
%Feb. 14, 1994. add control example.
%Feb. 23, 1994. add example 3.1.
%March 28, 1994. add metric regularity.
%April 11, 1994. modified title and introduction.
%April 14, 1994. modified the definition of $\beta$-derivative.
%April 17, 1994. modified to comply with the requirement of JMAA.
%April 28, 1994. Modified according to Ioffe's comments.
%May 12, 1994. Mior changes.
%May 20, 1994. Add stronger version of the fuzzy rule. Send to SICON.



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\begin{document}

\begin{center}
{\LARGE \bf {Viscosity Solutions and Viscosity Subderivatives in Smooth Banach 
Spaces  with  Applications to Metric Regularity}}\par
\vspace{2\baselineskip}
{\large Jonathan M. Borwein and Qiji Zhu\par
Department of Mathematics and Statistics\par
Simon Fraser University\par
Burnaby, BC V5A$\,$1S6\par
Canada }

\end{center}

\vspace{2\baselineskip}
\vspace{2\baselineskip}

Research of the first author was supported by NSERC and by the Shrum Endowment
at Simon Fraser University. 

\newpage
{\bf Viscosity Solutions and Subderivatives}
\bigskip

{\large Please direct all correspondence to: \par
\vspace{\baselineskip}
Jonathan M. Borwein\par
Department of Mathematics and Statistics\par
Simon Fraser University\par
Burnaby, BC V5A$\,$1S6\par
Canada}


\newpage

\begin{center}
{\LARGE \bf Viscosity Solutions and Viscosity Subderivatives in Smooth Banach 
Spaces  with  Applications to Metric Regularity}\par
\vspace{\baselineskip}
{\large Jonathan M. Borwein and Qiji Zhu}\par
\vspace{2\baselineskip}
\end{center}

\setlength{\baselineskip}{1.5\baselineskip}

{\em \bf Abstract.}
In Gateaux or Bornologically differentiable spaces there are two natural natural
generalizations of the concept of a Fr\' echet subderivative: In this
paper we study the viscosity subderivative (which is the 
more robust of the two) and establish a fuzzy sum rule for it in a 
smooth Banach space. This rule is applied to obtain existence and comparison 
results for viscosity solutions of the Hamilton-Jacobi-Bellman equation in 
$\beta$-smooth spaces. A unified treatment of metric regularity in 
smooth spaces completes the paper. This illustriates the flexibility of
viscosity subderivatives as a tool for analysis.

\vfill
{\bf Keywords}: Viscosity subderivative, fuzzy sum rule, viscosity
solutions,  Hamilton-Jacobi-Bellman equations, smooth spaces,
metric regularity.  \\

\vfill
{\bf AMS (1991) subject classification}: Primary 49J52,
Secondary 49L25, 49J40, 49J50, 58C20.

\newpage

%\setlength{\baselineskip}{1.5\baselineskip}

\section{Introduction}


It is well known that the proximal limit formula
for generalized derivatives  plays a crucial role
in Hilbert space nonsmooth and variational analysis. 
The reason is two-folds: many properties of a nonsmooth function are determined
by the (densely existing) proximal derivatives and proximal derivatives are 
easier to get hold of than various other generalized derivatives. 
However, the proximal
derivative concept depended crucially on the analysis of nearest points
and, therefore, relied heavily on the inner product structure of the 
underlying space. Recent research \cite{bi,bz} shows that, in fact,
what essential in this context is a ``smooth'' support function (rather than
a nearest point) that corresponding to the ``viscosity''
(as opposed to the limit)  subderivative concept. This makes it
possible to do nonsmooth and variational analysis in smooth Banach spaces
by using bornological subderivatives (compatible to the smoothness of the
underline space). Such a new technology is crucially important 
in studying problems with non Fr\' echet smooth underlying space
(see for example \cite{bz}).
The main purpose of  this paper is to
adapt a ``fuzzy'' summation rule given in \cite{bi} (which is always
important in developing generalized derivatives) to
the viscosity bornological subderivative and discuss its applications
to viscosity solutions of Hamilton-Jacobi-Bellman equations. 
For previous research on this topic we refer to Deville's paper
\cite{d2}. Also ``fuzzy'' summation rules for other types of 
subderivatives were discussed in Fabian \cite{f} and 
Ioffe \cite{i83,i84}. In discussing viscosity solutions, it turns out
that with different underlying spaces it is crucial to 
use bornologies other than Gateaux and Fr\' echet bornologies. This is 
also observed in our previous paper \cite{bz} for perturbed control problems. 
A crucial tool in proving our  ``fuzzy'' summation rules is the smooth
variational principle proved in \cite{bp} that requires the underlying
space has a smooth equivalent norm. In most of the following results
this condition can be reduced to that the space has a smooth bump function
by using the smooth variational principle proved by Deville, Godefroy
and Zizler \cite{DGZ}.
We also discuss 
the application of the ``fuzzy'' summation rule to the metric regularity 
problem. It is worth mentioning
that  in this problem using bornological subderivatives technique simplifies
the arguments considerably although similar (but somewhat weaker) results
can be derived without using this technique.



We introduce terminologies and prove our fuzzy summation rules 
(Theorems 2.1, 2.2, 2.3 and 2.4) in
Section 2.
Then, in Section 3 we  apply it to derive a
comparison result for  the viscosity
solution of a Hamilton-Jacobi-Bellman equation in a Banach space 
with a $\beta$-smooth norm. As an example we discuss a class of infinite 
horizon optimal control problem
in a Banach space 
with a $\beta$-smooth norm and prove that the  value function 
of this optimal control problem is the unique viscosity
solution of a so called Hamilton-Jacobi-Bellman equation 
(Theorems 3.1 and 3.3). These results in 
particular applies to such a problem in $L^1$  
which has  a weak Hadamard smooth equivalent norm 
as  shown in \cite{bf1}. 

In Section 4 we apply the fuzzy summation rule to to discuss metric regularity.
We prove  a dual sufficient condition for a function to be metrically regular
at a given point that is
weaker than the dual conditions given in Ioffe \cite{i84a} and Ginsburg and 
Ioffe \cite{gi} and  deduce a primal condition that improves 
a similar condition  in Borwein 
and Strojwas
\cite{bs}. Many authors have discussed primal sufficient conditions. 
The condition
we given below appears to be the  weakest up to now. Various primal conditions 
discussed in \cite{ac,af,b86,bs} are deduced as corollaries. We
also indicate how to handle regularity of multifunctions.

Finally let us remark that in  bornologically differentiable spaces
there are two natural generalizations of a  Fr\' echet subderivative.
It seems to us that in such spaces the viscosity bornology
subderivative (Definition 2.1) is  the right generalization to use
rather than the limit definition (Remark 2.2). 




\section{Viscosity Bornology Subderivatives and  
Fuzzy Summation Formulae}



Let $X$ be  a Banach space with closed unit ball $B$ and with
continuous real dual $X^{*}$.   A {\it bornology} $\beta$ of 
$X$ is a family of closed
bounded and centrally symmetric (convex) subsets of $X$ 
whose union is $X$, which is closed under multiplication
by positive scalars and is directed upwards (that is, the union of 
any two members of $\beta $ is contained in some member of $\beta$).
(It follows,
in particular, that any finite dimensional subspace belongs to the
subspace spanned by some element of the bornology.) We will denote
$X^*_\beta$ the dual space of $X$ endowed with the topology of uniform
convergence on $\beta$-sets. The most important bornologies are 
those formed by all (convex symmetric) bounded sets (Fr\'echet bornology
denoted by $F$), weak compact sets (weak Hadamard bornology denoted by
$WH$),  compact sets (Hadamard bornology denoted by
$H$) and finite dimensional sets (Gateaux bornology denoted by $G$). 
%The most important bornology in thispaper is 
%the {\it weak Hadamard} bornology (denoted by $WH$) 
%formed by all (convex symmetric)  weak compact sets (see \cite{bp,DGZ,Ph}).

By a {\it function} we always mean an {\it extended--real--valued}~
function usually lower (upper) semicontinuous and {\it proper} (that is to 
say not everywhere equal to
$+\infty$ ($-\infty$) and nowhere to $-\infty$ ($+\infty$)).
Given a function $f$ on $X$, we say
that $f$ is $\beta$-differentiable at $x$ and has a $\beta$-derivative
$\nabla^\beta f(x)$ if 
\[
t^{-1}(f(x+tu) - f(x) -
t\langle \nabla^\beta f(x),u\rangle )\rightarrow 0
\]
as $t\rightarrow
0$ uniformly in $u\in V$ for every $V\in \beta$.  We
say that a function $f$ is $\beta$-smooth if $\nabla^\beta f(x) :X\to
X^*_\beta$ is continuous. It is not hard to check that 
for a convex function, $\beta$-smooth and $\beta$-differentiable
on a convex set are equivalent.



\begin{definition} Let $f$ be a lower  semicontinuous function.
We say $f$ is $\beta$-{\bf viscosity subdifferentiable}  and
$x^*$  is a  $\beta$-{\bf viscosity subderivative}  of $f$ at $x$ if
there exists a locally Lipschitz function $g$ such that $g$ is
$\beta$-smooth in a neighbourhood of $x$, $\nabla^\beta
g(x)=x^*$
and $f-g$ attains a local minimum  at $x$. We denote the set
of all  $\beta$-viscosity subderivatives of $f$ at $x$ by
$D_\beta f(x)$.

\end{definition}


\begin{definition} Let $f$ be a upper semicontinuous function.
We say $f$ is  $\beta$-{\bf viscosity superdifferentiable} and
$x^*$  is a  $\beta$-{\bf viscosity superderivative} of $f$ at $x$ if
there exists a locally Lipschitz function $g$ such that $g$ is
$\beta$-smooth in a neighbourhood of $x$, $\nabla^\beta
g(x)=x^*$
and $f-g$ attains a local maximum at $x$. We denote the set
of all  $\beta$-viscosity superderivatives of $f$ at $x$ by
$D^\beta f(x)$.

\end{definition}

{\bf Remark 2.1} By shifting a constant we may always assume that the 
$\beta$ smooth function $g$ in the above definitions satisfies
$g(x)=f(x)$.


{\bf Remark 2.2} The limit definition of the $\beta$-subdifferential 
$\partial_\beta f(x)$  of $f$ at $x$ is as follows:
$x^*\in \partial_\beta f(x)$ if, for any $\varepsilon>0$ and
any $V\in \beta$, there exists a $\eta > 0$ such that 
$$
t^{-1}(f(x+th)-f(x))-\langle x^*,h\rangle >-\varepsilon 
\quad \forall t\in (0,\eta),~
h\in V.
$$
One can check that $D_\beta f(x) \subset \partial_\beta f(x)$.
It is proved in \cite{DGZ} that  
$D_F f(x) = \partial_F f(x)$. However, the two concepts are different
in general as shown by the following example.


{\bf Example 2.1} Let $f:R^N\to R$ ($n\ge 2$) be continuous and suppose $f$ is
Gateaux but not Fr\'echet (weak Hadamard) differentiable at $x$ (e.g.
$f:R^2\to R$ defined by $f(x,y)={{xy^3}\over {x^2+y^4}}$ when $(x,y)\neq (0,0)$
and $f(0,0)=0$ at point $(0,0)$).
Let $g(h):=-|f(x+h)-f(x)-\nabla^G f(x)h|$. Then

1) $\partial_G g(0)=\{0\}$;

2)  $D_G g(0)=\emptyset$;

and $g$ is locally uniformly continuous.
In fact, we can directly check that $\partial_G g(0)=\{0\}$. Since 
we always have $D_G g(0)\subset \partial_G g(0)$ if 2) is not true then
$D_G g(0)=\{0\}$. That is to say, upon shifting by  a constant if necessary,
there exists a locally Lipschitz  and Gateaux differentiable 
(and therefore Fr\'echet differentiable) function 
$k$ such that $k(0)=g(0)=0$; $\nabla^G k(0)=\nabla^G g(0)=0$ and 
$k\le g$ in a neighbourhood of $0$. 
Thus, 
$$
{{|f(x+h)-f(x)-\nabla^G f(x)h|}\over {\|h\|}}\le -{{k(h)-k(0)}\over {\|h\|}}
\le {{\|k(h)-k(0)\|}\over {\|h\|}}
$$
which implies that $f$ is Fr\' echet differentiable at $x$ a contradiction.
\smiley 


\begin{definition}\cite{bi} 
Let $(f_1,...,f_N)$ be lower semicontinuous functions
and $E$ a closed subset of $X$. We say that  $(f_1,...,f_N)$ is
{\bf uniformly lower semicontinuous} on $E$ if 
\begin{eqnarray*} 
&& \inf_{x\in E}\sum_{n=1}^N f_n(x) \\
&\le& \liminf_{\varepsilon\to
0}\{\sum_{n=1}^N f_n(x_n):  \|x_n-x_m\|\le \varepsilon,x_n, x_m\in E,
n,m=1,...,N\}.  
\end{eqnarray*} 
We say that
$(f_1,...,f_N)$ is {\bf locally uniformly lower semicontinuous} if, for any
$x\in \cap_{n=1}^N{\rm dom}(f_n)$, $(f_1,...,f_N)$ is uniformly lower
semicontinuous on  a closed ball centered at $x$.
\end{definition}

The next useful proposition follows directly from the definition:

\begin{proposition} Let $(f_1,...,f_N)$ be lower semicontinuous
functions on $X$ and $E$ a closed subset of $X$.  Then the following
conditions are sufficient for $(f_1,...,f_N)$ to be uniformly lower
semicontinuous on $E$:

1.) all but one of the functions are uniformly continuous on $E$;

2.) one of the functions has compact level sets when restricted to $E$;

3.) $X$ is finite dimensional.  \end{proposition}

The following lemma and theorem are  refined versions of Proposition 4
in \cite{bi}. It gives a ``fuzzy'' addition formula at the minimum
point of the summation of several functions in the graph sense rather than
in the domain sense in \cite{bi}.

\begin{lemma} Let $(f_1,...,f_N)$ be lower semicontinuous functions  and
$E$ a closed subset of $X$. Suppose that 
$(f_1,...,f_N)$ is  uniformly lower semicontinuous on $E$. Define, for
$t>0$, 
$$ 
M_t=\inf_{(x_1,...,x_N)\in E^N} [\sum_{n=1}^N
f_n(x_n)+t\sum_{n,m=1}^N\|x_n-x_m\|^2].  
$$ 
Assume that $(x_1^t,...,x_N^t)$ satisfies  
$$ \lim_{t\to
\infty}M_t- [\sum_{n=1}^N
f_n(x_n^t)+t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2]=0.  
$$ 
Then

(i) $\lim_{t\to \infty}t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2=0$;

(ii) $\lim_{t\to \infty}M_t=\inf_{x\in E}\sum_{n=1}^N f_n(x) $.
\end{lemma}

{\bf Proof.} Set $$ d_t:=M_t-[\sum_{n=1}^N
f_n(x_n^t)+t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2].  $$ Then $\lim_{t\to
\infty}d_t= 0$. Evidently $M_t$ increases with $t$ and $M_t\le
\inf_{x\in E}\sum_{n=1}^N f_n(x)$. Therefore, $M:=\lim_{t\to
\infty}M_t$ exists and $M\le \inf_{x\in E}\sum_{n=1}^N f_n(x)$.
Moreover, \begin{eqnarray*} M_{t/2} &\le & \sum_{n=1}^N
f_n(x_n^t)+{t\over 2}
		 \sum_{n,m=1}^N\|x_n^t-x_m^t\|^2\\
	&\le & \sum_{n=1}^N f_n(x_n^t)+t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2
	       -{t\over 2}\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2\\ &=&
	M_t-d_t-{t\over 2}\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2
\end{eqnarray*} or $$ 2(M_t-M_{t/2}-d_t)\ge
t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2.  $$ Thus, $$ \lim_{t\to
\infty}t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2=0.  $$ It remains to show that
$M\ge \inf_{x\in E}\sum_{n=1}^N f_n(x)$.  To do so, observe that
$\|x_n^t-x_m^t\|\to 0$ when $t\to 0$ and $(f_1,...,f_N)$ is 
uniformly lower semicontinuous on $E$. Therefore, \begin{eqnarray*} M
&=& \lim_{t\to \infty}\sum_{n=1}^N f_n(x_n^t)
      +t\sum_{n,m=1}^N\|x_n^t-x_m^t\|^2 \\ &\ge& \liminf_{t\to
  \infty}\sum_{n=1}^N f_n(x_n^t) \ge \inf_{x\in E}\sum_{n=1}^N f_n(x).
\end{eqnarray*} \smiley


\begin{theorem} Let $X$ be a Banach space with an equivalent
$\beta$-smooth norm and $(f_1,...,f_N)$ lower semicontinuous
functions on $X$.  Suppose that $(f_1,...,f_N)$ is 
locally uniformly lower semicontinuous and
$\sum_{n=1}^N f_n$ attains a local minimum at $x$.  Then, for any
$\varepsilon >0$, there exist $x_n\in x+ \varepsilon B, n=1,...,N$ such
that $|f_n(x_n)-f_n(x)|<\varepsilon $ and $$ 0\in  \sum_{n=1}^N D_\beta
f_n(x_n)  +  \varepsilon  B_{X^*}.  $$

\end{theorem}

{\bf Proof.} Let $\varepsilon >0$ be an arbitrary positive number.
Taking a smaller $\varepsilon$ if necessary we may assume that 
$(f_1,...,f_N)$ is lower uniformly continuous on $x+\varepsilon B$.
Define
$$
w_t(y_1,...,y_N):=\sum_{n=1}^N f_n(y_n)
      +t\sum_{n,m=1}^N\|y_n-y_m\|^2.
$$
Then, by Lemma 1.1,
\begin{eqnarray}
&& \lim_{t\to \infty}\inf\{w_t(y_1,...,y_N):
y_n\in  x+ \varepsilon B,n=1,..,N\}\nonumber \\
&=& \inf_{y\in  x+ \varepsilon B}\sum_{n=1}^N f_n(y)
=\sum_{n=1}^N f_n(x)=w_t(x,...,x).\label{e1}
\end{eqnarray}
Choose  $t_i\to \infty$ such that 
$$
w_{t_i}(x,...,x)<\inf\{w_{t_i}(y_1,...,y_N):y_n\in  x+ \varepsilon B,n=1,..,N\}
+{\varepsilon^2 \over Ni}.
$$
Then by the Borwein-Preiss Smooth Variational Principle, for each 
$i$ there exist a $\beta$ smooth function 
$\phi_i$ and $x^i_n, n=1,..,N$ such  that
$w_{t_i}-\phi_i$ attains a local minimum at $(x^i_1,...,x^i_N)$, 
$\|\nabla^\beta \phi_i\| <\varepsilon/N $, 
$\|x^i_n-x\|<\varepsilon /i$ and 
\begin{eqnarray}
w_{t_i}(x^i_1,...,x^i_N) <\inf\{w_{t_i}(y_1,...,y_N):
y_n\in  x+ \varepsilon B,n=1,..,N\}
+{\varepsilon^2 \over Ni}.\label{e2}
\end{eqnarray}
For each $n$, the function
\begin{eqnarray*}
y \to  w_{t_i}(x^i_1,...,x^i_{n-1}, y, x^i_{n+1},...,x^i_N)
	    -\phi_{i}(x^i_1,...,x^i_{n-1}, y, x^i_{n+1},...,x^i_N)
\end{eqnarray*}
attains a local minimum at $y=x^i_n$. Thus
$$
-\nabla^\beta_{x_n}\phi_{i}(x^i_1,...,x^i_N)
-4t_i\sum_{m=1}^N\|x^i_n-x^i_m\|\nabla^\beta \|\cdot \|\mid_{(x^i_n-x^i_m)}
\in D_\beta f_n(x^i_n).
$$
Summing these $N$ inclusions leads to 
$$
-\sum_{n=1}^N\nabla^\beta_{x_n}\phi_i(x^i_1,...,x^i_N)
-4t_i\sum_{n=1}^N\sum_{m=1}^N
\|x^i_n-x^i_m\|\nabla^\beta \|\cdot \|\mid_{(x^i_n-x^i_m)}
\in \sum_{n=1}^N D_\beta f_n(x^i_n).
$$
Observing that $\|-\sum_{n=1}^N\nabla^\beta_{x_n}\phi_i(x^i_1,...,x^i_N)\|\le 
\varepsilon$ and 
$$
\|x^i_n-x^i_m\|\nabla^\beta \|\cdot \|\mid_{(x^i_n-x^i_m)}
+\|x^i_m-x^i_n\|\nabla^\beta \|\cdot \|\mid_{(x^i_m-x^i_n)}=0
$$ 
so that
the double sum in the previous inclusion vanishes, we obtain
$$
0\in  \sum_{n=1}^N D_\beta f_n(x^i_n)+\varepsilon B_{X^*}.
$$
By (\ref{e1}) and (\ref{e2})
\begin{eqnarray*}
\sum_{n=1}^N f_n(x)
=\lim_{i\to \infty} \sum_{n=1}^N f_n(x_n^i).
\end{eqnarray*}
Since $f_n, n=1,...,N$ are lower semicontinuous
$$
\liminf_{i\to \infty}  f_n(x_n^i) =f_n(x), \quad n=1,...,N.
$$
Thus, when $i$ is sufficiently large $\|x_n^i-x\|<\varepsilon$ and 
$|f_n(x_n^i)-f_n(x)|<\varepsilon$ for $n=1,2,...,N$ and we are done.
\smiley




\begin{theorem} Let $X$ be a Banach space with an equivalent
$\beta$-smooth norm. Let $(f_1,...,f_N)$ be lower semicontinuous
functions and $x\in \cap_{n=1}^N{\rm dom}(f_n)$.  Then, for any
$\varepsilon >0$ and any weak-star neighbourhood $V$ of $0\in X^*$,
there exist $x_n\in x+ \varepsilon B, n=1,...,N$ such that
$|f_n(x_n)-f_n(x)|<\varepsilon $ and 
$$ 
D_\beta (\sum_{n=1}^N 
f_n)(x) \subset  \sum_{n=1}^N D_\beta f_n(x_n)  +  V.  
$$

\end{theorem}

{\bf Proof.} Let $\varepsilon>0$ and $V$ be a weak-star neighbourhood 
of $0\in X^*$. Fix $r>0$, $L$ a finite dimensional subspace of
$X$ containing $x$ and $V_\beta$ a neighbourhood of $0\in X^*_\beta$ such that
$V_\beta + L^\bot +r B_{X^*}\subset V$. Let 
$x^*\in D_\beta (\sum_{n=1}^N  f_n)(x)$. Then there exists
a $\beta$ smooth function $g$ such that $\nabla^\beta g(x)=x^*$
and $\sum_{n=1}^N f_n -g$ attains a local minimum at $x$.
Choose $0<\eta <\varepsilon$ such that $\|y-x\|< \eta<\varepsilon$ implies that
$\nabla^\beta g(y)-\nabla^\beta g(x)\in V_\beta$ and let 
$\delta_L$ be the indicator function of $L$. Then 
$\sum_{n=1}^N f_n -g+\delta_L$ attains a local minimum at $x$.
By Proposition 1.1 $(f_1,...,f_N,g,\delta_L)$ is locally uniformly
lower semicontinuous. 
Applying Theorem 1.1 yields that there exist $x_n, n=1,...,N+2$ such that 
$\|x_n-x\|<\eta<\varepsilon, n=1,...,N+2$, $|f_n(x_n)-f_n(x)|<\eta<\varepsilon, 
n=1,...,N$,
$|\delta_L(x_{N+2})-\delta_L(x)|<\eta$, i.e. $x_{N+2}\in L$
and 
$$
0\in \sum_{n=1}^N D_\beta f_n(x_n) -\nabla^\beta g(x_{N+1}) 
+D_\beta \delta_L(x_{N+2})+ r B_{X^*}.
$$
Since $D_\beta \delta_L(x_{N+2})\in L^\bot$ and 
$ x^*-\nabla^\beta g(x_{N+1})\in V_\beta$ we are done. \smiley

Using the method in \cite{bi}, we can prove the following stronger
result:





\begin{theorem} Let $X$ be a Banach space with an equivalent
$\beta$-smooth norm. Let $(f_1,...,f_N)$ be lower semicontinuous
functions and $x\in \cap_{n=1}^N{\rm dom}(f_n)$.  Then, for any
$\varepsilon >0$ and any weak-star neighbourhood $V$ of $0\in X^*$,
there exist $x_n\in x+ \varepsilon B, n=1,...,N$ such that
$|f_n(x_n)-f_n(x)|<\varepsilon $ and 
$$ 
\partial_\beta (\sum_{n=1}^N 
f_n)(x) \subset  \sum_{n=1}^N D_\beta f_n(x_n)  +  V.  
$$

\end{theorem}

{\bf Proof.} Let $\varepsilon>0$ and $V$ be a weak-star neighbourhood 
of $0\in X^*$. Fix $r>0$ and $L$ a finite dimensional subspace of
$X$ containing $x$  such that
$L^\bot +2r B_{X^*}\subset V$. Let 
$x^*\in \partial_\beta (\sum_{n=1}^N  f_n)(x)$. Then, for 
any $K\in \beta$,
\begin{eqnarray}
\lim_{t\to 0^+}\inf_{h\in tK}
t^{-1}[\sum_{n=1}^N(f_n(x+h)-f_n(x)-\langle x^*,h\rangle ] \ge 0.\label{e}
\end{eqnarray}
Choose a $K\in \beta$ containing the intersection of 
$L$ with a ball around zero. Then (\ref{e}) in particular implies
that
\begin{eqnarray*}
\lim_{t\to 0^+}\inf_{h\in tB\cap L}
t^{-1}[\sum_{n=1}^N(f_n(x+h)-f_n(x)-\langle x^*,h\rangle ] \ge 0.
\end{eqnarray*}
Since $L$ is a finite dimensional space, this is equivalent to
\begin{eqnarray*}
\lim_{\|y-x\|\to 0}\inf_{y-x\in L}
\|y-x\|^{-1}[\sum_{n=1}^N(f_n(y)-f_n(x)-\langle x^*,y-x\rangle ] \ge 0.
\end{eqnarray*}
Thus, there exists $\eta < r$ such that 
\begin{eqnarray*}
\sum_{n=1}^Nf_n(y)-\langle x^*,y\rangle + r\|y-x\| +\delta_L(y).
\end{eqnarray*}
attains a minimum at $x$ for $y\in x+\eta B$.


By Proposition 1.1 
$y\to (f_1(y),...,f_N(y),-\langle x^*,y\rangle,r\|y-x\|,\delta_L(y))$ 
is locally uniformly lower semicontinuous. 
Applying Theorem 1.1 yields that there exist $x_n, n=1,...,N+3$ such that 
$\|x_n-x\|<\eta<\varepsilon, n=1,...,N+3$, $|f_n(x_n)-f_n(x)|<\eta<\varepsilon, 
n=1,...,N$,
$|\delta_L(x_{N+3})-\delta_L(x)|<\eta$, i.e. $x_{N+3}\in L$
and 
$$
0\in \sum_{n=1}^N D_\beta f_n(x_n) -x^* +rD_\beta \|x_{N+2}-x\| 
+D_\beta \delta_L(x_{N+3})+ r B_{X^*}.
$$
Observing that  $D_\beta \delta_L(x_{N+3})\in L^\bot$ and 
$rD_\beta \|x_{N+2}-x\|\in rB_{X^*}$ we obtain
\begin{eqnarray*}
x^* &\in& \sum_{n=1}^N D_\beta f_n(x_n) +rD_\beta \|x_{N+2}-x\| 
+D_\beta \delta_L(x_{N+3})+ r B_{X^*}\\
&\in& \sum_{n=1}^N D_\beta f_n(x_n)  
+L^\bot+ 2r B_{X^*}\\
&\in& \sum_{n=1}^N D_\beta f_n(x_n) +V.
\end{eqnarray*} 
\smiley
















\begin{theorem} Let $X$ be a Banach space with an equivalent
$\beta$-smooth norm. Let $(f_1,...,f_N)$ be lower semicontinuous
functions, all but one of $f_n,n=1,...,N$ are locally uniformly
continuous and $x\in \cap_{n=1}^N{\rm dom}(f_n)$.  Then, for any
$\varepsilon >0$ and any neighbourhood $V$ of $0\in X^*_\beta$, there
exist $x_n\in x+ \varepsilon B, n=1,...,N$ such that
$|f_n(x_n)-f_n(x)|<\varepsilon $ and 
$$ D_\beta (\sum_{n=1}^N 
f_n)(x) \subset  \sum_{n=1}^N D_\beta f_n(x_n)  +  V.  
$$

\end{theorem}


{\bf Proof.} Let $\varepsilon>0$ and $V$ be a neighbourhood 
of $0\in X^*_\beta$. Let $r>0$, $U$ be a neighbourhood 
of $0\in X^*_\beta$ such that 
$U + r B_{X^*}\subset V$. Let 
$x^*\in D_\beta (\sum_{n=1}^N  f_n)(x)$. Then there exists
a $\beta$ smooth function $g$ such that $\nabla^\beta g(x)=x^*$
and $\sum_{n=1}^N f_n -g$ attains a local minimum at $x$.
Choose $0<\eta <\varepsilon$ such that $\|y-x\|< \eta<\varepsilon$ implies that
$\nabla^\beta g(y)-\nabla^\beta g(x)\in U$. 
By Proposition 1.1 $(f_1,...,f_N,g)$ is locally uniformly lower 
semicontinuous. 
Applying Theorem 1.1 yields that there exist $x_n, n=1,...,N+1$ such that 
$\|x_n-x\|<\eta<\varepsilon, n=1,...,N+1$, $|f_n(x_n)-f_n(x)|<\eta<\varepsilon, 
n=1,...,N$
and 
$$
0\in \sum_{n=1}^N D_\beta f_n(x_n) -\nabla^\beta g(x_{N+1}) + r B_{X^*}.
$$
Thus,
\begin{eqnarray*}
x^* &\in & \sum_{n=1}^N D_\beta f_n(x_n) +
	    \nabla^\beta g(x)-\nabla^\beta g(x_{N+1}) + r B_{X^*}\\
    &\in & \sum_{n=1}^N D_\beta f_n(x_n) + V.
\end{eqnarray*}
\smiley

{\bf Remark 2.3} As in \cite{bp} for a lower semicontinuous function
 $f: X\to [-\infty, \infty]$ we say that 
$f$ is {\it s-H\"older subdifferentiable}  at $x$ $(s\in (0,1])$ with 
subderivative $x^*\in X^*$ if there exists a positive constant
$C_x$ such that
$$
f(y)-f(x)-\langle x^*,y-x\rangle \ge -C_x\|y-x\|^{1+s}
$$
in a neighbourhood of $x$.
We denote the set of $s$-H\"older subderivatives of $f$ at $x$ by
$\partial^{H(s)}f(x)$. When $s=1$ such subderivatives
are called {\it Lipschitz smooth} and in Hilbert space they coincide
with Rockafellar's {\it proximal subderivatives}, written 
$\partial^\pi f(x)$ \cite{r}. 
Then all the above statements  and proofs will still hold
true in a Banach space with 
a power modulus of smoothness $t^{p}$ \cite{lt} 
if we replace `$\beta$' by `$H(p-1)$'. 

{\bf Remark 2.4} By the same argument of Lemma 1.3 \cite[Section VIII]{DGZ}
we can prove that 

\begin{lemma} Let $X$ be a Banach space that admits a bump function which
is Lipschitzian and $\beta$-smooth. Then  there exists a function $d:X\to R^+$
and $K>1$ such that 

i) $d$ is bounded, Lipschitzian on $X$ and $\beta$-smooth on 
$X\backslash \{0\}$.

ii) $\|x\|\le d(x) \le K\|x\|$ if $\|x\|\le 1$ and $d(x)=2$ if $\|x\|\ge 1$.
\end{lemma}

Then observing that we can replace the $\|\cdot \|^2$ term in Lemma 1.1,
Theorem 1.1 and their proofs by $d(\cdot)^2$ and use the 
Deville-Godefroy-Zizler Smooth Variational Principle (cf. \cite{DGZ,Ph})
in place of the Borwein-Preiss Smooth Variational Principle the condition that 
$X$ has a $\beta$ smooth norm can be replaced by the weaker condition that
$X$ has a $\beta$ smooth Lipschitzian bump function.


{\bf Remark 2.5} Because of the importance of the fuzzy addition rule in applying 
generalized derivatives, Ioffe \cite{i83,i84} named a Banach
space that has this property for lower semicontinuous functions for the
$\varepsilon$-Fr\'echet subderivative and the $\varepsilon$-Dini 
subderivative a trustworthy (T) and a weak trustworthy (WT) space, respectively. 
He proved that if a Banach space has a Fr\'echet or Gateaux smooth norm
then it has property (T) or (WT). It is also known that (T) or (WT)
is equivalent to the dense $\varepsilon$-Fr\'echet subdifferentiability (S)
or dense $\varepsilon$-Dini subdifferentiability (WS) of lower semicontinuous
functions \cite{f}. In this spirit we will call a Banach space 
$\beta$-trustworthy ($T_\beta$) if it has the property stated in the conclusion
of Theorem 2.2 and say a Banach space is a  (strong) dense 
$\beta$-subdifferentiability space if, for any lower semicontinuous function
$f$,  the set ($(x,f(x))\in graph(f)$) $x\in dom(f)$  for which 
$D_\beta f(x)\neq \emptyset $ is dense in the (graph) domain of $F$.
We will use notations $S^+_\beta$ and $S_\beta$ for  strong
$\beta$-subdifferentiability and $\beta$-subdifferentiability, respectively. 
We will
denote the properties `has a $\beta$-smooth norm' and `has a $\beta$-smooth 
Lipschitz bump function' by $H_\beta$ and $H_\beta^-$ respectively.
Then Theorem 2.2 and Remark 2.4 tell us
$H_\beta \Longrightarrow H_\beta^- \Longrightarrow T_\beta$.
It is obvious that $S^+_\beta \Longrightarrow S_\beta$.
Let $f$ be a lower semicontinuous function in a $T_\beta$ space.
Then, for any $x\in dom(f)$, applying the $T_\beta$ property to
$f+\delta_{\{x\}}$ yields $S^+_\beta$. Therefore, we have
$$
H_\beta \Longrightarrow H_\beta^- \Longrightarrow T_\beta
\Longrightarrow S^+_\beta \Longrightarrow S_\beta.
$$
In \cite{f} it is shown that $T\Longleftrightarrow S$ and 
$WT\Longleftrightarrow WS$.
Using similar  arguments we can show that 
$T_\beta\Longleftrightarrow S^+_\beta$. However, it is 
not clear if $S_\beta\Longleftrightarrow S^+_\beta$.
Since conditions $H_\beta$ and $H_\beta^-$ are easier to check 
than $S_\beta$ and $S_\beta^+$ we will not further pursue
this topic here.




\section{Viscosity Solutions to the HJB Equations}


Viscosity solutions were  introduced by Crandall and Lions for 
handling the non-classical solutions to the HJB equations. An important
class of such solutions is the value functions of certain
optimal control problems. It turns out that the uniqueness
of a continuous solution to the HJB equation is closely related to the 
fuzzy summation rule for subdifferentials. In this section we
prove a comparison theorem for a class of HJB equations in
a $\beta$-smooth space by applying the results
in the previous section. It yields a uniqueness theorem
for $\beta$-viscosity solution. We then apply the result to
show that the value function of  a class of infinite
horizon optimal control problems in a $\beta$-smooth
space is the unique $\beta$-viscosity solution of the corresponding  HJB equation. As 
Borwein and Fitzpatrick \cite{bf1} has shown that $L^1$ has a 
weak Hadamard equivalent norm, the result, in particular, applies to infinite
horizon optimal control problems in $L^1$ space.





Let $F:X\times R\times X^*_\beta\to R$. Assume that 

(H1) $F$ is uniformly 
continuous; 

and 

(H2) there exists a $\gamma>0$  such that 
$$
F(x,v,x^*)-F(x,u,x^*)\le \gamma (v-u),\quad 
\forall (x,x^*)\in X\times X^*_\beta.
$$
Consider 
\begin{eqnarray}
F(x,u,Du)=0. \label{hjb}
\end{eqnarray}

\begin{definition}
A function $u:X\to R$ is a $\beta$-{\bf viscosity 
subsolution}  of (\ref{hjb}) if u is upper semicontinuous
and, for every $x\in X$ and every $x^*\in D^\beta u(x)$:
$$
F(x,u(x),x^*)\le 0.
$$ 

A function $u:X\to R$ is a $\beta$-{\bf viscosity 
supersolution} of (\ref{hjb}) if u is lower semicontinuous
and, for every $x\in X$ and every $x^*\in D_\beta u(x)$ :
$$
F(x,u(x),x^*)\ge 0.
$$ 
We say that $u$ is a $\beta$-{\bf viscosity solution} if $u$ is both a 
$\beta$-viscosity 
subsolution and a $\beta$-viscosity supersolution.
\end{definition}

\begin{theorem} Let $X$ be a Banach space with an equivalent
$\beta$-smooth norm and $F$ a function satisfies assumptions
(H1) and (H2).
Let $u,v$ be two uniformly continuous functions such that
$v$ is lower bounded and $u$ is upper bounded.
If $u$ is a $\beta$-viscosity subsolution of (\ref{hjb})
and $v$ a $\beta$-viscosity supersolution of (\ref{hjb}), then $u\le v$.
\end{theorem}

{\bf Proof.} Let $\varepsilon >0$ be an arbitrary positive number.
Since  $F:X\times R\times  X^*_\beta\to R$ is uniformly continuous there
exist $\eta>0$ and a neighbourhood $V$ of $0\in X^*_\beta$
such that $\|x-y\|<\eta$ and $x^*-y^*\in V$ implies that
$$
|F(x,u,x^*)-F(y,u,y^*)|<\varepsilon,\quad \forall u\in R.
$$
The function $v-u$ is uniformly continuous and bounded below. 
By the smooth variational principle there exist $x\in X$ and 
$x^* \in D_\beta(v-u)(x)$ such that $x^*+{1\over 2}V\subset V$ and
$ (v-u)(x)<\inf_X(v-u)+\varepsilon$. Applying Theorem 2.4 with 
$f_1=v$ and $f_2=-u$, there exist $x_1, x_2\in X$, $x_1^*\in D_\beta v(x_1)$
and  $x_2^*\in D^\beta u(x_2)$ satisfying 

1) $\|x_1-x\|<\eta$ and $\|x_2-x\|<\eta$;

2) $|v(x_1)-v(x)|<\varepsilon$ and $|u(x_2)-u(x)|<\varepsilon$;

3) $x_1^*-x_2^*-x^*\in {1\over 2} V$.

Since the function $u$ is a viscosity subsolution of (\ref{hjb})
we have 
$$
F(x_1,u(x_1),x_1^*)\le 0.
$$
Similarly 
$$
F(x_2,v(x_2),x_2^*)\ge 0.
$$
Therefore, 
\begin{eqnarray*}
\inf_X(v-u) &>& (v-u)(x)-\varepsilon
	    > v(x_2)-u(x_1)-3\varepsilon\\
	    &\ge& \gamma^{-1}[F(x_1,v(x_2),x^*_1)-F(x_1,u(x_1),x^*_1)]
                     -3\varepsilon\\
            &=& \gamma^{-1}[F(x_1,v(x_2),x^*_1)-F(x_2,v(x_2),x^*_2)]\\
            &-& \gamma^{-1}[F(x_2,v(x_2),x^*_2)-F(x_1,u(x_1),x^*_1)]
		-3\varepsilon\\
            &=& \gamma^{-1}[F(x_1,v(x_2),x^*_1)-F(x_2,v(x_2),x^*_2)]-3\varepsilon
\end{eqnarray*}
Since $x_1^*-x_2^*\in  V$ we have 
$F(x_1,v(x_2),x^*_1)-F(x_2,v(x_2),x^*_2)\ge -\varepsilon$.
Thus,
\begin{eqnarray*}
\inf_X(v-u)>-(\gamma^{-1}+3)\varepsilon.
\end{eqnarray*}
As $\varepsilon $ is arbitrary $\inf_X(v-u)\ge 0$. \smiley


\begin{corollary} Let $X$ be a Banach space with an equivalent
$\beta$-smooth norm and $F$ a function that satisfies assumptions
(H1) and (H2). Then any uniformly continuous lower 
bounded $\beta$-viscosity solution
to (\ref{hjb}) is unique.
\end{corollary}

{\bf Remark 3.1} Assume that $\beta $ is at least as fine as  $\beta'$.
It is easy to see that  any $\beta$-viscosity solution to 
(\ref{hjb}) is also a $\beta'$-viscosity solution to 
(\ref{hjb}). It is natural to ask: does there exists an equation that has a 
$\beta$-viscosity solution but does not have a $\beta'$-viscosity solution?
(In fact, this question has been asked for Fr\'echet and Gateaux bornologies in 
\cite[Question VIII, 3]{DGZ} where it seems to us  that the two bornologies
were put in a wrong order.) The answer to this question is yes as shown
by the following trivial example.

{\bf Example 3.1} Let  $X$ be a $\beta'$-differentiable 
Banach space with a nowhere $\beta$-differentiable norm $\|\cdot \|$.
Consider 
\begin{eqnarray} 
1=0.\label{u1}
\end{eqnarray}
Then obviously equation (\ref{u}) does not have any
$\beta'$-viscosity solution but $u(x)=\|x\|$ is a uniformly 
continuous lower bounded 
$\beta$-viscosity solution of (\ref{u}). To see this, observe that 
$u$ is trivially a supersolution of (\ref{u1}). Since $u(x)$ is nowhere 
$\beta$-differentiable $D^\beta u(x)=\emptyset$ for all $x\in X$.
Thus, $u$ is also a $\beta$-viscosity subsolution and, therefore,
a $\beta$-viscosity solution of (\ref{u1}).

As concrete examples: 
\begin{itemize}
\item[a.] The sup norm of $C[0,1]$ is Gateaux smooth but nowhere
weak Hadamard differentiable \cite{bf1}.

\item[b.] $L_1[0,1]$ or $l_1(N)$ 
has an  equivalent weak Hadamard smooth norm \cite{bf1}
but is not Asplund and, therefore, has a nowhere Fr\'echet differentiable
norm.
\end{itemize} 




{\bf Remark 3.2} By Remark 3.1 when  $\beta $ is at least as fine as  $\beta'$,
a uniqueness result for a $\beta'$-viscosity solution 
implies that for a $\beta$-viscosity solution. Thus, in Corollary 3.1
we can always reduce the requirement on the smoothness of the space by
using a coarser bornology. However, the price to pay is we have to
impose stronger continuity conditions on $F$. This also leads to the 
following natural question: Assume that $\beta $ is finer than  $\beta'$. 
Does there exists an equation that has a 
unique $\beta'$-viscosity solution while whose
$\beta$-viscosity solutions are not unique? The next example gives a positive 
answer to this question.

{\bf Example 3.2} In Banach space $X$ consider 
\begin{eqnarray} 
u(x)+\|Du(x)\|=0.\label{u}
\end{eqnarray}
Assume that the norm $\|\cdot \|$ of $X$ is nowhere $\beta$-differentiable
and $X$ has an equivalent $\beta'$-smooth norm. Then 
\begin{itemize}
\item[1.] $u=0$ is the unique uniformly continuous lower  bounded 
$\beta'$-viscosity solution of (\ref{u});

\item[2.] $u(x)=\|x\|$ is a uniformly continuous lower bounded 
$\beta$-viscosity solution of (\ref{u}).
\end{itemize} 



{\bf Proof.} {\it Step 1.} If $u$ is a uniformly continuous lower  bounded 
$\beta'$-viscosity solution of (\ref{u}) then $u$ is nonnegative. In fact,
if $-\Delta =\inf_X u<0$ then since $X$ has an equivalent $\beta'$-smooth norm, 
by the Borwein-Preiss Smooth Variational
Principle, there exists an $x\in X$ and a $p\in X^*$ such that 
$$
u(x)<\inf_Xu + {\Delta \over 2},
$$
and 
$$
p\in D_{\beta'}u(x), \quad \|p\|<{\Delta \over 2}.
$$
This implies 
$$
u(x)+\|p\|<0
$$
so that  $u$ is not a $\beta'$-viscosity supersolution of (\ref{u}) which is 
absurd.

{\it Step 2.} If $u$ is a  nonnegative  
$\beta'$-viscosity solution of (\ref{u}) then $u$ is a  nonnegative  
$\beta'$-viscosity subsolution of (\ref{u}). Thus,
$D^{\beta'}u(x)=0$ whenever it is nonempty. By \cite[Theorem 6.3]{bz}
$u$ is a constant. It is then obvious that $u$ has to be $0$.
This completes the proof of 1.

{\it Step 3.}  We turn to prove 2. It is trivial to observe that 
$u(x)=\|x\|$ is a $\beta$-viscosity supersolution of (\ref{u}). 
Since $u(x)=\|x\|$ is nowhere $\beta$-differentiable 
$D^\beta u(x)=\emptyset $ for all $x\in X$. Thus, $u$
is also a $\beta$-viscosity subsolution and, therefore, 
a $\beta$-viscosity solution of (\ref{u}). \smiley


 
{\bf Remark 3.3} Notice that (\ref{u}) is the HJB equation corresponding to
the optimal control problem 
\begin{eqnarray*}
 {\rm minimize } && J(x,u):= \int_0^\infty e^{-s}f(x(s),u(s))ds  \\
	   {\rm subject~ to} && \dot x(s)=u(s)\\
			     && \quad x(0)=x, ~ u(s)\in B_X 
\end{eqnarray*}
with $f=0$. It is reasonable to expect  $0$ to be  the unique
viscosity solution of (\ref{u}). Remark 3.1 and Remark 3.2 shows that 
the $\beta$-viscosity solution is the right choice when $\beta$ is the 
finest bornology such that the space has a $\beta$-smooth norm. 




We now consider an infinite dimensional  
optimal control problem. Let $X$ be a Banach space with a $\beta$-smooth
norm and 
$U$ a metric space. Consider the following optimal control 
problem in $X$:
\begin{eqnarray*}
P(x) \quad {\rm minimize } && J(x,u):= \int_0^\infty e^{-\gamma s}f(x(s),u(s))ds  \\
	   {\rm subject~ to} && \dot x(s)=g(x(s),u(s))\\
			     && \quad x(0)=x, ~ u\in {\cal U}
\end{eqnarray*}
where $g:X\times U\to R$ is continuous, Lipschitz in $x$ uniformly in $U$,
for each $x\in X$, there exists $K_x\in \beta$ such that $g(x,U)\subset K_x$,
$f:X\times U\to R$ is continuous, bounded, continuous in x uniformly in $U$ and 
$$
{\cal U}:= \{u: u~{\rm is ~measurable ~and }~u(t)\in U, ~{\rm for}~
t\in [0,\infty) ~{\rm a.~ e.}\}.
$$

{\bf Remark 3.4} The condition requiring that $g(x,U)$ belongs to an appropriate
bornology is satisfied for the Hadamard bornology if $U$ is compact and
for the weak  Hadamard bornology if $U$ is weakly compact and $g$ is 
linear in the control.


It is easy to check that, for any $u\in {\cal U}$ there exists an
unique solution $x(\cdot,u)$ on $[0,\infty)$ to the control system.
Since $f$ is bounded the problem is well defined. We denote 
the value function of $P(x)$ by $V(x)$. Then we have the following:

\begin{theorem} (Dynamic Programming Principle) For any $t>0$,
$$
V(x)=\inf_{u\in {\cal U}}\{\int_0^t e^{-\gamma s}f(x(s,u),u(s))ds+
e^{-\gamma t}V(x(t,u))\}.
$$
\end{theorem}
This theorem is standard; for a proof see for example \cite{fr75}.

Define $H:X\times X^*\to R$ by 
$$
H(x,p)=\sup_{u\in U}\{-\langle p,g(x,u)\rangle-f(x,u)\}.
$$
We prove
\begin{theorem}
$V$ is the unique $\beta$-viscosity
solution of the following HJB equation
\begin{eqnarray}
\gamma V(x)+H(x,DV(x))=0.\label{h}
\end{eqnarray}
\end{theorem}

{\bf Proof.} 
Using Gronwall's inequality,  since  $f$ is continuous in $x$ uniformly in $U$
we can show that $V$ is uniformly continuous.
It is easy to see that $F(x,u,p):=\gamma u+H(x,p)$ satisfies the 
condition of Theorem 3.1. Therefore, uniqueness is a 
direct consequence of Corollary 3.1. 
We need only to show that $V$ is a $\beta$-viscosity solution of (\ref{h}).
By the Dynamic Programming Principle, For any $t>0$,
\begin{eqnarray}
V(x)=\inf_{u\in {\cal U}}\{\int_0^t e^{-\gamma s}f(x(s,u),u(s))ds+
e^{-\gamma t}V(x(t,u))\}.\label{d}
\end{eqnarray}

{\it a. Subsolution.} 
Let $y$ be an element of $X$  such that $p\in D^\beta V(y)$. Then there exists
a $\beta$-smooth function $w$ such that $\nabla^\beta w(y)=p$, $y$ is a
(local) maximum of $V-w$ and $0=(V-w)(y)$. Then by (\ref{d}), for 
any $u\in {\cal U}$ and $t>0$, 
\begin{eqnarray}
w(y)=V(y)\le \int_0^t e^{-\gamma s}f(x(s,u),u(s))ds+
e^{-\gamma t}V(x(t,u)).\label{d1}
\end{eqnarray}
where $x(t,u)$ is a solution corresponding to the control $u$ with
initial value $y$.
Observing that $\{x(s,u):s\in [0,t]\}$ is compact, we have 
$$
{d\over {dt}}[e^{-\gamma t}w(x(t,u))]=-e^{-\gamma t}\gamma w(x(t,u))+
e^{-\gamma t}\langle \nabla^\beta w(x(t,u)), g(x(t,u), u(t))\rangle.
$$
Therefore, we can write  (\ref{d1}) as 
\begin{eqnarray*}
t^{-1}\int_0^t e^{-\gamma s}[\gamma w(x(s,u))
-\langle \nabla^\beta w(x(s,u)), g(x(s,u), u(s))\rangle-f(x(s,u),u(s))]ds\le 0.
\end{eqnarray*}
Fixing an arbitrary $v\in U$ and setting $u(s)=v$ for all $s\in [0,t]$ yields
\begin{eqnarray*}
t^{-1}\int_0^t e^{-\gamma s}[\gamma w(x(s,u))
-\langle \nabla^\beta w(x(s,u)), g(x(s,u), v)\rangle-f(x(s,u),v)]ds\le 0.
\end{eqnarray*}
Taking limits when $t\to 0$, observing that the integrand is continuous in
$s$ and $x(0,u)=y$, we obtain
\begin{eqnarray*}
\gamma V(y)
-\langle p, g(y, v)\rangle-f(y,v)=
\gamma w(y)
-\langle \nabla^\beta w(y), g(y, v)\rangle-f(y,v)\le 0.
\end{eqnarray*}
Therefore,
$$
\gamma V(y)+H(y,p)\le 0,
$$
that is to say $V$ is a viscosity subsolution of (\ref{h}).

{\it b. Supersolution.} 
Now let $y$ be an element of $X$  such that $p\in D_\beta V(y)$. 
Then there exists
a $\beta$-smooth function $w$ such that $\nabla^\beta w(y)=p$, $y$ is a
(local) minimum of $V-w$ and $0=(V-w)(y)$. Then by (\ref{d}), there
exists $u^i\in {\cal U}$ such that
\begin{eqnarray}
w(y)+{1\over {i}} &=& V(y)+{1\over {i}}\nonumber \\
&\ge&  \int_0^{1/i} e^{-\gamma s}f(x(s,u^i),u^i(s))ds+
e^{-\gamma 1/i}V(x(1/i,u^i)).\label{d2}
\end{eqnarray}
Similar arguments as in the previous paragraph yield
\begin{eqnarray*}
{1\over {i}} &+& i\int_0^{1/i} e^{-\gamma s}[\gamma w(x(s,u^i))\\
&-& \langle \nabla^\beta w(x(s,u^i)), g(x(s,u^i), u^i(s))\rangle
-f(x(s,u^i),u^i(s))]ds\ge 0.
\end{eqnarray*}
We rewrite this inequality as 
\begin{eqnarray*}
\gamma w(y)
+i\int_0^{1/i}[-\langle \nabla^\beta w(y), g(y, u^i(s))\rangle
-f(y,u^i(s))]ds\ge h(i)
\end{eqnarray*}
where
$$
h(i)={1\over {i}}+h_1(i)+h_2(i)+h_3(i)
$$
and the $h_j's$ are defined as follows:
$$
h_1(i):=\gamma w(y)-i\int_0^{1/i} e^{-\gamma s}\gamma w(x(s,u^i)) ds,
$$
$$
h_2(i):=i\int_0^{1/i} [e^{-\gamma s}
\langle \nabla^\beta w(x(s,u^i)), g(x(s,u^i), u^i(s))\rangle
-\langle \nabla^\beta w(y), g(y, u^i(s))\rangle]ds
$$
and
$$
h_3(i):=i\int_0^{1/i} [e^{-\gamma s}f(x(s,u^i),u^i(s))
-f(y,u^i(s))]ds.
$$
It is obvious that  
$$-\langle \nabla^\beta w(y), g(y, u^i(s))\rangle
-f(y,u^i(s))\le H(y,p)
$$
and, therefore,
\begin{eqnarray}
\gamma V(y)+H(y,p)\ge h(i).\label{hl}
\end{eqnarray}
It is easy to see that 
$$
\sup \{\|x(s,u^i)-y\|:s\in[0,1/i]\}\to 0
$$ 
when $i\to \infty$. Thus, 
$\lim_{i\to \infty}h_1(i)=\lim_{i\to \infty}h_3(i)=0$. Observing
that $g$ is uniformly Lipschitz in $x$ and $g(y,u^i(s))$
is in the compact set $g(y,U)$, the $\beta$-smoothness of
$w$ yields $\lim_{i\to \infty}h_2(i)=0$. Therefore, 
$\lim_{i\to \infty} h(i) =0$. 
Sending $i$ to infinity in (\ref{hl}) we obtain that $V$ is a viscosity 
supersolution of  (\ref{h}) and, hence, a viscosity solution of  (\ref{h}).
\smiley 


\section{Metric Regularity}

It is well known that  metric regularity is closely related to the open
mapping property and plays an important role in studying exact penalty and
necessary conditions for constrained minimization problems. In this section
we apply the  fuzzy summation rule derived in Section 2 to show that a 
function is not metrically regular at a point corresponding  to a fuzzy
extremal condition at the same point. We then turn to sufficient conditions for
a function to be metrically regular at a given point. We deduce a dual condition 
weaker than the dual conditions given in Ioffe \cite{i84a} and Ginsburg and 
Ioffe \cite{gi} and a weaker version of the primal condition discussed in 
Borwein and Strojwas
\cite{bs}. Many authors have discussed primal sufficient conditions. The condition
we give below (Theorem 4.3) appears to be the  weakest up to now. 
Various primal conditions 
discussed in \cite{ac,af,b86,bs} are deduced as corollaries. To
avoid complications in the notations we consider metric regularity for a
single-valued function with respect to a closed set. (This is infact an
entirely general formulation (see Lemma 4.1 below).)
There are many discussions
about regularity of multifunctions (see for example \cite{m} and the reference
therein). We will indicate in the end of this section how to handle 
multifunctions with our results.



\begin{definition} Let $X$ and $Y$ be Banach spaces.
We say that $h:X\to Y$ is {\bf regular with respect to $S$ at $x_0$} if
there exist positive constants $\varepsilon, K$ such that,
for all $x\in S\cap (x_0+\varepsilon B)$ and $\xi \in h(x_0)+\varepsilon B$,

$$
d(x,S\cap h^{-1}(\xi))\leq K||h(x)-\xi||.
$$
\end{definition}


\begin{definition} Let $X$ and $Y$ be Banach spaces 
with $\beta$-smooth norms and 
$S$ a closed subset of   $X$. We say that $x_0$ is an 
{\bf extremal with respect to} 
$(h,S)$ if, for any $\eta>0$ there exist an unit vector $y^*$, $y\in 
(x_0+\eta B)\cap S$ and a constant $L$ such that
$$
0\in (\nabla h(x_0))^*y^* + LD_\beta d(y,S)+\eta B_{X^*}.
$$
\end{definition}


\begin{theorem}   Let $X$ and $Y$ be Banach spaces with  $\beta$-smooth norms, 
$S$ a closed subset of   $X$ and $x_0\in S$. 
Assume that $h:X\to Y$  is Lipschitz and 
strictly differentiable at $x_0$. Then $h$ is not regular with respect to 
$S$ at $x_0$ implies that
$x_0$ is an extremal with respect to 
$(h,S)$.
\end{theorem}


{\bf Proof. } 
Let $\eta$ be an arbitrary positive constant in $(0,1)$. Since $h$
is strictly differentiable at $x_0$, for any $d\in B_X$, there exists 
$\eta_1<\eta /4$ such that $\|x-x_0\|<\eta_1$ and $t\in (0,\eta_1)$ 
implies that
\begin{eqnarray}
\frac{h(x+td)-h(x)}t-\bigtriangledown h(x_0)d\in \frac{\eta}2 B_Y 
\label{st}
\end{eqnarray}
and 
$$
\|h(x)-h(x_0)\|< \frac{\eta^2}{128}.
$$
Suppose $h$ is not regular with respect to $S$ at $x_0$. Then there
exists an  $s\in S\cap (x_0+\frac{\eta_1}4 B)$ and a $\xi \in
h(x_0)+\frac{\eta^2}{128} B$ such that

\begin{eqnarray}
 d(s,S\cap h^{-1}(\xi))>\frac1{\eta}||h(s)-\xi||.\label{a} 
\end{eqnarray}

Apply Ekeland's Principle with

$$
f(x):=||h(x)-\xi||,\;\varepsilon :=||h(s)-\xi||\le \frac{\eta^2}{64}
$$

$$
\lambda :=\frac{\eta}8,\bar x:=s 
$$
Then there exist $v\in S$ such that

1. $||s-v||\leq \frac{\eta}8.$

2. $v$ minimizes
$$
x\to ||h(x)-\xi||+\frac{\varepsilon}{\lambda}||x-v|| 
$$
on $S$.

Let $L'$ be a local Lipschitz constant for  $h$ at $x_0$. Then $L:=L'+1$
is a local Lipschitz constant for 
$x\to ||h(x)||+ \frac{\varepsilon}{\lambda}||x-v||$.
Thus, we have

2'. $v$ locally minimizes
$$
x\to ||h(x)-\xi||+\frac{\varepsilon}{\lambda}||x-v||+L\cdot d(x,S).
$$

By Theorem 2.1 there exist $y,z$ and $w$ such that
\begin{eqnarray}
0 &\in& D _\beta ||h(\cdot )-\xi||(y)+\frac{\varepsilon}{\lambda}
                 D _\beta ||w-v||+LD _\beta
                    d(z,S)+\frac{\eta}8 B_{X^*} \nonumber \\ 
&\subset& D _\beta ||h(\cdot )-\xi||(y)
+LD _\beta d(z,S)+\frac{\eta}4 B_{X^*}\label{i}
\end{eqnarray}

where $\|y-v\|<\frac{\eta_1}4$ and 
$\|z-v\|<\frac{\eta_1}4$
and, therefore, $\|y-x_0\|<\eta_1$ and $\|z-x_0\|<\eta_1$.
By inequality (\ref{a}) $ d(s,S\cap h^{-1}(\xi))>0$. Therefore,
$\|h(s)-\xi\|>0$. Without loss of generality we may assume that 
$y$ were chosen close enough to $s$ such that $\|h(y)-\xi\|>0$.
Then, for any $d\in B$, when $t$ is small enough,
$y_{t}^{*}(d):=\bigtriangledown ^\beta ||.||(h(y+td)-\xi)$ exists. 
Since $\|y_{t}^{*}(d)\|=1$,
by (\ref{st}), for any $d\in B_X$, when $t$ is sufficiently small,


\begin{eqnarray}
\frac{\eta}2 &\geq& \langle y_{t}^{*}(d),
\frac{h(y+td)-h(y)}t-\bigtriangledown h(x_0)d\rangle \nonumber  \\ 
&=& \langle y_{t}^{*}(d),
\frac{(h(y+td)-\xi)-(h(y)-\xi)}t\rangle -\langle y_{t}^{*}(d),\bigtriangledown
h(x_0)d\rangle  \nonumber \\ 
&\geq& -\langle y_{t}^{*}(d),\bigtriangledown
h(x_0)d\rangle +\frac{||h(y+td)-\xi||-||h(y)-\xi||}t. \label{long}
\end{eqnarray}


By inclusion (\ref{i}) there exists $z^{*}\in
LD_\beta d(z,S)$ such that  $z^{*}=-u^{*}+v^{*}$ where  
$u^{*}\in D _\beta ||h(.)-\xi||(y)$ and $v^{*}\in \frac{\eta}4 B_{X^*}$.
Observe that, for a fixed $d$, there exists a $t_d$ such that 
when $t\in (0,t_d)$,
$$
\frac{||h(y+td)-\xi||-||h(y)-\xi||}t>\langle u^{*},d\rangle -\frac{\eta}4  
\ge \langle z^{*},d\rangle -\frac{\eta}2.  
$$
Combining this with (\ref{long}) we obtain that when $t\in (0,t_d)$ 
$$
\eta \geq -\langle y_{t}^{*}(d),\bigtriangledown h(x_0)d\rangle -\langle
z^{*},d\rangle . 
$$

Denote $y^{*}=\bigtriangledown ^\beta ||.||(h(y)-\xi)$. Then 
$y_{t}^{*}(d)=\bigtriangledown ^\beta ||.||(h(y+td)-\xi)\rightharpoondown _{w^*}
y^{*}$ as $t\to 0^+$ and $||y^{*}||=1$. (In fact, $y_{t}^{*}(d)$
converges to $y^{*}$ in $Y^*_\beta$.) Taking limits in the above inequality
as $t\rightarrow 0^+$ yields 

$$
\eta \geq -\langle y^{*},\bigtriangledown h(x_0)d\rangle -\langle
z^{*},d\rangle . 
$$
Since this is true for all $d\in B$ we obtain
$$
(\nabla h(x_0))^*y^*+z^*\in \eta B_{X^*}
$$
or 
$$
0\in (\nabla h(x_0))^*y^* + LD_\beta d(y,S)+\eta B_{X^*},
$$
as was to be shown. \smiley


We now turn to some sufficient conditions for $h$ to be regular
with respect to $S$ at $x_0$. We consider a dual condition first.
To do so we need the following definitions.


\begin{definition} Let $X$ be a Banach space with a $\beta$-smooth norm and 
$S$ a closed subset of   $X$. 
We define the $\beta$-{\bf normal cone} (denoted by $N_\beta (x,S)$) and the 
$\beta$-{\bf tangent cone} (denoted by $T_\beta (x,S)$)  of $S$ at $x\in  S$ by

$$
N_\beta (x,S):=\bigcup_{L>0} L\cdot 
D _\beta d\left( x,S\right)  
$$
and
$$
T_\beta (x,S):=(N_\beta (x,S))^0. 
$$
Here we adopt the convention that $(\emptyset)^0=X$.

\end{definition}

{\bf Remark 4.1.} Recall that the contingent cone $T_b(x,S)$ and 
the Clarke tangent cone $T_c(x,S)$ of $S$ at $x\in S$ are 
defined by 
$$
T_b(x,S):=\{u: \liminf_{t\to 0^+}\frac{d(x+tu,S)}t =0\}
$$
and 
$$
T_c(x,S):=\{u: \limsup_{t\to 0^+, y\to x}\frac{d(y+tu,S)}t =0\},
$$
respectively.
It is easy to see that $T_c(x,S)\subset T_b(x,S)\subset T_\beta (x,S)$ for any 
bornology $\beta$. Note also that $N_\beta(x,S)$ need not be closed.


We will also need the following equivalent definition of the $g$-normal cone
(cf. \cite[Theorem 1]{bi} and noticing that, by \cite[Definition 4]{bi}
and our definition of the viscosity subderivative, the set $N^k_\beta(S,x)$ of
$\beta$-normal of rank $k$ to $S$ at $x$ is the same as $kD_\beta d(x,S)$).

\begin{definition}
Let $X$ be a $\beta$ smooth Banach space and $S$ a closed subset of $X$.
For $x\in S$, we define the $g$-{\bf normal cone} of $S$ at $x$ to be
$$
N_g(x, S):=cl^* \bigcup_{k=1}^\infty \{
w^*-\lim_{n\to \infty} x_n^*: x_n^*\in kD_\beta d(x_n,S), ~
 x_n\to _S x\}.
$$ 

\end{definition}

It is known that \cite{bi} $T_c(x, S)=(N_g(x, S))^0$.


The next definition is a $\beta$-normal version of the 
finite codimension condition defined in \cite{gi}. In a $\beta$-smooth
space our condition is less arduous than the definition in \cite{gi}.


\begin{definition} We say that $f:X\to Y$ has the 
$\beta$-{\bf finite codimension 
property} with respect to $S$ at $x_0$ if there is a weak-star closed
subspace $V^*\subset X^*$ of finite codimension and constants 
$\varepsilon, c>0$ such that if
\begin{itemize}
\item[1.] $x\in S$ and $\|x-x_0\|<\varepsilon$;
\item[2.] $x^*\in N_\beta(x,S)$;
\item[3.] $\|y^*\|=1, d(y^*, V^*)<\varepsilon$;
\end{itemize}
then 
$$
\|(\nabla f(x_0))^*y^*+x^*\|\ge c.
$$
\end{definition}

The next theorem can be deduced from \cite[Theorem 2.7]{gi}. 
We give a self contained proof by using Theorem 4.1.

\begin{theorem}  Let $X$ be a Banach space with a $\beta$-smooth norm and 
$S$ a subset of   $X$. 
Assume that $S$   is closed, $h$  is strictly differentiable at $x_0$
and has the finite codimension 
property with respect to $S$ at $x_0$. Then
\begin{eqnarray}
cl \nabla h(x_0)T_c (x_0,S)= Y
\label{del1} 
\end{eqnarray}
implies that $h$ is regular
with respect to $S$.
\end{theorem}

{\bf Proof.} In light of Theorem 4.1 
we need only to show that $x_0$ is not an extremal with 
respect to $(h,S)$. Assume on the contrary that $x_0$ is an extremal with 
respect to $(h,S)$. Then, for each $n$, there exist an 
unit vector $y^*_n$ and $y_n\in (x_0+\frac 1n B)\cap S$ such that
\begin{eqnarray}
0\in (\nabla h(x_0))^*y^*_n + LD_\beta d(y_n,S) +\frac 1n B_{X^*}.
\label{e5}
\end{eqnarray}
We may assume that  $y^*_n\to_{w^*}y^*$. Then by the definition of 
the $g$-normal cone
$$
-(\nabla h(x_0))^*y^*\in N_g(x_0,S).
$$
In other words, $y^*$ is perpendicular to $\nabla h(x_0) T_c(x_0,S)$
(because $T_c(x_0,S)=N_g(x_0,S)^0$)
which, together with (\ref{del1}), yields $y^*=0$. Since $X^*=V^*+W^*$
where $W^*$ is a finite dimensional space $y^*_n=v^*_n+w^*_n$
with $v^*_n\in V^*$ and $w^*_n\in W^*$. Since $W^*$ is finite dimensional
and $\|w^*_n\|\le \|y^*_n\|=1$
without loss of generality we may assume that $w_n^*\to w^*$. Since 
$y^*=0$ we have  $w^*=0$. Thus, $d(V^*,y^*_n)\to 0$. Observing that
(\ref{e5})
implies that, for each $n$, there exists an $e_n^*\in \frac 1n B_{X^*}$ such that
$$ 
u^*_n:=-(\nabla h(x_0))^*y^*_n+e_n^*\in LD_\beta d(y_n,S)\subset
N_\beta(y_n,S).
$$ 
When $n$ is 
sufficiently large the finite codimension 
property of $h$ with respect to $S$ at $x_0$ implies that
$$
0<c\le \|(\nabla h(x_0))^*y^*_n +u^*_n\|=\|e_n^*\|\le \frac 1n, \quad \forall n
$$
which is a contradiction. \smiley

{\bf Remark 4.2} While the $\beta$ finite codimensional property is
less arduous than the codimention condition in \cite{gi} in a $\beta$-smooth
Banach space we should note that the result in \cite{gi} was proved without
the smoothness assumption on the underlying space.

Now we turn to a $\beta $-smooth space version of the 
primal condition discussed in \cite{bs}:





\begin{theorem}  Let $X$ and $Y$ be Banach spaces with  $\beta$-smooth norms and 
$S$ a closed subset of   $X$. Assume that $h:X\to Y$ is Lipschitz and strictly
differentiable at $x_0$. Suppose that:

(i) $cl \nabla h(x_0) T_c(x_0,S) =Y$,

(ii) there exists a nonempty compact set $K\subset Y$, and positive
numbers $\alpha, \gamma$ and $ \eta$ such that $\alpha < \gamma$ and
$$
\gamma B_Y\subset cl\{ \nabla  h(x_0) (T_\beta(x,S)\cap B_X)+\alpha B_Y\} + K,
$$
whenever $\|x-x_0\|\le \eta $.

Then $h$ is regular with respect to $S$ at $x_0$.
\end{theorem}

{\bf Proof.} We need only to show that the condition of this
theorem implies that of Theorem 4.2. Suppose conditions (i) and (ii)
are satisfied. By \cite[Lemma 2.1]{b87} we may assume that $K$ is  
finite dimensional. Let $V:=span(K)$ and 
$$
V^*:=\{ x^*\in X^* : x^* \bot V\}.
$$
Then $V^*$ is of finite codimension. Let 
$$
\varepsilon :=\min \{ \eta, \frac{\gamma -\alpha}{2 \sup \{\|k\|: k\in K\}}\}
$$
and $c:= \gamma -\varepsilon \sup \{\|k\|: k\in K\} -\alpha.$
Consider $x, x^*$ and $y^*$ satisfy $1., 2.$ and $3.$ in Definition 4.5.
Let $\xi $ be an arbitrary element of $\gamma B_Y$. Then by 
condition (ii)
$$
\xi = \nabla h(x_0) u +\alpha b +k
$$
where $u\in T_\beta(x,S)\cap B_X, b\in B_Y$ and $k\in K$.
Then 
\begin{eqnarray*}
\langle y^*, -\xi \rangle &=& \langle (\nabla h(x_0))^*y^*, -u \rangle
+ \alpha \langle y^*, -b \rangle +\langle y^*, -k \rangle\\
&=& \langle (\nabla h(x_0))^*y^*+x^*, -u \rangle+\langle x^*, u \rangle 
+ \alpha \langle y^*, -b \rangle +\langle y^*, -k \rangle\\
&\le & \|(\nabla h(x_0))^*y^*+x^*\| + \alpha + 
\varepsilon \sup \{\|k\|: k\in K\}.
\end{eqnarray*}
Taking the supremum on the left hand side leads to
$$
\gamma \le \|(\nabla h(x_0))^*y^*+x^*\| + \alpha + 
\varepsilon \sup \{\|k\|: k\in K\}
$$
or 
$$
c \le \|(\nabla h(x_0))^*y^*+x^*\|.
$$
\smiley



{\bf Remark 4.3} Since $T_c(x,S)\subset T_b(x,S)\subset T_\beta (x,S)$
the condition of this theorem is weaker than that of \cite[Theorem 4.1]{bs}
in $\beta$-smooth spaces. We should note that the results of 
\cite[Theorem 4.1]{bs} are proved without the $\beta$-smooth assumption.
The same remark also applies to Corollaries below. 




As shown in Borwein and Strojwas \cite{bs} conditions
(i) and (ii) are weaker than the sufficient conditions for regularity
given in Borwein \cite{b86} for the cases of (i) $Y$ is a finite dimensional
space, (ii) $S$ is convex and (iii) $S$ is (compactly) epi-Lipschitzian (see 
\cite{bs} for the definition). 
Therefore
we have the following corollaries.


\begin{corollary} \cite{b86} Let $X$ and $Y$ be Banach spaces with  $\beta$-smooth norms and 
$S$ a closed convex subset of   $X$. 
Assume that $h:X\to Y$ is Lipschitz and strictly
differentiable at $x_0\in S$. Suppose that:

$$0\in {\rm core} \nabla h(x_0) (S-x_0).$$


Then $h$ is regular with respect to $S$ at $x_0$.

\end{corollary}

\begin{corollary} \cite{b86} Let $X$  be a Banach space with  $\beta$-smooth 
norms, $Y$ a finite dimensional Banach space and 
$S$ a closed subset of   $X$. 
Assume that $h:X\to Y$ is Lipschitz and strictly
differentiable at $x_0\in S$. Suppose that:

$$ \nabla h(x_0) T_c(x_0,S)=Y.$$


Then $h$ is regular with respect to $S$ at $x_0$.

\end{corollary}


{\bf Remark 4.4} Corollary 4.2 and the following example are helpful
for understanding the nature of the finite codimension property 
of definition 4.5.


{\bf Example.} Let $S:=X:=l_2$ and $h:X\to R^n$ be defined by 
$h((x_i)_{i=1}^\infty)=(x_1,...,x_n)$. For any $x$ and 
any weak star closed subspace $V^*\subset X^*$
with finite codimension, $N_\beta(x,S)=\{0\}$
and we can pick a unit vector $y^*\in V^*$ of the form
$y^*=(0,..,0,y^*_{n+1},...)$. Then $\nabla h(x_0))^*y^*=0$. Therefore,
$h:X\to Y$ does not have the $\beta$-finite codimension 
property with respect to $S$ at $x_0$ for any $x_0\in X$.
Nonetheless, by Corollary 4.2 $h$ is regular at arbitrary $x_0\in X$.

This shows that the  codimension property defined in definition 4.5
is related to both $f$ and $S$ and is by nature a condition
for infinite dimensional spaces. The image space $Y$ of $h$ being of finite
dimension does not guarantee that $h:X\to Y$ has the $\beta$-finite codimension 
property with respect to arbitrary $S$ at $x_0$. 

\begin{corollary} \cite{bs} Let $X$ and $Y$ be Banach spaces with  $\beta$-smooth norms and 
$S$ a compactly epi-Lipschitzian subset of   $X$. 
Assume that $h:X\to Y$ is Lipschitz and strictly
differentiable at $x_0$. Suppose that:

$$ \nabla h(x_0) T_c(x_0,S)=Y.$$


Then $h$ is regular with respect to $S$ at $x_0$.

\end{corollary}



Next we show that the sufficient condition for regularity 
in terms of uniform sleekness of the set $S$ given by Aubin 
and Frankowska \cite{af} is also a direct consequence of Theorem 4.3.
First we introduce the  definition of sleekness and uniform sleekness.





 

\begin{definition} We say $S$ is $\beta $-{\bf sleek} at $x$ if
$$
\liminf_{y\to x} T_\beta (y,S)\supset T_b (x,S).
$$

We say $S$ is {\bf uniformly $\beta $-sleek} at $x$ if, for $u\in T_b
(x,S)\cap B$,

$$
\lim_{y\rightarrow x}d(u,T_\beta (y,S))=0 
$$

uniformly.

\end{definition}

{\bf Remark 4.5} (a.) By Remark 4.1 $\beta $-sleekness is weaker than sleekness
defined in \cite{af}.

(b.) Borwein and Ioffe \cite{bi} have shown that
$$
T_c (x,S)\supset \liminf_{y\to x} T_\beta (y,S)
$$
where $T_c (x,S)$ signifies the Clarke tangent cone. It is well known
that $T_c(x,S)\subset T_b (x,S)$.
Thus, if $S$ is $\beta $-sleek at $x$ then $T_c(x,S)= T_b (x,S)$
(i.e. $S$ is $b$ tangentially regular at $x$).



\begin{corollary}   Let $X$ and $Y$ be  Banach spaces with  $\beta$-smooth norms 
and  $S$ a subset of   $X$. 
Assume that $S$   is closed and uniformly $\beta $-sleek at 
$x_0$ and $h:X\to Y$  is strictly differentiable at $x_0\in S$. Then
\begin{eqnarray}
cl \nabla h(x_0)T_c (x_0,S)=Y 
\end{eqnarray}
implies that $h$ is regular
with respect to $S$.
\end{corollary}

{\bf Proof. } We need only to show that $h$ and $ S$  satisfy condition 
(ii) in Theorem 4.2.  By the Robinson-Ursescu  Theorem (see \cite{af})
there exists a 
$\gamma>0$ such that 
$$
\gamma B_Y\subset \nabla h(x_0)(T_c(x_0, S)\cap \frac 12 B_X).
$$
Take an $\varepsilon< \frac 12$ such that $\alpha:= \varepsilon \|h(x_0)\|< \gamma$.
The uniform sleekness of $S$ at $x_0$ implies that there exists an
$\eta $ such that $\|x-x_0\|\le \eta$ implies that 
$$
T_c(x_0, S)\cap \frac 12 B_X \subset T_\beta(x_0, S)\cap B_X + \varepsilon B_X.
$$ 
Therefore 
$$
\gamma B_Y\subset \nabla h(x_0)(T_c(x_0, S)\cap \frac 12 B_X)
\subset  cl\{ \nabla  h(x_0) (T_\beta(x,S)\cap B_X)+\alpha B_Y\}
$$
as was to be shown. \smiley



Recently Az\' e and Chou \cite{ac} derived a primal sufficient condition in terms of 
equi-circatangent cones. It turns out this condition can also be deduced
directly from Theorem 4.3. 

\begin{definition} \cite{ac}
A cone $K\subset X$ is said to be {\bf equi-circatangent} to $S$ at $x\in S$
if 
$$
\lim_{t\to 0^+,y\to _S x} e(C\cap B_X, t^{-1}(S-y))=0
$$
where $e(A,S):=\inf\{\lambda >0: A\in S+\lambda B_X\}$.
\end{definition}


The primal condition discussed in \cite{ac} 
(reduced to a single-valued function form)
can be stated as:

\begin{corollary} Let $X$ and $Y$ be Banach spaces with  $\beta$-smooth norms and 
$S$ a closed convex subset of   $X$. 
Assume that $h:X\to Y$ is Lipschitz and strictly
differentiable at $x_0$. Let $K$ be an equi-circatangent cone of $S$ at $x_0$.
Suppose that 
$$
\nabla h(x_0) K=Y.
$$
Then $h$ is regular with respect to $S$ at $x_0$. 


\end{corollary}

{\bf Proof.} We show that the conditions of this corollary imply that of 
Theorem 4.3. Condition (i) is obvious. To show condition (ii), invoking
Robinson-Ursescu Theorem there exists 
a positive constant $\gamma $ such that 
$$
\gamma B_X\subset \nabla h(x_0) (K\cap \frac 12 B_X).
$$ 
By the definition of $K$ there exists a $\eta$ such that,
for all $t\in (0,\eta)$ and $y\in (x_0+\eta B_X)\cap S$,  
$$
K\cap B_X  \subset \frac{S-x}t + \varepsilon B_X
$$
where $\varepsilon := \min (\frac \gamma {2 \|\nabla h(x_0)\|}, \frac 12)$.
Then 
$$
K\cap \frac 12 B_X \subset T_\beta (x,S)\cap B_X + \varepsilon B_X.
$$
Therefore 
\begin{eqnarray*}
\gamma B_Y &\subset & \nabla h(x_0) (K\cap \frac 12 B_X )\\
&\subset & \nabla h(x_0) (T_\beta (x,S)\cap  B_X ) + 
\varepsilon \nabla h(x_0) B_X \\
&\subset & \nabla h(x_0) (T_\beta (x,S)\cap  B_X ) + 
\frac{\gamma}2 B_X.
\end{eqnarray*}
\smiley 
 

Let us turn finally to discuss regularity for multifunctions.

\begin{definition}
Let $X$ and $Y$ be Banach spaces and $F$ a multifunction from $X$ to $Y$.
We say that $F$  is {\bf regular} at $(x_0,y_0)\in Gr(F)$ if there exist
positive constants
$\varepsilon$ and $K$ such that, for all $x\in x_0+\varepsilon B$ and 
$y\in y_0+\varepsilon B$,
$$
d(x, F^{-1}(y))\le Kd(y, F(x)).
$$
\end{definition}

The following lemma will enable us to 
connect the regularity of a multifunction to 
that of a single-valued function and derive sufficient
conditions for a multifunction to be regular through various of the 
results discussed before.

\begin{lemma} Let $p:X\times Y\to Y$ be defined by $p(x,y)=y$. 
Then a multifunction $F$  from $X$ to $Y$ is regular at $(x_0,y_0)\in Gr(F)$ 
as soon as $p$ is regular with respect to $Gr(F)$ at $(x_0,y_0)$.
\end{lemma}

{\bf Proof.} Since $p$ is regular with respect to $Gr(F)$ at $(x_0,y_0)$
there exist positive constants $\varepsilon$ and $K$ such that, 
for all $(x,z)\in ((x_0,y_0)+\varepsilon B)\cap Gr(F)$,
$$
d((x,z), Gr(F)\cap p^{-1}(y))\le K \|z-y\|.
$$
Taking a larger $K$ if necessary we may assume that $K>1$.
Taking the infimum with respect to $z\in F(x)$ yields
\begin{eqnarray*}
Kd(y,F(x)) &\ge& \inf_{z\in F(x)} d((x,z), Gr(F)\cap p^{-1}(y))\\
 &=& \inf_{z\in F(x)} \inf \{\|x-x'\|+\|z-y'\| :
x'\in F^{-1}(y'), y'=y\} \\
&=& \inf_{z\in F(x)} \{d(x, F^{-1}(y)) +\|z-y\| \} \\
&=& d(x, F^{-1}(y)) +d(y, F(x)).
\end{eqnarray*}
Thus, 
$$
d(x, F^{-1}(y)) \le (K-1)d(y, F(x)).
$$
\smiley

Using this lemma and the various sufficient conditions we discussed before
will lead to sufficient conditions for a multifunction to be regular. 
For example, combining Lemma 4.1 with Theorem 4.3 we get:


\begin{theorem}  Let $X$ and $Y$ be Banach spaces with  $\beta$-smooth norms and 
$F$ a multifunction from   $X$ to $Y$.  Suppose that:

(i) $cl Im (D_c F(x_0,y_0) )=Y$,

(ii) there exists a nonempty compact set $K\subset Y$, and positive
numbers $\alpha, \gamma$ and $ \eta$ such that $\alpha < \gamma$ and
$$
\gamma B_Y\subset cl\{ Im(D_cF(x,y))\cap B_X)+\alpha B_Y\} + K,
$$
whenever $\|x-x_0\|\le \eta $ and $\|y-y_0\|\le \eta $.

Then $F$ is regular at $(x_0,y_0)$.
\end{theorem}


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\end{document}





