MEAN VALUE, UNIVALENCE,
AND IMPLICIT FUNCTION THEOREMS
MIHAI CRISTEA
We establish some mean value theorems concerning the generalized derivative on
a direction in the sense of Clarke, in connection with a mean value theorem of
Lebourg [14] and Pourciau [18] for locally lipschitzian maps. We use the results
to generalize the lipschitzian local inversion theorem of Clarke [2] and give global
univalence results of Hadamard-Levy-John type, extending earlier results from
[4] and [9]. We prove some extensions of some known univalence theorems of
Warschawski and Reade from complex univalence theory. Our extensions hold
for a class of mappings defined by a generalized ACL property, containing the
locally lipschitzian mappings, the quasiregular mappings, and the space of Sobolev
1,1
mappings Wloc
(D, Rn ) ∩ C(D, Rn ). We also give in this class some implicit
function theorems.
AMS 2000 Subject Classification: 30C45, 26B10, 30C65.
Key words: mean value, local and global univalence, implicit function theorem.
1. INTRODUCTION
An extensive literature has been devoted in the last 30 years to the socalled generalized derivative of Clarke, whose natural setting is in the class
of locally lipschitzian mappings f : D → Rm , with D ⊂ Rn open. Such
mappings are a.e. differentiable and if E ⊂ D is such that mn (E) = 0 and
f is differentiable on D \ E, the generalized derivative ∂E f (x) of f at x is
defined as
co{A ∈ L(Rn , Rn ) | there exists xp → x, xp ∈ D \ E such that f ‘(xp ) → A}.
Here, mq is the q-Hausdorff measure in Rn . A set A ⊂ Rn has q∞
S
dimensional measure if A =
Ap with mq (Ap ) < ∞ for every p ∈ N. The
p=1
generalized derivative ∂E f (x) is defined at all points x ∈ D, although f is only
a.e. differentiable. However, ∂E f (x) does not usually reduce to the ordinary
derivative f ‘(x) because f ‘ may be discontinuous at x.
REV. ROUMAINE MATH. PURES APPL., 54 (2009), 2, 131–145
132
Mihai Cristea
2
If v ∈ S n = {x ∈ Rn | kxk = 1}, D ⊂ Rn is open, x ∈ D and
f : D → Rm is a map, we define
f (x + tp v) − f (x)
m
→w ,
Df,v (x) = w ∈ R | there exists tp → 0 so that
tp
the derivative set ofS
the map f at the point x on the direction v, and if A ⊂ D,
we set Df,v (A) =
Df,v (x). If Df,v (x) ⊂ Rm and card Df,v (x) = 1, then
x∈A
f (x+tv)−f (x)
= ∂f
t
∂v (x), the directional derivative of
t→0
(x) +
the direction v. We put Df,v
(x) = lim sup f (x+tv)−f
. If E ⊂
t
t→0
that mn (E) = 0 and ∂f
∂v exists on D \ E, we define the generalized
there exists lim
f in x on
D is such
derivative
of f at x on the direction v as
n
∂f
∂E
(x) = co w ∈ Rm | there exists xp ∈ D \ E, xp → x
∂v
o
∂f
(xp ) → w ,
such that
∂v
S
∂f
∂f
and if A ⊂ D, we put ∂E ( ∂v )(A) = co
∂E ( ∂f
∂v )(x). We see that if ∂v is
x∈A
m
bounded near x, then ∂E ( ∂f
∂v )(x) is a compact convex subset of R .
For maps f : D ⊂ Rn → Rm with D ⊂ Rn open and E ⊂ D with
mn (E) = 0 such that f is differentiable on D \ E, we can also define the
generalized derivative of f at x in the sense of Clarke as
∂E f (x) = co{A ∈ L(Rn , Rm ) | there exists xp ∈ D \ E, xp → x
such that f ‘(xp ) → A},
since the definition is consistent even if f ‘ is not bounded near x. But if f ‘
is bounded near x, then ∂E f (x) S
is a compact convex subset of L(Rn , Rm ).
∂E f (x). If D ⊂ Rn is open, x ∈ D and
If A ⊂ D, we put ∂E f (A) = co
x∈A
f : D → Rm is a map, we put
kf (y) − f (x)k
D+ f (x) = lim sup
,
ky − xk
y→x
D− f (x) = lim inf
y→x
kf (y) − f (x)k
.
ky − xk
If D ⊂ Rn is open, v ∈ S n and f : D → Rm is continuous, we say
that f is v-ACL (absolutely continuous on the direction v) if there exists
B ⊂ Hv = {x ∈ Rn | hx, vi = 0} with mn−1 (B) = 0 such that f |Ix : Ix → Rm
is absolutely continuous for every compact interval Ix ⊂ P −1 (x) ∩ D and every
x ∈ Hv \ B, where P : Rn → Hv is the projection on Hv . If e1 , . . . , en is the
canonical base in Rn and f is ei -ACL for i = 1, . . . , n, we say as in [22, page 88]
that f is ACL, and if f is v-ACL for every v ∈ S n , we say as in [9] that f
is a GACL map. Using [20, page 6], we see that a continuous map from the
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Mean value, univalence, and implicit function theorems
133
1,1
Sobolev space Wloc
(D, Rm ) is a GACL map. We can also easily see that a
locally lipschitzian map is GACL. If A ∈ L(Rn , Rm ), we put
kAk = sup kA(x)k,
kxk=1
l(A) = inf kA(x)k.
kxk=1
We shall prove the following basic mean value theorem, extending some
results from [7] and [8].
Theorem 1. Let D ⊂ Rn be open, v ∈ S n , E ⊂ D with mn (E) = 0,
f : D → Rm continuous on D, H ⊂ Rm convex, U ⊂ D open, [a, b] ⊂ U such
b−a
that v = kb−ak
, Df,v ((D \ E) ∩ U ) ⊂ H and Df,v (x) compact in Rm for every
x ∈ (D \ E) ∩ U . Suppose that one of the following conditions holds:
1) f is v-ACL.
2) ∂f
∂v is locally integrable on U and m1 (f (E)) = 0.
3) E is of (n − 1)-dimensional measure.
Then for every > 0 there exist v ∈ H and θ ∈ Rm with kθ k < such that f (b) − f (a) = v kb − ak + θ , hence there exists λ ∈ H̄ such that
f (b) − f (a) = λkb − ak.
The following consequences of Theorem 1 are obvious.
Theorem 2. Let D ⊂ Rn be open, v ∈ S n , E ⊂ D with mn (E) = 0,
f : D → Rm continuous on D such that ∂f
∂v exists on D \ E, U ⊂ D open,
b−a
m
[a, b] ⊂ U such that v = kb−ak , H ⊂ R convex such that ∂f
∂v ((D\E)∩U ) ⊂ H.
Suppose that one of the following conditions holds:
1) f is v-ACL,
2) ∂f
∂v is locally integrable on U and m1 (f (E)) = 0.
3) E is of (n − 1)-dimensional measure.
Then for every > 0 there exists v ∈ H and θ ∈ Rm with kθ k ≤ such that f (b) − f (a) = v kb − ak + θ , hence there exists λ ∈ H̄ such that
f (b) − f (a) = λkb − ak.
Theorem 3. Let D ⊂ Rn be open, v ∈ S n , E ⊂ D with mn (E) = 0,
+
f : D → Rm continuous such that there exists Lv > 0 with Df,v
(x) ≤ Lv on
D \ E and suppose that one of the following conditions holds:
1) f is v-ACL.
2) m1 (f (E)) = 0.
3) E is of (n − 1)-dimensional measure.
b−a
Then if [a, b] ⊂ D is such that v = kb−ak
we have kf (b) − f (a)k ≤
Lv kb − ak.
Theorem 4. Let D ⊂ Rn be open, v ∈ S n , E ⊂ D with mn (E) = 0,
f : D → Rm continuous such that ∂f
∂v exists on D \ E and is locally bounded
134
Mihai Cristea
4
b−a
on D, [a, b] ⊂ D such that v = kb−ak
and suppose that one of the following
conditions holds:
1) f is v-ACL.
2) ∂f
∂v is locally integrable on D and m1 (f (E)) = 0.
3) E is of (n − 1)-dimensional measure.
Then there exists λ ∈ ∂E ( ∂f
∂v )([a, b]) such that f (b) − f (a) = λkb − ak.
Theorem 5. Let D ⊂ Rn be open, v ∈ S n , E ⊂ D with mn (E) = 0,
m
f : D → Rm continuous such that ∂f
∂v exists on D \E and there exists H ⊂ R
b−a
convex for which ∂E ( ∂f
∂v )([a, b]) ⊂ H for every [a, b] ⊂ D with v = kb−ak .
Suppose that one of the following conditions holds:
1) f is v-ACL.
2) ∂f
∂v is locally integrable and m1 (f (E)) = 0.
3) E is of (n − 1)-dimensional measure.
b−a
Then if [a, b] ⊂ D is such that v = kb−ak
, there exists λ ∈ H̄ with
f (b) − f (a) = λkb − ak.
A known mean value theorem of Lebourg [14] and Pourciau [18] says that
if D ⊂ Rn is open, f : D → Rm is locally lipschitzian, E ⊂ D is such that
mn (E) = 0 and f is differentiable on D \ E, then, for [a, b] ⊂ D, there exists
A ∈ ∂E f ([a, b]) such that f (b) − f (a) = A(b − a). The preceding theorems are
the corresponding versions for v-ACL mappings. We notice that in Theorem 5
we do not ask the derivative ∂f
∂v to be locally bounded. We also have
Theorem 6. Let D ⊂ Rn be open, v ∈ S n , E ⊂ D with mn (E) = 0,
f : D → Rm continuous on D and differentiable on D \ E, U ⊂ D open,
b−a
Q = co(f ‘((D \ E) ∩ U ), [a, b]) ⊂ U such that v = kb−ak
and suppose that one
of the following conditions holds:
1) f is v-ACL.
2) ∂f
∂v is locally integrable on D and m1 (f (E)) = 0.
3) E is of (n − 1)-dimensional measure.
Then for every > 0 there exists A ∈ Q and θ ∈ Rm with kθ k ≤ such that f (b) − f (a) = A (b − a) + θ . If f 0 is locally bounded on D, we can
find A ∈ ∂E f ([a, b]) such that f (b) − f (a) = A(b − a).
We know that f : D ⊂ Rn → Rm is locally lipschitzian on D if and only
if f is GACL and f 0 exists a.e. and is locally bounded on D. Our Theorem 6
brings some new information if f 0 is locally bounded and f is not a GACL
map, and this may happen in case 1), when we ask f to be v-ACL only on
the direction v, and in case 3), when we only ask the “singular” set E to be
“thin” enough, i.e., to be of (n − 1)-dimensional measure.
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Mean value, univalence, and implicit function theorems
135
The following generalization of Denjoi-Bourbaki’s theorem can be proved
using the classical proof:
b−a
Theorem 7. Let E, F be normed spaces, a, b ∈ E, v = kb−ak
, K ⊂ [a, b]
at most countable, f : [a, b] → F continuous such that there exists M > 0 with
+
Df,v
(x) ≤ M for every x ∈ [a, b] \ K. Then kf (b) − f (a)k ≤ M kb − ak.
Using Theorem 7 can prove the following infinite dimensional version of
Theorem 3.
Theorem 8. Let E be an infinite dimensional Banach space, v ∈ E
∞
S
with kvk = 1, F a normed space, D ⊂ E open, K =
Kn with Kn compact
n=1
sets for n ∈ N, f : D → F continuous such that there exists Lv > 0 with
+
b−a
Df,v
(x) ≤ Lv on D \ K. Then if [a, b] ⊂ D is such that v = kb−ak
, we have
kf (b) − f (a)k ≤ Lv kb − ak.
The second aim of this paper is to use the preceding mean value theorems to prove some univalence and local univalence results. A known theorem
concerning the theory of the generalized derivative in the sense of Clarke is
the lipschitzian local inversion theorem of Clarke. This theorem says that if
D ⊂ Rn is open, x0 ∈ D, f : D → Rn is locally lipschitizian on D, E ⊂ D is
such that mn (E) = 0 and f is differentiable on D \ E such that det A 6= 0 for
every A ∈ ∂E f (x0 ) (this last condition implies that 0 ∈
/ ∂E ( ∂f
∂v )(x0 ) for every
n
v ∈ S ), then f is a local homeomorphism at x0 . We denote for u, v ∈ Rn \{0}
by a(u, v) the angle between u and v which is less or equal to π, and if v ∈ S n
and 0 ≤ ϕ < π we set Cv,ϕ = {w ∈ Rn | a(v, w) < ϕ}, the cone of direction v
and angle ϕ, centered at 0.
We can easily see that there exist continuous and not locally lipschitzian
mappings f : D ⊂ Rn → Rm such that there exists v ∈ S n and Lv > 0
+
with Df,v
(x) ≤ Lv on D. If for such a mapping the condition “det A 6= 0
for every A ∈ ∂E f (x0 )” is satisfied at a point x0 ∈ D, we use the fact that
m
∂E ( ∂f
/ ∂E ( ∂f
∂v )(x) is a compact subset of R for x ∈ D, that 0 ∈
∂v )(x0 ) and the
∂f
upper continuity of the multivalued map x → ∂E ( ∂v )(x) to see that there exist
rx0 > 0, w ∈ S n and δ > 0 such that ∂E ( ∂f
∂v )(B(x0 , rx0 )) ⊂ δw + Cw,π . This
remark shows that the next theorem is an extension of Clarke’s lipschitzian
local inversion theorem for v-ACL mappings (and also an extension of a result
from [9]).
Theorem 9. Let D ⊂ Rn be a domain, E ⊂ D with mn (E) = 0,
x0 ∈ D, f : D → Rm a GACL map such that ∂f
∂v exists on D \ E for every
v ∈ S n , and suppose that there exists rx0 > 0 such that B(x0 , rx0 ) ⊂ D and
136
Mihai Cristea
6
that for every v ∈ S n there exist w ∈ S n and δ > 0 depending on v such that
∂E ( ∂f
∂v )(B(x0 , rx0 )) ⊂ δw + Cw,π .
Then f is injective on B(x0 , rx0 ) and if δ = δx0 does not depend on
v ∈ S n , then kf (b) − f (a)k ≥ δx0 kb − ak for every a, b ∈ B(x0 , rx0 ).
A known global inversion theorem of Hadamard, Levy and John [4], [12]
says that if E, F are Banach spaces and f : E → F is a local homeomorphism
1
such that there exists ω : [0, ∞) → [0, ∞) continuous with D− f (x) ≥ ω(kxk)
for
every x ∈ E, then f : E → F is a homeomorphism. Cristea [9] gave a version
for a.e. differentiable GACL mappings, extending a result of Pourciau [18].
Another known global inversion theorem of Banach, Mazur and Stoilow [3]
says that if E, F are pathwise connected Hausdorff spaces, F simply connected
and f : E → F is a local homeomorphism which is a proper or a closed map,
then f : E → F is a homeomorphism. A version of this theorem for a.e.
differentiable GACL mappings can be found in [9]. We prove here a version
for GACL mappings not necessarily a.e. differentiable.
Theorem 10. Let E ⊂ Rn be such that mn (E) = 0, f : Rn → Rn a
on D \ E for every v ∈ S n and let ω : [0, ∞) →
GACL map such that ∂f
∂v exists R
∞ ds
[0, ∞) be continuous such that 1 ω(s)
= ∞. Suppose that for every x0 ∈ Rn
there exists rx0 > 0 such that for every v ∈ S n there exists w ∈ S n depending
1
n
n
on v such that ∂E ( ∂f
∂v )(B(x0 , rx0 )) ⊂ ω(kx0 k) w + Cw,π . Then f : R → R is
a homeomorphism.
Theorem 11. Let D, F be domains in Rn , F simply connected, E ⊂ D
with mn (E) = 0, f : D → F a GACL map which is closed or proper such
n
that ∂f
∂v exists on D \ E for every v ∈ S . Suppose that for every x0 ∈ D
there exists rx0 > 0 such that B(x0 , rx0 ) ⊂ D and for every v ∈ S n there exist
w ∈ S n and δ > 0 depending on v such that ∂E ( ∂f
∂v )(B(x0 , rx0 )) ⊂ δw + Cω,π .
Then f : D → F is a homeomorphism.
A basic complex univalence theorem of Warshawski says that if D ⊂ C
is a convex domain and f ∈ H(D) is such that Re f ‘(z) > 0 on D, then f is
univalent on D. The result was generalized by Reade [19], who showed that
if D ⊂ C is a ϕ-angular convex domain with 0 ≤ ϕ < π and f ∈ H(D) is
such that |arg f ‘(z)| < π−ϕ
2 on D, then f is univalent on D. Here, a domain
D ⊂ Rn is ϕ-angular convex, with 0 ≤ ϕ < π, if for every z1 , z2 ∈ D there
exists z3 ∈ D such that [z1 , z3 ] ∪ [z2 , z3 ] ⊂ D and a(z1 − z3 , z2 − z3 ) ≥ π − ϕ,
and we see that a 0-angular convex domain is a convex domain. Mocanu [17,
16] extended these results to C 1 mappings and Cristea [6], [7] and Gabriela
Kohr [13] gave some extensions to continuous mappings. However, in [7] the
sets Df,v (x) are supposed to be compact in Rm for every x ∈ D. The theorem
7
Mean value, univalence, and implicit function theorems
137
of Rademacher and Stepanow shows that there exists E ⊂ D with mn (E) = 0
such that ∂f
∂v exists on D \ E. The compactness of the sets Df,v (x) for every
∂f
x ∈ D implies that ∂f
∂v is locally bounded on D, hence the sets ∂E ( ∂v )(x) are
compact in Rm for every x ∈ D. We shall prove a version of these results in
which we do not suppose the locally boundedness of the derivative ∂f
∂v on D
∂f
and for which the sets ∂E ( ∂v )(x) may be unbounded for some points x ∈ D.
n
Theorem 12. Let 0 < ϕ < π, ψ = π−ϕ
2 , D ⊂ R a ϕ-angular convex
domain, E ⊂ D with mn (E) = 0, f : D → Rm a GACL map such that ∂f
∂v
exists on D \ E for every v ∈ S n , and suppose that for every v ∈ S n there
exists δ > 0 only depending on v such that ∂E ( ∂f
∂v )([a, b]) ⊂ δv + Cv,ψ for every
b−a
[a, b] ⊂ D with v = kb−ak
. Then f is injective on D.
The usefulness of the preceding theorems is that they are valid in the
class of GACL mappings while such maps are not always locally lipschitzian,
neither a.e. differentiable, although the directional derivatives ∂f
∂v exist a.e.
on D for every v ∈ S n (but may be not locally bounded on D). One of the
main subclass of the class of GACL mappings is the important class of con1,1
(D, Rm ) (see [20, page 6]) and its well known
tinuous Sobolev maps from Wloc
subclass of quasiregular mappings (see [20] for a basic monograph regarding
quasiregular mappings), hence our results hold in this classes of mappings.
Also, Theorem 9, which extends the lipschitzian local inversion theorem of
Clarke, holds for mappings f : D ⊂ Rn → Rm with m 6= n. Also, we can replace in Theorems 9, 10, 11, 12 the condition “f is a GACL map” by one of the
n
conditions “ ∂f
∂v is locally integrable on D for every v ∈ S and m1 (f (E)) = 0”
or “E is of (n−1)-dimensional measure”, since in their proofs we use the mean
value Theorem 1.
Finally, we shall use the mean value result from Theorem 6 to prove some
implicit function theorems.
Theorem 13. Let U ⊂ Rn and V ⊂ Rm be open, E ⊂ U × V such that
mn+m (E) = 0, f : U × V → Rm continuous on U × V and differentiable on
(U × V ) \ E such that for every z = (x, y) ∈ U × V there exist α > 0 with
B̄(z, α) ⊂ U × V and m, M > 0 such that k ∂f
∂x (u)k ≤ M on ((U × V ) \ E) ∩
∂f
B(z, α), and l(C) ≥ m for every C ∈ co( ∂y ((U × V ) \ E) ∩ B(z, α)). Suppose
that either f is GACL, or that E is of (m + n − 1)-dimensional measure. Then
for every z = (a, b) ∈ U × V there exist r, δ > 0 and a unique lipschitzian map
ϕ : B(a, r) → B(b, δ) such that ϕ(a) = b and f (x, ϕ(x)) = f (a, b) for every
x ∈ B(a, r).
Theorem 14. Let U ⊂ Rn and V ⊂ Rm be open, E ⊂ U × V such that
mn+m (E) = 0, f : U × V → Rm continuous on U × V and differentiable on
138
Mihai Cristea
(U ×V )\E such that
∂f
∂x
and
∂f
∂y
8
are locally bounded on U ×V and det C 6= 0 for
∂E ( ∂f
∂y )(z)
every C ∈
and every z ∈ U × V . Suppose that either f is GACL, or
that E is of (m+n−1)-dimensional measure. Then for every z = (a, b) ∈ U ×V
there exist r, δ > 0 and a unique lipschitzian map ϕ : B(a, r) → B(b, δ) such
that ϕ(a) = b and f (x, ϕ(x)) = f (a, b) for every x ∈ B(a, r).
Theorems 13 and 14 can be connected to some earlier results of Cristea
[5]. For some other recent results in this area see [10], [15], [23]. We end with
an implicit function theorem for Sobolev mappings.
1,m+n
Theorem 15. Let U ⊂ Rn and V ⊂ Rm be open, f ∈ Wloc
(U ×
∂f
m
V, R ) continuous such that det( ∂y (z)) > 0 a.e. on U × V . Then for a.e.
(a, b) ∈ U × V there exist r, δ > 0 and a unique continuous map ϕ : B(a, r) →
1,1
B(b, δ) such that ϕ(a) = b, ϕ ∈ Wloc
(B(a, r), Rm ) and f (x, ϕ(x)) = f (a, b)
for every x ∈ B(a, r).
2. PROOFS
1
, kf (b)k <
Proof of Theorem 1. Let 0 < < 21 be such that kf (a)k < 16
1
16 . We can find a ∈ B(a, ), b ∈ B(b, ) such that kf (a ) − f (a)k < 32 ,
b −a
kf (b ) − f (b)k < 32 , [a , b ] ⊂ U , kb − a k = kb − ak, v = kb −a k and
m1 ([a , b ] ∩ E) = 0, and let g : [0, 1] → [a , b ], g (t) = a + t(b − a ) for
t ∈ [0, 1] and h = f ◦ g .
Suppose first that f is v-ACL. Then we can choose a , b such that
f |[a , b ] : [a , b ] → Rm is absolutely continuous. Hence we can find 0 <
δ < 2 such that if 0 ≤ a0 < b0 < a1 < b1 <, . . . , < am < bm = 1 are such
m
m
P
P
kh (bq ) − h (aq )k < 16
(bq − aq ) < δ , then
that
.
q=0
q=0
Since m1 ([a , b ]∩E) = 0 and Df,v (x) is compact for every x ∈ [a , b ]\E,
we can use Lemma 1 from [7] to find intervals Iq = (cq , dq ), q = 0, 1, . . . , m
m
P
such that 1− (dq −cq ) < δ2 and every interval Iq can be covered by intervals
q=0
Iqk = (tqk −qk , tqk +qk ) such that (f (g (tqk )+tv)−f (g (tqk ))/t ∈ B(H, 8kb−ak
)
for 0 < |t| < qk , k = 0, 1, . . . , k(q), q = 0, 1, . . . , m, and 0 ≤ c0 < d0 <
c1 < d1 <, . . . , < cm < dm ≤ 1. We can also suppose that we can find at
least a point αqk ∈ Iqk ∩ Iq(k−1) for k = 1, . . . , k(q), q = 0, 1, . . . , m. Let
sq2k = tqk , k = 0, 1, . . . , k(q), q = 0, 1, . . . , m, sq2k−1 = αqk , k = 1, . . . , k(q),
q = 0, 1, . . . , m. We can suppose that sqk+1 − sqk ≥ 0 for k = 0, 1, . . . , 2k(q) − 1,
q = 0, 1, . . . , m, and that cq = tq0 = sq0 , dq = tqk(q) = sq2k(q) , q = 0, 1, . . . , m.
9
Mean value, univalence, and implicit function theorems
h (sqk+1 )−h (sqk )
sqk+1 −sqk
2k(q)−1
P (sqk+1 −sqk )
dq −cq Zqk
k=0
Let Zqk =
Zq =
139
for q = 0, 1, . . . , 2k(q) − 1, q = 0, 1, . . . , m, and let
for q = 0, 1, . . . , m. Then Zqk ∈ kb − akB(H, 8kb−ak
)
for k = 0, 1, . . . , 2k(q) − 1, q = 0, 1, . . . , m, hence Zq ∈ kb − akB(H, 8kb−ak
) for
q = 0, 1, . . . , m.
P
−1
m−1
m
P
. Since
Let ρ =
(h (cq+1 ) − h (dq )) and λ =
(dq − cq )
q=0
m
P
q=0
dq −cq
λ Zq
kµ k <
q=0
∈ kb − akB(H,
8kb−ak
and 1 < λ <
such that
1
1−δ
<
m
P
q=0
1
<
1−2
8kb−ak ),
dq −cq
λ Zq
we can find v ∈ H and µ ∈ Rm with
= kb − ak(v + µ ). We see that kρ k <
(a )k
+ kf (a)−f
≤
λ
f (b )−f (a )−ρ
+
λ
(λ −1)
λ −1
λ kf (b)k + λ kf (a)k +
1 + 22 . Take θ = f (b) − f (a) −
(a )
kb − akµ . We have kf (b) − f (a) − f (b )−f
k≤
λ
kf (b)−f (b )k
λ
16
8
+ 8 + 16
+ 16
< 2 , and this implies that kθ k < .
m
m−1
P
P
We have f (b ) − f (a ) =
(h (dq ) − h (cq )) +
(h (cq+1 ) − h (dq )) =
q=0
q=0
P
m
dq −cq
λ
Z
+ ρ = λ v kb − ak + λ µ kb − ak + ρ , hence f (b) − f (a) =
q
λ
q=0
v kb − ak + θ .
Suppose now that condition 2) holds. Let Hv = {x ∈ Rn | hx, vi = 0}
and P : Rn → Hv the projection on Hv . By the theorem of Fubini, there exists
+
(x) < ∞ on
B ⊂ Hv with mn−1 (B) = 0, so that m1 (P −1 (y) ∩ E) = 0 and Df,v
P −1 (y)\E for every y ∈ Hv \B, and the theorem of Rademacher and Stepanov
−1 (y) ∩ D for every y ∈ H \ B. This implies
implies that ∂f
v
∂v exists a.e. in P
∂f
that if F = {x ∈ D | ∂v does not exist at x}, then mn (F ) = 0. Since ∂f
∂v is
locally integrable on D, we can choose a , b as before such that in addition
m1 ([a , b ] ∩ F ) = 0 and ∂f
∂v is integrable on [a , b ], i.e., h‘ is integrable on
kf (x+tv)−f (x)k
[0, 1]. Since lim sup
< ∞ on [a , b ] \ E, we have m1 (f (A)) = 0
t
t→0
for every A ⊂ [a , b ] \ E with m1 (A) = 0, and since m1 (f (E)) = 0, we have
m1 (h (A)) = 0 for every A ⊂ [0, 1] with m1 (A) = 0 (i.e. h satisfies condition
(N ) on [0, 1]). We use now a theorem of Bary [21, page 285] to deduce that h
is absolutely continuous on [a , b ]. We use from now on the same argument
as before.
Suppose now that condition 3) holds. For A ⊂ D we denote by ω(f, A) =
sup f (x) − inf f (x) the oscillation of f on the set A. Using a theorem of Gross
x∈A
x∈A
[22, page 106], we can take a , b such that the set E = [a , b ] ∩ E is at most
140
Mihai Cristea
10
countable for > 0. Let E = (ai )i∈N , E = g (F ), with ai = g (ti ), i ∈ N.
Let Ji be open intervals centered at ai , so that ω(f, Ji ) < 2i+7
for every
∞
P
i ∈ N and
l(Ji ) ≤ 2 kb − ak. Let Ii ⊂ [0, 1] be such that Ji = g (Ii ) for
i=1
every i ∈ N.
If t ∈
/ F , since Df,v (x) is compact on [a , b ]\E, we can use Lemma 1 from
(g (t))
[7] to find αt > 0 such that f (g (t)+sv)−f
∈ B(H, 8kb−ak
) for 0 < |s| < αt
s
S
and let B =
(t − αt , t + αt ). Then [0, 1] \ B is a compact subset of
t∈[0,1]\F
F , hence [0, 1] \ B ⊂
∞
S
Ii . Extracting if
i=1
d0 < c1 < d1 <, . . . , < cm
necessary a finite subcovering, we
can find 0 < c0 <
< dm ≤ 1 such that every interval
[dq , cq+1 ] is the union of a finite number of intervals Ii for q = 0, 1, . . . , m − 1,
m−1
∞
∞
S
P
P
and [0, 1] \ B ⊂
[dq , cq+1 ]. Since
ω(f, Ji ) ≤
< 16
, we have
2i+7
q=0
m−1
P
kh (cq+1 ) − h (dq )k ≤
q=0
1−
m
P
q=0
(dq −cq ) ≤
∞
P
i=1
16 .
i=1
Let Iq = (cq , dq ) for q = 0, 1, . . . , m. Then
l(Ii ) < 2 and every interval Iq can be covered by intervals
i=1
f (q (t
)+tv)−f (g (t
))
qk
qk
Iqk = (tqk − qk , tqk + qk ) such that
) for
∈ B(H, 8kb−ak
t
0 < |t| < qk , k = 1, . . . , k(q), q = 0, 1, . . . , m. We use from now on the same
argument as before. Proof of Theorem 4. Let H = ∂E ( ∂f
∂v ([a, b])). Then H is a compact
convex subset of Rm and for > 0, we can find δ > 0 such that ∂f
∂v ((D \
∂f
E) ∩ B([a, b], δ )) ⊂ B(H, ). Let H = co( ∂v ((D \ E) ∩ B([a, b], δ ))) for > 0.
Then H ⊂ B(H, ) for > 0. By Theorem 1, we can find v ∈ H and θ ∈ Rm
with kθ k < such that f (b) − f (a) = v kb − ak + θ and, letting → 0, we
can find λ ∈ H such that f (b) − f (a) = λkb − ak.
Example 1. Let f : [0, 1] → [0, 1] be continuous with f (0) = 0, f (1) = 1
and f ‘(t) = 0 a.e. in [0, 1] and let F : [0, 1]2 → R2 be defined by F (x, y) =
(f (x), 0) for x, y ∈ [0, 1]. Then F is not e1 -ACL and there exists E ⊂ [0, 1]2
with m2 (E) = 0, so that F is differentiable on [0, 1]2 \ E and F ‘(z) = 0 on
[0, 1]2 \ E. We see that ∂E F ([0, 1]2 ) = {0} and kF (1, 0) − F (0, 0)k = 1, hence
conditions 2) and 3) imposed in Theorems 1, 2, 3, 4, 5, 6 are necessary.
Proof of Theorem 6. Let H = Q(v) = {w ∈ Rm | there exists A ∈ Q such
that w = A(v)}. We use Theorem 1 to see that for > 0 there exist A ∈ Q
and θ ∈ Rm with kθ k < such that f (b) − f (a) = A (b − a) + θ for [a, b] ⊂ D
b−a
. If f ‘ is locally bounded on D, we use Theorem 4. with v = kb−ak
11
Mean value, univalence, and implicit function theorems
141
Proof of Theorem
∈ [0, 1].
n 7. Let K = {rn }n∈N and αs = a+s(b−a) for s P
1
Let > 0 and A = t ∈ [0, 1] | kf (αs )−f (a)k ≤ (M +)kαs −ak+
2n
rn ∈[a,αs )
o
for every s ∈ [0, t) . Then A 6= 0, A is an interval and let c = sup A .
P
1
Then kf (αc ) − f (a)k ≤ (M + )kαc − ak + 2n , hence A = [0, c].
rn ∈[a,αc )
Suppose that 0 < c < 1. If αc ∈
/ K, we can find δ > 0 such that c <
c + δ < 1 and kf (αt ) − f (αc )k ≤ (M + )kαt − αc k for c ≤ t < c + δ, hence
kf (αt )−f (a)k
P ≤1kf (αt )−f (αc )k+kf (αc )−f (a)k
P ≤1(M +)(kαt −αc k+kαc −
ak) + ≤
(M
+
)kα
−
ak
+
n
t
2
2n for every t ∈ [c, c + δ).
rn ∈[a,αc )
rn ∈[a,αt )
If αc ∈ K, αc = rm , we use the continuity of f at αc to find δ > 0 such
that c < c + δ < 1 and kf (αt ) − f (αc )k ≤ 2m for t ∈ [c, c + δ). Then
kf (αt ) − f (a)k ≤ kf (αt ) − f (αc )k + kf (αc ) − f (a)k ≤ 2m + (M + )kαc − ak +
P
P
1
1
2n ≤ (M + )kαt − ak + 2n .
rn ∈[a,αt )
rn ∈[a,αc )
We obtained in both cases that t ∈ A for every t ∈ [c, c + δ) and this
contradicts the definition of c = sup A . It follows that c = 1 and letting → 0
we get kf (b) − f (a)k ≤ M kb − ak. Proof of Theorem 8. Let M = {w ∈ E | there exists x ∈ K and t ∈ R
such that w = x + tv}. Then M also is a countable union of compact sets and
since dim E = ∞, we have int M = ∅. Let p ∈ N and ap ∈ B(a, p1 ) \ M , bp ∈
b −a
B(b, p1 ) \ M be such that [ap , bp ] ⊂ D and v = kbpp −app k . Then [ap , bp ] ∩ K = ∅
and using Theorem 7 on [ap , bp ] we find that kf (bp ) − f (ap )k ≤ Lv kbp − ap k.
Letting p → ∞, we obtain kf (b) − f (a)k ≤ Lv kb − ak. Remark 1. We can easily obtain some Lipschitz conditions using Theorem 3 or Theorem 8. Suppose that D is a c-convex domain (i.e. for every
a, b ∈ D there exists γ : [0, 1] → D rectifiable such that γ(0) = a, γ(1) = b
+
and l(γ) ≤ ckb − ak), and that Df,v
(x) ≤ L on D \ E for every v with kvk = 1.
If the map f is as in Theorems 3 or 8, than f is cL-Lipschitz on D. Indeed,
let a, b ∈ D and γ : [0, 1] → D a rectifiable path such that γ(0) = a, γ(1) = b
and l(γ) ≤ ckb − ak, and let ∆ = (0 = t0 < t1 <, . . . , < tm = 1) ∈ D([0, 1]) be
such that [γ(tk ), γ(tk+1 )] ⊂ D for k = 0, 1, . . . , m − 1. Then kf (b) − f (a)k ≤
m−1
m−1
P
P
kf (γtk+1 ) − f (γtk )k ≤ L
kγ(tk+1 ) − γ(tk )k ≤ Ll(γ) ≤ Lckb − ak.
k=0
k=0
Also, if in Theorem 3 we take D = {x ∈ Rn | αi < xi < βi , i = 1, . . . , n}
+
and there exists L > 0 such that Df,e
(x) ≤ L on D \ E for i = 1, . . . , n, then f
i
√
is nL-Lipschitz on D. Indeed, let a, b ∈ D, a = (a1 , . . . , an ), b = (b1 , . . . , bn ),
and let zi = (b1 , . . . , bi−1 , bi , ai+1 , . . . , an ) for i = 0, 1, . . . , n, with z0 = a,
142
Mihai Cristea
12
−zi
zn = b. Then zi ∈ D for i = 0, 1, . . . , n − 1, ei+1 = kzzi+1
for i = 0, 1, . . . , n1
i+1 −zi k
n−1
n−1
P
P
kzi+1 − zi k =
kf (zi+1 − f (zi )k ≤ L
and we have kf (b) − f (a)k ≤
L
n
P
|bi − ai | ≤
√
i=0
nLkb − ak.
i=0
i=1
Proof of Theorem 9. Let a, b ∈ B(x0 , rx0 ) be such that v =
Sn
b−a
kb−ak
and let
∂E ( ∂f
∂v )(B(x0 , rx0 ))
⊂ δw + Cw,π and let H =
w∈
and δ > 0 be such that
δw + Cw,π . By Theorem 5 we can find λ ∈ H̄ such that f (b) − f (a) = λkb − ak,
hence kf (b) − f (a)k ≥ δkb − ak. This implies that f is injective on B(x0 , rx0 ).
If δ does not depend on the direction v, it is obvious that D− f (x0 ) ≥ δx0 . Proof of Theorem 10. It follows from Theorem 9 that f is a local homeo1
morphism and D− f (x) ≥ ω(kxk)
for every x ∈ Rn . By Theorem 6 from [4],
n
n
f : R → R is a homeomorphism. Proof of Theorem 11. It follows from Theorem 9 that f is a local homeomorphism which is a proper or a closed map, and we apply Banach-Stoilow’s
theorem (see [3] for a proof). Proof of Theorem 12. Let z1 , z2 ∈ D, z1 6= z2 be such that f (z1 ) =
f (z2 ). Since D is ϕ-angular-convex, there exists z3 ∈ D such that [z1 , z3 ] ∪
z2 −z3
3
[z2 , z3 ] ⊂ D and a(z2 − z3 , z1 − z3 ) ≥ π − ϕ. Let u = kzz11 −z
−z3 k , v = kz2 −z3 k .
The hypothesis and Theorem 5 imply that there exist δu , δv > 0 such that
(z3 )
f (z1 )−f (z3 )
∈ δu u + Cu,ψ ⊂ Cu,ψ and f (zkz22)−f
∈ δv v + Cv,ψ ⊂ Cv,ψ . Then
kz1 −z3 k
−z3 k
f (z1 ) − f (z3 ) = f (z2 ) − f (z3 ) ∈ Cu,ψ ∩ Cv,ψ . This implies that a(z1 − z3 , z2 −
z3 ) = a(u, v) < 2ψ = π − ϕ, and we reached a contradiction. It follows that
f (z1 ) 6= f (z2 ) for every z1 , z2 ∈ D, hence f is injective on D. Proof of Theorem 13. Let z = (a, b), α, m, M > 0 be such that B̄(z, α) ⊂
≤ M for every u ∈ (((U × V ) \ E) ∩ B(z, α)) and l(C) ≥ m for
D,
n
m
every C ∈ co( ∂f
∂y ((U ×V )\E)∩B(z, α)). Let F : U ×V → R ×R be defined
by F (x, y) = (x, f (x, y) + b − f (a, b)) for (x, y) ∈ U × V . Then F is continuous
on U ×V , differentiable on (U ×V )\E, and a GACL map if f is a GACL map.
We show that there exists l > 0 such that l(A) ≥ l for every A ∈ co(F ‘((U ×
V ) \ E) ∩ B(z, α)). Indeed, let A ∈ co(F ‘((U × V ) \ E) ∩ B(z, α)). Then there
∂f
exist B ∈ co( ∂f
∂x ((U × V ) \ E) ∩ B(z, α)) and C ∈ co( ∂y ((U × V ) \ E) ∩ B(z, α))
such that
IdRn 0
A=
.
B
C
k ∂f
∂x (u)k
Let (u, v) ∈ Rn × Rm be such that kuk2 + kvk2 = 1. Then kA(u, v)k2 =
√ and l = min{, , √1 }. If kuk ≥ ,
kuk2 + kB(u) + C(v)k2 . Let 0 < < 2m
M
M
2
2
13
Mean value, univalence, and implicit function theorems
143
2
we see that kA(u, v)k2 ≥ kuk2 ≥ M 2 ≥ l2 , hence kA(u, v)k ≥ l. Suppose now
that kuk ≤ M . We have |kB(u) + C(v)k − kC(v)k| ≤ kB(u)k ≤ M kuk ≤ ,
hence kB(u) + C(v)k ≥ kC(v)k − ≥ mkvk − . In the case kvk ≤ 2
m , we
1
42
2
2
2
2
have kA(u, v)k ≥ kuk = 1 − kvk ≥ 1 − m2 ≥ 2 ≥ l , hence kA(u, v)k ≥ l.
2
2
2
In the case kvk > 2
m , we have kA(u, v)k ≥ kB(u) + C(v)k ≥ (mkvk − ) ≥
2
2
2
2
(m m − ) = ≥ l , and we also have kA(u, v)k ≥ l. It follows that l(A) ≥ l
for every A ∈ co(F ‘((U × V ) \ E) ∩ B(z, α)).
We show now that F is a local homeomorphism at z. Let x, y ∈ B(z, α),
x 6= y, such that F (x) = F (y) and let 0 < < lky − xk. Using Theorem 6,
we can find A ∈ co(F ‘((U × V ) \ E) ∩ B(z, α)) and θ ∈ Rn+m such that
kθ k < and F (y) − F (x) = A (y − x) + θ . It follows that kF (y) − F (x)k =
kA (y − x) + θ k ≥ kA (y − x)k − kθ k ≥ lky − xk − > 0, hence F (y) 6= F (x).
We proved that F is injective on B(z, α) and also that D− F (u) ≥ l on B(z, α).
Let now W ∈ V(z) and δ > 0 such that B(a, δ) ⊂ U, B(b, δ) ⊂ V and
F : B(a, δ) × B(b, δ) → W is a homeomorphism, and let g = (g1 , g2 ) : W →
B(a, δ)×B(b, δ) be its inverse. Let l > 0 be such that B(a, l)×B(b, l) ⊂ W and
r = min{l, δ}. We have (x, y) = F (g(x, y)) = (g1 (x, y), f (g1 (x, y), g2 (x, y)) +
b − f (a, b)) for every (x, y) ∈ B(a, r) × B(b, δ) and we deduce that x = g1 (x, y)
and f (x, g2 (x, y)) = y − b + f (a, b) for every x ∈ B(a, r) and y ∈ B(b, δ).
Define ϕ : B(a, r) → B(b, δ) by ϕ(x) = g2 (x, b) for x ∈ B(a, r). We see
that f (x, ϕ(x)) = f (a, b) for every x ∈ B(a, r). We also see that F (a, b) =
(a, b) = (a, f (a, g2 (a, b)) + b − f (a, b)) = F (a, g2 (a, b)) = F (a, ϕ(a)), and using
the injectivity of the map F on B(a, r) × B(b, δ), we see that ϕ(a) = b. If
ψ : B(a, r) → B(b, δ) is a map such that ψ(a) = b and f (x, ψ(x)) = f (a, b) for
every x ∈ B(a, r), we have F (x, ϕ(x)) = (x, f (x, ϕ(x)) + b − f (a, b)) = (x, b) =
(x, f (x, ψ(x)) + b − f (a, b)) = F (x, ψ(x)) for every x ∈ B(a, r). Using again
the injectivity of the map F on B(a, r) × B(b, δ), we deduce that ϕ(x) = ψ(x)
for every x ∈ B(a, r). Since D− F (u) ≥ l on B(a, r) × B(b, δ), the mapping g
is 1l lipschitzian, hence ϕ is 1l -lipschitzian. Proof of Theorem 14. Let z(= a, b) and α > 0 be such that there exists
∂f
M > 0 with k ∂f
∂x (u)k ≤ M , k ∂y (u)k ≤ M for every u ∈ ((U ×V )\E)∩B(z, α).
Let Q = {A ∈ L(Rm , Rm ) | det A 6= 0}. Then Q is open in L(Rm , Rm )
∂f
∂f
and since ∂f
∂x and ∂y are bounded near z, the set ∂E ( ∂y )(z) is a compact,
convex subset of Q. We can choose α > 0 as before such that there exists
∂f
∂f
δ > 0 with B(∂E ( ∂f
∂y )(z), δ) ⊂ Q and ∂E ( ∂y )(B(z, α)) ⊂ B(∂E ( ∂y )(z), δ) ⊂ Q.
Let F : U × V → Rm × Rm , F (x, y) = (x, f (x, y) + b − f (a, b)) for
(x, y) ∈ U × V . We show that F is injective on B(z, α). Let z1 , z2 ∈ B(z, α).
Since F is continuous on U ×V , differentiable on (U ×V )\E, and a GACL map
if f is and a GACL map and F 0 is bounded on B(z, α), we can use Theorem 6
144
Mihai Cristea
14
to find A ∈ ∂E F ([z1 , z2 ]) such that F (z2 ) − F (z1 ) = A(z2 − z1 ). Then
A=
IdRn
B
0
C
,
∂f
∂f
where B ∈ ∂E ( ∂f
∂x ([z1 , z2 ]), C ∈ ∂E ( ∂y )([z1 , z2 ]), hence C ∈ ∂E ( ∂y )(B(z, α)) ⊂
Q. It follows that detA = det C 6= 0 and this implies that F (z2 ) 6= F (z1 ). We
proved that F is injective on B(z, α) and we argue now as in the proof of
Theorem 13. Proof of Theorem 15. Let F : U × V → Rn × Rm , F (x, y) = (x, f (x, y))
1,m+n
for (x, y) ∈ U × V . Then JF (z) > 0 a.e. in U × V , F ∈ Wloc
(U × V, Rn+m ).
By Theorem 6.1 in [11, page 150], F is locally invertible with a local inverse
1,1
in the Sobolev class Wloc
around a.e. points z ∈ U × V . We argue now as in
the proofs of Theorem 13 and 14. REFERENCES
[1] F.H. Clarke, Generalized gradients and applications. Trans. Amer. Math. Soc. 205
(1975), 247–262.
[2] F.H. Clarke, On the inverse function theorem. Pacific. J. Math. 64 (1976), 97–102.
[3] M. Cristea, Some properties of interior mappings. Banach-Mazur’s theorem. Rev.
Roumaine Math. Pures Appl. 32 (1987), 211–214.
[4] M. Cristea, Some conditions for the openness, local injectivity and global injectivity of a
mapping between two Banach spaces. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 35
(1991), 67–79.
[5] M. Cristea, Local inversion theorems and implicit function theorems without assuming
differentiability. Bull. Math. Soc. Sci. Roumanie (N.S.) 36 (1992), 227–236.
[6] M. Cristea, A generalization of a theorem of P.T. Mocanu. Rev. Roumaine Math. Pures
Appl. 43 (1998), 355–359.
[7] M. Cristea, A condition of injectivity on a ϕ-angular convex domain, An. Univ. Bucureşti Mat. 49 (2000), 127–132.
[8] M. Cristea, Some conditions of injectivity of the sum of two mappings. Mathematica
(Cluj) 43(66) (2001), 23–34.
[9] M. Cristea, A generalization of some theorems of F.H. Clarke and B.H. Pourciau. Rev.
Roumaine Math. Pures Appl. 50 (2005), 137–151.
[10] M. Cristea, A note on global implicit function theorem. JIPAM J. Inequal. Pure Appl.
Math. 8 (2007), 4, Article 100, 15 pp.
[11] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford Univ.
Press, 1995.
[12] F. John, On quasiisometric mappings, I. Comm. Pure Appl. Math. 21 (1968), 77–110.
[13] G. Kohr, Certain sufficient conditions of injectivity in Cn . Demonstratio Math. 31
(1998), 395–404.
[14] G. Lebourg, Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. A
281 (1975), 795–797.
15
Mean value, univalence, and implicit function theorems
145
[15] V.M. Miklyukov, On maps almost quasi-conformally close to quasi-isometries, Preprint
425, Univ. of Helsinki, 2005.
[16] P.T. Mocanu, Starlikeness and convexity for non-analytic functions in the unit disk.
Mathematica (Cluj) 22 (1980), 77–83.
[17] P.T. Mocanu, A sufficient condition for injectivity in the complex plane. Pure Math.
Appl. 6 (1995), 2, 231–238.
[18] B.H. Pourciau, Hadamard’s theorem for locally Lipschitz maps. J. Math. Anal. Appl.
85 (1982), 279–285.
[19] M.O. Reade, On Umezawa’s criteria for univalence, II. J. Math. Soc. Japan 10 (1958),
255–258.
[20] S. Rickman, Quasiregular Mappings. Ergeb. Math. Grenzgeb. (3) 26. Springer-Verlag,
Berlin, 1993.
[21] S. Saks, Theory of the Integral. Dover Publications, New York, 1964.
[22] J. Väisälä, Lectures on n-dimensional Quasiconformal Mappings. Lecture Notes in
Math. 229. Springer-Verlag, 1971.
[23] I.V. Zhuravlev, A.Iu. Igumnov and V.M. Miklyukov, On an implicit function theorem,
Preprint 346, Univ. of Helsinki, 2003.
Received June 2008
University of Bucharest
Faculty of Mathematics and Computer Sciences
Str. Academiei 14
010014 Bucharest, Romania,
[email protected]