WSAA 14
Marián J. Fabián; David Preiss
On the Clarke’s generalized jacobian
In: Zdeněk Frolík and Vladimír Souček and Marián J. Fabián (eds.): Proceedings of the 14th Winter
School on Abstract Analysis. Circolo Matematico di Palermo, Palermo, 1987. Rendiconti del Circolo
Matematico di Palermo, Serie II, Supplemento No. 14. pp. [305]--307.
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ON.THE CLARKE S GENERALIZED JACOBIAN
M. Fabian and D. Preiss
n
k
Let f: R — * R
be a locally Lipschitz function defined on
an open ball B(x,r) CZ Rn centered at x and of radius r > O .
According to the Rademacher's theorem [5] there exists a set
E CZ R n of Lebesgue measure zero such that the Gateaux derivative
Df(y) exists whenever ycB(x,r)\E . Using this fact Clarke [2]
introduced the generalized Jacobian 3f(x) as the closed convex
hull of all possible limits
lim Df(y. ), where y. €B(x,r)\E .
y±-+y
i
i
o
n
Similarly, if E
is replaced by a null set E C R
containing
E Q , one can define 3-gfU). Thus 9 £ f(x) = 3f(x). For k = 1
Clarke [l] showed that 3gf(x) = df(x) for any null set E containing E
and asked in [2] if the equality remains true for
k > 1. In what follows we answer this question affirmatively by
showing
5
Theorem.
£^-<) = <>f (x)
for all k and for all null sets E including
E .
m
Proof. All the spaces R
are considered with the Euclidean
norm |f.|(. The symbol <•, .> denotes the usual.inner product.
The space of linear mappings from R n to R
as well as its dual
nk
will be identified with R . Since clearly 3 £ f(x)C 3f(x), it
remains to prove the converse. By contradiction, let us assume
that this inclusion is proper. Then there exist a functional A in
R nk and U e R such that
sup [<AfL>:
L€ Pgf (x)}
^ ^ < sup {VA,L>:
The definition of 5^f (x) yields an
<A,Df(y)) < U
whenever
Indeed, otherwise we could find
€> 0
L£ <?f(x)] .
such that
y€B(x,£)\E .
y. 6 B(x,l/i) \ E with
This paper is in final form and no version of it will be
submitted for publication elsewhere.
306
M. FABIAN, D. PREISS
<A,Df(y.)V "gr oL
and hence there would exist an L6 A-f (x)
such
that ^A,L)?.^, which is impossible. Also, according to the definition of <?f(x), there is y = (y lf ... f y n ) <? B(x, ff) \ E Q with
<* -<<A,Df(y)> .
By joining the last two inequalities we get
< ot < <A,Df (y ))
<A,Df (y)>
Denoting
whenever y c B(x,
A = ( a ^ ) , i = l,...,k,
+ ••• t a k j f ^i
J = lf.|ni
we can
j = l,...,n,
e) \ E .
and
gj
= a-.^ +
write the above inequality in
the form
ZgAy)
n
21s — j b —
n
dg'Ay)
< * ^ Z J —%T:—
whenever
yeB(x,£)\E .
Let C(s) be the n-dimensional cube with apices (y-.±s,...,y is).
Whenever s > 0 is so small that C(s) C B ( x , e ) , the above inequality holds almost everywhere in C(s) and, consequently,
n
C
JgAy)
y <(2s)
•
h !> ^ r ^-* "
n
Let us denote
£ ( s ) = sup f l g ^ y ) - g , ( y ) - D g . ( y ) ( y - y . ) | :
J
J
J
ll J
and
c (s)
j
J
"*
& ^T •
max ( y j - y j <%. s'J ,
i=l,....,n
= £<y.i.....yj-i»yj+i»'"»yn):
(yi,...,yj_i»yj»yj+i»...»yn)6C(s)} »
Using the Fubini theorem, we get
*
? ^(y)
' C(Js)-%-dyi,,,dy"=
=
n -Pg.(y)
<2s)
n
Z_»
C ( s ) <?= ±1
J
J =
-»...»n»
"
s
J -
•
•
^ gJi ( y 1 , . . . , y i _ 1 , yJ i + « s , yJ i + 1 , . . . , y
-
^°
) x
--
c •
xdy1...dyj_1dyJ+1...dyn =
=
c^s)U'§i < S [ S j ( y i , , , , , y j - 1 , y j + '' S , y j + 1 , , , , , y n ) " Sj(y) "
J
V
_
- 1
y
,
y
y )
- Dg J (y)(y 1 -yi»....yj_i-y < 3..i' » j+i" j+i' -*' n- n J
tfs y
X_y 1 ...dy J . 1 dy- J+1 ...dy n 3.
Hence (x) implies
r. 1 *
^UB)*1"1^)
+ 2s
^g4(y)
^i(y)\
-ay"-/*
n 1
* 2s - - J - - (2s) n
J
.
ON THE CLARKE'S GENERALIZED JACOBIAN
307
(2s) n £ — J (2s) n 21 ^,(JB)/8 <c(2s) n * < ( 2 s ) n £
—J—y
j=l
> j
j=l J
J=l <^j
if s > 0 is sufficiently small. Let us note that <T.(s)/s
*0
as s i O since Gateaux and Fr£chet differentiability in finitedimensional spaces coincide for Lipschitz functions. Thus, dividing the above inequality by (2s) n and letting s go to zero, we
obtain, a wrong inequality. This contradiction finishes the proof.
Remark 1. For f and x as above Pourciau [4] considered
a generalized Jacobian which in our notation is equal to ^™ f(x)
x
with
E
l ~ E 0 U fy ^B(x,r) \ E Q : y is not a Lebesgue point of Dfj .
As f is locally Lipschitz,
*v f(x) = <?f(x).
E-. is a null set. Hence by Theorem
Remark 2. The reader probably noticed that the above proof
is actually based on the Gauss - Green theorem. In fact, this theorem shows a "Denjoy property for derivatives of mappings between
R n and R k " suggested by [2, Remark 5*1.
REFERENCES
1. CLARKE F.H. "Geneгalized gгadients and applications", Tгans.
Amer. Math. S o c , 205(1975), 247-262.
2. CLARKE F.H. "On the inverse function theorem", Pacific J. Math.,
64(1976), 97-102.
3. HIRIART-URRUTY J.B. "Chaгacterizations of the plenary hull of
the geneгalized Jacobian matrix", Math. Pгog. Study, 17(1982),
1-12.
4. POURCIAU B.H. "Analysis and optimization of Lipschitz continuous
mappings", J. Opt. Theoгy Appl., 22(1977), 311-351.
5. STEIN E.M. "Singulaг integгals and diffeгentiability properties
of functions", Pгinceton Univeгsity Press, New Jersey 1970.
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