数理解析研究所講究録
第 1984 巻 2016 年 132-142

132

QUASI VARIATIONAL INEQUALITIES INECONOMIC GROWTH MODELS

WITH TECHNOLOGICAL DEVELOPMENT
Nobuyuki Kenmochi
School of Education (Mathematics), Bukkyo University
kenmochi@bukkyo-u.ac.jp

1. Introduction
This is a report on the recent work [5] with A. Kadoya and M. Niezgodka. The main
objective of this paper is to reconsider economic growth models (cf. [7,8]) in the macroeconomics from a viewpoint of the mathematical theory on quasi-variational inequalities.
As many economists pointed out, the technological innovation brings various changes to
production systems. Especially it enables to get big output by rather small labor force
and this is a very important point for our aging society in the future.
In this paper we propose new economic growth models, taking account of technologicaldevelopment, in which we investigate its influence on the growth of economics. Moreover,
we discuss it in a closed system between major economic elements which are capital,technological level, labor force and output; in the classical growth model due to R. M.
Solow [8] the most important one was “capital” as well as its dynamics on condition thatthe evolution of technological level and labor force are prescribed independently eachother and the output is prescribed by the other three elements. However, in a complex
structure of our future society it is quite natural to suppose that these elements dependon each other and is expected that the production system is formulated as a closed loop
between them. Along such a direction we shall propose a simplified model promoting the
economic growth.
2. Formulation and theorems
We consider an economic model that includes the formation and dynamics of knowledgetechnological (K&T) region in the system:$w’(t)+$ 伽 ( ) $=\sigma P(L\cdot A(t))w(t)^{\alpha},$
$t$

(1)

$t>0$ ,

$r_{k}’(t)+\partial\psi_{k}(r_{k}(t))\ni f_{k}(w(t), r_{k}(t)) , t>-\tau_{0}, k=1, 2, \cdots, N$$A’(t)+\partial I_{K_{0}(r(t))}(A(t))\ni g(r(t);w(t), A(t))$

in$R^{N},$,(2)$t>0,$(3)

with $r(t):=(r_{1}(t), r_{2}(t), \cdots, r_{N}(t))$ ,$w(t)=w_{0}(t)$for $t\in[-\tau_{0}, 0],$$A(O)=A_{0},$$r(-\tau_{0})=0$

.

(4)

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In the macro-economics, $w(t)$ is the capital, $K_{0}(r(t))$ is the knowledge-technology (K&T)
region cultivated by making continuous investiment, and $A(t)$ is the technological level.
The objective of this paper is to construct a global in time solution $\{w, r, A\}$ of (1) $-(4)$which posseses some properties from the economic point of view; for instance,(e1) the capital $w(t)$ is non-decreasing indecreasing in , too,$t$

and the eﬀective labor $L\cdot A(t)$ is non-$t$(e2) the K&T region $K_{0}(r(t))$ is non-decreasing in(e3) $w(t)$ , $r(t)$ and $L\cdot A(t)$ converges as $tarrow\infty$ to
point
of $A(t)$ .

$t$and $A(t)\in K_{0}(r(t))$ for all$w_{\infty},$$r_{\infty}$

and$L\cdot A_{\infty}$$t,$

for any cluster$A_{\infty}$

In this work one of important questions is how to set up some specified classes of func, $g(r;w, A)$ and $K_{0}(r)$ so that the solution posseses the above mentioned
tions
properties.
$f_{k}(w, r_{k})$Our problem is treated under the following assumptions:(A1) $b>0,$ $0<\sigma<1,$ $0<\alpha<1,$ $\tau_{0}>0$ are constants, $L$ $:=(L_{1}, L_{2}, \cdots, L_{N})$ with
$L_{k}>0(1\leq k\leq N)$ and $P(r)$ is a smooth function onsuch that$R_{+}$$P( O)=0, P’(r)>0, \forall r>0, \lim_{r\downarrow 0}P’(r)=\infty.$(A2) To each vector$R_{+}^{N}$$r$$:=(r_{1}, r_{2}, \cdots, r_{N})\in R_{+}^{N}$a compact and convex subset $K_{0}(r)$ ofis asigned so thatInt.

$K_{0}(r)\neq\emptyset,$$\forall r\in R_{+}^{N}$

with $r_{k}>0,$ $k=1$ , 2,$\cdots,$$N,$and the mapping $rarrow K_{0}(r)$ is Lipschitz continuous in the sense of Hausdorﬀ
distance inand monotone increasing in the sense that $K_{0}(r)\subset K_{0}(r’)$ if$r_{k}’(1\leq k\leq N)$ for all $r=(r_{1}, r_{2}, \cdots, r_{N})$ and $r’=(r_{1}’, r_{2}’, \cdots, r_{N}’)\in R_{+}^{N}.$$R^{N}$(A3)$r_{k}\leq$is a proper l.s. , non-negative convex function on $R$ such that $\psi_{k}(r)=0$ for
all $r\leq r_{0}$ with a fixed positive number
andis bounded from above, sayfor a positive finite number ; hence
for $r<r_{0}$ and$\psi_{k}$$c.$$r_{0}$$D(\psi_{k})$$D(\psi_{k})\subset(-\infty, \gamma_{k}] or (-\infty, \gamma_{k})$$\gamma_{k}$$\partial\psi_{k}(r)=0$$R(\partial\psi_{k})=R+\cdot$$N,$(A4) For each $k=1$ , 2,
) is a positive, non-decreasing (in each variable) and
Lipschitz continuous function on. If $A$ $:=(A_{1}, A_{2}, \cdots, A_{N})$ with $0<A_{k}<\gamma_{k}$and $1\leq k\leq N$ and if is a positive number satisfying $bw=\sigma P(L\cdot A)w^{\alpha}$ , then$f_{k}$$\cdots,$$R_{+}^{2}$$w$$f_{k}(w, A_{k})> \sup\partial\psi_{k}(A_{k})$.

(A5) $g(r;w, A)$ $:=(g_{1}(r;w, A), g_{2}(r;w, A), \cdots, g_{N}(r;w, A))$ is a Lipschitz continuous. Iffunction fromand $A=(A_{1}, A_{2}, \cdots, A_{N})\in$
into$N$ , thenwith $A_{k}>0$ for all $k=1$ , 2,
$R_{+}^{N}\cross R_{+}\cross R_{+}^{N}$$\partial K_{0}(r)$$R_{+}^{N}$$r\in R_{+}^{N}$$\cdots,$$N\cdot L\geq 0, \forall N\in N_{c}(A)$

,

134
and
$\{g(r;w, A)-(_{N}\max_{\in N_{c}(A)}g(r;w, A)\cdot N)^{+}N\}\cdot L\geq 0,$$\forall w\geq 0,$

where $N_{c}(A)$ is the unit normal cone of $K_{0}(r)$ at $A$ , namely$N_{c}(A) :=\{N\in R^{N}||N|=1, N\cdot(x-A)\leq 0, \forall x\in K_{0}(r)\}.$

Then we have:
Theorem 1. For the initial dataand non-iecreasing onand$w_{0}\in W_{+}^{1,2}(-\tau_{0},0)$

and$A_{0}$

assume that$w_{0}$is positive$[-\tau_{0}, 0]$$A_{0}\in Int.K_{0}(r(O), bw_{0}(0)<\sigma P(L\cdot A_{0})w_{0}(0)^{\alpha}.$Furthermore suppose that$g(r;w, A)\cdot L>0,$$\forall w\geq 0,$$\forall r:=(r_{1}, r_{2}, \cdots, r_{N})\in R_{+}^{N}$$\forall A=(A_{1}, A_{2}, \cdots, A_{N})\in K_{0}(r)$with $r_{k}>0,$ $k=1$ , 2,with $A_{k}>0,$ $k=1$ , 2,$\cdots,$$\cdots,$

$N,$

(5)
$N.$

Then problem (1)-(4) admits at least one global in time solution $\{w, r, A\}$ such that
$w,$

$r$

and

$L\cdot$$A$

are $non-$ decreasing on$[0, \infty$).

Theorem 2. Under the same assumptions as in Theorem 1, let $\{w, r, A\}$ be any global
in time solution of (1)-(4). Then we have:(i)$w_{\infty}$$:= \lim_{tarrow\infty}w(t)$$K_{0}(r(t))$(ii) Let$A_{\infty}$,$r_{\backslash \infty}$

converges to$:= \lim_{tarrow\infty}r(t)$$K_{0}(r_{\infty})$and$\ell_{\infty}$$:= \lim_{tarrow\infty}L\cdot A(t)$exist. Moreover,in the sense of Hausdorﬀ distance as

be any cluster point of $A(t)$ as
we have:$tarrow\infty$. Then$\ell_{\infty}=L\cdot A_{\infty}$$tarrow\infty.$. Moreover, with

$r_{\infty}=(r_{1\infty}, r_{2\infty}, \cdots, r_{N\infty})$$bw_{\infty}=\sigma P(L\cdot A_{\infty})w_{\infty}^{\alpha},$$f_{k}(w_{\infty}, r_{k\infty})\in\partial\psi_{k}(r_{k\infty}) , k=1, 2, \cdots, N,$$g(r_{\infty};w_{\infty}, A_{\infty})\cdot L\in\partial I_{K_{0}(r_{\infty})}(A_{\infty})\cdot L,$

where$\partial I_{K_{0}(r_{\infty})}(A_{\infty})\cdot L=\{r_{\infty}^{*} .

L|r_{\infty}^{*}\in\partial I_{K_{0}(r_{\infty})}(A_{\infty})\}.$

Remark.1 In general, it is not guaranteed that $A(t)$ converges inas $tarrow\infty$ . The$-(4)$
uniqueness question of solutions to problem (1)remains open.. See [2,3,4] for related
works.
$R^{N}$

Remark 2. (Quasivariational structure) Let $\{r, w, A\}$ be a solution of our problem (1)(4). Now we denote by $w=QA$ the solution of (1) uniquely determined by $A$ and by
$-$

135
$r=Rw$ the solution of of (2) uniquely determined by $w$ . With these operator $Q$ and$R$ , system (1) $-(3)$ can be written in one evolution inclusion$r$$A’(t)+\partial I_{K_{0}(RQA(t))}(A(t))\ni g(RQA(t);QA(t), A(t))$.

We should note that the convex constraint $K_{0}(RQA(t))$ depends on the unknown $A(t)$ .
In this sence, system (1) $-(3)$ includes the quasivariational structure and it is called aquasivariationl problem. For the general theory on quasivariational evolution inclusions,see [6].

3. Examples
In this section we give some illustrative examples of data $K_{0}(r)$ ,$g(r;w, A)$ .

$\psi_{k}$$f_{k}(w, r_{k})$and(Example of $K_{0}(r)$ and $g(r;w, A)$ )For a finite number $p>1$ and alarge positive constant $M> \max\{\gamma_{k}|k=1, 2, \cdots, N\},$we put$K_{0}(r)$$:=\{r’=(r_{1}’, r_{2}’, \cdots, r_{N}’)\in R_{+}^{N}$

Then it is clear thatcondition (A2). Next, let
See Fig. 3.

$\sum_{i=1}^{N}\frac{r_{k}^{;p}}{\min\{(r_{k})^{p},M^{p}\}}\leq 1\},$$\forall r\in R_{+}^{N}$.

(6)

is bounded and the mapping $rarrow K_{0}(r)$ satisfies
be the hyperplane cantaining , which is orthogonal to $L.$$\bigcup_{r\in R_{+}^{N}}K_{0}(r)$$T_{L}$

As is easily seen, for each vector

$0$

$r$

point $A_{r}=(A_{r1}, A_{r2}, \cdots, A_{rN})$ on. We now defineat$\partial K_{0}(r)$, there is one and only one
meets withand a hyperplane parallel to$:=(r_{1}, r_{2}, \cdots, r_{N})\in R_{+}^{N}$$\partial K_{0}(r)$$T_{L}$$A_{r}$$g(r;w, A) :=c_{1}(w)L+c_{2}(w)(A_{r}-A) , \forall r\in R_{+}^{N}, \forall w\geq0, \forall A\in R_{+}^{N}$

,

(7)

136
where
and
are positive, globally bounded and Lipschitz continuous functions on. The mapping $rarrow K_{0}(r)$ and the vector field defined by (6) and (7) satisfy (A5).
To check it we observe that$c_{1}$$c_{2}$$R_{+}$$g$$A_{r} \cdot L= \max A\cdot L= \max A\cdot L,$$A\in\partial K_{0}(r).

A\in K_{0}(r)$whence$g(r;w, A)\cdot L=c_{1}(w)|L|^{2}+c_{2}(w)(A_{r}-A)\cdot L>0,$$\forall r\in R_{+}^{N}, \forall w\geq 0, \forall A\in K_{0}(r)$.

$N$ , andAlso, for any point $A=(A_{1}, A_{2}, \cdots, A_{N})$ ofwith $A_{k}>0,$ $k=1$ , 2,
any $N\in N_{c}(A)$ we see from (6) that $N\cdot L\geq 0$ . Since $(A_{r}-A)\cdot N\leq 0$ , it turns out
that$\partial K_{0}(r)$$\cdots,$$(c_{1}(w)L+c_{2}(w)(A_{r}-A))\cdot L-\{(c_{1}(w)L+c_{2}(w)(A_{r}-A))\cdot N\}N\cdot L$$= c_{1}(w)(|L|^{2}-|L\cdot N|^{2})+c_{2}(w)((A_{r}-A)\cdot L)-(A_{r}-A)\cdot N(N\cdot L))$$\geq$O.

Thus (A5) was checked.
Example $of \psi_{k} and f_{k}(w, r_{k})$ )$N$ , we give an example of
For each $k=1$ , 2,appearing in (2) accompaniedwith scientific innovation. Consider the proper, l.s. . convex function
on $R$ givenby$($$\psi_{k}$$\cdots,$$c$$\psi_{k}$$\psi_{k}(r):=\{\begin{array}{ll}0, for r\leq\gamma_{k1},\mu_{k1}(r-\gamma_{k1}) , for \gamma_{k1}<r\leq\gamma_{k2},\mu_{k2}(r-\gamma_{k2})+\mu_{k1}(\gamma_{k2}-\gamma_{k1}) , for \gamma_{k2}<r\leq\gamma_{k},\infty, for r>\gamma_{k},\end{array}$In this case the graph of the subdiﬀerential$\psi_{k}$is given by Fig.1:

(8)

137
where

$\gamma_{k1},$$\gamma_{k2},$$\mu_{k1}$and$\mu_{k2}$are constants such that$0<\gamma_{k1}<\gamma_{k2}<\gamma_{k}<\infty, 0<\mu_{k1}<\mu_{k2}<\infty.$

Moreover consider a positive, Lipschitz continuous and non-decreasing function $f_{k}(w, r)$such thaton,(9)$R_{+}^{2}$$f_{k}(w, r)\geq\epsilon_{0}, \forall w\geq 0, \forall r\in[0, \gamma_{k1})$$f_{k}(w, r)\geq\mu_{k1}+\epsilon_{0}, \forall w\geq 0, \forall r\in[\gamma_{k1}, \gamma_{k2})$$f_{k}(w, r)\geq\mu_{k2}+\epsilon_{0}, \forall w\geq 0, \forall r\in[\gamma_{k2}, \gamma_{k}]$$f_{k}(\mu k1, \gamma k1)>\mu_{k1}, f_{k}(\mu_{k2}, \gamma_{k2})>\mu k2$

(10)

,

(11)

,

.

(12)

for a positive number . It is easy to see (A4) from (9) $-(12)$ . As a concrete example of$f_{k}(w, r)$ there is the follwoing function:$\epsilon_{0}$$f_{k}(w, r)=\epsilon_{0}w+\mu_{k1}f_{k1}(w, r)+(\mu_{k2}-\mu_{k1})f_{k2}(w, r)$

,

where
$f_{k1}(w, r)=\{\begin{array}{ll}0, for r\leq r_{k1}-\epsilon_{1},\frac{1}{\epsilon_{1}}(r-r_{k1})+1, for r_{k1}-\epsilon_{1}<r<r_{k1},1, for r\geq r_{k1},\end{array}$$f_{k2}(w, r)=\{\begin{array}{ll}0, for r\leq r_{k2}-\epsilon_{1},\frac{1}{\epsilon_{1}}(r-r_{k2})+1, for r_{k2}-\epsilon_{1}<r<r_{k2},1, for r\geq r_{k2}.\end{array}$See Fig.5 which shows the graph of $y=f_{k}(w, r)$ for each fixed $w>0.$$y\triangleleft ktw_{t}r) FB\cdot\S$138
Remark 1. In the above Fig. 1 and 5, it is illustrated that there happen scientificinnovations atand $r=\gamma_{k2}$ on the subject
and it costs a greate deal torecompose their knowledges obtained by innovations as industrial technologies. Also,
there is a infinite scientific wall at, which is the limit of knowledge and one has no
ideas how to get it over.
$r=\gamma_{k1}$$r_{k}$$r=\gamma_{k}$Remark 2. Even in the above example, the uniqueness question of solutions to problem(1) $-(4)$ remains open.
4. Outline of the proof
In this section we mention the outline of the proofs of Theorems 1 and 2; for thecomplete proofs we refer to the paper [5].
(Existence proof)

We construct a local in time solution of (1) $-(4)$ by the fixed point arguement. Let
andbe as in the statement of Theorem 1, and for a finite time $T>0$ and a$L,$ $w_{0}$$A_{0}$

positive constant$C_{1}>|w_{0}|_{W^{1,2}(-\tau 0,0)},$put$X_{T}(w_{0}, C_{1})$$:=\{w\in W_{+}^{1,2}(-\tau_{0}, T)|w=w_{0} on [-\tau_{0}, 0], |w|_{W^{1,2}(-\tau 0,T)}\leq C_{1}\}$

.

(13)

Now, for each $w\in X_{T}(w_{0}, C_{1})$ , solve the problem$\{\begin{array}{l}r_{k}’(t)+\partial\psi_{k}(r_{k}(t))\ni f_{k}(w(t), r_{k}(t)) , for a.e.

t\in(-\tau_{0}, T) , r_{k}(-\tau_{0})=0,A’(t)+\partial I_{K_{0}(r(t))}(A(t))\ni g(r(t);w(t), A(t)) for a.e.

t\in(O, T) ,A(O)=A_{0},\end{array}$(14)

where $r(t)$ $:=(r_{1}(t), r_{2}(t), \cdots, r_{N}(t))$ ; in fact, problem (14) can be solved by the general
theory on evolution equations generated by time-dependent subdiﬀerentials, since $tarrow$$K_{0}(r(t))$ and $tarrow g(r(t);w(t), A(t))$ are regular enough in . Moreover, we see that there$T,$is a positive constantdepending only on(Lipschitz constant of ) and the$A$initial datasuch that the solutions and satisfy the following inequality:$t$$C_{2}$$w_{0},$$A_{0}$$C_{1},$$L_{g}$$g$$r$$|r|_{W^{1,2}(0,T;R^{N})}+|A|_{W^{1,2}(0,T;R^{N})}\leq C_{2}$

,

(15)

as long as $w\in X_{T}(w_{0}, C_{1})$ . By the way, there is a positive constantonand such that
$C_{2}$$C_{3}$

depending only$T$$\sup_{0\leq t\leq T}L\cdot A(t)\leq C_{3}$

,

(16)

as long as $w\in X_{T}(w_{0}, C_{1})$ .
Now, consider the Cauchy problem associated with the solutions$\tilde{w}’(t)+b\tilde{w}(t)=\sigma P(L\cdot A(t))\tilde{w}(t)^{\alpha},$$t\in[O, T],$$\tilde{w}=w_{0}$

$r$

on

and $A$ of (14)$[-\tau_{0}, 0].$

139
Then, this problem has one and only one$C^{1}$-solution$\tilde{w}$, and$0< \tilde{w}(t)\leq\max\{w_{0}(0) , \{\frac{\sigma P(C_{3})}{b}\}^{\frac{1}{1-\alpha}}\}=:C_{4}, t\in[0, T]$

(17)

.

These inequalities are obtained as follows. Since $\sigma P(L\cdot A(t))\leq\sigma P(C_{3})$ by (16), comparing with the solution of$\hat{w}$$\tilde{w}$$\hat{w}’(t)+b\hat{w}(t)=\sigma P(C_{3})\hat{w}(t)^{\alpha},$$t\geq 0,$$\tilde{w}=w_{0}$on$[-\tau_{0}, 0],$we get by virtue of the usual comparison results that$\tilde{w}(t)\leq\hat{w}(t)\leq\max\{w_{0}(0) , \{\frac{\sigma P(C_{3})}{b}\}^{\frac{1}{1-\alpha}}\}=:C_{5}$

(18)

;

note thatis the positive root of equation $\sigma P(C_{3})X^{\alpha}-bX=0$ of $X.$
Thus (18) holds. We see immediately from (17) and (18) that$\xi$$:= \{\frac{\sigma P(C_{3})}{b}\}^{\frac{1}{1-\alpha}}$$\tilde{w}’(t)\leq\sigma P(C_{3})C_{5}^{\alpha}=:C_{6}, \forall t\in[0, T].$

Accordingly, with a small time $T>0$ satisfying $|w_{0}|_{W^{1,2}(-\tau 0,0)}+(C_{5}+C_{6})\sqrt{T}\leq C_{1}$ we
see that$| \tilde{w}|_{W^{1,2}(-\tau 0,T)} = (|w_{0}|_{W^{1,2}(-\tau_{0},0)}^{2}+\int_{0}^{T}(|\tilde{w}(t)|^{2}+|\tilde{w}’(t)|^{2})dt)^{\frac{1}{2}}$$\leq |w_{0}|_{W^{1,2}(-\tau 0,0)}+(\int_{0}^{T}|\tilde{w}(t)|^{2}dt)^{\frac{1}{2}}+(\int_{0}^{T}|\tilde{w}’(t)|^{2}dt)^{\frac{1}{2}}$$\leq |w_{0}|_{W^{1,2}(-\tau 0,0)}+(C_{5}+C_{6})\sqrt{T}\leq C_{1}.$This shows that the mapping $S$ , which is defined by $S(w)=\tilde{w}$ via the solution $A$ of (3),maps
into itself. Moreover it is not dificult. to derive that the set $X(w_{0};C_{1})$is non-empty, compact and convex in $C([-\tau_{0}, T])$ and $S$ is continuous in it with respect
to the topology of $C([-\tau_{0},$We now apply the Schauder’s fixed point theorem to the mapping $S$ in $X_{T}(w_{0}, C_{1})$ to
find at least one fixed point $w\in X_{T}(w_{0}, C_{1})$ of $S$ , namely $w=S(w)$ . By the definitionof it is easy to see that the tripret $\{w, r, A\}$ , with the solutions and $A$ of (14), gives
solutions of (1) $-(3)$ on
or $[0, T]$ with (4). Furthermore, it is a standard work to)) orextend this local in time solution on the whole time interval$X_{T}(w_{0}, C_{1})$

$T$

$S$

$r$

$[-\tau_{0}, T]$$[-\tau_{0}, \infty$$[0, \infty$

monotonicity properties $of w, r and L\cdot A in$ time)
Let $\{w, r, A\}$ be a global in time solution of (1) $-(4)$ . In our proof the main point is
to show that $L\cdot A(t)$ is non-decreasing in $t\geq 0.$
First of all, we prepare the statement:$(The$

is non-decreasing on an interval
non-decreasing on

Lemma 1. If$w$$[0,$ $t$$[0,$ $t$

then the solution$r_{k}$of (2) is also

140
We shall use this below.
(Step 1) Now, we put$t_{0}$$:= \sup\{t\geq 0|L\cdot$ $A$

is non–decreasing on $[0,$$t$

is given in the interior of $K_{0}(r(O))$ and $tarrow K_{0}(r(t))$ is continuous in the sense of
Since
Hausdorﬀ distance (cf. (A2)), we see that $A(t)$ is in the interior of $K_{0}(r(t))$ for an small$t’>0$ , so that $\partial I_{K_{0}(r(t))}(A(t))=0$ for all $t\in[0,$time interval [This implies by$A’(t)=g(t)$(3) that$:=g(r(t);w(t), A(t))$ for a.e. $t\in[0,$From our assumption we
have that $g(t)\cdot L\geq 0$ , whence $A’(t)\cdot L\geq 0$ for a.e. $t\in[O,$This implies that $A(t)\cdot L$
is non-decreasing on
Hence $t_{0}>$ O. Our claim is to show that $t_{0}=\infty$ . For acontradiction, suppose that
. In this case we have
. Otherwise,$A(t_{0})\in
Int.K_{0}(r(t_{0}))$since, by repeating the same argument as above we deduce that
$L\cdot A(t)$ is non-decreasing on an interval [, t\’o],. This contradicts the definitionofSince $\sigma P(L\cdot A(t))$ is non-decreasing onby (A1), it follows from the usualcomparison result that is non-decreasing and of, namely $w’\geq 0$ on
on
If $w’(t_{0})>0$ , then is non-decreasing
for some positive number . Therefore$N$ , of (2), namely $r(t)$it follows from Lemma 1 that the solutions, $k=1$ , 2,
isnon-decreasing on. This implies by (A2) that the mapping $tarrow K_{0}(r(t))$ is
non-decreasing inwith respect to. In another case of $w’(t_{0})=0$ , it$bw(t_{0})=\sigma
P(L\cdotA(t_{0}))w(t_{0})^{\alpha}$
holds thatand
. Therefore, by (A4),$A_{0}$$0,$ $t$

$t$

$t$

$t$

$[0,$ $t$$t_{0}<\infty$$A(t_{0})\in\partial K_{0}(r(t_{0}))$$t\’{o}>t_{0}$$t_{0}$$t_{0}.$$[0, t_{0}]$$C^{1}$$w$$[0, t_{0}].$$[0, t_{0}]$$[-\tau_{0}, t_{0}+\delta_{0}]$$w$$\delta_{0}$$r_{k}(t)$$\cdots,$$[-\tau_{0}, t_{0}+\delta_{0}]$$t\in[-\tau_{0}, t_{0}+\delta_{0}]$$R_{+}^{N}$$A(t_{0})\in\partial K_{0}(r(t_{0}))$(19).

$f_{k}(w(t_{0}), r_{k}(t_{0}))> \sup\partial\psi_{k}(r_{k}(t_{0}))$Here, we apply Theorem 3.5 in [1] to see that the right-derivative
and$\frac{d^{+}}{dt}r_{k}(t)$exists at every$t\geq 0$$\frac{d^{+}}{dt}r_{k}(t)=\inf_{\xi\in\partial\psi_{k}(r_{k}(t_{0}))}|f_{k}(w(t), r_{k}(t))-\xi|.$

In the present case, by (19) it turns out that$\frac{d^{+}}{dt}r_{k}(t_{0})=\inf_{\xi\in\partial\psi_{k}(r_{k}(t_{0}))}(f_{k}(w(t_{0}), r_{k}(t_{0}))-\xi)>0,$$k=$
which implies thatis increasing on an interval
for a small$N$, 2,. As a consequence, we observe that $r(t)$ is non-decreasing on, and
so is $tarrow K_{0}(r(t))$ on
As was seen above, in any case the mapping $tarrow K_{0}(r(t))$ is non-decreasing inonfor a small positive number$r_{k}(t)$$[t_{0}, t_{0}+\delta_{0}]$$\delta_{0}>0,$$1$$[0, t_{0}+\delta_{0}]$$\cdots,$$[0, t_{0}+\delta_{0}].$$R_{+}^{N}$$[0, t_{0}+\delta_{0}]$$\delta_{0}>0.$(Step 2) Now, put$E=\{t\cdot\in[0, t_{0}+\delta_{0}]|A(t)\in\partial K_{0}(r(t))\}.$

We pay our attention to the equation of $A$ which is written in the form:$A^{*}(t)$$:=g(r(t);w(t), A(t))-A’(t)\in\partial I_{K_{0}(r(t))}(A(t))$

for a.e.

$t\in[0, t_{0}+\delta_{0}],$141
subject to the initial condition $A(O)=A_{0}$ . Here we note from the definition of subdiﬀerentials of indicator functions that$\partial I_{K_{0}(r(t))}(A)=\{\begin{array}{ll}0, if A\in Int.K_{0}(r(t)) ,\{cN|c\in R_{+}, N\in N_{c}(A)\}, if A\in\partial K_{0}(r(t)) .\end{array}$

Therefore$A^{*}(t)$should be of the form:(20)$A^{*}(t)=\{\begin{array}{ll}0, if t\neq E,c(t)N(t) , if t\in E,\end{array}$where $c(t)$ is non-negative function of $t\in E$ and $N(t)$ is an element in the normal cone
$N_{c}(A(t))$ for $t\in E$ ; note that $E$ is a closed in $R+andc(\cdot)N(\cdot)\in L^{2}(E;R^{N})$ .
Next, at each point $A(t)\in\partial K_{0}(r(t))$ we decompose the forcing term into the normal
and tangetial components:$g$$g(t):=g(r(t);w(t), A(t))=g_{N}(t)N(t)+g_{T}(t)$ ,where $N(t)$ is the same normal vector as in (20) and$g_{T}(t):=g(t)-g_{N}(t)N(t)$ ,$g_{N}(t)$$:=g(t)\cdot N(t)$.

Lemma 2. We have that $0\leq c(t)\leq g_{N}(t)$ for $a.e.$ $t\in E.$
Proof. At each point $A(t)\in\partial K_{0}(r(t))$ we observe that $A^{*}(t)\cdot A’(t)\geq 0$ . In fact, since$K_{0}(r(t))$ is non-decreasing, it follows from the definition of subdiﬀerential ofthat$I_{K_{0}(r(t))}$$A^{*}(t)\cdot(A(t)-A(t-\delta))\geq 0$

for all$\delta>0$. Hence, by deviding the both sides by and taking the limit as$\delta$$\delta\downarrow 0$, we get$A^{*}(t)\cdot A’(t)\geq 0.$

Next, we multiply equation (3) by$A^{*}(t)$to obtain$|A^{*}(t)|^{2}\leq(g_{N}(t)N(t)+g_{T}(t))\cdot A^{*}(t)$for any $t\in E$ . Since $A^{*}(t)=c(t)N(t)$ and $g(t)\cdot A^{*}(t)=0$ , it follows from the above
inequality that $c(t)^{2}\leq g_{N}(t)c(t)$ , hence $0\leq c(t)\leq g_{N}(t)$ . ◇Here, taking the inner product between$L$and the both sides of$A’(t)=g_{T}(t)+(g_{N}(t)-c(t))N(t)$at any point $A(t)$ with $t\in E$ , we derive from contition (A5) and Lemma 2 that$A’(t)\cdot L=g_{T}(t)\cdot L+(g_{N}(t)-c(t))N(t)\cdot L\geq 0.$

Also, at any point $A(t)$ , $t\neq E$ , namely $A(t)\in Int.K_{0}(r(t))$ we have by (5) that$A’(t)\cdot L=g(t)\cdot L\geq 0.$

As a cosequence the inequality $A’(t)\cdot L\geq 0$ holds for a.e.
. This contradicts the definition ofis non-decreasing on$t\in[0, t_{0}+\delta_{0}]$$[0, t_{0}+\delta_{0}]$$t_{0}$, and thus $A\cdot L$. Thus$t_{0}=\infty$

holds.

142(Step 3) Finally we show that is non-decreasing on). In fact, since $L\cdot A(t)$is non-decreasing on), the coeﬃcient $\sigma P(L\cdot A(t))$ of the equation$[0, \infty$$w$$[0, \infty$$w’(t)+bw(t)=\sigma P(L\cdot A(t))w(t)^{\alpha}$

is non-decreasing on$[0, \infty$), too. Hence,$w$

is non-decreasing on$[0, \infty$).

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