MEAN VALUE, UNIVALENCE, AND IMPLICIT FUNCTION THEOREMS

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1 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 [] 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 mappings W 1,1 loc (D, Rn ) C(D, R n ). We also give in this class some implicit function theorems. AMS 000 Subject Classification: 30C45, 6B10, 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 R m, with D R n open. Such mappings are a.e. differentiable and if E D is such that m n (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(R n, R n ) there exists x p x, x p D \ E such that f (x p ) A}. Here, m q is the q-hausdorff measure in R n. A set A R n has q- dimensional measure if A = A p with m q (A p ) < 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 (009),,

2 13 Mihai Cristea If v S n = {x R n x = 1}, D R n is open, x D and f : D R m is a map, we define { D f,v (x) = w R m there exists t p 0 so that f(x + t } pv) f(x) w, t p the derivative set of the map f at the point x on the direction v, and if A D, we set D f,v (A) = D f,v (x). If D f,v (x) R m and card D f,v (x) = 1, then x A f(x+tv) f(x) there exists lim t 0 t the direction v. We put D + f,v = f (x), the directional derivative of f in x on. If E D is such (x) = lim sup t t 0 exists on D \ E, we define the generalized derivative f(x+tv) f(x) that m n (E) = 0 and f of f at x on the direction v as ( ) f { E (x) = co w R m there exists x p D \ E, x p x such that f } (x p) w, and if A D, we put E ( f )(A) = co x A E ( f f )(x). We see that if bounded near x, then E ( f )(x) is a compact convex subset of Rm. For maps f : D R n R m with D R n open and E D with m n (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(R n, R m ) there exists x p D \ E, x p x such that f (x p ) 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) is a compact convex subset of L(R n, R m ). If A D, we put E f(a) = co E f(x). If D R n is open, x D and f : D R m is a map, we put D + f(x) = lim sup y x x A f(y) f(x), D f(x) = lim inf y x y x f(y) f(x). y x If D R n is open, v S n and f : D R m is continuous, we say that f is v-acl (absolutely continuous on the direction v) if there exists B H v = {x R n x, v = 0} with m n 1 (B) = 0 such that f I x : I x R m is absolutely continuous for every compact interval I x P 1 (x) D and every x H v \ B, where P : R n H v is the projection on H v. If e 1,..., e n is the canonical base in R n and f is e i -ACL for i = 1,..., n, we say as in [, 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 [0, page 6], we see that a continuous map from the is

3 3 Mean value, univalence, and implicit function theorems 133 Sobolev space W 1,1 loc (D, Rm ) is a GACL map. We can also easily see that a locally lipschitzian map is GACL. If A L(R n, R m ), we put A = sup A(x), l(a) = inf A(x). x =1 x =1 We shall prove the following basic mean value theorem, extending some results from [7] and [8]. Theorem 1. Let D R n be open, v S n, E D with m n (E) = 0, f : D R m continuous on D, H R m convex, U D open, [a, b] U such that v =, D f,v((d \ E) U) H and D f,v (x) compact in R m for every x (D \ E) U. Suppose that one of the following conditions holds: 1) f is v-acl. ) f is locally integrable on U and m 1(f(E)) = 0. 3) E is of (n 1)-dimensional measure. Then for every > 0 there exist v H and θ R m with θ < such that f(b) f(a) = v b a + θ, hence there exists λ H such that f(b) f(a) = λ b a. The following consequences of Theorem 1 are obvious. Theorem. Let D R n be open, v S n, E D with m n (E) = 0, f : D R m continuous on D such that f exists on D \ E, U D open, [a, b] U such that v =, H Rm convex such that f ((D\E) U) H. Suppose that one of the following conditions holds: 1) f is v-acl, ) f is locally integrable on U and m 1(f(E)) = 0. 3) E is of (n 1)-dimensional measure. Then for every > 0 there exists v H and θ R m with θ such that f(b) f(a) = v b a + θ, hence there exists λ H such that f(b) f(a) = λ b a. Theorem 3. Let D R n be open, v S n, E D with m n (E) = 0, f : D R m continuous such that there exists L v > 0 with D + f,v (x) L v on D \ E and suppose that one of the following conditions holds: 1) f is v-acl. ) m 1 (f(e)) = 0. 3) E is of (n 1)-dimensional measure. Then if [a, b] D is such that v = L v b a. we have f(b) f(a) Theorem 4. Let D R n be open, v S n, E D with m n (E) = 0, f : D R m continuous such that f exists on D \ E and is locally bounded

4 134 Mihai Cristea 4 on D, [a, b] D such that v = conditions holds: 1) f is v-acl. ) f is locally integrable on D and m 1(f(E)) = 0. 3) E is of (n 1)-dimensional measure. Then there exists λ E ( f and suppose that one of the following )([a, b]) such that f(b) f(a) = λ b a. Theorem 5. Let D R n be open, v S n, E D with m n (E) = 0, f : D R m continuous such that f exists on D\E and there exists H Rm convex for which E ( f )([a, b]) H for every [a, b] D with v =. Suppose that one of the following conditions holds: 1) f is v-acl. ) f is locally integrable and m 1(f(E)) = 0. 3) E is of (n 1)-dimensional measure. Then if [a, b] D is such that v = f(b) f(a) = λ b a., there exists λ H with A known mean value theorem of Lebourg [14] and Pourciau [18] says that if D R n is open, f : D R m is locally lipschitzian, E D is such that m n (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 to be locally bounded. We also have Theorem 6. Let D R n be open, v S n, E D with m n (E) = 0, f : D R m continuous on D and differentiable on D \ E, U D open, Q = co(f ((D \ E) U), [a, b]) U such that v = and suppose that one of the following conditions holds: 1) f is v-acl. ) f is locally integrable on D and m 1(f(E)) = 0. 3) E is of (n 1)-dimensional measure. Then for every > 0 there exists A Q and θ R m with θ such that f(b) f(a) = A (b a) + θ. If f 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 R n R m is locally lipschitzian on D if and only if f is GACL and f exists a.e. and is locally bounded on D. Our Theorem 6 brings some new information if f 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.

5 5 Mean value, univalence, and implicit function theorems 135 The following generalization of Denjoi-Bourbaki s theorem can be proved using the classical proof: Theorem 7. Let E, F be normed spaces, a, b E, v =, K [a, b] at most countable, f : [a, b] F continuous such that there exists M > 0 with D + f,v (x) M for every x [a, b] \ K. Then f(b) f(a) M b a. 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 with v = 1, F a normed space, D E open, K = K n with K n compact sets for n N, f : D F continuous such that there exists L v > 0 with D + f,v (x) L v on D \ K. Then if [a, b] D is such that v =, we have f(b) f(a) L v b a. 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 R n is open, x 0 D, f : D R n is locally lipschitizian on D, E D is such that m n (E) = 0 and f is differentiable on D \ E such that det A 0 for every A E f(x 0 ) (this last condition implies that 0 / E ( f )(x 0) for every v S n ), then f is a local homeomorphism at x 0. We denote for u, v R n \{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 C v,ϕ = {w R n 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 R n R m such that there exists v S n and L v > 0 with D + f,v (x) L v on D. If for such a mapping the condition det A 0 for every A E f(x 0 ) is satisfied at a point x 0 D, we use the fact that E ( f )(x) is a compact subset of Rm for x D, that 0 / E ( f )(x 0) and the upper continuity of the multivalued map x E ( f )(x) to see that there exist r x0 > 0, w S n and δ > 0 such that E ( f )(B(x 0, r x0 )) δw + C w,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 R n be a domain, E D with m n (E) = 0, x 0 D, f : D R m a GACL map such that f exists on D \ E for every v S n, and suppose that there exists r x0 > 0 such that B(x 0, r x0 ) D and n=1

6 136 Mihai Cristea 6 that for every v S n there exist w S n and δ > 0 depending on v such that E ( f )(B(x 0, r x0 )) δw + C w,π. Then f is injective on B(x 0, r x0 ) and if δ = δ x0 does not depend on v S n, then f(b) f(a) δ x0 b a for every a, b B(x 0, r x0 ). A known global inversion theorem of Hadamard, Levy and John [4], [1] says that if E, F are Banach spaces and f : E F is a local homeomorphism such that there exists ω : [0, ) [0, ) continuous with D f(x) 1 ω( x ) 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 R n be such that m n (E) = 0, f : R n R n a GACL map such that f exists on D \E for every v Sn and let ω : [0, ) [0, ) be continuous such that ds 1 ω(s) =. Suppose that for every x 0 R n there exists r x0 > 0 such that for every v S n there exists w S n depending on v such that E ( f )(B(x 1 0, r x0 )) ω( x 0 ) w + C w,π. Then f : R n R n is a homeomorphism. Theorem 11. Let D, F be domains in R n, F simply connected, E D with m n (E) = 0, f : D F a GACL map which is closed or proper such that f exists on D \ E for every v Sn. Suppose that for every x 0 D there exists r x0 > 0 such that B(x 0, r x0 ) D and for every v S n there exist w S n and δ > 0 depending on v such that E ( f )(B(x 0, r x0 )) δw + C ω,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 on D, then f is univalent on D. Here, a domain D R n is ϕ-angular convex, with 0 ϕ < π, if for every z 1, z D there exists z 3 D such that [z 1, z 3 ] [z, z 3 ] D and a(z 1 z 3, z z 3 ) 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 D f,v (x) are supposed to be compact in R m for every x D. The theorem

7 7 Mean value, univalence, and implicit function theorems 137 of Rademacher and Stepanow shows that there exists E D with m n (E) = 0 such that f exists on D \ E. The compactness of the sets D f,v(x) for every x D implies that f is locally bounded on D, hence the sets E( f )(x) are compact in R m 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 on D and for which the sets E ( f )(x) may be unbounded for some points x D. Theorem 1. Let 0 < ϕ < π, ψ = π ϕ, D Rn a ϕ-angular convex domain, E D with m n (E) = 0, f : D R m a GACL map such that f 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 [a, b] D with v = )([a, b]) δv + C v,ψ for every. 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 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 continuous Sobolev maps from W 1,1 loc (D, Rm ) (see [0, page 6]) and its well known subclass of quasiregular mappings (see [0] 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 R n R m with m n. Also, we can replace in Theorems 9, 10, 11, 1 the condition f is a GACL map by one of the conditions f is locally integrable on D for every v Sn and m 1 (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 R n and V R m be open, E U V such that m n+m (E) = 0, f : U V R m continuous on U V and differentiable on (U V ) \ E such that for every z = (x, y) U V there exist α > 0 with B(z, α) U V and m, M > 0 such that f x (u) M on ((U V ) \ E) B(z, α), and l(c) m for every C co( f ((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 R n and V R m be open, E U V such that m n+m (E) = 0, f : U V R m continuous on U V and differentiable on

8 138 Mihai Cristea 8 (U V )\E such that f f x and are locally bounded on U V and det C 0 for every C E ( f )(z) 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], [3]. We end with an implicit function theorem for Sobolev mappings. Theorem 15. Let U R n and V R m be open, f W 1,m+n loc (U V, R m ) continuous such that det( f (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) B(b, δ) such that ϕ(a) = b, ϕ W 1,1 loc (B(a, r), Rm ) and f(x, ϕ(x)) = f(a, b) for every x B(a, r).. PROOFS Proof of Theorem 1. Let 0 < < 1 1 be such that f(a) < 16, f(b) < We can find a B(a, ), b B(b, ) such that f(a ) f(a) < 3, f(b ) f(b) < 3, [a, b ] U, b a = b a, v = b a b and a m 1 ([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 ] R m is absolutely continuous. Hence we can find 0 < δ < such that if 0 a 0 < b 0 < a 1 < b 1 <,..., < a m < b m = 1 are such that m (b q a q ) < δ, then m h (b q ) h (a q ) < 16. Since m 1 ([a, b ] E) = 0 and D f,v (x) is compact for every x [a, b ]\E, we can use Lemma 1 from [7] to find intervals I q = (c q, d q ), q = 0, 1,..., m such that 1 m (d q c q ) < δ and every interval I q can be covered by intervals 8 ) I qk = (t qk qk, t qk + qk ) such that (f(g (t qk )+tv) f(g (t qk ))/t B(H, for 0 < t < qk, k = 0, 1,..., k(q), q = 0, 1,..., m, and 0 c 0 < d 0 < c 1 < d 1 <,..., < c m < d m 1. We can also suppose that we can find at least a point α qk I qk I q(k 1) for k = 1,..., k(q), q = 0, 1,..., m. Let s q k = t qk, k = 0, 1,..., k(q), q = 0, 1,..., m, s q k 1 = α qk, k = 1,..., k(q), q = 0, 1,..., m. We can suppose that s q k+1 sq k 0 for k = 0, 1,..., k(q) 1, q = 0, 1,..., m, and that c q = t q0 = s q 0, d q = t qk(q) = s q k(q), q = 0, 1,..., m.

9 9 Mean value, univalence, and implicit function theorems 139 Let Z qk = h(sq k+1 ) h(sq k ) s q k+1 sq k Z q = k(q) 1 k=0 for q = 0, 1,..., k(q) 1, q = 0, 1,..., m, and let (s q k+1 sq k ) d q c q Z qk for q = 0, 1,..., m. Then Z qk b a B(H, 8 ) for k = 0, 1,..., k(q) 1, q = 0, 1,..., m, hence Z q b a B(H, q = 0, 1,..., m. Let ρ = m 1 (h (c q+1 ) h (d q )) and λ = m µ < 8 ) for ( m 1. (d q c q )) Since d q c q λ Z q b a B(H, 8 ), we can find v H and µ R m with 8 such that m and 1 < λ < 1 1 δ < 1 1 b a µ. We have f(b) f(a) f(b) f(a) f(b) f(b ) λ + f(a) f(a) λ We have f(b ) f(a ) = m d q c q λ Z q = (v +µ ). We see that ρ < 16 < 1 +. Take θ = f(b) f(a) f(b) f(a) ρ λ + λ (λ 1 ) λ f(b) + λ 1 λ f(a) + 16 <, and this implies that θ <. (h (d q ) h (c q )) + m 1 (h (c q+1 ) h (d q )) = ( m ) d λ q c q λ Z q + ρ = λ v b a + λ µ b a + ρ, hence f(b) f(a) = v b a + θ. Suppose now that condition ) holds. Let H v = {x R n x, v = 0} and P : R n H v the projection on H v. By the theorem of Fubini, there exists B H v with m n 1 (B) = 0, so that m 1 (P 1 (y) E) = 0 and D + f,v (x) < on P 1 (y)\e for every y H v \B, and the theorem of Rademacher and Stepanov implies that f exists a.e. in P 1 (y) D for every y H v \ B. This implies that if F = {x D f does not exist at x}, then m n(f ) = 0. Since f is locally integrable on D, we can choose a, b as before such that in addition m 1 ([a, b ] F ) = 0 and f is integrable on [a, b ], i.e., h is integrable on f(x+tv) f(x) [0, 1]. Since lim sup t < on [a, b ] \ E, we have m 1 (f(a)) = 0 t 0 for every A [a, b ] \ E with m 1 (A) = 0, and since m 1 (f(e)) = 0, we have m 1 (h (A)) = 0 for every A [0, 1] with m 1 (A) = 0 (i.e. h satisfies condition (N) on [0, 1]). We use now a theorem of Bary [1, page 85] 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 [, page 106], we can take a, b such that the set E = [a, b ] E is at most

10 140 Mihai Cristea 10 countable for > 0. Let E = (a i ) i N, E = g (F ), with a i = g (t i ), i N. Let J i be open intervals centered at a i, so that ω(f, J i ) < for every i+7 i N and l(j i ) b a. Let I i [0, 1] be such that J i = g (I i ) for i=1 every i N. If t / F, since D f,v (x) is compact on [a, b ]\E, we can use Lemma 1 from [7] to find α t > 0 such that f(g(t)+sv) f(g(t)) s B(H, 8 ) for 0 < s < α t and let B = (t α t, t + α t ). Then [0, 1] \ B is a compact subset of t [0,1]\F F, hence [0, 1] \ B I i. Extracting if necessary a finite subcovering, we i=1 can find 0 < c 0 < d 0 < c 1 < d 1 <,..., < c m < d m 1 such that every interval [d q, c q+1 ] is the union of a finite number of intervals I i for q = 0, 1,..., m 1, and [0, 1] \ B m 1 [d q, c q+1 ]. Since ω(f, J i ) < i+7 16, we have m 1 h (c q+1 ) h (d q ) i=1 i=1 16. Let I q = (c q, d q ) for q = 0, 1,..., m. Then 1 m (d q c q ) l(i i ) < and every interval I q can be covered by intervals i=1 I qk = (t qk qk, t qk + qk ) such that f(q(t qk)+tv) f(g (t qk )) t B(H, 8 ) for 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 ([a, b])). Then H is a compact convex subset of R m and for > 0, we can find δ > 0 such that f ((D \ E) B([a, b], δ )) B(H, ). Let H = co( f ((D \ E) B([a, b], δ ))) for > 0. Then H B(H, ) for > 0. By Theorem 1, we can find v H and θ R m with θ < such that f(b) f(a) = v b a + θ and, letting 0, we can find λ H such that f(b) f(a) = λ b a. 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] R be defined by F (x, y) = (f(x), 0) for x, y [0, 1]. Then F is not e 1 -ACL and there exists E [0, 1] with m (E) = 0, so that F is differentiable on [0, 1] \ E and F (z) = 0 on [0, 1] \ E. We see that E F ([0, 1] ) = {0} and F (1, 0) F (0, 0) = 1, hence conditions ) and 3) imposed in Theorems 1,, 3, 4, 5, 6 are necessary. Proof of Theorem 6. Let H = Q(v) = {w R m there exists A Q such that w = A(v)}. We use Theorem 1 to see that for > 0 there exist A Q and θ R m with θ < such that f(b) f(a) = A ()+θ for [a, b] D with v =. If f is locally bounded on D, we use Theorem 4.

11 11 Mean value, univalence, and implicit function theorems 141 Proof of Theorem { 7. Let K = {r n } n N and α s = a+s() for s [0, 1]. 1 Let > 0 and A = t [0, 1] f(α s ) f(a) (M +) α s a + n r } n [a,α s) for every s [0, t). Then A 0, A is an interval and let c = sup A. 1 Then f(α c ) f(a) (M + ) α c a +, hence A n = [0, c]. r n [a,α c) Suppose that 0 < c < 1. If α c / K, we can find δ > 0 such that c < c + δ < 1 and f(α t ) f(α c ) (M + ) α t α c for c t < c + δ, hence f(α t ) f(a) f(α t ) f(α c ) + f(α c ) f(a) (M +)( α t α c + α c 1 1 a ) + (M + ) α n t a + for every t [c, c + δ). n r n [a,α c) r n [a,α t) If α c K, α c = r m, we use the continuity of f at α c to find δ > 0 such that c < c + δ < 1 and f(α t ) f(α c ) for t [c, c + δ). Then m f(α t ) f(a) f(α t ) f(α c ) + f(α c ) f(a) + (M + ) α m c a (M + ) α n t a +. n r n [a,α c) r n [a,α t) 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 f(b) f(a) M b a. 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 a p B(a, 1 p ) \ M, b p B(b, 1 p ) \ M be such that [a p, b p ] D and v = bp ap b. Then [a p a p p, b p ] K = and using Theorem 7 on [a p, b p ] we find that f(b p ) f(a p ) L v b p a p. Letting p, we obtain f(b) f(a) L v b a. 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) c b a ), and that D + f,v (x) L on D \ E for every v with v = 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) c b a, and let = (0 = t 0 < t 1 <,..., < t m = 1) D([0, 1]) be such that [γ(t k ), γ(t k+1 )] D for k = 0, 1,..., m 1. Then f(b) f(a) m 1 f(γ tk+1 ) f(γ tk ) L m 1 γ(t k+1 ) γ(t k ) Ll(γ) Lc b a. k=0 k=0 Also, if in Theorem 3 we take D = {x R n α i < x i < β i, i = 1,..., n} and there exists L > 0 such that D + f,e i (x) L on D \E for i = 1,..., n, then f is nl-lipschitz on D. Indeed, let a, b D, a = (a 1,..., a n ), b = (b 1,..., b n ), and let z i = (b 1,..., b i 1, b i, a i+1,..., a n ) for i = 0, 1,..., n, with z 0 = a,

12 14 Mihai Cristea 1 z n = b. Then z i D for i = 0, 1,..., n 1, e i+1 = and we have f(b) f(a) n 1 L n b i a i nl b a. i=1 i=0 f(z i+1 f(z i ) L n 1 Proof of Theorem 9. Let a, b B(x 0, r x0 ) be such that v = z i+1 z i z i+1 z i for i = 0, 1,..., n 1 z i+1 z i = i=0 and let w S n and δ > 0 be such that E ( f )(B(x 0, r x0 )) δw + C w,π and let H = δw +C w,π. By Theorem 5 we can find λ H such that f(b) f(a) = λ, hence f(b) f(a) δ b a. This implies that f is injective on B(x 0, r x0 ). If δ does not depend on the direction v, it is obvious that D f(x 0 ) δ x0. Proof of Theorem 10. It follows from Theorem 9 that f is a local homeomorphism and D f(x) 1 ω( x ) for every x Rn. By Theorem 6 from [4], f : R n R n 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 1. Let z 1, z D, z 1 z be such that f(z 1 ) = f(z ). Since D is ϕ-angular-convex, there exists z 3 D such that [z 1, z 3 ] [z, z 3 ] D and a(z z 3, z 1 z 3 ) π ϕ. Let u = z 1 z 3 z 1 z 3, v = z z 3 z z 3. The hypothesis and Theorem 5 imply that there exist δ u, δ v > 0 such that f(z 1 ) f(z 3 ) z 1 z 3 δ u u + C u,ψ C u,ψ and f(z ) f(z 3 ) z z 3 δ v v + C v,ψ C v,ψ. Then f(z 1 ) f(z 3 ) = f(z ) f(z 3 ) C u,ψ C v,ψ. This implies that a(z 1 z 3, z z 3 ) = a(u, v) < ψ = π ϕ, and we reached a contradiction. It follows that f(z 1 ) f(z ) for every z 1, z D, hence f is injective on D. Proof of Theorem 13. Let z = (a, b), α, m, M > 0 be such that B(z, α) D, f x (u) M for every u (((U V ) \ E) B(z, α)) and l(c) m for every C co( f ((U V )\E) B(z, α)). Let F : U V Rn R m 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 exist B co( f f x ((U V )\E) B(z, α)) and C co( ((U V )\E) B(z, α)) such that ( ) IdR n 0 A =. B C Let (u, v) R n R m be such that u + v = 1. Then A(u, v) = u + B(u) + C(v). Let 0 < < m and l = min{, M, 1 }. If u M,

13 13 Mean value, univalence, and implicit function theorems 143 we see that A(u, v) u l, hence A(u, v) l. Suppose now M that u M. We have B(u) + C(v) C(v) B(u) M u, hence B(u) + C(v) C(v) m v. In the case v m, we have A(u, v) u = 1 v m l, hence A(u, v) l. In the case v > m, we have A(u, v) B(u) + C(v) (m v ) (m m ) = l, and we also have A(u, v) 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 y, such that F (x) = F (y) and let 0 < < l y x. Using Theorem 6, we can find A co(f ((U V ) \ E) B(z, α)) and θ R n+m such that θ < and F (y) F (x) = A (y x) + θ. It follows that F (y) F (x) = A (y x) + θ A (y x) θ l y x > 0, hence F (y) 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 = (g 1, g ) : 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)) = (g 1 (x, y), f(g 1 (x, y), g (x, y)) + b f(a, b)) for every (x, y) B(a, r) B(b, δ) and we deduce that x = g 1 (x, y) and f(x, g (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) = g (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, g (a, b)) + b f(a, b)) = F (a, g (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 1 l lipschitzian, hence ϕ is 1 l -lipschitzian. Proof of Theorem 14. Let z(= a, b) and α > 0 be such that there exists M > 0 with f f x (u) M, (u) M for every u ((U V )\E) B(z, α). Let Q = {A L(R m, R m ) det A 0}. Then Q is open in L(R m, R m ) and since f f x and are bounded near z, the set E( f )(z) is a compact, convex subset of Q. We can choose α > 0 as before such that there exists δ > 0 with B( E ( f )(z), δ) Q and E( f )(B(z, α)) B( E( f )(z), δ) Q. Let F : U V R m R m, 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 z 1, z 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 is bounded on B(z, α), we can use Theorem 6

14 144 Mihai Cristea 14 to find A E F ([z 1, z ]) such that F (z ) F (z 1 ) = A(z z 1 ). Then ( IdR n 0 A = B C where B E ( f x ([z 1, z ]), C E ( f )([z 1, z ]), hence C E ( f )(B(z, α)) Q. It follows that deta = det C 0 and this implies that F (z ) F (z 1 ). 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 R n R m, F (x, y) = (x, f(x, y)) for (x, y) U V. Then J F (z) > 0 a.e. in U V, F W 1,m+n loc (U V, R n+m ). By Theorem 6.1 in [11, page 150], F is locally invertible with a local inverse in the Sobolev class W 1,1 loc 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. 05 (1975), [] F.H. Clarke, On the inverse function theorem. Pacific. J. Math. 64 (1976), [3] M. Cristea, Some properties of interior mappings. Banach-Mazur s theorem. Rev. Roumaine Math. Pures Appl. 3 (1987), [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), [5] M. Cristea, Local inversion theorems and implicit function theorems without assuming differentiability. Bull. Math. Soc. Sci. Roumanie (N.S.) 36 (199), [6] M. Cristea, A generalization of a theorem of P.T. Mocanu. Rev. Roumaine Math. Pures Appl. 43 (1998), [7] M. Cristea, A condition of injectivity on a ϕ-angular convex domain, An. Univ. Bucureşti Mat. 49 (000), [8] M. Cristea, Some conditions of injectivity of the sum of two mappings. Mathematica (Cluj) 43(66) (001), [9] M. Cristea, A generalization of some theorems of F.H. Clarke and B.H. Pourciau. Rev. Roumaine Math. Pures Appl. 50 (005), [10] M. Cristea, A note on global implicit function theorem. JIPAM J. Inequal. Pure Appl. Math. 8 (007), 4, Article 100, 15 pp. [11] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford Univ. Press, [1] F. John, On quasiisometric mappings, I. Comm. Pure Appl. Math. 1 (1968), [13] G. Kohr, Certain sufficient conditions of injectivity in C n. Demonstratio Math. 31 (1998), [14] G. Lebourg, Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. A 81 (1975),

15 15 Mean value, univalence, and implicit function theorems 145 [15] V.M. Miklyukov, On maps almost quasi-conformally close to quasi-isometries, Preprint 45, Univ. of Helsinki, 005. [16] P.T. Mocanu, Starlikeness and convexity for non-analytic functions in the unit disk. Mathematica (Cluj) (1980), [17] P.T. Mocanu, A sufficient condition for injectivity in the complex plane. Pure Math. Appl. 6 (1995),, [18] B.H. Pourciau, Hadamard s theorem for locally Lipschitz maps. J. Math. Anal. Appl. 85 (198), [19] M.O. Reade, On Umezawa s criteria for univalence, II. J. Math. Soc. Japan 10 (1958), [0] S. Rickman, Quasiregular Mappings. Ergeb. Math. Grenzgeb. (3) 6. Springer-Verlag, Berlin, [1] S. Saks, Theory of the Integral. Dover Publications, New York, [] J. Väisälä, Lectures on n-dimensional Quasiconformal Mappings. Lecture Notes in Math. 9. Springer-Verlag, [3] I.V. Zhuravlev, A.Iu. Igumnov and V.M. Miklyukov, On an implicit function theorem, Preprint 346, Univ. of Helsinki, 003. Received June 008 University of Bucharest Faculty of Mathematics and Computer Sciences Str. Academiei Bucharest, Romania, mcristea@fmi.unibuc.ro