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Metric regularity, openness and Lipschitzian behavior of multifunctions
Nonlinear Analysis. Theory. .Merhods & Applicarions. Printed m Great Britam.
METRIC
REGULARITY,
Vol. 13. No. 6. pp. 629443,
1989. 0
OPENNESS AND LIPSCHITZIAN OF MULTIFUNCTIONS JEAN-PAUL
0362-%6X/89 S3.00+ .OO 1989 Pcrgamon Press plc
BEHAVIOR
PENOT
Faculte des Sciences, Avenue de I’Universitk 6400 Pau, France (Received
I1 December
1987; received for publication
Key words and phruses: Gage, inverse mapping,
regular multifunction,
19 April 1988)
Lipschitzian multifunction,
modulus, open mapping,
uniform continuity.
1. INTRODUCTION
of regular multifunction has emerged out of the pioneering work of Ljusternik and Sobolev 1201 through the efforts of several workers. In its initial form it was conceived as a mean for solving equalities or inequalities under more or less classical surjectivity assumptions on some approximants [9, 10, 14,21,22,24,25]. Robinson [30] stressed the key estimate on the distance to the solution set which is characteristic of what is called the (metric) regularity property we formally define in the following way among several slight variants. THE NOTION
1.1. A multifunction F: X =t Y between two metric spaces is said to be metrically regular around (x0, y,,) with x0 E X, y, E F(x,,) if there exist positive numbers CY,8, E such that
Definition
d(x, F_‘(Y)) whenever x E B(x,,
a),
5 cW(x),
(1.1)
Y)
/I) with d(F(x), y) I E.
y E B(y,,
Here we denote by B(z, r) the open ball with center z and radius r, and for a subset A of a metric space (M, d) and b E M we set d(A, b) = d(b, A) = inf[d(a, b) : a E A). The special case in which X = IR”, Y = W and F is given by F(X) = [YE lR’:Yi
for some
m E
i =
=f;(x)
1, . . . . m,
[ 1, p] has received a particular
d(x,F-‘(~))
Yj Z&(X)
attention.
with t+ = max(t, 0). In particular, penalization term
m
i
j=m+l
+
1, . . . . p)
i
C&(X) - Y$’
j=m+l
for y = 0, the distance d(x, F-‘(O))
c ii IfiCx)l+ C i=l
=
The estimate (1.1) can then be written
5 c f IL(X) - YiI + c i=l
j
is estimated by the
fi(x)+-
This connection with penalization theory has been deeply used by Clarke IS] for treating mathematical programming problems and optimal control problems. 629
630
J.-P. PENOT
A great impetus has been given to the concept of regularity by Ioffe [15-171 who showed that the Ekeland’s variational principle could be used to give verifiable criteria for regularity (see also [3, 41). It has been shown [4, 12, 24, 251, that regularity is a useful property for computing tangent cones to sets given as inverse images or defined be equalities and inequalities. Simultaneously the infinitesimal properties ensuring regularity have been refined [4, 12, 16, 17, 19, 27, 331. Lipschitzian properties of multifunctions have been recognized to be a useful tool for several topics in nonlinear analysis as fixed point theory [23, 251, differential inclusions [2, 5, 131 implicit function theorems [l], mathematical programming [6, 71. The following result shows that convexity entails a Lipschitzian behavior. PROPOSITION 1.2. ([18, lemma III Bl], (25, theorem 5.11). Let Xand Y be two normed vector spaces with unit balls Bx and BY respectively. Let F: X 3 Y be a multifunction with convex graph such that for some p > 0 and some open subset U of X one has F(x) fl pBy # @ for each x E U. Then for each x0 E U there exist CY> 0, m > 0, p > 0 such that for each r E IR,
F(x,) n rB, c F(x,) + (mr + P)&,
x2)&
for any x,, x2 in x0 + aBx.
This kind of behavior is a special case of what has been called a sub-Lipschitzian behavior by Rockafellar in his thorough study [31]. A more common property of nonconvex multifunctions has been discovered by Aubin [l] in connection with inverse mapping theorems. Definition 1.3. A multifunction F: X =t Y between two metric spaces is said to be pseudoLipschitzian around (x0, yo), where y. E F(x,), if there exist neighborhoods U of x0, Vof y. and c E IR, such that &Y, 3F(xd)
-( c4q
for any x,, x2 E U, y, E VII F(x,).
9~2)
The word regularity has been used in a different sense by Robinson (291 for studying openness properties of convex multifunctions. The precise property he deals with can be defined as follows. Definition 1.4. A multifunction F: X =I Y is said to be open at a linear rate around (x0, yo), with x0 E X, y. E F(x,), if there exist positive numbers (Y,/I, p such that for any x E B(x,, a), y E B(y,, P), r E [0, p] with y E F(x) one has NY,
cd
C F(B(x, r)).
The classical open mapping theorem and its extension to convex multifunctions as in [29] are instances in which such a property occurs. The purpose of the present paper is to show that the three preceding concepts are equivalent. THEOREM 1.5. Let F: X s Y be a multifunction and let (x0, y,,) E X x Y with y. E F(x,). Then the following assertions are equivalent: (a) F is metrically regular around (x0, yo); (b) F-’ is pseudo-Lipschitzian around (yo, x0); (c) F is open around (x0, yo) at a linear rate.
Metric regularity, openness and Lipschitzian behavior
631
This result has been announced in [26, 271. A simplified form of the implication (a) * (c) is contained in [4, theorem 5.1) with several verifiable criteria and striking applications. Here this equivalence is extended to properties which go beyond pseudo-Lipschitzian behavior, linear openness and metric regularity but still retain some uniformity properties. In particular, in Section 4 we introduce the notion of uniform continuity of a multifunction; Section 3 deals with uniform openness. Uniform regularity is considered in Section 5. The extension of theorem 1.5 to such properties does not require much more effort; some useful material about generalized inverses of one variable functions adapted from [28] is given in Section 2. This extension is justified by the fact that in some concrete situations the Lipschitz behavior has to be replaced by a less demanding property as an Holderian behavior. It is known for instance that the Lipschitz property of a moving closed convex subset of a Hilbert space is lost by the associated projection of a given point but that an Holderian behavior still remains. Miscellaneous related properties are gathered in Section 6, completing the studies made in [I, 4, 311. 2. PRELIMINARIES:
GAGES
AND MODULUS
In the sequel we denote by P the set of positive real numbers and we set IR, = P U 101, the role of IP could be replaced by any open interval of IR. Let N be the set of nondecreasing mappings from R, into IR, and let G be the set of gages i.e. the set of nondecreasing mappings from P into IP U (+ 00) taking at least one finite value. Any element g of G is considered as an element of N, extending g to lR+ by setting g(0) = 0, g(+ m) = supg(P). We denote by M the set of modulus i.e. the set of m E N such that inf m(lP) = 0. Obviously these definitions depend only on the germ at 0 of the mappings: if f, g E N have the same germ at 0 (i.e. iffand g coincide on some neighborhood of 0 in R,) then f E G (resp. M) iff g E G (resp. M). For f: R+ -+ rii, we set R+ = IR,U I+ a]. In most of our preliminaries
E(f)
= ((r, s) E P x IP:j-(r) 5 s),
H(f)
= ((r, s) E P x P : s 5 f(r)].
We denote by E,(f) (resp. H,(f)) the sets obtained by replacing the inequality by a strict inequality in what precedes. These sets correspond to the epigraph (resp. hypograph) off and its strict analogue when one uses an increasing bijection for identifying IP with IF?. We denote by g the family of subsets E of P x IP such that (r, t) E E whenever t 1 s for some s E IP with (r, s) E E; we call g the family of pseudo-epigraphs; the family of pseudohypographs is the family X of subsets H of IP x IP such that (r, t) E H whenever (r, s) E H, t TVIPwith t I s. We denote by &+the family of E E $ such that for any (r, s) E E and any 4 E iP with 4 5 r (q, s) belongs to E. Similarly, X, is the family of H E 3.2such that for any (r, s) E H and any t E P with t 1 r one has (t, s) E H. Obviously, for any f E RT (resp. N) E(f) and E,(f) belong to & (resp. &+) while H(f) and H,(f) belong to X (resp. X,). Conversely, for any E E & (resp. &+) there exists a unique f E I!?:(resp. N) such that E,(f) c E c E(f); we denote it by (Pi and we observe that it is given by v)~(T) = infls E IP: (r, s) E E),
J.-P. PENOT
632
with the convention
inf 0 = + co, sup 0 = 0. Moreover pE is the greatest f E rii”,such that
E c E(f): g 5
VE
e+ E
(2.1)
C E(g).
Similarly, with any H E 3C (resp. X,) we associate pH E Ri”,(resp. N): pH(r) = supls E P: (r,s) E H), so that HS(pH) C H C H(pH)
and for any g E WYwe have g 2 pH e H c H(g).
In the sequel we always identify a multifunction A c IP x IP we set
(2.2)
(or relation) with its graph. In particular for
A-’ = ((s, r): (r,s) E A) and A-’ is the inverse relation of A. It is obvious, but important for what follows, to observe that a subset A of P2 belongs to 8, iff A-’ belongs to X, . Applying this observation to epigraphs and hypographs we get the following refinements of some concepts of inverse functions introduced in [28]. Definitions 2.1. (a) An element g E Nis said to be a semi-epi-inverse inverse off) if E(g) C H(f)-’ (resp. E,(g) c H(f)-‘).
(b) It is said to be a semi-hypo-inverse
off E N (resp. a quasi-epi-
off (resp. a quasi-hypo-inverse
off)
if H(g) c E(f)-’
(resp. H,(g) c E(f )-‘). (c) It is a quasi-inverse off if it is both a quasi-epi-inverse
off and a quasi-hypo-inverse off. (d) It is an epi-inverse off if it is a semi-epi-inverse off and a quasi-inverse off: H,(f)-’ C E(g) C H(f)-‘. It is an hypo-inverse off if it is a semi-hypo-inverse off and a quasi-inverse of J E,(f)-’ c H(g) c E(f)-‘. We observe that g is a quasi-epi-inverse off iff f is a quasi-epi-inverse of g. A similar equivalence holds for quasi-hypo-inverses but is not true for semi-epi-inverses. The following lemma shows that the situation is much simplified when some continuity property is at hand. LEMMA 2.2. (a) If f E Nis U.S.C. then g E Iv is a semi-epi-inverse
off. (b) If f E N is 1.s.c. then g E N is a semi-hypo-inverse
off iff g is a quasi-epi-inverse
off iff g is a quasi-hypo-inverse
off.
Proof. When f is U.S.C.H(f) is closed and H(f)-’ is closed too. Then, as for any g E IV, E(g) is the vertical closure of E,(g), E(g) is contained in the closure of E,(g), so that E,(g) C H(f)-’ iff E(g) c H(f)-‘. The proof of the second assertion is similar. I
One disposes of two canonical generalized inverses off E N, obtained by using E = H(f)-’ or H,(f)-‘, H = E(f)-’ or E,(f)-’ in the process we described above: f’
= pH*-,
= lpEdJ-J-‘,
fh = $Q&-)_, = @(fJ-‘:
f e(.3) = inflr E IP: s 5 f(r)]
= suplt E IP:f(t)
< s],
f h(s) = inflr E IP: s
= sup(t E P: f(t)
I s).
Metric regularity, openness and Lipschitzian behavior
These mappings have extremal properties and can serve for a characterization Moreover they have regularity properties whatever f E N is.
633
of quasi-inverses.
LEMMA 2.3. For eachf E N,fe is lower semicontinuous (1.s.c.) and f” is upper semicontinuous (u.s.c.). In fact H,(fe) = int E(f)-’ = int E,(f)-’ and E,(fh) = int H(f)-’ = int H,(f)-‘.
Proof. Let (s, r) E H,(f): there exists t E IP with t > r, f(t) < s so that IO, t[ x If(t), + co[ is a neighborhood of (r, S) contained in E,(f). Conversely if (s, r) E int E(f)-’ and if (s’, r’) belongs to some neighborhood of (s, r) contained in E(f)-’ and is such that s’ < S, r’ > r we have f’(s) 1 r’ > r hence (.s, r) E H,cf’). The proof of the assertion with fh is similar. H
Let us give a characterization
of quasi-inverses off E N.
PROPOSITION2.4. (a) For any f, g in N, g is a quasi-epi-inverse off iff g 5 f e iff f 1 g’. (b) For any f, g in N, g is a quasi-hypo-inverse off iff g I fh iff f 5 gh. (c) For any f, g in N, g is a quasi-inverse off ifff’ s g I fh. Proof. (a) Asfe = (Pi with H = E,(f)-‘, (2.2) ensures that g 2 f’ iff E,(f)-’ C H(g) iff E,(g) C H(f)-‘. As f and g have symmetric roles g 2 f’ iff f 1 ge. (b) The proof is similar: as f” = pE with E = Hs(f)-I, (2.1) ensures that g I f" iff H,( f )-’ C E(g) iff H,(g) C E( f )-I; again f and g can be interchanged. Assertion (c) is a consequence of assertions (a) and (b). n
COROLLARY 2.5. For any f E N, f’ and fh are quasi-inverses off; moreover f is an hypo-inverse off’ and an epi-inverse off h. Proof. The first assertion is a special case of proposition 2.4 (c). The second one follows from the fact that f’ (resp. f*) is I.s.c. (resp. u.s.c.) so that f is not only a quasi-hypo-inverse (resp. quasi-epi-inverse) of f’ (resp. fh) but a semi-hypo-inverse (resp. semi-epi-inverse) of f (resp. fh). n
We are now in a position to characterize
semi-inverses.
PROPOSITION2.6. (a) For any f, g in N, g is a semi-epi-inverse off iff f 2 gh. (b) For any f, g in N, g is a semi-hypo-inverse off iff f I ge. Proof. Let us suppose g E N is a semi-epi-inverse of f. Then E(g) C H(f)-‘. Now as H,(gh) C E(g)-’ we get HS(gh) C H(f), hence gh of. Conversely, suppose that f, g in N are such that gh I f. Then as g is an epi-inverse of gh by corollary 2.5 we have E(g) c H(gh)-’ c H(f)-’ and g is a semi-epi-inverse off. The proof of the second assertion is similar. W
In [28] the role of quasi-inverses in the regularization process of a real-valued function was pointed out. Here we focus our attention on the interplay of the notions of generalized inverses with the concepts of gage and modulus. Let us first note the following elementary facts (see [28] for the first assertions).
J.-P. PENO~
634
LEM%~A 2.7. For any f E N the 1.s.c. hull fof f and the U.S.C. hullfof J(r) = suplf(q): f(r) = inf(f(s): If f E G then fand
f are given by
4 E p, 4 < r1, s E ip, s > r].
f^ belong to G. If f E M then fand
j; belong to M.
PROPOSITION 2.8. Let f and g be elements of N taking at least one finite value. (a) If g is a quasi-epi-inverse off and if f E M then g E G.
(b) If f is a quasi-hypo-inverse of g and if g E G then f E M. (c) For each g E G one has ge E M, gh E M. (d) For each f E M one has f’ E G, f hE G.
Proof. (a) By assumption we have E,(g) c H(f)-‘. If g 4 G there exists s E IP with g(s) = 0 so that {s] x [P C N(f)-‘: for each r E IP we have s s f(r), hence inff(iP) 2 s > 0 and f $ M. (b) Let us suppose H,(f) C E(g)-’ and g E G. Then for any s E IP we have g(s) > 0 so that we can find r E Ip with (s, r) E H,(g) hence (r, s) E E(f): f(r) 5 s. This shows that inff(lP) = 0 and f E M. By proposition 2.4 assertions (c) and (d) are special cases of assertions (b) and (a). H Remark 2.9. It is not true that if f E M and if g is a quasi-hypo-inverse instance if f 3 0, g = 0 is quasi-hypo-inverse off and g B G.
off then g E G. For
3. UNIFORM OPENNESS denoted by d for each space. Most topological vector spaces, of what follows could be extended to uniform spaces, in particular using a family of extended semi-distances but we refrain to do so for Ithe sake of simplicity. The
In the sequel W, X, Y are metric spaces, with a distance
use of closed balls instead of open balls would necessitate some changes which are left to the reader; in general open balls seem to be more convenient except in the case the closed balls are compact. The family of neighborhoods of a point x E X is denoted by 92(x). The following definition brings some uniformization in the notion of open multifunction. Definition 3.1. A multifunction F: X =t Y is said to be uniformly open relatively to subsets U of X, V of Y if for any r E IP there exists $ E 1psuch that for any (u, u) E F n (U x V) one has WV, s) c F(Hu, r)). A gage y E G such that Ku, v(r)) C F(B(u, r))
for any r E IP fl y-‘(Ip), (u, u) E Fn (U x V)
(3.1)
is said to be a gage of uniform openness of F relatively to (I and V, then F is said to be yuniformly open relatively to U and V. Given (x0, yo) E X x Y, F is said to be y-uniformly open around (x,, , yO) if there exists U E %(x,,), V E 3t(y,) such that (3.1) holds. A related notion is the concept of modulus of uniform openness. An element p of the set M of modulus is said to be a modulus of uniform openness of F relatively to I/ and V if B(u, s) C 0&u,
p(s))
for any s E IP fl p-‘(fP), (u, u) E F fl (U x V).
(3.2)
Metric regularity, openness and Lipschitzian behavior
635
For the sake of brevity we often omit the mention “relatively to U and I/” in the sequel when there is no risk of confusion. 3.2. If y is a gage of uniform openness of F then any semi-epi-inverse p of y is a modulus of uniform openness of F. Conversely, if P is a modulus of uniform openness of F then any semi-hypo-inverse y of p is a gage of uniform openness of F. In particular, given an U.S.C. y E G (resp. a I.s.c. ,U E M) we take p = yh (resp. y = p’) above.
PROPOSITION
Proof.
Let us set A = ((r, s) E [P X IP: V (u, II) E (u
X v) I-7 F
B(u, s) C F(B(u, r))).
Then y E G (resp. P E M) is a gage (resp. modulus) of uniform openness of F (relatively to U and V) iff H(y) C A (resp. E(p)-’ C A). The inclusion E(p)-’ C H(y) (resp. H(y) C E(p)-‘) which express the assumption of the first (resp. second) assertion then clearly implies its conclusion, taking proposition 2.8 into account. n Remark. As we are using open balls, we observe that in fact any quasi-hypo-inverse y of a modulus of uniform openness of F is a gage of uniform openness of F if it is a gage: if for any r, s in IP with s < y(r) and any (u, IJ) E (U x V) n F we have B(v, s) C F(B(u, r)), then we also have B(u, y(r)) c F(B(u, r)) for any r E P fl y-‘(P), (u, V) E (I/ x V) n F. n
It follows from this remark and propositions 2.4, 2.8 and 3.2 that for fl E Mb is a modulus of uniform openness of F iff ph is a gage of uniform openness of F. Example 3.3. Suppose X and Y are Banach spaces and F: X + Y is a surjective continuous linear mapping. Then F is uniformly open relatively to X and Y with a linear rate: one can find c E IP such that o given by w(t) = ct, is a modulus of uniform openness and y given by y(r) = c-rr is a gage of uniform openness of F. However, in most cases as the one given in the following result it is not easy to determine explicitly a gage or a modulus of openness. PROPOSITION 3.4. Let U and V be compact subsets of X and Y respectively and let F: X =I Y be a multifunction such that F n (U x V) is closed. Then if Fis open at each (x, y) E F fl (I/ x V) it is uniformly open relatively to U and V.
Proof. Suppose the contrary. Then we can find r E IP, a sequence (s,) C IP with limit 0 and a sequence ((x,, y,)) E F fl (U x V) such that B(y,, s,) qJ F(B(x,, r)). Without loss of generality we may assume ((x,, , y,)) has a limit (x, y) in U x V and in fact in F n (U x V) as this is a closed subset of U x V. As F is open at (x, y), for any q E Ip with q < r we can find p E Tpsuch that B(y, p) c F(B(x, 4)). Now for n large enough we have B(y,, s,) C B(y, p) and B(x, q) C B(x,, r), hence B(y,, s,) c F(B(x, q)) C F(B(x,, , r)), a contradiction. n
636
J.-P. PENOT
Slightly generalizing definition 2.1 one says that F: X 2 Y is y-uniformly a subset C of X x Y if 0,
W)) c WW,
r))
open relatively to
for each r E ip fl y-‘(IP) and each (x,y) E Fn C.
Then one gets the following obvious composition
(3.3)
result.
PROPOSITION3.5. Suppose F: X 3 Y is y-uniformly open relatively to C C X x Y and G: Y =t Z is &uniformly open relatively to D C Y x Z. Then H = G 0 F is 6 0 y-uniformly open relatively to
4. UNIFORM CONTINUITY AND UNIFORM PSEUDO-CONTINUITY The following definition multifunction.
gives a uniform version of the notion of lower continuity
of a
Definition 4.1. A multifunction E: W 3 X between two metric spaces is said to be uniformly continuous relatively to a subset Z of W x X if for any r E IP there exists s E IP such that for any (w, X) E E fl Z and any y E B(w, s) one has E(y) n B(x, r) z 0. Usually Z is taken to be a product definition.
V x U. A special case is contained
(4.1) in the following
Definifion 4.2. A multifunction E: W 3 X is said to be pseudo-uniformly continuous (P.u.c.) around ( wO,x0) E E if E is uniformly continuous relatively to some neighborhood Z of (w,, x0) in WxX. A gage y such that E(y) n B(x, r) f 0
for any r E IP fl y-‘([P), (w, x) E E fl Z, y E B(w, v(r))
(4.2)
is said to be a gage of uniform continuity of E relatively to Z (resp. around (w,, , x0) if Z is taken as in definition 4.2). Let us observe that if E is uniformly continuous relatively to Z there exists a greatest gage of uniform continuity relatively to Z given by Y&) = sup(s E II’: v (w, x) E E fI Z, v y E B(w, s) E(y) f3 B(x, r) # @ 1. A modulus ,u such that E(y) n B(x, p(s)) # 0
for any s E IP n p”-‘(IP), (w, x) E E fl Z, y E B(w, s)
(4.3)
is said to be a modulus of uniform continuity of E relatively to Z. The links of the preceding notion with the previous concepts of uniform openness are rather obvious. PROPOSITION4.3. A multifunction E: W 3 X is uniformly continuous relatively to Z C W x X iff F = E-‘: X =t W is uniformly open relatively to Z-‘. Moreover y E G (resp. ,u E M) is a gage (resp. a modulus) of uniform continuity of E relatively to Z iff it is a gage (resp. a modulus) of uniform openness of F relatively to Z-‘.
Metric
Proof. coincide:
regularity,
Since E(y) (7 B(u, r) f 0
openness
is equivalent
A = {(r, S) E Ip2: v (u, u) E F B = ((r, s) E P2:
and Lipschitzian
v (v, u) E
n z-’
E n Z,
637
behavior
to y E E-‘(B(u,
r)) the following
sets
B(u,s)c E-‘(B(u, f))), v y E B(u, s)
E(y) n B(u, r) # 0 1.
Since y E G (resp. p E M) is a gage (resp. a modulus) of uniform continuity of E iff its graph is a subset of B (resp. B-‘) and since y (resp. p) is a gage (resp. a modulus) of uniform openness of F iff its graph is a subset of A (resp. A-‘) the result follows. W The preceding proposition enables one to deduce from proposition 3.2. a way to pass from gages of uniform continuity to modulus of uniform continuity using semi-inverses. When Z is a product, Z = I/ x II (and clearly there is no loss of generality in assuming that when one considers pseudo-uniform continuity around some point ( wO,x0) of W x X) one can express the uniform continuity property in terms of the Hausdorff-Pompeiu excess e given for two subsets A and B of X by e(A, B) = sup(d(a, B): a E A], where d(a, B) = inf(d(a, b): b E Bj, keeping our convention fact, when Z = V x U relation (4.2) can be written d(x, E(Y)) -=cr
inf 0 = + m, sup 0 = 0. In
for any w E V, x E E(w) n U, y E B(w, y(r))
which implies that for any r E [P rl y-‘(I?) for any w E Vn E-‘(U),
e(E(w) n U. E(Y)) 5 r
y E B(w, y(r)).
(4.4)
When this relation holds for any r E ip n y-‘(IP) y is called a metric gage of uniform continuity of E relatively to I/ x U. Conversely, if y is a metric gage of uniform continuity of E relatively to V x U its lower semicontinuous hull jj is a gage of uniform continuity of E relatively to Vx UasforanyrEiPny-‘(lP),forany(w,x)EEn(Vx U)andanyyEB(w,y(r))wecan find some p E 10, r[ with y E B(w, y(p)), hence d(x, E(Y)) 5 P < r and E(y) n B(x, r) # 0. In a similar way we see that for any modulus of uniform continuity p of E relatively to V x U its upper semicontinuous hull A = fi is a metric modulus of uniform continuity of E relatively to V x U in the following sense: e(E(w) n U, E(Y)) 5 WWYN
for any w E V n E-‘(U),
y E W.
(4.5)
In fact, for any s > d(w, y) and any x E E(w) fl U one has d(x, E(y)) 5 p(s). Conversely, if A is such that (4.5) holds then for anyfl E Msuch that p(t) > I(s) for any (s, t) E P2 with t > s, where 1 is the lower semicontinuous hull of A, p is a modulus of uniform continuity of E relatively to V x U since for any s E Ip n p-‘(IP), any (w, x) E E n (V x U), and any y E B(w, S) we can find some Q E IP with q < s, d(w, y) < q, hence d(x, E(Y)) 5 W(~.Y)) so that E(y) n B(x,j#))
f
0.
5 l(q)
< 1.44
638
J.-P. PENOT
Using propositions 3.2 and 4.3 and the preceding remarks one could deduce connections between metric gages and metric modulus of uniform continuity. We prefer to deal with this question in a more direct way. PROPOSITION 4.4.
Let E: W 3 X be a multifunction which is uniformly continuous relatively to some subset V x U of W x X. If y E G (resp. P E M) is a metric gage (resp. modulus) of uniform continuity of E relatively to V x I/ then for any quasi-inverse ,u (resp. y) of y (resp. p) fi (resp. y) is a modulus (resp. gage) of uniform continuity of E relatively to V x U. In particular yh is a modulus of uniform continuity of E relatively to V x I/ iff y is a gage of uniform continuity of E relatively to V x U. Proof.
In the sequel uniform continuity means uniform continuity relatively to V x U. Let
c = ((r, S) E P2:
V w E vfl
E-‘(U),
vy
E B(w,s)
e@(w)
Let y E G be a metric gage of uniform continuity of E and let fl fi E M by 2.7 and 2.8(b) and E,(p)-’ C H(y) c C. Therefore, for y E W with &d(w, y)) < + 00 and for any (r, S) E lP2 with r (r, s) E ES(~)-’ C C hence e(E(w) fl U, E(y)) 5 r. Using lemma
fl U,
E(y))
5 r).
be a quasi-inverse of y. Then any w E V fl E-‘(U), for any > p(s), s > d(w, y) we have 2.7 we get
e@(w) n K E(Y)) 5 P(d(w, u)). Conversely let P E M be a metric modulus of uniform continuity of E and let y be any quasiinverse of ,u. Then y E G by 2.8 (a). Moreover, for any s E P rl p-‘(P) we have (&), s) E C, so that E(p)-’ C C. Therefore, for any r E P n y-‘(F)) and any w E Vrl E-‘(U), y E B(w, y(r)) we can find s E IP with d(w, y) < s < y(r) so that (r, s) E H,(y) C E(A)-l C C and e(E(w) n U, E(y)) I r: y is a metric gage of uniform continuity of E relatively to vxu. n The following lemma explains our terminology: our concept of pseudo-uniform continuity carries a sort of upper semicontinuity property as well as an obvious lower semicontinuity property. LEMMA 4.5. Let E: W =t X be a multifunction which is pseudo-uniformly continuous around (w,, x0) E E. Then the germ of ,u E M is the germ of a metric modulus of pseudo-uniform continuity of E around (we, x0) iff there exist p E IP and U E 92(x,,) such that for any w, , w, in B( w,,, p) one has
e(E(w,) n USE(w,)) 5 N(w,,
w2N < + 00.
(4.6)
If in particular ,u is linear we recover the notion of pseudo-Lipschitzian multifunction [l, 3 11. Another case of interest is the pseudo-Holderian case for which A(s) = cs” with c E P, (YE IO, I[. V E Z(w,,) and 1,~ E M be such that (4.5) holds and Proof. Let U E 92(x,), P ) [0, a] = A ) [0, CY]for some (Y> 0 with I < + 00. Let us first observe that E is 1.s.c. at (w,, x0): for any E E iP we can find 6 E P such that ~(6) < E so that for any y E B(w,, 6) d(x,, E(Y))
-( Cc(d(w,, Y)) < E
639
Metric regularity, openness and Lipschitzian behavior
and E(y) n &x0, E) # 0. In particular, taking E such p E P with p < +a, B(w,,p) c V and E(y) rl B(x,, B(w,, p) C E-‘(&x0, E)). Thus B(w,, p) C V fl E-‘(U) with w,. w2 E B(w,, p) we get (4.6). Conversely, if (4.6) holds, choosing Q E P with o < p,
that B(x,, E) C I/ we can find some E) # @ for each y E B(w,,p), i.e. and taking w = w,, y = w2 in (4.5)
and defining A E M by 1(s) = p(s) for s E [0, a], A(s) = + 00 for s > o we get that for any w E Vn E-‘(U) with V = B(w,,p - a) and for any y E W with A(d(w, y)) < + 00 we have y E B(w,, p) hence e@(w) n I/, E(Y)) 5 N(w,Y)) When ~(d(w,y))
= 44~~)).
= + 00 the inequality (4.5) is obvious.
It is well known (and easy to show) that a multifunction k iff for any x E X the function dE,x(*) given by
n E: W =t X is Lipschitzian with rate
d~,x(w) = d(x, E(w)) is Lipschitzian with rate k. A similar characterization holds for a pseudo-Lipschitzian function and for a pseudo-uniformly continuous multifunction.
multi-
PROPOSITION 4.6. Let E: W =t X be a multifunction and let (w,, x,,) E E. The following assertions are equivalent: (a) E is pseudo-uniformly continuous at (we, x0); (b) there exists some cr, j? in P and some P E M such that for any w,. w2 E B(w,, OL),any x E B(%,P)
I&.&%)
- &,x(%)l
5 P(d(W,, w,)).
When p is linear the preceding property is equivalent to a Lipschitz property of dE: (w, x) + d(x, E(w)) on some neighborhood of (w,, , x0). The proof of this result being a mimic of the proof of [31] theorem 2.3 is omitted. 5. REGULARITY
The following introduction.
definition
extends the concept
of a regular multifunction
given in the
Definition 5.1. A multifunction F: X =t Y is said to be uniformly regular around (x0, yO) E F with modulus P E M if there exist (Y,p in P such that for any x E B(x,, , a), y E B(y, , /I) one has 0,
F-‘(Y))
5 &W(x),
Y)).
(5.1)
Then fi is called a modulus of regularity of F around (x0, yO), and F is said to be p-regular around (x0, yO). Taking p(t) = ct for t E [0, E], p(t) = + 00 for t > E we obtain metric regularity as in definition 1.1. The following result extends the connection between regularity and pseudo-Lipschitzian behavior pointed out in the introduction. F: X =! Y is uniformly regular around (x,,,y,J E F iff F-’ is continuous around (yO, x0). More precisely, the germ of an U.S.C. modulus p
THEOREM 5.2. A multifunction
pseudo-uniformly
J.-P. PENOT
640
is the germ of a modulus of regularity of F around (x0, yo) iff it is the germ of a metric modulus of uniform continuity of F-’ around (Ye, x0). Proof. Let p be a modulus of regularity of F around (x,, , yo) and let CY,p E IP be as in definition 5.1. Let E E JO,p[ and let E, E M be given by A(r) = p(t) for t E [O, E], A(t) = + 03 for t > E. Let us set U = B(x,,cr), V= B(y,,/3 - E), W = Y, wO = yO, E = F-‘. Then for any w E Vtl E-‘(U) and any y E W with d(w,y) E dom 1 we have d(w,Y) I E and d(y,y,) 5 d(Y, w) + d(w, y,,) < E + (p - E) = p hence for each x E E(w) II U d(x, F-‘(Y))
5 ~c(W’(x), Y)) 5 /4&w, Y)) = A(d(w, Y))
as w E F(w). Taking the supremum over x E E(w) fl C?we get M(w)
n U E(Y)) 5 WWYN
for any w E V fl E-‘(U), y E W with d(w, y) E dom A; the case A(d(w, y)) = f 03 is obvious. (b) Conversely let E = F-’ be pseudo-uniformly continuous around (Ye, x0) with modulus A; let lJ E X(x,), V E ‘X(y,) be such that for any w E Vfl E-‘(U) and any y E Y one has e(F-‘(w) n U, F-‘(y)) s A(d(w,y)). We may suppose U = B(x,, p), V = B(y,, a) for some p, o in IP. Let T E iP be such that r 5 CTand A(7) c p. Then, choosing w = ye above, we get for any Y E B(Y,, r) d(xo, F-‘(Y))
I w-‘(Yo)
f-7u, F-‘(Y))
5 A(?) < P
SO that B(x,,, p) t7 F-‘(y) # @ and B(y,, T) c F(U) n V. Let cy = p, P = i7 and let x E B(x,,, a), y E B(y,, 8). Let us define p E M by p(t) = + 00 for t 2 /3, p(t) = X(t) for t E [0, j?[, where f is the U.S.C.hull of A. The inequality
d(x, F-‘(Y))
5 &W(x),
~1)
is obvious when F(x) = (2, or when d(F(x), y) 2 /I. Therefore we may suppose there exists w E F(x) with d(w, y) < p, hence w E B(y,, r). Now for any z E F(x) with z $ &ye, r) we have d(y, z) 2 T - /I = p so that d@‘(x), Y) = d(F(x) n B(Y, 9 r), Y). Taking w E F(x) fl B(y,, 7)arbitrary the inequality e(F-‘(w) using the right continuity of 1, d(x, F-‘(y))
rl U, F-‘(y))
I inflA(d(w, y)): w E F(x) n B(Y,, dl = &W(x),
5 A(d(w, y)) yields, ~1).
and F is p-regular around (x0, yO). n Using proposition 4.3 and the remarks following it one can deduce from theorem 5.2 connections between uniform openness and uniform regularity. It is also quite simple to proceed directly. PROPOSITION5.3.(a) If F: X =! Y is uniformly open around (xc, ye) E F with modulus P then F is uniformly regular around (x0, yo) with a modulus o whose germ at 0 is the germ of the upper-semi-continuous hull p of P. (b) Conversely, if F is uniformly regular around (x0, ye) E F with modulus ,u and if 6 E G is an hypo-semi-inverse of ,Uthen there exists E E P such that F is uniformly open around (x0, ye) with gage y = min(6, E).
Metric regularity, openness and Lipschitzian behavior
641
Proof. (a) Let us suppose (3.2) holds and let us show (5.1) is satisfied with fi replaced by w with w ) [0, E[ = ,i 1[0, E[, o(r) = + COfor r 2 E, with E, 01, B chosen so that B(x,, CY)C I/, B(y,, p + E) c V. Let us suppose on the contrary that there exist x E B(x,, a), y E B(y,, P) with t := d(x, F-‘(y)) > o(r) for r := d(F(x),y). Then, as o(r) < + 00 we have r < E; using lemma 2.7 we can find s > r with s < E, cl(s) < t. Let II E F(x) be such that d(u,y) < s. Then u belongs to B(y,, /3 + E) c V and by (3.2) we have y E B(u, s) c F(B(x, p(s))), so that F-‘(y) n B(x, p(s)) f 0, a contradiction with t > p(s). (b) Let us suppose now (5.1) holds. Let E E IP be such that E < /? and let U = B(x,, a), V = B(y,, p - E). Let us show that for any (u, u) E (I/ x V) n F, for any r E IP f~ y-‘(P) and any y E B(u, y(r)) we have y E F(B(u, r)). First let us note that y E B(y,, /I) since d(y,, u) < /3 - E, d(u,y) c E. As t := d(y, F(u)) s d(y, u) < y(r) I 6(r) and as H(P) C E(6)-’ we have (t, r) E H,(6)-’ C E,(p) or r > p(t). As (5.1) yields p(t) 2 d(u, F-‘(y)) we can find some x E F-‘(y) with d(u, x) < r, so that y E F(B(u, r)). n 6. MISCELLANEOUS
LIPSCHITZIAN
PROPERTIES
OF MULTIFUNCTIONS
Let us us supplement the preceding study with some related remarks. The first one is a slight extension of [ 11, corollary 9.21 and [3 1, theorem 4. l] giving conditions ensuring that closedness and pseudo-Lipschitzian behavior are preserved under composition. Let us recall that a multifunction F: X =t Y between two metric spaces is said to be closed at x0 E X if for any converging sequence ((x,, y,)), _,o of F with x0 = lim x,, one has lim y, E F&J; it is closed on A c X if it is closed at each x,, E A. For A = X this amounts to the closedness of the graph of F. Let us call F lower semicontinuous at (x0, Y,), where x,, E X, Y, c Y if for each sequence (x,,) with limit x0 in X there exists y,, E Ye, an infinite subset J of N and a sequence CYj>jE J with limit y. such that yj E F(Xj) for each j E J. When Ye is a singleton this corresponds to a known concept; when Y, = F(x,,) we simply say that F is semicontinuous at x0. For Y = lRd F is semicontinuous at (xr,, Y) iff lim supIIF(x)II < + 00 where IIF(x)jl = inf( ])y]]:y E F(x))); of x,.
this condition is in particular satisfied if F is bounded on a neighborhood
PROPOSITION 6.1. Let X, Y, 2 be metric spaces, let F: X =t Y, G: Y =t Z be multifunctions and letK:XxZ=t YbegivenbyK(x,z)=F(x)nG-‘(z). (a) If F is closed at x0, if G-r is closed at z. and if K is lower semicontinuous at ((x0, zo), Y) for any z. E Z then H = G 0 F is closed at x0. (b) If F is closed at x0, if G-’ is closed at z. for some z. E H(x,), if K is lower semicontinuous at ((x0, zo), Y) and if F (resp. G) is pseudo-Lipschitzian at (x0, yo) (resp. (yo, zo)) for each y. E K(x, ,z,) then H = G 0 F is pseudo-Lipschitzian at (x0, zo).
Proof. (a) Let ((x, , z,)) be a sequence of H with limit (x0, zo) and let us show z. E H(x,). Since K is lower semicontinuous at ((x0, z,), Y) can find y. c Y, an infinite subset J of N and a sequence with limit y. such that yj E K(xj, .Q) for each j E J. As F is closed at x0 and G-’ is closed at z. we have y, E K(x,, zo). Thus K is semicontinuous at (x0, zo) and z. E G(Y,) C G(F(xo)). (b) Let us show there exists E E P and c E iP such that (Yj)j
E
J
d(z, H(x’))
5 cd(x, x’)
for any x, x’ E B(x, , E), z E H(x) fl B(zO , E).
J.-P. PENOT
642
If this is not the case we can find sequences (x,), (XL), (z,) with limits x0, x0 and z0 respectively with d(t, , H(x;)) z n&x,, XL) > 0 and z, E H(x,,) for each n E IN. Then by what precedes we can find y0 E K(_u,, z,,), an infinite subset J of N and a sequence (yJi EJ with limit y,, such that yj E K(x_, zj) for eachj E .I. Then y, E F(x,,), z0 E G(y,) SO that for some C, C’ in P and for each j E J large enough we have d(y,, F(xj)) I Cd(Xj, xj). Similarly, taking yj E F(x)) with d(yj, yj) I (C + l)d(Xj, Xj’) and observing that (yjl) --*yo, we have d(Zjp G(y,!)) I c’d(yj, yj) I H c’(c + l)d(Xj, x,!), a contradiction with our choice of (x,), (x;), (z,). We mentioned in the introduction that the Clarke strict tangent cone is a key tool for criteria of metric regularity and pseudo-Lipschitzian behavior. Let us observe as a final remark that conversely a pseudo-Lipschitzian behavior simplifies the expression of the strict (Clarke) generalized derivative of a multifunction. Recall that given a multifunction F: X =t Y between two normed vector spaces and (x0, yo) E F, the strict generalized derivative of F at (x0, yo) is given by D’F(XO,YON~
= tu E Y: (~9 u) E G$Xo,
YON,
where $(x0, yo) is the (Clarke) strict tangent cone to F at (x0, yo) given by T>(x,, yo) = lim inf t-‘(F - (x, y)) (see [I]). kYG.y.Ya.uo)
PROPOSITION
6.2. If F: X =I Y is pseudo-Lipschitzian D’F(x, , ye)(u) =
lim inf
at (x0, yo) then F(x + tu) - y
(x.Yt 5 &o.YO)
t
*
Therefore DF’(x,, yo) is defined as a limit of a differential quotient as in the case of a mapping. The proof is similar to the first part of the proof of [32, theorem 3.21. Proof. Let u E D’F(x,, ye)(u) and let ((x,, y,)) 3 (x0, yo), (t,) -+ 0,. By definition of r&x,, yo) we can find a sequence ((u,, u,)) with limit (u, u) such that (x,,, y,) + tn(un, u,) E F for each n E N. As F is pseudo-Lipschitzian at (x0, yo) we can find c E IR, and no E N such that for n r no d(y, + tnu,,, F(x, + t,u)) I cd(x, + r,u,,x,,
Therefore
+ t,u).
we can find (w,,) with IIu,, - w,)) I c)Iu,, - ~11+ t,, hence (w,) -, u, and y, + t, w,, E lim inf t-‘(F(x + tu) - y. The reverse in-
F(w, + 1, u) for each n E N. This shows u E
elusion is obvious.
n
(+.Y)tyo.YO) - +
Remark. Contrary to what occurs when F is a locally Lipschitzian mapping [32] the cone Ti(x,, yo) is not always a vector subspace of X x Y, as simple examples show. REFERENCES 1.
AUBIN
J.-P.,
Lipschitz behavior of solutions to convex minimization problems, Math. Operur. Res. 9, 87-111
(1984). 2. 3.
AUBIN, J.-P. & CELLINA AUSLENDER A., Stability
239-254 (1984).
A., Dif/erenfiai Inclusions. Springer, Heidelberg (1984). in mathematical programming with nondifferentiable data, SIAM J. Control Opfim. 22,
hletric regularity, openness and Lipschitzian behavior
643
4. BORWEINJ., Stability and regular points of inequality systems, J. Opfim. Theor. Applic. 48, 9-52 (1986). 5. CLARKE F., Optimization and Nonsmooth Analysis. John Wiley, New York (1983). 6. CORNET B. & LAROQUEG., Lipschitz properties of solutions in mathematical programming, J. Optim. Theor. Applic. (to appear). 7. CORNETB & VIALJ.-P., Lipschitzian solutions of perturbed nonlinear programming problems, SIAM J. Control Opfim. 24, 1123-1138 (1986).
8.
DMITRUCK A. V., MILYUTIN A. A. & OSMOLOVSKIN.-P.,
Ljusternik’s theorem and the theory of extrema, Russian
Math. Surveys 35, 11-51 (1980).
9. DUONG P. C. & TUY H., Stability, surjectivity and local invertibility of non-differentiable mappings, Acfa Viefnam. 3, 89-103 (1978). 10. DOLECKI S., A general theory of necessary optimality conditions, J. math. Analysis Applic. 78, 267-308 (1980). 11. DOLECKIS., Compactoid and compact filters, Pucif. J. Mulh. 117, 69-98 (1985). 12. GINER E., Etudes sur les fonctionnelles integrales, These, Universite of Pau (1985). C.-J. & VAN VLECKF.-S., Lipschitzian generalized differential equations, Rend. Sem. Mat. Univ. 13. H~MMELBERG Pudovu 48, 159-169 (1972).
14.
HOFFMAN A.,
On approximate solutions of systems of linear inequalities, J. Res. n&n. Bur. Stand. B 49, 263-265
(1952).
15. IOFFEA.-D., Regular points of Lipschitz functions, Trans. Am. moth. Sot. 251, 61-69 (1979). 16. IOFFEA.-D., Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Analysis 8, 517-539 (1984).
17. IOFFEA.-D., Approximate subdifferentials
and applications.
I: The finite dimensional theory, 7runs. Am. math.
Sot. 281, 389-416 (1984).
18. 19. 20. 21.
JANINR., Sur la dualite et la sensibiliti dans les problirmes de programmation mathematique, These, Paris (1974). LEBOURGG., Thesis (to appear). LJUSTERNIKL.-A. & SOBOLEV V.-I., Elements of Functional Analysis. Nauka, Moscow (1965). MICHELP., Probl&ne des inigalites, Applications & la programmation differentielle dans les espaces de Banach, Cr.
hebd. S&nc. Acud. Sci. Paris A 345, 345-348 (1972).
22. MKHEL P.. Probleme des inegalites. Applications g la programmation et au contrBle optimal, Bull. Sot. Math. Fr. 101, 413-434 (1973). 23. NADLER S.-B., JR., Multivalued contraction mappings, Pacif. J. Moth. ?O, 475-488 (1969). 24. PENOTJ.-P., Invers,ion &droiie d’applications non lintaires applications,Foll. Nut. Anulyse Numk., Lu C&e SW Loup 58-59 (1973); C.r. hPbd. Stknc. Acud. SC;. Paris 290, 997-1000
(1980).
25. PENOTJ.-P., On regularity conditions in mathematical programming, Math. rog. Study 19, 167-199 (1982). 26. PENOTJ.-P.,/Penalization and regularization methods fo~~onsmpoth analysi ;P unpublished lecture,.‘workshop on 16-17 (1983). Nonlinear Anulysis and Optimizufion, Core, Louvain-La-Neuve,iJune 27. PENOT J.-P., About linearization, conization, calmness, openness and regularity, in Nonlinear Analysis und Applicufions (Edited by V. LAKSHMIKANTHAM), pp. 439-450. Marcel Dekker, New York (1987). 28. PENO~J.-P. & VOLLEM.. Inversion of real-valued functions and applications, 2. Operut. Res. (to appear). 29. ROBINSON S. M., Regularity and stability for convex multivalued functions, M&h. Operut. Res. 1, 130-143 (1976). 30. ROBINSON S. M., Stability theory for systems of inequalities, part II: differentiable nonlinear systems, SIAM J. numer. Analysis 13, 497-513 (1976).
31. ROCKAFELLAR R. T.. Lipschitzian properties of multifunctions, Nonlinear Anulysis 9, 867-885 (1985). 32. ROCKAFELLAR R. T.. Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. Henri PoincurP 2, 167-184 (1985).
33. ROCKAFELLAR R. T., Extensions of subgradient calculus with applications to optimization, Nonlinear Analysis 9, 665-698 (1985).
METRIC
REGULARITY,
Vol. 13. No. 6. pp. 629443,
1989. 0
OPENNESS AND LIPSCHITZIAN OF MULTIFUNCTIONS JEAN-PAUL
0362-%6X/89 S3.00+ .OO 1989 Pcrgamon Press plc
BEHAVIOR
PENOT
Faculte des Sciences, Avenue de I’Universitk 6400 Pau, France (Received
I1 December
1987; received for publication
Key words and phruses: Gage, inverse mapping,
regular multifunction,
19 April 1988)
Lipschitzian multifunction,
modulus, open mapping,
uniform continuity.
1. INTRODUCTION
of regular multifunction has emerged out of the pioneering work of Ljusternik and Sobolev 1201 through the efforts of several workers. In its initial form it was conceived as a mean for solving equalities or inequalities under more or less classical surjectivity assumptions on some approximants [9, 10, 14,21,22,24,25]. Robinson [30] stressed the key estimate on the distance to the solution set which is characteristic of what is called the (metric) regularity property we formally define in the following way among several slight variants. THE NOTION
1.1. A multifunction F: X =t Y between two metric spaces is said to be metrically regular around (x0, y,,) with x0 E X, y, E F(x,,) if there exist positive numbers CY,8, E such that
Definition
d(x, F_‘(Y)) whenever x E B(x,,
a),
5 cW(x),
(1.1)
Y)
/I) with d(F(x), y) I E.
y E B(y,,
Here we denote by B(z, r) the open ball with center z and radius r, and for a subset A of a metric space (M, d) and b E M we set d(A, b) = d(b, A) = inf[d(a, b) : a E A). The special case in which X = IR”, Y = W and F is given by F(X) = [YE lR’:Yi
for some
m E
i =
=f;(x)
1, . . . . m,
[ 1, p] has received a particular
d(x,F-‘(~))
Yj Z&(X)
attention.
with t+ = max(t, 0). In particular, penalization term
m
i
j=m+l
+
1, . . . . p)
i
C&(X) - Y$’
j=m+l
for y = 0, the distance d(x, F-‘(O))
c ii IfiCx)l+ C i=l
=
The estimate (1.1) can then be written
5 c f IL(X) - YiI + c i=l
j
is estimated by the
fi(x)+-
This connection with penalization theory has been deeply used by Clarke IS] for treating mathematical programming problems and optimal control problems. 629
630
J.-P. PENOT
A great impetus has been given to the concept of regularity by Ioffe [15-171 who showed that the Ekeland’s variational principle could be used to give verifiable criteria for regularity (see also [3, 41). It has been shown [4, 12, 24, 251, that regularity is a useful property for computing tangent cones to sets given as inverse images or defined be equalities and inequalities. Simultaneously the infinitesimal properties ensuring regularity have been refined [4, 12, 16, 17, 19, 27, 331. Lipschitzian properties of multifunctions have been recognized to be a useful tool for several topics in nonlinear analysis as fixed point theory [23, 251, differential inclusions [2, 5, 131 implicit function theorems [l], mathematical programming [6, 71. The following result shows that convexity entails a Lipschitzian behavior. PROPOSITION 1.2. ([18, lemma III Bl], (25, theorem 5.11). Let Xand Y be two normed vector spaces with unit balls Bx and BY respectively. Let F: X 3 Y be a multifunction with convex graph such that for some p > 0 and some open subset U of X one has F(x) fl pBy # @ for each x E U. Then for each x0 E U there exist CY> 0, m > 0, p > 0 such that for each r E IR,
F(x,) n rB, c F(x,) + (mr + P)&,
x2)&
for any x,, x2 in x0 + aBx.
This kind of behavior is a special case of what has been called a sub-Lipschitzian behavior by Rockafellar in his thorough study [31]. A more common property of nonconvex multifunctions has been discovered by Aubin [l] in connection with inverse mapping theorems. Definition 1.3. A multifunction F: X =t Y between two metric spaces is said to be pseudoLipschitzian around (x0, yo), where y. E F(x,), if there exist neighborhoods U of x0, Vof y. and c E IR, such that &Y, 3F(xd)
-( c4q
for any x,, x2 E U, y, E VII F(x,).
9~2)
The word regularity has been used in a different sense by Robinson (291 for studying openness properties of convex multifunctions. The precise property he deals with can be defined as follows. Definition 1.4. A multifunction F: X =I Y is said to be open at a linear rate around (x0, yo), with x0 E X, y. E F(x,), if there exist positive numbers (Y,/I, p such that for any x E B(x,, a), y E B(y,, P), r E [0, p] with y E F(x) one has NY,
cd
C F(B(x, r)).
The classical open mapping theorem and its extension to convex multifunctions as in [29] are instances in which such a property occurs. The purpose of the present paper is to show that the three preceding concepts are equivalent. THEOREM 1.5. Let F: X s Y be a multifunction and let (x0, y,,) E X x Y with y. E F(x,). Then the following assertions are equivalent: (a) F is metrically regular around (x0, yo); (b) F-’ is pseudo-Lipschitzian around (yo, x0); (c) F is open around (x0, yo) at a linear rate.
Metric regularity, openness and Lipschitzian behavior
631
This result has been announced in [26, 271. A simplified form of the implication (a) * (c) is contained in [4, theorem 5.1) with several verifiable criteria and striking applications. Here this equivalence is extended to properties which go beyond pseudo-Lipschitzian behavior, linear openness and metric regularity but still retain some uniformity properties. In particular, in Section 4 we introduce the notion of uniform continuity of a multifunction; Section 3 deals with uniform openness. Uniform regularity is considered in Section 5. The extension of theorem 1.5 to such properties does not require much more effort; some useful material about generalized inverses of one variable functions adapted from [28] is given in Section 2. This extension is justified by the fact that in some concrete situations the Lipschitz behavior has to be replaced by a less demanding property as an Holderian behavior. It is known for instance that the Lipschitz property of a moving closed convex subset of a Hilbert space is lost by the associated projection of a given point but that an Holderian behavior still remains. Miscellaneous related properties are gathered in Section 6, completing the studies made in [I, 4, 311. 2. PRELIMINARIES:
GAGES
AND MODULUS
In the sequel we denote by P the set of positive real numbers and we set IR, = P U 101, the role of IP could be replaced by any open interval of IR. Let N be the set of nondecreasing mappings from R, into IR, and let G be the set of gages i.e. the set of nondecreasing mappings from P into IP U (+ 00) taking at least one finite value. Any element g of G is considered as an element of N, extending g to lR+ by setting g(0) = 0, g(+ m) = supg(P). We denote by M the set of modulus i.e. the set of m E N such that inf m(lP) = 0. Obviously these definitions depend only on the germ at 0 of the mappings: if f, g E N have the same germ at 0 (i.e. iffand g coincide on some neighborhood of 0 in R,) then f E G (resp. M) iff g E G (resp. M). For f: R+ -+ rii, we set R+ = IR,U I+ a]. In most of our preliminaries
E(f)
= ((r, s) E P x IP:j-(r) 5 s),
H(f)
= ((r, s) E P x P : s 5 f(r)].
We denote by E,(f) (resp. H,(f)) the sets obtained by replacing the inequality by a strict inequality in what precedes. These sets correspond to the epigraph (resp. hypograph) off and its strict analogue when one uses an increasing bijection for identifying IP with IF?. We denote by g the family of subsets E of P x IP such that (r, t) E E whenever t 1 s for some s E IP with (r, s) E E; we call g the family of pseudo-epigraphs; the family of pseudohypographs is the family X of subsets H of IP x IP such that (r, t) E H whenever (r, s) E H, t TVIPwith t I s. We denote by &+the family of E E $ such that for any (r, s) E E and any 4 E iP with 4 5 r (q, s) belongs to E. Similarly, X, is the family of H E 3.2such that for any (r, s) E H and any t E P with t 1 r one has (t, s) E H. Obviously, for any f E RT (resp. N) E(f) and E,(f) belong to & (resp. &+) while H(f) and H,(f) belong to X (resp. X,). Conversely, for any E E & (resp. &+) there exists a unique f E I!?:(resp. N) such that E,(f) c E c E(f); we denote it by (Pi and we observe that it is given by v)~(T) = infls E IP: (r, s) E E),
J.-P. PENOT
632
with the convention
inf 0 = + co, sup 0 = 0. Moreover pE is the greatest f E rii”,such that
E c E(f): g 5
VE
e+ E
(2.1)
C E(g).
Similarly, with any H E 3C (resp. X,) we associate pH E Ri”,(resp. N): pH(r) = supls E P: (r,s) E H), so that HS(pH) C H C H(pH)
and for any g E WYwe have g 2 pH e H c H(g).
In the sequel we always identify a multifunction A c IP x IP we set
(2.2)
(or relation) with its graph. In particular for
A-’ = ((s, r): (r,s) E A) and A-’ is the inverse relation of A. It is obvious, but important for what follows, to observe that a subset A of P2 belongs to 8, iff A-’ belongs to X, . Applying this observation to epigraphs and hypographs we get the following refinements of some concepts of inverse functions introduced in [28]. Definitions 2.1. (a) An element g E Nis said to be a semi-epi-inverse inverse off) if E(g) C H(f)-’ (resp. E,(g) c H(f)-‘).
(b) It is said to be a semi-hypo-inverse
off E N (resp. a quasi-epi-
off (resp. a quasi-hypo-inverse
off)
if H(g) c E(f)-’
(resp. H,(g) c E(f )-‘). (c) It is a quasi-inverse off if it is both a quasi-epi-inverse
off and a quasi-hypo-inverse off. (d) It is an epi-inverse off if it is a semi-epi-inverse off and a quasi-inverse off: H,(f)-’ C E(g) C H(f)-‘. It is an hypo-inverse off if it is a semi-hypo-inverse off and a quasi-inverse of J E,(f)-’ c H(g) c E(f)-‘. We observe that g is a quasi-epi-inverse off iff f is a quasi-epi-inverse of g. A similar equivalence holds for quasi-hypo-inverses but is not true for semi-epi-inverses. The following lemma shows that the situation is much simplified when some continuity property is at hand. LEMMA 2.2. (a) If f E Nis U.S.C. then g E Iv is a semi-epi-inverse
off. (b) If f E N is 1.s.c. then g E N is a semi-hypo-inverse
off iff g is a quasi-epi-inverse
off iff g is a quasi-hypo-inverse
off.
Proof. When f is U.S.C.H(f) is closed and H(f)-’ is closed too. Then, as for any g E IV, E(g) is the vertical closure of E,(g), E(g) is contained in the closure of E,(g), so that E,(g) C H(f)-’ iff E(g) c H(f)-‘. The proof of the second assertion is similar. I
One disposes of two canonical generalized inverses off E N, obtained by using E = H(f)-’ or H,(f)-‘, H = E(f)-’ or E,(f)-’ in the process we described above: f’
= pH*-,
= lpEdJ-J-‘,
fh = $Q&-)_, = @(fJ-‘:
f e(.3) = inflr E IP: s 5 f(r)]
= suplt E IP:f(t)
< s],
f h(s) = inflr E IP: s
= sup(t E P: f(t)
I s).
Metric regularity, openness and Lipschitzian behavior
These mappings have extremal properties and can serve for a characterization Moreover they have regularity properties whatever f E N is.
633
of quasi-inverses.
LEMMA 2.3. For eachf E N,fe is lower semicontinuous (1.s.c.) and f” is upper semicontinuous (u.s.c.). In fact H,(fe) = int E(f)-’ = int E,(f)-’ and E,(fh) = int H(f)-’ = int H,(f)-‘.
Proof. Let (s, r) E H,(f): there exists t E IP with t > r, f(t) < s so that IO, t[ x If(t), + co[ is a neighborhood of (r, S) contained in E,(f). Conversely if (s, r) E int E(f)-’ and if (s’, r’) belongs to some neighborhood of (s, r) contained in E(f)-’ and is such that s’ < S, r’ > r we have f’(s) 1 r’ > r hence (.s, r) E H,cf’). The proof of the assertion with fh is similar. H
Let us give a characterization
of quasi-inverses off E N.
PROPOSITION2.4. (a) For any f, g in N, g is a quasi-epi-inverse off iff g 5 f e iff f 1 g’. (b) For any f, g in N, g is a quasi-hypo-inverse off iff g I fh iff f 5 gh. (c) For any f, g in N, g is a quasi-inverse off ifff’ s g I fh. Proof. (a) Asfe = (Pi with H = E,(f)-‘, (2.2) ensures that g 2 f’ iff E,(f)-’ C H(g) iff E,(g) C H(f)-‘. As f and g have symmetric roles g 2 f’ iff f 1 ge. (b) The proof is similar: as f” = pE with E = Hs(f)-I, (2.1) ensures that g I f" iff H,( f )-’ C E(g) iff H,(g) C E( f )-I; again f and g can be interchanged. Assertion (c) is a consequence of assertions (a) and (b). n
COROLLARY 2.5. For any f E N, f’ and fh are quasi-inverses off; moreover f is an hypo-inverse off’ and an epi-inverse off h. Proof. The first assertion is a special case of proposition 2.4 (c). The second one follows from the fact that f’ (resp. f*) is I.s.c. (resp. u.s.c.) so that f is not only a quasi-hypo-inverse (resp. quasi-epi-inverse) of f’ (resp. fh) but a semi-hypo-inverse (resp. semi-epi-inverse) of f (resp. fh). n
We are now in a position to characterize
semi-inverses.
PROPOSITION2.6. (a) For any f, g in N, g is a semi-epi-inverse off iff f 2 gh. (b) For any f, g in N, g is a semi-hypo-inverse off iff f I ge. Proof. Let us suppose g E N is a semi-epi-inverse of f. Then E(g) C H(f)-‘. Now as H,(gh) C E(g)-’ we get HS(gh) C H(f), hence gh of. Conversely, suppose that f, g in N are such that gh I f. Then as g is an epi-inverse of gh by corollary 2.5 we have E(g) c H(gh)-’ c H(f)-’ and g is a semi-epi-inverse off. The proof of the second assertion is similar. W
In [28] the role of quasi-inverses in the regularization process of a real-valued function was pointed out. Here we focus our attention on the interplay of the notions of generalized inverses with the concepts of gage and modulus. Let us first note the following elementary facts (see [28] for the first assertions).
J.-P. PENO~
634
LEM%~A 2.7. For any f E N the 1.s.c. hull fof f and the U.S.C. hullfof J(r) = suplf(q): f(r) = inf(f(s): If f E G then fand
f are given by
4 E p, 4 < r1, s E ip, s > r].
f^ belong to G. If f E M then fand
j; belong to M.
PROPOSITION 2.8. Let f and g be elements of N taking at least one finite value. (a) If g is a quasi-epi-inverse off and if f E M then g E G.
(b) If f is a quasi-hypo-inverse of g and if g E G then f E M. (c) For each g E G one has ge E M, gh E M. (d) For each f E M one has f’ E G, f hE G.
Proof. (a) By assumption we have E,(g) c H(f)-‘. If g 4 G there exists s E IP with g(s) = 0 so that {s] x [P C N(f)-‘: for each r E IP we have s s f(r), hence inff(iP) 2 s > 0 and f $ M. (b) Let us suppose H,(f) C E(g)-’ and g E G. Then for any s E IP we have g(s) > 0 so that we can find r E Ip with (s, r) E H,(g) hence (r, s) E E(f): f(r) 5 s. This shows that inff(lP) = 0 and f E M. By proposition 2.4 assertions (c) and (d) are special cases of assertions (b) and (a). H Remark 2.9. It is not true that if f E M and if g is a quasi-hypo-inverse instance if f 3 0, g = 0 is quasi-hypo-inverse off and g B G.
off then g E G. For
3. UNIFORM OPENNESS denoted by d for each space. Most topological vector spaces, of what follows could be extended to uniform spaces, in particular using a family of extended semi-distances but we refrain to do so for Ithe sake of simplicity. The
In the sequel W, X, Y are metric spaces, with a distance
use of closed balls instead of open balls would necessitate some changes which are left to the reader; in general open balls seem to be more convenient except in the case the closed balls are compact. The family of neighborhoods of a point x E X is denoted by 92(x). The following definition brings some uniformization in the notion of open multifunction. Definition 3.1. A multifunction F: X =t Y is said to be uniformly open relatively to subsets U of X, V of Y if for any r E IP there exists $ E 1psuch that for any (u, u) E F n (U x V) one has WV, s) c F(Hu, r)). A gage y E G such that Ku, v(r)) C F(B(u, r))
for any r E IP fl y-‘(Ip), (u, u) E Fn (U x V)
(3.1)
is said to be a gage of uniform openness of F relatively to (I and V, then F is said to be yuniformly open relatively to U and V. Given (x0, yo) E X x Y, F is said to be y-uniformly open around (x,, , yO) if there exists U E %(x,,), V E 3t(y,) such that (3.1) holds. A related notion is the concept of modulus of uniform openness. An element p of the set M of modulus is said to be a modulus of uniform openness of F relatively to I/ and V if B(u, s) C 0&u,
p(s))
for any s E IP fl p-‘(fP), (u, u) E F fl (U x V).
(3.2)
Metric regularity, openness and Lipschitzian behavior
635
For the sake of brevity we often omit the mention “relatively to U and I/” in the sequel when there is no risk of confusion. 3.2. If y is a gage of uniform openness of F then any semi-epi-inverse p of y is a modulus of uniform openness of F. Conversely, if P is a modulus of uniform openness of F then any semi-hypo-inverse y of p is a gage of uniform openness of F. In particular, given an U.S.C. y E G (resp. a I.s.c. ,U E M) we take p = yh (resp. y = p’) above.
PROPOSITION
Proof.
Let us set A = ((r, s) E [P X IP: V (u, II) E (u
X v) I-7 F
B(u, s) C F(B(u, r))).
Then y E G (resp. P E M) is a gage (resp. modulus) of uniform openness of F (relatively to U and V) iff H(y) C A (resp. E(p)-’ C A). The inclusion E(p)-’ C H(y) (resp. H(y) C E(p)-‘) which express the assumption of the first (resp. second) assertion then clearly implies its conclusion, taking proposition 2.8 into account. n Remark. As we are using open balls, we observe that in fact any quasi-hypo-inverse y of a modulus of uniform openness of F is a gage of uniform openness of F if it is a gage: if for any r, s in IP with s < y(r) and any (u, IJ) E (U x V) n F we have B(v, s) C F(B(u, r)), then we also have B(u, y(r)) c F(B(u, r)) for any r E P fl y-‘(P), (u, V) E (I/ x V) n F. n
It follows from this remark and propositions 2.4, 2.8 and 3.2 that for fl E Mb is a modulus of uniform openness of F iff ph is a gage of uniform openness of F. Example 3.3. Suppose X and Y are Banach spaces and F: X + Y is a surjective continuous linear mapping. Then F is uniformly open relatively to X and Y with a linear rate: one can find c E IP such that o given by w(t) = ct, is a modulus of uniform openness and y given by y(r) = c-rr is a gage of uniform openness of F. However, in most cases as the one given in the following result it is not easy to determine explicitly a gage or a modulus of openness. PROPOSITION 3.4. Let U and V be compact subsets of X and Y respectively and let F: X =I Y be a multifunction such that F n (U x V) is closed. Then if Fis open at each (x, y) E F fl (I/ x V) it is uniformly open relatively to U and V.
Proof. Suppose the contrary. Then we can find r E IP, a sequence (s,) C IP with limit 0 and a sequence ((x,, y,)) E F fl (U x V) such that B(y,, s,) qJ F(B(x,, r)). Without loss of generality we may assume ((x,, , y,)) has a limit (x, y) in U x V and in fact in F n (U x V) as this is a closed subset of U x V. As F is open at (x, y), for any q E Ip with q < r we can find p E Tpsuch that B(y, p) c F(B(x, 4)). Now for n large enough we have B(y,, s,) C B(y, p) and B(x, q) C B(x,, r), hence B(y,, s,) c F(B(x, q)) C F(B(x,, , r)), a contradiction. n
636
J.-P. PENOT
Slightly generalizing definition 2.1 one says that F: X 2 Y is y-uniformly a subset C of X x Y if 0,
W)) c WW,
r))
open relatively to
for each r E ip fl y-‘(IP) and each (x,y) E Fn C.
Then one gets the following obvious composition
(3.3)
result.
PROPOSITION3.5. Suppose F: X 3 Y is y-uniformly open relatively to C C X x Y and G: Y =t Z is &uniformly open relatively to D C Y x Z. Then H = G 0 F is 6 0 y-uniformly open relatively to
4. UNIFORM CONTINUITY AND UNIFORM PSEUDO-CONTINUITY The following definition multifunction.
gives a uniform version of the notion of lower continuity
of a
Definition 4.1. A multifunction E: W 3 X between two metric spaces is said to be uniformly continuous relatively to a subset Z of W x X if for any r E IP there exists s E IP such that for any (w, X) E E fl Z and any y E B(w, s) one has E(y) n B(x, r) z 0. Usually Z is taken to be a product definition.
V x U. A special case is contained
(4.1) in the following
Definifion 4.2. A multifunction E: W 3 X is said to be pseudo-uniformly continuous (P.u.c.) around ( wO,x0) E E if E is uniformly continuous relatively to some neighborhood Z of (w,, x0) in WxX. A gage y such that E(y) n B(x, r) f 0
for any r E IP fl y-‘([P), (w, x) E E fl Z, y E B(w, v(r))
(4.2)
is said to be a gage of uniform continuity of E relatively to Z (resp. around (w,, , x0) if Z is taken as in definition 4.2). Let us observe that if E is uniformly continuous relatively to Z there exists a greatest gage of uniform continuity relatively to Z given by Y&) = sup(s E II’: v (w, x) E E fI Z, v y E B(w, s) E(y) f3 B(x, r) # @ 1. A modulus ,u such that E(y) n B(x, p(s)) # 0
for any s E IP n p”-‘(IP), (w, x) E E fl Z, y E B(w, s)
(4.3)
is said to be a modulus of uniform continuity of E relatively to Z. The links of the preceding notion with the previous concepts of uniform openness are rather obvious. PROPOSITION4.3. A multifunction E: W 3 X is uniformly continuous relatively to Z C W x X iff F = E-‘: X =t W is uniformly open relatively to Z-‘. Moreover y E G (resp. ,u E M) is a gage (resp. a modulus) of uniform continuity of E relatively to Z iff it is a gage (resp. a modulus) of uniform openness of F relatively to Z-‘.
Metric
Proof. coincide:
regularity,
Since E(y) (7 B(u, r) f 0
openness
is equivalent
A = {(r, S) E Ip2: v (u, u) E F B = ((r, s) E P2:
and Lipschitzian
v (v, u) E
n z-’
E n Z,
637
behavior
to y E E-‘(B(u,
r)) the following
sets
B(u,s)c E-‘(B(u, f))), v y E B(u, s)
E(y) n B(u, r) # 0 1.
Since y E G (resp. p E M) is a gage (resp. a modulus) of uniform continuity of E iff its graph is a subset of B (resp. B-‘) and since y (resp. p) is a gage (resp. a modulus) of uniform openness of F iff its graph is a subset of A (resp. A-‘) the result follows. W The preceding proposition enables one to deduce from proposition 3.2. a way to pass from gages of uniform continuity to modulus of uniform continuity using semi-inverses. When Z is a product, Z = I/ x II (and clearly there is no loss of generality in assuming that when one considers pseudo-uniform continuity around some point ( wO,x0) of W x X) one can express the uniform continuity property in terms of the Hausdorff-Pompeiu excess e given for two subsets A and B of X by e(A, B) = sup(d(a, B): a E A], where d(a, B) = inf(d(a, b): b E Bj, keeping our convention fact, when Z = V x U relation (4.2) can be written d(x, E(Y)) -=cr
inf 0 = + m, sup 0 = 0. In
for any w E V, x E E(w) n U, y E B(w, y(r))
which implies that for any r E [P rl y-‘(I?) for any w E Vn E-‘(U),
e(E(w) n U. E(Y)) 5 r
y E B(w, y(r)).
(4.4)
When this relation holds for any r E ip n y-‘(IP) y is called a metric gage of uniform continuity of E relatively to I/ x U. Conversely, if y is a metric gage of uniform continuity of E relatively to V x U its lower semicontinuous hull jj is a gage of uniform continuity of E relatively to Vx UasforanyrEiPny-‘(lP),forany(w,x)EEn(Vx U)andanyyEB(w,y(r))wecan find some p E 10, r[ with y E B(w, y(p)), hence d(x, E(Y)) 5 P < r and E(y) n B(x, r) # 0. In a similar way we see that for any modulus of uniform continuity p of E relatively to V x U its upper semicontinuous hull A = fi is a metric modulus of uniform continuity of E relatively to V x U in the following sense: e(E(w) n U, E(Y)) 5 WWYN
for any w E V n E-‘(U),
y E W.
(4.5)
In fact, for any s > d(w, y) and any x E E(w) fl U one has d(x, E(y)) 5 p(s). Conversely, if A is such that (4.5) holds then for anyfl E Msuch that p(t) > I(s) for any (s, t) E P2 with t > s, where 1 is the lower semicontinuous hull of A, p is a modulus of uniform continuity of E relatively to V x U since for any s E Ip n p-‘(IP), any (w, x) E E n (V x U), and any y E B(w, S) we can find some Q E IP with q < s, d(w, y) < q, hence d(x, E(Y)) 5 W(~.Y)) so that E(y) n B(x,j#))
f
0.
5 l(q)
< 1.44
638
J.-P. PENOT
Using propositions 3.2 and 4.3 and the preceding remarks one could deduce connections between metric gages and metric modulus of uniform continuity. We prefer to deal with this question in a more direct way. PROPOSITION 4.4.
Let E: W 3 X be a multifunction which is uniformly continuous relatively to some subset V x U of W x X. If y E G (resp. P E M) is a metric gage (resp. modulus) of uniform continuity of E relatively to V x I/ then for any quasi-inverse ,u (resp. y) of y (resp. p) fi (resp. y) is a modulus (resp. gage) of uniform continuity of E relatively to V x U. In particular yh is a modulus of uniform continuity of E relatively to V x I/ iff y is a gage of uniform continuity of E relatively to V x U. Proof.
In the sequel uniform continuity means uniform continuity relatively to V x U. Let
c = ((r, S) E P2:
V w E vfl
E-‘(U),
vy
E B(w,s)
e@(w)
Let y E G be a metric gage of uniform continuity of E and let fl fi E M by 2.7 and 2.8(b) and E,(p)-’ C H(y) c C. Therefore, for y E W with &d(w, y)) < + 00 and for any (r, S) E lP2 with r (r, s) E ES(~)-’ C C hence e(E(w) fl U, E(y)) 5 r. Using lemma
fl U,
E(y))
5 r).
be a quasi-inverse of y. Then any w E V fl E-‘(U), for any > p(s), s > d(w, y) we have 2.7 we get
e@(w) n K E(Y)) 5 P(d(w, u)). Conversely let P E M be a metric modulus of uniform continuity of E and let y be any quasiinverse of ,u. Then y E G by 2.8 (a). Moreover, for any s E P rl p-‘(P) we have (&), s) E C, so that E(p)-’ C C. Therefore, for any r E P n y-‘(F)) and any w E Vrl E-‘(U), y E B(w, y(r)) we can find s E IP with d(w, y) < s < y(r) so that (r, s) E H,(y) C E(A)-l C C and e(E(w) n U, E(y)) I r: y is a metric gage of uniform continuity of E relatively to vxu. n The following lemma explains our terminology: our concept of pseudo-uniform continuity carries a sort of upper semicontinuity property as well as an obvious lower semicontinuity property. LEMMA 4.5. Let E: W =t X be a multifunction which is pseudo-uniformly continuous around (w,, x0) E E. Then the germ of ,u E M is the germ of a metric modulus of pseudo-uniform continuity of E around (we, x0) iff there exist p E IP and U E 92(x,,) such that for any w, , w, in B( w,,, p) one has
e(E(w,) n USE(w,)) 5 N(w,,
w2N < + 00.
(4.6)
If in particular ,u is linear we recover the notion of pseudo-Lipschitzian multifunction [l, 3 11. Another case of interest is the pseudo-Holderian case for which A(s) = cs” with c E P, (YE IO, I[. V E Z(w,,) and 1,~ E M be such that (4.5) holds and Proof. Let U E 92(x,), P ) [0, a] = A ) [0, CY]for some (Y> 0 with I < + 00. Let us first observe that E is 1.s.c. at (w,, x0): for any E E iP we can find 6 E P such that ~(6) < E so that for any y E B(w,, 6) d(x,, E(Y))
-( Cc(d(w,, Y)) < E
639
Metric regularity, openness and Lipschitzian behavior
and E(y) n &x0, E) # 0. In particular, taking E such p E P with p < +a, B(w,,p) c V and E(y) rl B(x,, B(w,, p) C E-‘(&x0, E)). Thus B(w,, p) C V fl E-‘(U) with w,. w2 E B(w,, p) we get (4.6). Conversely, if (4.6) holds, choosing Q E P with o < p,
that B(x,, E) C I/ we can find some E) # @ for each y E B(w,,p), i.e. and taking w = w,, y = w2 in (4.5)
and defining A E M by 1(s) = p(s) for s E [0, a], A(s) = + 00 for s > o we get that for any w E Vn E-‘(U) with V = B(w,,p - a) and for any y E W with A(d(w, y)) < + 00 we have y E B(w,, p) hence e@(w) n I/, E(Y)) 5 N(w,Y)) When ~(d(w,y))
= 44~~)).
= + 00 the inequality (4.5) is obvious.
It is well known (and easy to show) that a multifunction k iff for any x E X the function dE,x(*) given by
n E: W =t X is Lipschitzian with rate
d~,x(w) = d(x, E(w)) is Lipschitzian with rate k. A similar characterization holds for a pseudo-Lipschitzian function and for a pseudo-uniformly continuous multifunction.
multi-
PROPOSITION 4.6. Let E: W =t X be a multifunction and let (w,, x,,) E E. The following assertions are equivalent: (a) E is pseudo-uniformly continuous at (we, x0); (b) there exists some cr, j? in P and some P E M such that for any w,. w2 E B(w,, OL),any x E B(%,P)
I&.&%)
- &,x(%)l
5 P(d(W,, w,)).
When p is linear the preceding property is equivalent to a Lipschitz property of dE: (w, x) + d(x, E(w)) on some neighborhood of (w,, , x0). The proof of this result being a mimic of the proof of [31] theorem 2.3 is omitted. 5. REGULARITY
The following introduction.
definition
extends the concept
of a regular multifunction
given in the
Definition 5.1. A multifunction F: X =t Y is said to be uniformly regular around (x0, yO) E F with modulus P E M if there exist (Y,p in P such that for any x E B(x,, , a), y E B(y, , /I) one has 0,
F-‘(Y))
5 &W(x),
Y)).
(5.1)
Then fi is called a modulus of regularity of F around (x0, yO), and F is said to be p-regular around (x0, yO). Taking p(t) = ct for t E [0, E], p(t) = + 00 for t > E we obtain metric regularity as in definition 1.1. The following result extends the connection between regularity and pseudo-Lipschitzian behavior pointed out in the introduction. F: X =! Y is uniformly regular around (x,,,y,J E F iff F-’ is continuous around (yO, x0). More precisely, the germ of an U.S.C. modulus p
THEOREM 5.2. A multifunction
pseudo-uniformly
J.-P. PENOT
640
is the germ of a modulus of regularity of F around (x0, yo) iff it is the germ of a metric modulus of uniform continuity of F-’ around (Ye, x0). Proof. Let p be a modulus of regularity of F around (x,, , yo) and let CY,p E IP be as in definition 5.1. Let E E JO,p[ and let E, E M be given by A(r) = p(t) for t E [O, E], A(t) = + 03 for t > E. Let us set U = B(x,,cr), V= B(y,,/3 - E), W = Y, wO = yO, E = F-‘. Then for any w E Vtl E-‘(U) and any y E W with d(w,y) E dom 1 we have d(w,Y) I E and d(y,y,) 5 d(Y, w) + d(w, y,,) < E + (p - E) = p hence for each x E E(w) II U d(x, F-‘(Y))
5 ~c(W’(x), Y)) 5 /4&w, Y)) = A(d(w, Y))
as w E F(w). Taking the supremum over x E E(w) fl C?we get M(w)
n U E(Y)) 5 WWYN
for any w E V fl E-‘(U), y E W with d(w, y) E dom A; the case A(d(w, y)) = f 03 is obvious. (b) Conversely let E = F-’ be pseudo-uniformly continuous around (Ye, x0) with modulus A; let lJ E X(x,), V E ‘X(y,) be such that for any w E Vfl E-‘(U) and any y E Y one has e(F-‘(w) n U, F-‘(y)) s A(d(w,y)). We may suppose U = B(x,, p), V = B(y,, a) for some p, o in IP. Let T E iP be such that r 5 CTand A(7) c p. Then, choosing w = ye above, we get for any Y E B(Y,, r) d(xo, F-‘(Y))
I w-‘(Yo)
f-7u, F-‘(Y))
5 A(?) < P
SO that B(x,,, p) t7 F-‘(y) # @ and B(y,, T) c F(U) n V. Let cy = p, P = i7 and let x E B(x,,, a), y E B(y,, 8). Let us define p E M by p(t) = + 00 for t 2 /3, p(t) = X(t) for t E [0, j?[, where f is the U.S.C.hull of A. The inequality
d(x, F-‘(Y))
5 &W(x),
~1)
is obvious when F(x) = (2, or when d(F(x), y) 2 /I. Therefore we may suppose there exists w E F(x) with d(w, y) < p, hence w E B(y,, r). Now for any z E F(x) with z $ &ye, r) we have d(y, z) 2 T - /I = p so that d@‘(x), Y) = d(F(x) n B(Y, 9 r), Y). Taking w E F(x) fl B(y,, 7)arbitrary the inequality e(F-‘(w) using the right continuity of 1, d(x, F-‘(y))
rl U, F-‘(y))
I inflA(d(w, y)): w E F(x) n B(Y,, dl = &W(x),
5 A(d(w, y)) yields, ~1).
and F is p-regular around (x0, yO). n Using proposition 4.3 and the remarks following it one can deduce from theorem 5.2 connections between uniform openness and uniform regularity. It is also quite simple to proceed directly. PROPOSITION5.3.(a) If F: X =! Y is uniformly open around (xc, ye) E F with modulus P then F is uniformly regular around (x0, yo) with a modulus o whose germ at 0 is the germ of the upper-semi-continuous hull p of P. (b) Conversely, if F is uniformly regular around (x0, ye) E F with modulus ,u and if 6 E G is an hypo-semi-inverse of ,Uthen there exists E E P such that F is uniformly open around (x0, ye) with gage y = min(6, E).
Metric regularity, openness and Lipschitzian behavior
641
Proof. (a) Let us suppose (3.2) holds and let us show (5.1) is satisfied with fi replaced by w with w ) [0, E[ = ,i 1[0, E[, o(r) = + COfor r 2 E, with E, 01, B chosen so that B(x,, CY)C I/, B(y,, p + E) c V. Let us suppose on the contrary that there exist x E B(x,, a), y E B(y,, P) with t := d(x, F-‘(y)) > o(r) for r := d(F(x),y). Then, as o(r) < + 00 we have r < E; using lemma 2.7 we can find s > r with s < E, cl(s) < t. Let II E F(x) be such that d(u,y) < s. Then u belongs to B(y,, /3 + E) c V and by (3.2) we have y E B(u, s) c F(B(x, p(s))), so that F-‘(y) n B(x, p(s)) f 0, a contradiction with t > p(s). (b) Let us suppose now (5.1) holds. Let E E IP be such that E < /? and let U = B(x,, a), V = B(y,, p - E). Let us show that for any (u, u) E (I/ x V) n F, for any r E IP f~ y-‘(P) and any y E B(u, y(r)) we have y E F(B(u, r)). First let us note that y E B(y,, /I) since d(y,, u) < /3 - E, d(u,y) c E. As t := d(y, F(u)) s d(y, u) < y(r) I 6(r) and as H(P) C E(6)-’ we have (t, r) E H,(6)-’ C E,(p) or r > p(t). As (5.1) yields p(t) 2 d(u, F-‘(y)) we can find some x E F-‘(y) with d(u, x) < r, so that y E F(B(u, r)). n 6. MISCELLANEOUS
LIPSCHITZIAN
PROPERTIES
OF MULTIFUNCTIONS
Let us us supplement the preceding study with some related remarks. The first one is a slight extension of [ 11, corollary 9.21 and [3 1, theorem 4. l] giving conditions ensuring that closedness and pseudo-Lipschitzian behavior are preserved under composition. Let us recall that a multifunction F: X =t Y between two metric spaces is said to be closed at x0 E X if for any converging sequence ((x,, y,)), _,o of F with x0 = lim x,, one has lim y, E F&J; it is closed on A c X if it is closed at each x,, E A. For A = X this amounts to the closedness of the graph of F. Let us call F lower semicontinuous at (x0, Y,), where x,, E X, Y, c Y if for each sequence (x,,) with limit x0 in X there exists y,, E Ye, an infinite subset J of N and a sequence CYj>jE J with limit y. such that yj E F(Xj) for each j E J. When Ye is a singleton this corresponds to a known concept; when Y, = F(x,,) we simply say that F is semicontinuous at x0. For Y = lRd F is semicontinuous at (xr,, Y) iff lim supIIF(x)II < + 00 where IIF(x)jl = inf( ])y]]:y E F(x))); of x,.
this condition is in particular satisfied if F is bounded on a neighborhood
PROPOSITION 6.1. Let X, Y, 2 be metric spaces, let F: X =t Y, G: Y =t Z be multifunctions and letK:XxZ=t YbegivenbyK(x,z)=F(x)nG-‘(z). (a) If F is closed at x0, if G-r is closed at z. and if K is lower semicontinuous at ((x0, zo), Y) for any z. E Z then H = G 0 F is closed at x0. (b) If F is closed at x0, if G-’ is closed at z. for some z. E H(x,), if K is lower semicontinuous at ((x0, zo), Y) and if F (resp. G) is pseudo-Lipschitzian at (x0, yo) (resp. (yo, zo)) for each y. E K(x, ,z,) then H = G 0 F is pseudo-Lipschitzian at (x0, zo).
Proof. (a) Let ((x, , z,)) be a sequence of H with limit (x0, zo) and let us show z. E H(x,). Since K is lower semicontinuous at ((x0, z,), Y) can find y. c Y, an infinite subset J of N and a sequence with limit y. such that yj E K(xj, .Q) for each j E J. As F is closed at x0 and G-’ is closed at z. we have y, E K(x,, zo). Thus K is semicontinuous at (x0, zo) and z. E G(Y,) C G(F(xo)). (b) Let us show there exists E E P and c E iP such that (Yj)j
E
J
d(z, H(x’))
5 cd(x, x’)
for any x, x’ E B(x, , E), z E H(x) fl B(zO , E).
J.-P. PENOT
642
If this is not the case we can find sequences (x,), (XL), (z,) with limits x0, x0 and z0 respectively with d(t, , H(x;)) z n&x,, XL) > 0 and z, E H(x,,) for each n E IN. Then by what precedes we can find y0 E K(_u,, z,,), an infinite subset J of N and a sequence (yJi EJ with limit y,, such that yj E K(x_, zj) for eachj E .I. Then y, E F(x,,), z0 E G(y,) SO that for some C, C’ in P and for each j E J large enough we have d(y,, F(xj)) I Cd(Xj, xj). Similarly, taking yj E F(x)) with d(yj, yj) I (C + l)d(Xj, Xj’) and observing that (yjl) --*yo, we have d(Zjp G(y,!)) I c’d(yj, yj) I H c’(c + l)d(Xj, x,!), a contradiction with our choice of (x,), (x;), (z,). We mentioned in the introduction that the Clarke strict tangent cone is a key tool for criteria of metric regularity and pseudo-Lipschitzian behavior. Let us observe as a final remark that conversely a pseudo-Lipschitzian behavior simplifies the expression of the strict (Clarke) generalized derivative of a multifunction. Recall that given a multifunction F: X =t Y between two normed vector spaces and (x0, yo) E F, the strict generalized derivative of F at (x0, yo) is given by D’F(XO,YON~
= tu E Y: (~9 u) E G$Xo,
YON,
where $(x0, yo) is the (Clarke) strict tangent cone to F at (x0, yo) given by T>(x,, yo) = lim inf t-‘(F - (x, y)) (see [I]). kYG.y.Ya.uo)
PROPOSITION
6.2. If F: X =I Y is pseudo-Lipschitzian D’F(x, , ye)(u) =
lim inf
at (x0, yo) then F(x + tu) - y
(x.Yt 5 &o.YO)
t
*
Therefore DF’(x,, yo) is defined as a limit of a differential quotient as in the case of a mapping. The proof is similar to the first part of the proof of [32, theorem 3.21. Proof. Let u E D’F(x,, ye)(u) and let ((x,, y,)) 3 (x0, yo), (t,) -+ 0,. By definition of r&x,, yo) we can find a sequence ((u,, u,)) with limit (u, u) such that (x,,, y,) + tn(un, u,) E F for each n E N. As F is pseudo-Lipschitzian at (x0, yo) we can find c E IR, and no E N such that for n r no d(y, + tnu,,, F(x, + t,u)) I cd(x, + r,u,,x,,
Therefore
+ t,u).
we can find (w,,) with IIu,, - w,)) I c)Iu,, - ~11+ t,, hence (w,) -, u, and y, + t, w,, E lim inf t-‘(F(x + tu) - y. The reverse in-
F(w, + 1, u) for each n E N. This shows u E
elusion is obvious.
n
(+.Y)tyo.YO) - +
Remark. Contrary to what occurs when F is a locally Lipschitzian mapping [32] the cone Ti(x,, yo) is not always a vector subspace of X x Y, as simple examples show. REFERENCES 1.
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