global univalence and global inversion theorems in banach spaces

.%onl,neorAno/.vsrs. Theorv, Mrrhods d Applrrolrons. Vol. 13. No. 5, pp. 539-553. IYSY Prcnred in Great Bnrain.

0362-516X/89 $3.00~ IN F 1989 Pergamon Press plc

GLOBAL UNIVALENCE AND GLOBAL INVERSION THEOREMS IN BANACH SPACES

SORIN RLDULESCU

Institutul de Matematica, Str. Academiei 14, R-70.109 Bucuresti, Romania

and

MARIUS RLDULESCU

Centre of Mathematical Statistics, Bd. Magheru 22, R-70.158, Bucurevti, Romania

(Received 7 March 1988; received for publication 27 April 1988)

Key words andphrases: Global univalence theorems, global inversion theorems, global implicit function theorems, logarithmic norm.

1. INTRODUCTION

A GLOBAL univalence theorem is a theorem which provides conditions for a function to be one- to-one on its domain.

A global inversion theorem is a theorem which provides precise conditions for a local homeomorphism (diffeomorphism of Cp class for some p I 1) to be a global one. Here by a global homeomorphism we understand a homeomorphism which is onto.

Global inversion theorems represent a powerful tool for establishing existence and uniqueness theorems for nonlinear equations in Banach spaces.

These theorems may also be used for obtaining elegant proofs of global existence and uniqueness theorems for differential equations [l-5, 9, 18, 28-31, 391.

Applications of global univalence and global inversion theorems may also be found in: mathematical economy [6, 14, 24, 35, 361, algebra [14], statistics [23, 241, mathematical pro- gramming [ 191, general theory of optimisation [17], numerical analysis [ 10, 20, 371, nonlinear circuit theory [15, 33, 341, stability theory [21], differential geometry [3], etc.

The book of Parthasarathy [24] represents an excellent reference for global univalence. Recent results on this topic may be found in [25, 261.

The reader interested in global inversion theorems and their applications may consult [3, 27-31, 37, 391.

In Section 3 of this paper we establish some new global inversion theorems, we generalize a global inversion theorem belonging to Desoer and Haneda [lo] and we use it to obtain some global variants of the implicit function theorem.

In Section 4 of the paper, using topological degree we prove a generalization of Hurwitz’s theorem on sequences of univalent analytic functions. Several global univalence theorems are obtained using this generalization.

Another result of this section extends a global univalence theorem belonging to Gale and Nikaido [13], from the finite dimensional case to the case of Banach spaces of arbitrary dimension,

Section 5 of the paper contains a generalization of a recent result belonging to Vidossich [40] concerning the problem of global univalence.

539

540 S. R~LXJLESCU and M. R.&DULESCU

2. BASIC PROPERTIES OF THE FUNCTIONAL [.,-I+ AND OF THE

LOGARITHMIC NORM

We shall use throughout the paper the following notations. We denote by R the field of real numbers, by Q the set lx E E?: x > 0) and by IK the field IR or the field of complex numbers. If E and Fare two linear normed spaces then Isom(E, F) denotes the set of all linear continuous isomorphisms of E onto F. Let E be a linear normed space. For every x, y E E we define

[x,yl+ = lim 11,~ + *1’ - Ml . f10

Since the map t ++ 11 x + f_ylj is convex it follows that [x, _Y]+ exists for every x, y E E.

If x,y,y,,y, E E then the following relations hold:

- IIYII 5 kYl+ 5 IlYll9

[XY, + Yzl, 5 LGYJ+ + kYzl+,

Lx, 01, = 0, P,Yl+ = Ilull,

[xv tu1+ = w, Yl, 9 t 2 0,

1x9 -Yl+ = I---GYl+,

[x,x1+ = ll-4~ IkY*l+ - kYzl+I 5 IIY, - Y*ll.

In the case where E is a pre-Hilbert space we have

i

(x9 Y> .__ [x, y], = llxll

ifx#O

llrll if x = 0.

Let E be a linear normed space over the field IK. The logarithmic norm ,B on L(E, E) is defined by

,@) = lim 111 + tAII - 1 A E L(E, E).

II0 t ’

Note that if we denote by [. , -1, the functional corresponding to the linear normed space L(E, E), then

P(A) = (1, Al+ f A E L(E, E).

The functional ,LJ has the following properties:

IP(A)I 5 ]IAll,

III(~) = o&4), CY 2 0,

AA + B) 5 P(A) + P(B),

,u(A + CYI) = p(A) + Re CX, ci E IK,

b(A) - N)l 5 llA - Bill

k A-4, 5 ~64) * btl. x E E.

Global univalence and inversion theorems 541

Notice that the functional p is not a norm, since it may take negative values. In the case where E = IK” and (a,), 5 i,jan is the matrix associated with the operator

A E L(E, E) in the canonic base of IK”, considering on E the norms I’, l2 and I”, we have

/c*(A) = max Re ~j Ibjsn

where cri, a2, . . . , cr, denote the eigenvalues of the matrix (A + A*)/2,

MA) = ,zFzn[ Reaii + iiIa;jI].

If A E L(R”, R”) then ,u(-A) < 0 implies det A > 0.

In the case where E is a Hilbert space, we have

p(A) = sup Re(Ah, h). llhll = I

Other useful properties of the logarithmic norm can be found in [8] and [12]. Let E be a linear normed space over IK and A E L(E, E). Denote by a(A) the spectrum of

A and define the functional

v(A) = sup Re a(A).

Note that the following relations hold:

- v(- A) = inf Re a(A)

-p(-A) I - v(-A) I v(A) I p(A),

v(A + al) = v(A) + Re 01, a! E IK,

v(A) = lim r(Z + tA) - 1

fl0 t

where r(Z + tA) denotes the spectral radius of the operator I + tA.

If E is a Hilbert space then p(A) = v((A + A*)/2). Consequently p(A) = v(A) if A is self-adjoint.

In the case where E = IK” and A E L(E, E) satisfies v(A) I 0, we note that

IdetA] z- Iv(A)]“.

3. GLOBAL INVERSION THEOREMS AND GLOBAL IMPLICIT FUNCTION THEOREMS

In [lo], Desoer and Haneda proved the following.

THEOREM 3.1. Let f: F’ -, /R” be a C’ map. If there exists a continuous map c: IR, + lR: such that

3

(0 c(s)dS = +a, (3.1)

0

S. RADULESCU and M. R;~DULESCU

Pu(f’@)) 5 -CM), x E I?“, (3.2)

then f is a C’ global diffeomorphism. The proof given in [lo] is based on the following global inversion theorem belonging to

Hadamard-Palais.

THEOREM 3.2 [22]. Letf: E!” + K?” be a C’ map. Then f is a C’ global diffeomorphism if and only if the following conditions hold:

Wf’(xN f 0, x E R”, (3.3)

lim ]If(x)ll = +oo. (3.4) 11.~11 - m

In the sequel we shall apply a global inversion theorem belonging to Hadamard and Levy to obtain a generalization of Desoer-Haneda’s theorem.

We shall use the following results.

THEOREM 3.3 [29]. Let E, F be two Banach spaces and f: E -+ F a C’ map such that

f’(x) E Isom(E, F), x E E.

If there exists a continuous map w: R, ---t IRz such that

i

“ds -=+m

0 MS)

IILf’(-w’I1 5 dllxll)9 x E E,

then f is a C’ global diffeomorphism.

(3.5)

(3.6)

(3.7)

THEOREM 3.4 [27]. Let E, F be two Banach spaces andf: E -+ F be a C’ map with properties

(3.4) and (3.5). If there exists a continuous map w: lR+ -+ lRr with property (3.7), then f is a C’ global diffeomorphism.

Theorem 3.3 is known as the Hadamard-Levy’s theorem. Here we presented a somewhat more general statement, cf. [9].

LEMMA 3.5. Let E be a linear normed space and A E L(E, E) an operator such that p(A) < 0. Then A is invertible and

(3.8)

Proof. It follows from the definition of the logarithmic norm that for every E E (0, -p(A)) there exists 6, > 0 such that

111 + fAII - 1

t < &I) + & < 0 if 0 c t < a,,

whence llZ+ tAI[ < 1 if t E (0,6,). By tA = (I + tA) - Zand 111 + tAI[ < 1 we have that tA is


invertible for every t E (0,6,). Consequently, A is invertible. If x E E and t E (0,6,) then

_ ,lAx,, = (Il.4 - tll4l) - llxll I IIX + t4l - Ml I III + t‘4ll - 1 llxll I (/@) + E)llxll t t t

Therefore /Ax/i 2 -(p(A) + &)I] ]I f x or every x E E. Since A is invertible we have

llxll 2 - MA) + 8) . lb-‘XII, x E E,

whence llA-‘]l 5 -(l/@(A) + E)). Since E was taken arbitrarily in the interval (0, -p(A)), inequality (3.8) holds.

THEOREM 3.6. Let E be a Banach space,) E + E a C’ map and c: IR+ -+ F?Z a continuous map

with properties (3.1) and (3.2). Then f is a C’ gobal diffeomorphism.

Proof. By (3.2) and lemma 3.5 it follows thatfis a C’ local diffeomorphism. Let o = l/c. Then we have

IIu’wl-‘II 5 ---& 5 -&) = dllxll), x E E. (3.9)

By theorem 3.3. we obtain that f is a C’ global diffeomorphism.

COROLLARY 3.7. Let H be a Hilbert space and f: H + Ha C’ map. If there exists a continuous

map c: E?+ -+ IRf which satisfies condition (3.1) and the condition

IReV’( &I 2 4lxll) * Il~ll*, x, h E H, (3.10)

then f is a global C’ diffeomorphism.

COROLLARY 3.8. Let J IK” ---* IK” a C’ map. If there exists a continuous map c: IF?+ --* lR1; for which condition (3.1) holds and if we have

x E IK”, i E (1,2, . . . . n) (3.11)

the f is a C’ global diffeomorphism. Here IK” is endowed with P-norm.

Proof. Let J = (1,2, . . . , n). By (3.11) we have that Reaf;/axi keeps constant sign on IK” for i E J. Let ei = sgn[Re afi/ax,(x)], x E IK”, i E J. Define gi = - Eifi, i E J and

g = (g,,g,, . . . . g,). By (3.11) we obtain

when p.&‘(x)) -( -44, x E IK”. By theorem 3.6 g is a C’ global diffeomorphism which implies f is a C’ global diffeomorphism.

54-l S. R~DULESCU and M. RADULESCC

THEOREM 3.9. Let E be a Banach space and J E ---t E a C’ map with property (3.4). If there exists a continuous map c: I??, + IR: which satisfies (3.2) thenfis a C’ global diffeomorphism.

Proof. By (3.9) and theorem 3.4 the conclusion of the theorem follows.

THEOREM 3.10. Let f: 1R” --+ iR” be a map of C’ class. If there exists three continuous maps (Iii: il?, --+ F?*,, i = 1, 2, 3 for which:

wj = w;” * co;-‘, (3.12)

-ccl

!

ds

owJo=+oo~ (3.13)

v(f’W) 5 - ~,(ll~ll), X E iR”, (3.14)

Ilf’(X)II 5 ~,tll-a x E I?“, (3.15)

then f is a global C’ diffeomorphism.

Proof. If x E F?” let A,(X), A,(x), . . . . A,(x) denote the eigenvalues of the matrixf’(x). Since for every k E (1, 2, . . . . n) we have

- l&M 5 ReA&) 5 v(f’W) 5 -~,(llxll),

it follows that

]det[f’(x)ll = fi I&A 2 ~~(llxll). k=l

Now we use the following result (see [ll, p. 10201) there exists a constant c, > 0 such that

l/~-i]] s c,]detAj-’ - ll~ll”-’

for every invertible matrix A E L(ll?“, F?“). This enables us to write

IILf’wl-‘II 5 c,. ldetf’(x)l-’ * Ilf’(-4l”-’ 5 c, * ~;“~ll~ll~llf’~~~ll”~’

5 cn * dllxll), XE IF?”

and by theorem 3.3 we obtain that f is a C’ global diffeomorphism. The following theorem is a global variant of the implicit function theorem.

THEOREM 3.11. Let E, F be two Banach spaces and f: E x F + F a C’ map. If there exists a continuous map c: E?, + lR*, which satisfies (3.1) and

P(.&XX,Y)) 5 -CdlYllh (x, Y> E E x F,

then there exists a unique C’ map 9: E x F --+ F such that

Ax, V(X z)) = z (x, z) E E x F.

(3.16)

(3.17)

Proof. By theorem 3.6, the mapf(x, *) is a C’ global diffeomorphism for every x E E.

Consider the map g: E x F x F -+ F, g(x, y, z) = f(x, y) - z, (x, y, z) E E x F x F.


Note that g is a C’ map and g; = f;. By (3.16) and theorem 3.6 there exists a unique map (D: E x F --t F such that g(x, rp(x, z), z) = 0. From the implicit function theorem we obtain cp is a C’ map. Note that rp satisfies (3.17).

COROLLARY 3.12. Let f: IK" x IK" --+ IK" be a C’ map. Suppose that there exists a continuous map c: iR+ -+ R; which satisfies (3.1) and verifies the inequality

(3.18)

for every (x,y) E IK" x IKm and i E (1.2, . . ., m), then there exists a unique C’ map, rp: IK" -+ IK" such that f(x, p(x)) = 0, x E IK".

Here IK” is endowed with /“-norm.

Proof. Let J = (1,2, . . . . ml. By (3.18) the maps Re afii/ayi, i E J have constant sign on IK" x IK'". Let ei = sgn[Re afi/ayi(x, y)], (x, y) E IK” x IK", i E J. Define. gi = - Eifi , i E J and

g = (g, 9 g,, . . . , g,). By (3.18) we have

Re$(x,y) + i %(x,Y) 5 -c(Ilrll), I I 5;; aYj

(x,y) E IK" x IK”, iEJ I

whence UgJx, Y)) 5 - ctllrll), k Y) E IK” x IK”. By theorem 3.11, there exists a unique map (4: iK” -+ IK”’ of C’ class such that g(x, p(x)) = 0, x E IK" whencef(x, v(x)) = 0, x E IK”.

By theorem 3.9 one can easily establish the following global variant of the implicit function theorem.

THEOREM 3.13. Let E, F be two Banach spaces and J E x F + F a C’ map such that

lj~~pJll = +a, x E E. (3.19)

If there exists a continuous map c: IR, + IR, which satisfies (3.16), then there exists a unique C’ map P: E x F --+ F such that (3.17) holds.

Other global variants of the implicit function theorem may be found in [7, 37, 381.

4. GLOBAL UNIVALENCE THEOREMS

First, it is necessary to recall a few known facts.

Definition. A topological space X is said to be zero-dimension if every point of it has a funda- mental system consisting of closed-open neighborhoods. By convention, the empty set has dimension - 1.

One can easily see that every discrete set or countable set (endowed with the relative topology) is zero-dimensional.

Definition. Let X, Y be two topological spaces and let f: X -+ Y be a continuous map. The map f is said to be light if for every y Ed the set f -l(y) is zero-dimensional.

546 S. R~DDVLESCU and hl. RADULESCU

One can easily see that every local homeomorphism is light. Iffis a continuous map such that the inverse image of each point is at most countable set, then f is light. Iff: R” -+ IF?” is a C’ map and the set Z, = lx E R”: det[f’(x)] = O] is discrete then f is light.

Let E be a Banach space. If A S E, put M(A) = (f: A -+ E: (3) g: A + E continuous and

condensing such that f(x) = x + g(x), x E A].

If U is an open bounded subset of E, f E M(O) and y E E - f(aU), then d(f, I/, y) will denote the topological degree off relative to U and y, cf. Lloyd [16].

Let D be an open subset of E. If E is finite dimensional then one can easily see that M(D) = C(D, E).

The following theorem is a generalization of Hurwitz’s theorem from analytic function

theory.

THEOREM 4.1. Let E be a Banach space and D an open subset of E. If f, f,, E M(D), n 2 1, and the following conditions hold.

1” All the maps f,, n L 1, are one-to-one. 2” The sequence (fn),5, converges uniformly on every bounded set to f.

3” f is light. then f is globally univalent.

Proof. Suppose that f is not one-to-one. Then there exists distinct x,, _u, E D such that f(x,) = f(xz) = y. One can easily see that there exists two open, bounded, nonempty subsets U, and r/z of D such that Xi E c/i, i = 1,2 andy E E - (f(aU,) U f(aU,)). Since the set f(aU,) U f(aU,) is closed, there exists r > 0 such that B(y, r) c E - (f(aU,) U f(aU,)). Therefore d(f, U, y) = d(f, Vi, z) for every z E B(y, r) and i = 1, 2.

By 2” there exists n, 2 1 such that B(y, r) rl (f,(aU,) U f,(aU,)) = @ for every n L n,. Note that there exists m I n, such that for every n L m and i E (1,2) we have

d(f, Ui 9 Y) = d(fn 9 vi, Y) = d(fn 7 Ui > fn(xi)) # 0.

Hence, for every n L m, the equation f,(x) = y has at least a solution in U, and at least a solution in U,.

This contradicts condition 1 O. Therefore f is globally univalent.

COROLLARY 4.2. Let D be an open set of lR” and let f,& E C’(D, R”), j 1 1. Suppose that the following conditions hold.

(i) All the maps&, j 2 1 are one-to-one. (ii) The sequence (jJjz, converges uniformly on every compact of D.

(iii) The set Z, = lx E D: det[f ‘(x)] = 0) is at most countable. Then f is globally univalent.

THEOREM 4.3. Let E be a real Hilbert space, D an open set of E, and f E M(D) a light map. If f is monotone, then f is globally univalent.

Proof. Since f E M(D) then there exists a continuous, condensing g: D -+ E such that

f(x) = x + g(x) for every x E D. Consider the sequence f,,(x) = x + (n/(n + l))g(x), x E D, n L 1 and note that f, E M(D) and ((n + l)/n)f,(x) = (l/n)x + f(x), x E D, nl 1. Since f is monotone it follows that ((n + l)/n)f, is strongly monotone, consequently f,, is strongly monotone.


One can easily see that the sequence (f,), z , converges uniformly on every bounded set of D tof, and f, is one-to-one for every n 2 1. Note thatfis light. Then, by theorem 4.1 ,fis globally univalent.

COROLLARY 4.4. Let E be a real Hilbert space, D an open subset of E, and f E M(D) a monotone map. Iffis a local homeomorphism, then f is globally univalent.

THEOREM 4.5. Let E be a real Hilbert space, D an open convex subset of E and f E M(D) a C’ map. Suppose that we have the following.

1” (f’(x)(h), h) > 0, x E D, h E E. 2” The set Z, = lx E D: f’(x) $ Isom(E, E)) is discrete or at most countable. Then f is globally univalent.

Proof. By 1” and the convexity of D the map f is monotone and by 2” it is light. Then, by theorem 4.3, f is globally univalent.

THEOREM 4.6. Let E be a real Hilbert space and let f E M(E) be a monotone light map. Suppose that

lim IIf(x)ll = + to. (4.1) llxll -m

The f is a global homeomorphism.

Proof. By (4.1) we have that f is proper, hence f(E) is closed. It follows from theorem 4.3 that f is globally univalent therefore f (E) is an open set cf. [41]. Since E is connected, f (E) = E. Therefore f is a global homeomorphism.

THEOREM 4.7. Let E be a Banach space, D an open convex subset of E and let J D + E be a Frechet differentiable map such that

df’(x)) < 0, x E D. (4.2)

Then f is globally univalent.

Proof. Let x,y E D, x # y. Let U: [0, l] -+ D, u(t) = fx + (1 - t)y, t E [0, l] and u: [0, l] --t E, v = f 0 u.

Consider the operator -4 = {A f ‘(u(s)) ds. If f is a C’ map then

[x - Y,f(X) -f(Y)]+ = [x - Y, v(l) - ~(W+ = [x-Yy,~~vl(sws]+

= [ s

x - Y, If ‘(W)(x - Y) ds I

= Ix - Y, A(x - Y)l+ 0

+

52 IIX - Y II * P(A) 5 IIX - YII s

’ df ‘(de)) ds < 0, 0

when f (x) # f(y), i.e. f is globally univalent.

548 S. R~DULESCU and M. RADULESCU

In the case where f is Frechet differentiable but not continuously differentiable the equation u(1) - u(0) = 1; u’(s) d.s may not hold, and the proof thatfis globally univalent is a little more complicated.

Let 01: [O, 1) --+ (0, i) be a map with the following properties:

0 < a(t) 5 1 - t t E [O, 11 (4.3)

II “y T;(r) - u’(t) < -cl(f’(w))llx - Yll, II t E [O, 1) and s E (f, t + a(t)). (4.4)

Let t, = 0, J, = [to + CY(~~),~], Jk = (1 - 21Vk, 1 - 2-k], k L 2. By considering minimal coverings of the compact intervals Jk , k L 1, with open intervals of

the type (t, t + al(t)), t E [0, l), we can construct a sequence (tJrrbO with the following properties

lim t,z = 1 (4.5) n-m

tk+, E @k > tk + dtk))r k > 0. (4.6)

Now the global univalence off follows from (4.4), (4.6) and from the following sequence of inequalities:

ix - Y,fW -f(Y)l+ = Ix - Y> u(l) - ml+

x - y, i u(fk+,) - dfk)

k=O 1 + 5 kf/ - Y, U(fk+,) - dfk)l+

5 kco[x - y, u(tk+,) - U(fk) - (fk+, - tk)U’(tk)l+

m

+ ,;,[x - Y, (fk+, - fk)U’(tk)l+

m

5 c (fk+l - fk)

k=O (II

‘(‘;;;f 1 ;(“) - d(tk) + /Ix - _+(f’(u(fk)))

k II >

< 0.

COROLLARY 4.8. Let D be an open convex subset of IR” and let J D -+ R” be a Frechet differentiable map. If for every x E D and i E (1,2, . . . , n) one has:

then f is globally univalent. Here R” is endowed with P-norm.

(4.7)

Proof. Let J = (1, 2, . . ., n). By (4.7) the maps ~?Jfi/G’x, i E J have constant sign on D. Let

Ei = sgn afii/axi(x), x E D, i E J.


Define gj = - eifi, i E J and g = (g, , g, , . . . , g,). By (4.7) we have

x E D, ieJ (4.8)

when p_(g’(x)) c 0, x E D. By theorem 4.7 g is globally univalent whence f is globally univalent.

LEMMA 4.9. Let a > 0. Let E. F be two Banach spaces and D an open subset of E. Suppose that f, g: D + Fare two C’ maps with the following properties.

1’ f + lg is one-to-one for every t E (0, a).

2” f is a C’ local diffeomorphism. Then f is globally univalent.

Proof. Let xl, x2 E D, xl # x2 such that f(x,) = f (x2) = y. Note that h: IR x D -, F,

h(t, x) = f(x) + tg(x), (t,x) E IR x D is a C’ map and h:(t, x) = f’(x) + [g’(x) for (t, x) E R x D.

By 2” there exists b E (0, a) such that h:(t, x) E Isom(E, F) for k E (1,2) and ItI < b. By implicit function theorem there exist c > 0, two open disjoint neighborhoods D, and D,

of x, respectively x2 and C’ maps ui: (- c, c) + Di, i = 1,2 such that u,(O) = Xi, h(t, ui(t)) = y for i = 1,2 and I E (--c, c). But this fact contradicts 1”. Therefore f is globally univalent.

THEOREM 4.10. Let E be a Banach space, D an open convex subset of E, and J D + E a C’ local diffeomorphism such that

df’(x)) 5 0, x E D. (4.9)


Proof. Let g(x) = -x, x E D. Define h: IR: x D + E, h(t, x) = f(x) + fg(x), x E D, t E IT?:. Note that

N:(r, 4) = lu(f ‘(x)) - t < 0 for (t, x) E Rf x D.

By theorem 4.7 we have that h(t, -) is globally univalent for every t E I?*, . By applying lemma 4.9 one has f is globally univalent.

THEOREM 4.11. Let E be a Banach space, D an open convex subset of E and f E M(D) a Frechet differentiable map. Suppose that f-is light and that

H-f’(x)) 5 0. (4.10)


Proof. Since f E M(D) there exists a continuous condensing map g: D + E such that f(x) = x + g(x), x E D. Consider the sequence f,(x) = x + (n/(n + l))g(x), x E D, n L 1 and

550 S. R~DULESCU and M. R~DULESCU

note that f, E M(D) and ((n + l)/n)f,,(x) = (l/n)x + f(x), x E D, n 2 1. By (4.10) we have

( n+l iu- y-f;(x)

> = - ; + id-f’(x)) 5 - ; < 0.

By theorem 4.7 all the maps ((n + l)/n)f, are one-to-one, whence f,, n 2 1 are one-to-one. One can easily see that the sequence (f,),, tI converges uniformly on every bounded set of D to f. Then by theorem 4.1 f is globally univalent.

COROLLARY 4.12 (Gale and Nikaido [13]). Let D be an open convex subset of IR” and let J D -+ m” be a Frechet differentiable map such that we have the following.

1” det[ah/ax,(x) + ajj/aXi(X)], ai,j5k L 0 for every x E D and k E (1,2, . . . . n]. 2” det[f’(x)] > 0, x E D. Then f is globally univalent.

Proof. By lo the matrix t[f ‘(x) + f’(x)*] is semipositive definite for every x E D whence p(-f’(x)) I 0, x E D. Note that condition 2” implies f is light. By theorem 4.11 f is globally univalent.

5. THE GLOBAL UNIVALENCE PROBLEM

In this section we consider an old problem of Olech on the global stability of an autonomous system on the plane which is equivalent to a global univalence problem in IR’.

Global univalence problem Let f: IT?’ -+ lR2 be a C’ map such that

v(f’(x)) < 0,

Is then f globally univalent? It is not known whether the global univalence

problem is equivalent to the following problem of

Global stability problem

x E R2. (5.1)

problem has an affirmative answer. This Olech [21].

Let J K?’ -+ IR2 be a C’ map which verifies condition (5.1). Suppose that x = 0 is a critical point of the system (S) x = f(x).

Is then the solution x = 0 of the system (S) globally asymptotically stable or, in other words, does each solution curve of the system approach the critical point zero as t + oo?

In [21] it is shown that x = 0 is globally asymptotically stable for the system (S) if f is globally univalent and satisfies condition (5.1).

Useful references about global univalence problem and its connections with global stability may be found in Parthasarathy [24].

Note that if condition (5.1) holds, then tr[f’(x)] c 0, x E lR2. In [40], Vidossich, assuming the boundedness of the derivative off, obtains a multidimen-

sional generalization for the global univalence problem.


THEOREM 5.1. Let a, b > 0. Let J R” + R” be a C’ map such that the following conditions hold:

V(x)) < 0, XE R”, (5.2)

tr(f’(x)] 5 -a, x E I?“, (5.3)

Ilf’(x)II 5 b, x E rn”. (5.4)

Then f is globally univalent. The following theorem is new and represents a generalization of the above result since we give

up condition (5.3) and relax conditions (5.2) and (5.4).

THEOREM 5.2. Let f: R” + I?” be a C’ map such that

v(f’W) 5 0, x E I?“.

Suppose that f is light and there exists a continuous increasing map o: IR, following conditions hold:

Ilf'Wll 5 4lxll), x E ll?“, * ds ow”-‘o=+cQ*

(5.5)

IT?: such that the

(5.6)

(5.7)


Proof. Consider the sequence (fi)j,, , A: R” -+ I?“, defined byfi(x) j 2 1. Note that

and

= f(x) - (l/j)x, x E R”,

1 -Y J

Ilfmll = llfTx1 - ;z/i (: Ilf’Wll + f 5 4lxll) + ; 9 x E rn”, jz 1.

By theorem 3.10 we have that allfj, j 2 1, are global diffeomorphisms. By applying theorem 4.1 we obtain that f is globally univalent.

Remark 5.3. If f satisfies condition (5.2), then f is light.

Remark 5.4. The conclusions of theorems 3.10 and 5.2 hold if we replace the hypothesis that f is a C’ map by the weaker hypothesis that f is Frtchet differentiable.

This fact follows at once from the following.

THEOREM 5.5 [32]. Let D be an open subset of IT?” and f: D + I?” a Frechet differentiable map such that

Wf’Wl f 0, x E D. (5.8)

Then f is a local diffeomorphism on D.

552 S. R~~DULESCLT and M. RKDULESCU

THEOREM 5.6. Let f: I?” -+ I?” be a Frkhet differentiable map which satisfies condition (5.5) and let (tj)j_, be a sequence of positive real numbers which converges to zero. Suppose that f is light and satisfies the following condition:

lim IIf - tjXll = + 00, j2 1. (5.9) llxil - 9


Proof. Consider the sequence (fj)jz 1 defined by fj(X) = f(x) - tjX, x E iR”, jr 1, and note that

V(fi’(X)) = V(f’(X) - tj’) = V(f’(X)) - tj < 0, x E R”, j2 1.

By (5.9) we have that ,,~li~_~~~(x)~~ = +a, j L 1.

By applying theorems 3.2 and 5.5 we obtain that 4, j L 1, are global homeomorphisms. Since (fj)j~, converges uniformly to f on every bounded subset of IR”, it follows from theorem 4.1 that f is globally univalent.

COROLLARY 5.7. Letf: iR” -, IR” be a Frtchet differentiable map which satisfies condition (5.2) and the condition

lim ‘If(x)ll _ 0, llxll

(5.10) llxll -m


Proof. By (5.2) f is a local homeomorphism. Note that (5.10) implies (5.9). Now the conclusion of the corollary follows from theorem 5.6.

1.

2. 3. 4.

5.

6.

7.

8. 9.

10.

II.

12. 13. 14.

IS.

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global univalence and global inversion theorems in banach spaces

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