\lambda)X+\epsilon_{i}^{\lambda}$

128

where
$\epsilon_{i}^{\lambda}=\frac{1}{\lambda}\int_{()\lambda}^{i\lambda}i-1)(A\tau(r)X-AT(i\lambda)xdr$

for $i=1,2,$$\ldots$. By the dissipativity of$A$

we have$(||x_{i}^{\lambda}-\tau(i\lambda)x||-||x_{i}^{\lambda}-\tau(i\lambda)_{X}-\lambda(Ax_{i}-\lambda AT(i\lambda)x)||)/\lambda\leq 0$,

which implies thatfor $i=0,1,2,$ . Since AT:$X$ is continuous, it is concluded thatuniformly on every
compact subinterval of. Assertion (ii) is proved by a density argument, since
is dense in $X$ .
$||x_{i}^{\lambda}-T(i \lambda)x||\leq\lambda\sum_{k=1}^{i}||\mathcal{E}_{k}^{\lambda}||$$[0, \infty)arrow$$(\cdot)x$$\ldots$$\lim_{\lambda\downarrow 0}(I-\lambda A_{0)^{-}}1t/\lambda 1_{X}=T(t)x$$[0, \infty)$$D(A_{0}^{2})$Application 2. We next give an application of Theorem 1.2 to the Cauchy problem forquasi-linear wave equation with damping$u_{t}=v_{x}$

(1.7)

$\{$

$v_{t}=\varphi(\prime u)x-\nu v$

where $\nu>0$ and$\varphi\in C^{4}(\mathrm{R})$,satisfies $\varphi(0)=\varphi’(0)=0$ and $\varphi’’(r)\geq c_{0}>0$ for$r\in \mathrm{R}$.

Remark. The problem of existence and uniqueness of global solutions of (1.7) has
been studied in different ways, by several authors (for example, see [20]). Our approach isoperator-theoretic and based on Theorem 1.2.
Theorem 1.3$H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$$([8,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5.1])$

with. There is an $r_{0}>0$ such that for each$||(u_{0,0}v)||_{H^{2}\cross H}2\leq r_{0}$$(u_{0}, v\mathrm{o})\in$, the problem (1.7) has a unique solution $(u, v)$

in the class$C^{1}([0, \infty);L^{2}(\mathrm{R})\cross L^{2}(\mathrm{R}))\cap L^{\infty}(0, \infty;H2(\mathrm{R})\cross H^{2}(\mathrm{R}))$satisfying the initial condition $(u, v)|_{t=0}=(u_{0}, v\mathrm{o})$ . Moreover, there exist constants $M\geq 1$
andsuch that if $(u, v)$ and \^u, are solutions with initial dataandsatisfying
andrespectively, thenwe have$\omega\geq 0$$($$H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$$(u_{0}, v_{0})$$\hat{v})$$||(u_{0}, v_{0})||H2\cross H2\leq r_{0}$$||(\hat{u}_{0},\hat{v}0)||_{H^{2}\cross H^{2}}\leq r_{0}$$||(u(t, \cdot),v(t, \cdot))-(\hat{u}(t, \cdot),\hat{v}(t, \cdot))||_{L\cross}2L^{2}\leq Me^{\omega t}||(u_{0}, v\mathrm{o})-(\hat{u}_{0},\hat{v}_{0})||L2\cross L2$

Now, let $X$ be a real Hilbert space
It is shown that a nonnegative functional$V((u, v),$

$($

\^u,

$\hat{v})$)for$t\geq 0$.

with norm $||(u, v)||=(||u||_{L}2|2+|v||^{2}L2)^{1/2}$ .
on $X\cross X$ defined by$L^{2}(\mathrm{R})\cross L^{2}(\mathrm{R})$$V$$(\hat{u}_{0},\hat{v}_{0})\in$$=( \int_{-\infty}^{\infty}(\int_{u}^{\hat{u}}\sqrt{\varphi’’(r)}dr\mathrm{I}^{2}+(\hat{v}-v)^{2}dx)^{1/}2$

129

is Lipschitz continuous and satisfies property (V). We use a functional $H$ defined by$H(u,v)= \frac{1}{2}\int_{-\infty}^{\infty}v^{2}+|\nu u+\partial_{x}v|^{2}+|\nu\partial_{x}u+\partial_{x}^{2}v|^{2}d_{X}$$+ \frac{1}{2}\int_{-\infty}^{\infty}\varphi’(u)(|\partial’|^{2}xu+|\partial_{x}^{2}u|^{2})dX+\int_{-\infty}^{\infty}\varphi(u)dx$

for$(u, v)\in H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$.

Theorem 1.3 is deduced from the following theorem by virtue of Theorem 1.2.
Theorem 1.4
in $X$ defined by$([8,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5.2])$$A$. There is an $r_{0}>0$ such that a nonlinear operator$A(u, v)=(\partial_{x}v, \partial_{x}\varphi^{;}(u)-\nu v)$$\{$

for $(u, v)\in D$$D=\{(u, v)\in H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R});H(u, v)\leq r_{0}\}$satisfies the following four properties:(a) The set $D$ is closed in X.

-(b) The operator $A:Darrow X$ is continuous.

(c) There exists$\omega\in \mathrm{R}$such that

$A$

\^u,

$\hat{v})$$D_{+}V((u, v),$

for $(u, v),$ \^u,
$($

(d)

$\hat{v})\in D$

$($

is

$\omega$-dissipative with respect to) $(A(u, v),$ $A($ \^u,$\hat{v})$$V$; namely) $\leq\omega V((u,v),$ \^u,$($$\hat{v})$).

$\lim_{\lambda\downarrow}\inf_{0}d((u,v)+\lambda A(u,v),$ $D)/\lambda=0$

for $(u,v)\in D$ .

We explain only the way to choose a number
properties (a) through (d) of Theorem 1.4.

$r_{0}$and to find a functional$V$so that four

Suppose that the problem (1.7) has a global smooth solution. A number$v(t, \cdot))\in D$ for $t\geq 0$ , since the subtangential$r_{0}>0$ must be chosen such thatcondition (A1) is deduced from this property. For this purpose, it is first proved that there
exist $c>0$ and a continuous function satisfying $\rho(0)=0$ such that$(u(t, \cdot),$ $v(t, \cdot))$$(u(t, \cdot),$$\rho$$(d/dt)H(u(t, \cdot),$ $v(t, \cdot))+\{c-\rho(H(u(t, \cdot), v(t, \cdot)))\}H(u(t, \cdot),$ $v(t, \cdot))\leq 0$

for $t\geq 0$ . The property that$(u(t, \cdot),$ $v(t, \cdot))\in D$

for $t\geq 0$

130
is satisfied, if $r_{0}>0$ is chosen such that $H(u, v)\leq r_{0}$ implies $\rho(H(u, v))<c$ . Indeed, if$(u_{0}, v\mathrm{o})\in D$then$(d/dt)H(u(t, \cdot),$ $v(t, \cdot))\leq 0$

for $t\geq 0$ ; hence $H(u(t, \cdot),$$v(t, \cdot))\leq H(u_{0}, v\mathrm{o})\leq r_{0}$, namely$(u(t, \cdot),$ $v(t, \cdot))\in D$

for $t\geq 0$ .

According to the abstract idea mentioned before, let us find a functional associatedwith the differential equation (1.7). For this purpose, let us consider the equation$V$$\dot{u}_{t}=\dot{v}_{x}$$\{$$\dot{v}_{t}=(\varphi^{\prime J}(u)\dot{u})x-\nu\dot{v}$

where

,

denotes the derivative of with respect to . Note that the set $D$ is bounded in, and so the set $\{A(u, v);(u,v)\in D\}$ is bounded in.
For $(w, z)\in D$ , let us define a linear operator $dA(w, z)$ inby$\dot{u}$$u$$\theta$$H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$$E:=H^{1}(\mathrm{R})\cross H^{1}(\mathrm{R})$$L^{2}(\mathrm{R})\cross L^{2}(\mathrm{R})$$dA(w, z)(u, v)=(v_{x}, (\varphi^{J\prime}(w)u)_{x}-\nu v)$

forThen, the operator $dA(w, z)$ generates a contractive
the norm defined by$(u, v)\in H^{1}(\mathrm{R})\cross H^{1}(\mathrm{R})$.

semigroup on $X$ equipped with$(C_{0})$$||(u, v)||_{(w},z)=( \int_{-\infty}^{\infty}\varphi’(\prime w)u^{2}+v^{2}d_{X})^{1}/2$

Since$||(u, v)||_{(z}w,)-||(u, v)||_{(} \hat{w},\hat{z})\leq(\int_{-\infty}^{\infty}(\sqrt{\varphi’’(w)}-\sqrt{\varphi’’(\hat{w})})2)^{1}u^{2}dx/2$

by Minkowski’s inequality, we have by the boundedness of $D$ in$H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$$||(u, v)||_{()}w,z-||(u, v)||_{(\hat{w}},\hat{z})\leq L||(u, v)||_{(}\hat{w},\hat{z})||w-\hat{w}||_{H^{1}}$,which implies that the norm $||(u, v)||_{(w,Z})$ depends Lipschitz continuously on $(w, z)$ with
$E=H^{1}(\mathrm{R})\cross H^{1}(\mathrm{R})$. Therefore, the functional $V((u, v),$ \^u,$($$\hat{v})$) given by$\inf_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}\{\int_{0}^{1}(\int_{-}^{\infty}\infty u\varphi(J’)\dot{u}+\dot{v}^{2}dX\mathrm{I}2\}1/2d\theta$

is a desired one. By an easy computation we find$\int_{0}^{1}(\int_{-\infty}^{\infty}\varphi’(\prime u)\dot{u}2+\dot{v}^{2}dx)^{1/2}d\theta$$= \int_{0}^{1}(\int_{-\infty}^{\infty}(\frac{\partial}{\partial\theta}\int_{0}^{u}\sqrt{\varphi’’(r)}dr)^{2}+\dot{v}^{2}dx\mathrm{I}1/2d\theta$$\geq(\int_{-\infty}^{\infty}(\int_{0}^{1}\frac{\partial}{\partial\theta}(\int_{0}^{u}\sqrt{\varphi’’(r)}dr\mathrm{I}^{d}\theta)^{2}+(\int_{0}^{1}\dot{v}d\theta)^{2}dx)^{1/}2$$=( \int_{-\infty}^{\infty}(\int_{u(0)}^{u(}1)d\sqrt{\varphi’’(r)}r\mathrm{I}^{2}+(v(1)-v(\mathrm{O}))^{2}d_{X}\mathrm{I}^{1}/2$

131

is strictly increasing and satisfies. Now, let
and $\lim_{rarrow-\infty^{\psi}}(r)=-\infty$ , because $\varphi’’(r)\geq c_{0}>0$ for

On the other hand, the function$\lim_{rarrow\infty}\psi(r)=\infty$$u_{0},$$u_{1}\in L^{2}(\mathrm{R})$$\psi(r):=\int_{0}^{r}\sqrt{\varphi’’(S)}ds$$r\in \mathrm{R}$and define$u(x, \theta)$by$\psi(u(X, \theta))=\psi(u_{0}(X))+\theta(\psi(u_{1}(X))-\psi(u0(x)))$for$\theta\in[0,1]$.

Then we have $\psi’(u)\dot{u}=^{\psi}(u1)-\psi(u_{0})$ , namely$\sqrt{\varphi’’(u)}\dot{u}=\int_{u_{0}}^{u_{1}}\sqrt{\varphi’’(r)}$dr.

, the function defined by $v(x, \theta)=v_{0}(x)+\theta(v_{1}(x)-v\mathrm{o}(x))$ satisfies$\dot{v}=v_{1}-v_{0}$ . Under an appropriate condition, the desired functional
is thus given byFor$v_{0},$$v_{1}\in L^{2}(\mathrm{R})$

$v$

$V$

$V((u, v),$

$($

\^u,

$\hat{v})$)$=( \int_{-\infty}^{\infty}(\int_{\hat{u}}^{u}\sqrt{\varphi’’(r)}dr)^{2}+(v-\hat{v})^{2}dX)^{1/}2$

2. Approximation of Semigroups of Lipschitz Operators
The space $X$ is assumed to be approximated by a sequence
the sense that for each there exists $P_{n}\in B(X, X_{n})$ such that$\{X_{n}\}$of Banach spaces in

$n$

(2.1)

$\lim_{narrow\infty}||P_{n}x||_{n}=||x||$

for $x\in X$ .

This notion was introduced by botter [19] (see also [11]). Note that $X$ is a function
is a space of sequences which is identified with a space of discrete functions
space and
$X_{n}$

defined only at certain grid points, in most applications.
Moreover, the set $D$ is assumed to be approximated by a sequence
for $n\geq 1$ , in the following sense:is closed insuch that $\lim_{narrow\infty}||x_{n}-P_{n}X||_{n}=0$ .
For any $x\in D$ , there exist$\{D_{n}\}$$D_{n}$of sets where$X_{n}$$x_{n}\in D_{n}$Note that there is no difficulty in verifying this assumption because the property that
$P_{n}(D)\subset D_{n}$holds for $n\geq 1$ , in most applications.

An approximation of a semigroup of Lipschitz operators by a sequence of discrete parameter semigroups is discussed. An approximation theorem of Lipschitz operators is givenby the following theorem.

132
Theorem 2.1
For each $n\geq 1$ , letcondition$C_{n}$. Let

be a null sequence of positive numbers.
be a Lipschitz operator from
into itself satisfying the stability$([9,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}4.5])$$\{h_{n}\}$$D_{n}$$||c_{n}^{\iota}X-c^{\mathrm{t}}y|n|n\leq Me|\omega h_{n}l|_{X}-y||_{n}$for $x,y\in D_{n}$ and $l=1,2,$
Assume that$A$

where $M\geq 1$ andare independent of .
is a continuous operator from $D$ into $X$ satisfying the property$\omega\geq 0$$\ldots f$

if $x_{n}\in D_{ny}x\in D$ and$n$$\lim_{narrow\infty}||x_{n}-P_{n}x||_{n}=0$

then$\lim_{narrow\infty}||(C_{n}x_{n}-xn)/h_{n}-P_{n}AX||n=0$,

and the subtangential condition$\lim_{h\downarrow 0}\inf d(x+hAX, D)/h=0$

for $x\in D$ .

Then is the infinitesimal generator of a semigroup $\{T(t);t\geq 0\}$ of Lipschitz operators
on D. Moreover, forand $x\in D$ with $\lim_{narrow\infty}||x_{n}-P_{n}x||_{n}=0$ , we have$A$

$x_{n}\in D_{n}$$\lim_{narrow\infty}||C_{nn}^{[}t/h_{n}1_{Xn}-PT(t)x||n=0$,

and the convergence is uniform on every compact subinterval of
the integer part of $r\geq 0$ .

$[0, \infty)$. Here$[r]$denotes

The existence problem of the global solution of the following quasi-linear wave equationof Kirchhoff type is discussed by using a finite difference scheme of Lax-Riedrichs type, as
an application of Theorem 2.1.
$\partial_{t}u=\partial_{x}v$

(2.2)

$\{$

$\partial_{t}v=\beta’(||u||_{L}2)2\partial_{x}u-\nu v$$u|_{t=0=u_{0}}$,$v|_{t=0}=v_{0}$.

Here $\nu>0$ andis assumed to be a convex function
ing$\beta(0)=0$ and $\beta’(s)\geq m_{0}>0$ for $s\geq 0$ .
For this purpose, let
andbe two null sequences of positive numbers such that$h_{n}/k_{n}=r$ , whereis an appropriate positive constant determined later.
$\beta\in C^{2}([\mathrm{o}, \infty);[0, \infty))$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{h}^{\Gamma}$$\{h_{n}\}$$\{k_{n}\}$$r$

Let us consider a difference scheme of Lax-Riedrichs type$(u_{l+1,i^{-}}(u_{l,i+1}+u\iota,i-1)/2)/h_{n}=(v\iota,i+1^{-}v_{l,i-1})/2k_{n}$,$(v_{\iota+1,i^{-}}(v_{\mathrm{t},i1}++vl,i-1)/2)/h_{n}$

(2.3)

$\{$

$=\beta’(||\{ul+1,i\}||2n)(u\iota,i+1^{-}ul,i-1)/2k_{n}-\nu(v\iota,i+1+v\iota,i-1)/2$$u_{0,i}= \frac{1}{k_{n}}\int_{(/2)k_{n}}i-1d(i+1/2)knu\mathrm{o}(x)x$,,$v_{0,i}= \frac{1}{k_{n}}\int_{i}^{(/)}(-1/2)k_{n}vi+12kn\mathrm{o}(X)d_{X}$133

for $l=0,1,2,$
by$\ldots$

and $i=0,$$\pm 1,$ $\pm 2,$$\ldots$, where the symbol$||\cdot||_{n}$is the norm in$l^{2}$

defined$||u||_{n}=(_{i=} \sum_{-\infty}^{\infty}uk2)^{1}in/2$

Let $R>0$ and set $E_{R}=\beta(R)+\beta’(R)R$ , and choose $r>0$ such that
(2.4)

Let

$r \cdot\sup\{\sqrt{\beta’(\xi)};\xi\in[0, E_{R}/m_{0}]\}<1$

be a real Banach space$X_{n}$$l^{2}\cross l^{2}$.

equipped with the norm defined by.

$||(u, v)||_{n}=(||u||_{n}2+||v||_{n}2)^{1/2}$

is approximated by a sequence of
It is well-known thatdefined bysense of (2.1), by considering the operator$X=L^{2}(\mathrm{R})\mathrm{X}L^{2}(\mathrm{R})$$\{X_{n}\}$in the$P_{n}$$\frac{1}{k_{n}}\int_{(i}-1/2)k_{n}v(x)dx\}$$P_{n}(u,v)= \{(\frac{1}{k_{n}}\int_{(i-1/2)k_{n}}^{(1}i+/2)kn(i+1/2)kn)u(X)dx,$.

We begin by showing that a family of solution operators for the difference scheme of
Lax-Riedrichs type forms a discrete parameter semigroup on a certain closed set satisfying
the stability condition. Rearranging the equation (2.3) we have$u_{l+1,i}=(u\iota,i+1+u_{l,i-1})/2+r(v_{l,i+1^{-}}v_{l,i-1})/2$

,

$\{$

$v_{\mathrm{t}+1,i}=(v_{\iota},i+1+v_{l,i-1})/2+r\beta’(||\{u\iota+1,i\}||2n)(u_{l},i+1^{-}u_{\mathrm{t}},i-1)/2-\nu h_{n}(v_{\iota,i+1}+v_{l,i-1})/2$

We now consider a mapping
if and only if$C_{n}$

from$D_{n}$

into$X_{n}$.

by the following relation: $(w, z)=C_{n}(u, v)$$w_{i}=(u_{i+1}+u_{i-1})/2+r(v_{i+1}-v_{i-}1)/2$ ,$\{$$z_{i}=(v_{i+1}+v_{i-1})/2+r\beta’(||w||_{n}2)(ui+1-u_{i-}1)/2-\nu h_{n}(v_{i+1}+v_{i-1})/2$

for $i=0,$$\pm 1,$ $\pm 2,$$\ldots$. Here$D_{n}$

is a closed subset of$X_{n}$,of the form

$D_{n}=\{(u, v);H_{n}(u,v)\leq s\}$,where $H_{n}(u,v)$ is defined by$H_{n}(u,v)=||u||_{n}2+|| \delta_{n}0u||_{n}^{2}+||\delta_{n}0\delta 0nu||_{n}2+\frac{1}{\beta’(||u||_{n}^{2})}(||v||_{n}2+||\delta_{n}0v||_{n}2|+|\delta_{nn}^{00}\delta v||^{2}n)$where $\delta_{n}^{0}u=\{(u_{i+}1-u_{i-}1)/2k_{n}\}$ . The number $s>0$ can be chosen such that the operator
into itself and that the discrete semigroup $\{C_{n}^{l};\iota=0,1,2, \ldots\}$ is stable in themaps$C_{n}$$D_{n}$

134
following sense, by condition (2.4): There exist $M\geq 1$ andthat( v)– \^u,$\omega\geq 0$$||c_{n}^{\iota}(u, v)-C_{n}l(\hat{u},\hat{v})||_{n}\leq Me^{\omega h_{n}}\iota||$for $(u, v),$ \^u,
The operator$($$\hat{v})\in X_{n},$

$A$

in

$n=1,2,$$X$$\ldots$

and $l=1,2,$$\ldots$

$u,$

$($

independent of

$n$

such

$\hat{v})||_{n}$.

defined by$A(u, v)=(\partial_{x}v,\beta’(||u||_{L}2)2\partial xu-\mathcal{U}v)$$\{$

for $(u, v)\in D$$D=\{(u,v)\in H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R});H(u, v)\leq r_{0}\}$is continuous on $D$ , by using the Landau inequality. Here $H(u, v)$ is defined by$H(u, v)=||u||_{L^{2}}2+||u|x|2L2+||u_{xx}||2L2+ \frac{1}{\beta’(||u||_{L^{2}}2)}(||v||_{L}2|2+|v_{x}||2L2+||vxx||_{L^{2}}2)$for. The number $r_{0}>0$ can be chosen such that $P_{n}(D)\subset D_{n}$ for
all $n\geq 1$ and that the subtangential condition is satisfied.
$(u, v)\in H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$Consequently, it is deduced from Theorem 2.1 that the solution which are computed
recursively by Lax-Riedrichs scheme (2.3) converges to the smooth solution of Kirchhoff
equation (2.2).
Theorem 2.2
assertions hold for each$([9,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5.4])$.

There exists an $r_{0}>0$ such that the following
satisfying:$(u_{0}, v\mathrm{o})\in H^{2}(\mathrm{R})\cross H^{2}(\mathrm{R})$$||(u_{0}, V0)||H^{2}\cross H^{2}\leq r_{0}$(i) Problem (2.2) has a unique solution $(u(t), v(t))$ in the classsatisfying the initial condition.

$C^{1}([\mathrm{o}, \infty);L^{2}(\mathrm{R})\cross L^{2}(\mathrm{R}))$$(u(\mathrm{O}), v(\mathrm{O}))=(u_{0}, v\mathrm{o})$(ii) The solution $(u(t), v(t))$ is approximated by a sequence of solutionsof (2.3) in the sense that$\{(\{u_{l,i}\}, \{v_{l,i}\})\}$$\lim_{narrow\infty}||P_{n}(u(t, \cdot),v(t, \cdot))-\{(u_{[/]}th_{n},i, v[t/h_{n}1^{i},)\}||n=0$

uniformly on every compact subinterval of$[0, \infty)$, where$P_{n}$

is defined by$P_{n}(u,v)= \{(\frac{1}{k_{n}}\int_{(i-}(i+1/2)k_{n}u(x)dX,$$\frac{1}{k_{n}}\int_{(1}1/2)k_{n}i(i+1/2)knv(-/2)k_{n}X)dX)\}$.

References[1] A. Bressan, T. P. Liu and T. Yang,stability estimates for $n\cross n$ conservation
laws, Arch. Ration. Mech. Anal. 149 (1999) 1-22.
$L^{1}$

135[2] H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear
operators in Banach spaces, J. Funct. Anal. 9 (1972) 63-74.
[3] P. R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal.
2 (1968) 238-242.

[4] P. R. A. J. Chorin, T. J. R. Hughes, M. F. McCracken and J. E. Marsden, Productformulas and nume rical algorithms, Comm. Pure Appl. Math. 31 (1978) 205-256.
[5] M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators,

Proc. Sympos. Pure Math. 45 (1986) 305-337.
[6] T. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hy-

perbolic systems with applications to nonlinear elastodynamics and general relativity,
Arch. Ration. Mech. Anal. 63 (1977) 273-294.
[7] Y. Kobayashi, S. Oharu and N. Tanaka, An approximation theorem of Lax type forsemigroups of Lipschitz operators, Progress in Nonlinear Differential Equations andTheir Applications 42 (2000) 159-166.
[8] Y. Kobayashi and N. Tanaka, Semigroups of Lipschitz operators, Advances in Dif-

ferential Equations (in press).
[9] Y. Kobayashi and N. Tanaka, Convergence and approximation of semigroups of
Lipschitz operators, (in preparation).
[10] Y.

$\mathrm{K}_{\overline{\mathrm{O}}\mathrm{m}}\mathrm{u}\mathrm{r}\mathrm{a}$, Nonlinear semigroups in Hilbert space, J. Math. Soc. Japan 19 (1967)

493-507.
[11] T. G. Kurtz, Extension of Trotter’s operator semigroup approximation theorems, J.

Funct. Anal. 3 (1969) 354-375.
[12] T. G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with
an application to Gas kinetics, bans. Amer. Math. Soc. 186 (1973) 259-272.
[13] V. Lakshmikantham, A. R. Mitchell and R. W. Mitchell, Differential equations on
closed subsets of a Banach space, Rans. Amer. Math. Soc. 220 (1976) 103-113.
[14] J. E. Marsden, On product formulsa for nonlinear semigroups, J. Funct. Anal. 13(1973) 51-72.
[15] R. H. Martin Jr., Differential equations on closed subsets of a Banach space,
Amer. Math. Soc. 179 (1973) 399-414.

$r_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{s}}$.

136[16] I. Miyadera and Y. Kobayashi, Convergence and approximation of nonlinear semigroups, Proceedings of Japan-France Seminar on Functional Analysis and Numerical

Analysis (1978) 277-295.
[17] I. Miyadera and S. Oharu, Approximation of semi-groups of nonlinear operators,
T\^ohoku Math. J. 22 (1970) 24-47.

-contractive metrics for systems of conservation laws, ’bans.
[18] B. Temple, No
Amer. Math. Soc. 288 (1985) 471-480.
$L^{1}$[19] H. F. Rotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958)

887-919.
[20] Y. Yamada, Quasilinear wave equations and related nonlinear evolution equations,
Nagoya Math. J. 84 (1981) 31-83.