MEAN VALUE, UNIVALENCE, AND IMPLICIT FUNCTION THEOREMS

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

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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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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 such that f (b) − f (a) = v kb − ak + θ , hence there exists λ ∈ H 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 such that f (b) − f (a) = v kb − ak + θ , hence there exists λ ∈ H 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

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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 ¯ with Then if [a, b] ⊂ D is such that v = kb−ak , there exists λ ∈ H 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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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

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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

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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 ¯ α) ⊂ U × V and m, M > 0 such that k ∂f (u)k ≤ M on ((U × V ) \ E) ∩ B(z, ∂x ((U × V ) \ E) ∩ B(z, α)). Suppose B(z, α), and l(C) ≥ m for every C ∈ co( ∂f ∂y 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

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(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 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

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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 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,

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−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 ¯ such that f (b) − f (a) = λkb − ak, δw + Cw,π . By Theorem 5 we can find λ ∈ H 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. ¯ α) ⊂ Proof of Theorem 13. Let z = (a, b), α, m, M > 0 be such that B(z, D, k ∂f ∂x (u)k ≤ M for every u ∈ (((U × V ) \ E) ∩ B(z, α)) and l(C) ≥ m for 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 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


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

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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. Bucure¸sti 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 g´ en´ eralis´ e. C. R. Acad. Sci. Paris S´er. A 281 (1975), 795–797.

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[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. V¨ ais¨ al¨ a, 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]