Nonlrnear Analysis, Theory, Methods & Ap~licarrons, Vol. 20, No. 9, pp. 1069-1077, 1993. 0362-546X193 $6.00+ .oO

Printed in Great Britain. 0 1993 Pergamon Press Lfd

AN EXTENSION TO HADAMARD GLOBAL INVERSE FUNCTION THEOREM IN THE PLANE

M . SABATINI

UniversitA di Trento, I-38050 Povo (TN), Italy

(Received 1 November 1991; received for publication 1 August 1992)

Key words andphrases: Jacobian conjecture, global invertibility, global asymptotic stability, polynomial maps.

1. INTRODUCTION

IN THIS paper we are concerned with two classical problems known as “Jacobian conjectures”. The oldest one, formulated by Keller in 1939 (see [l] for a review on the subject), states that any polynomial map with a nonvanishing, constant Jacobian determinant is globally invertible, and has a polynomial inverse. The second one conjectures that a critical point of an autonomous plane differential system

i = H(z), z E R2, HE C3’(R2, R2), (9

is globally asymptotically stable if, for any z E R2, det JH(z) > 0 and tr JH(z) < 0, where J&Z) is the Jacobian matrix of H at z (see [2] for a recent review of results on the subject). These two problems seem not to be related to each other, but can be considered as different aspects of the same question, since, under the above conditions on J&z), the global asymptotic stability of a critical point is equivalent to the injectivity of H (see [2, 31). The two conjectures are still unsolved (Keller’s one for any dimension but 1). Many partial results were proved under suitable additional conditions. In particular, the global invertibility of a nonsingular map of the plane into itself has been proved in what can be considered the intersection of the two conjectures: if a polynomial map satisfies det J,(z) > 0 and tr J&z) < 0 at any point of the plane, then it is injective [4], hence a diffeomorphism of R2 onto itself [5, 61.

A characterization of globally invertible continuous maps in finite dimensional spaces was given by Hadamard (see [7; 8, p. 3831). He showed that a locally invertible continuous map H: RN + RN is a homeomorphism of RN onto itself if and only if it is proper, that is if and only if H-‘(K) is compact for any compact K c R N. Hadamard’s theorem, and its generalizations to abstract spaces (see [9] for an account of its applications in functional analysis), can be applied when some information on the divergence of the sequences [H(z,)l, for (z,( + 00, is available. In general, this is not the case for the Jacobian conjectures, for which it seems desirable to have conditions involving the partial derivatives of H, rather than its module.

In this paper we show that, if H = (f, g) E C&(R2, R2), the injectivity of H can be obtained weakening Hadamard’s hypothesis to request that lim,,,If(z,)/ = 00 only on sequences of the

type 2, = I rk)l v t, + &co, where y(t) is a solution of the differential system

Vf (2) z=lvfol’ <Vf,

1069

1070 M SABATINI

This happens if, for instance, the following integral condition is satisfied

” +m

! inf \Vf(z)l dr = $00. 0 Izl=r

(0

Condition (Z) is used to prove the parallelizability of the solutions of the Hamiltonian system associated to j’. This gives the connectedness of the level sets off, and the injectivity of H.

The weakened hypothesis does not imply the properness of H, since it does not give any information on the behaviour of the second component of H. In general, such a condition is not sufficient to ensure the surjectivity of H even when both its components satisfy it, so that they are both unbounded from above and from below. On the other hand, by replacing (V,) with the Hamiltonian system associated to g, we can show that iff is upper and lower unbounded on the level sets of g and vice versa, then H is a diffeomorphism of R2 onto itself. This comes out to also be a necessary condition for H to be a diffeomorphism, so giving a different characteriza- tion of the diffeomorphisms of the plane onto itself. Again, we can give a sufficient condition involving only the partial derivatives of H in order to ensure that such a property holds

+a0 inf idetJdz)l dr = +CO inf ldetJ&)I dr = +cx,

0 IzI=r IVf(z)l i 0 IZI =r Ivml (J)

The results obtained are applied to prove the global asymptotic stability of a class of weakly dissipative differential systems in the plane, and to give some conditions for a polynomial map to be globally invertible.

2. DEFINITIONS AND RESULTS

We consider maps with locally Lipschitzian derivatives, HE C&(Rz, R2), H(z) := (f(z), g(z)), z := (x, y). We denote by

J,(z) := %f(z) ayf(z)

a,&) a, s(z) >

the Jacobian matrix of H at z, and by det J&z) its determinant. Throughout the rest of this paper, we assume that H is nonsingular, that is det JH(z) # 0, v z E R2.

In the sequel we shall study the solutions of the Hamiltonian and gradient systems associated tof. We reparameterize the time in order to have global existence of the solutions of the systems

L xc--’ IA a f Y

3=-&pf

r x=--’ IA a f x

_i = & . a,f. <Vf>

A similar notation will be used for systems obtained in the same way from g. We denote by

Hadamard global inverse function theorem 1071

r+,(t) (y,(t)) the solution of (H_) (resp. (Vf)) such that 4,(O) = z (y,(O) = z). & and yz denote the corresponding orbits, 4; and y: (4; and y;) the positive (negative) semi-orbits

4; := (w E R2: w = c&(t) t L 0) c$; := {w E R2: w = c&(t) t 5 0).

If f is nonsingular, then (H_) and (Vf) have no critical points, so that any orbit is positively and negatively unbounded. This allows us to give a definition of parallel neighbourhood equivalent to but simpler than that introduced in [lo].

Definition 2.1. Let 6 be an orbit of a plane differentiable dynamical system TC without equilibrium points. Let r be a differentiable curve intersecting 6 and transversal to the vector field associated to n. The union of the orbits intersecting r is said to be a parallel neighbour- hood of 6.

Definition 2.2. We say that two nontrivial orbits 6i, & of a plane dynamical system are inseparable if, for any couple of parallel neighbourhoods U, of 6, and U, of &, one has ur fl u, # 0.

Definition 2.3. For any z E R2, we define the strip associated to z as the set

S[z] := (w E R’: 4, n yz # 01.

The positive semi-strip associated to z is the set

S+[z] := [w E R2: 4, fl y: # 01.

Strips are open, (Hf)-invariant and connected. Semi-strips are (Hf)-invariant and connected. In a similar way, we can define the orthogonalstrip SL[z], by exchanging the roles of (Hf) and (Vf).

Definition 2.4. A differential system in the plane is said to be parallelizable if it is topologically equivalent to the system

L

x= 1

j = 0.

In the next lemma we collect some elementary facts.

LEMMA 2.1. (i) f is increasing along the solutions of (V,); (ii) every orbit 4 of (Hf) intersects an orbit y of (V’) at most once;

(iii) every orbit of (Hf) coincides with a connected component off -‘(a) for some a E R2; (iv) every orbit C#J of (Hf) has a parallel neighbourhood U = U- U $J U Ut such that (I- fl

Ui = 0, U-, CT’ are connected, and v z- E U-, Vz E $, vz+ E U’:f(C) <f(z) <f(z’).

We denote by R the set R U (-co, +a).

Definition 2.5. Let w E aS [z] and let {w,) c S [z] be a sequence such that w, + w as n -, 00. Let t, be the unique parameter such that y,(t,) E $I,,,,. Then (by proposition 1.3 in [12]) {t,) has a unique accumulation point T E i?. In this case w is said to belong to the T-boundary arS[z] of S [z], and T is said to be a boundary parameter.

1072 M. SABATINI

The boundary of a strip S[z] is the disjoint union of its T-boundaries, for T E I?. Every T-boundary is the disjoint union of, at most countable, a set of (Hf)-orbits.

Remark 2.1. If S [z] has a T-boundary a%[.~], then any (Hs)-orbit C$ contained in #S[z] is inseparable from any other (Hf)-orbit contained in arS[z]. If T # +co, then 4 is also inseparable from the (H_)-orbit passing through y,(T).

LEMMA 2.2. Let f E t!:,,(R2, R) have a nonvanishing gradient. If there exists a E R2 such that f-‘(a) has two distinct connected components, then (Hf) has two inseparable orbits.

Proof. Let &,, $i the orbits of (H_) corresponding to the connected components off -‘(a). Since (Hf) is nonsingular, there exists a segment

Z := (a(s) := .szi + (1 - s)zo, s E [O, 11)

intersecting pi only in Zi, i = 0, 1. The function c(s) := f(a(s)) is not constant on C, otherwise +i U C U c$~ would be contained in a single connected component of f-'(a). Since c(O) = C(l) = a, [ has an extremum, say a minimum m, different from a. Let z, = o(s,) be a point where such a (not necessarily proper) minimum is achieved. The point zm divides X into two closed subsegments C, , C, , containing zO, z, , respectively. The (H,)-orbit I$,,, starting at z,,, is tangent to X at z,, and at any other point of intersection with C, otherwise every orbit in a parallel neighbourhood of 4, would cross C, and-by lemma 2.1-z, would not be a point of minimum. Hence 4, lies in a part of the plane bounded by a semi-orbit of r$,,, by C, and by a semi-orbit of C#Q . The (V,)-orbit yrn passing through z, crosses Z transversally at zm. By the continuous dependence on initial data, there exists a parallel neighbourhood U = U- U 4, U U+ of c#J,, such that every (H,)-orbit in U+ meets C at least twice. This implies that the positive semi-strip S’[z,] contains points of both X0 and Zi . Since f is increasing along the solutions of (V,), &, and ~$i cannot both be contained in S’[z,]. Then two cases are possible.

Case 1. &,, +i $ S’[z,]. Then XS’[z,] n Ci # 0, for i = 0, 1. Let Wi be a point of %‘[z,] fl Xi, i = 0, 1. If one of the WiS belongs to a T-boundary, with T # +a, then it is inseparable from some other (HI-)-orbit, by remark 2.1. If both are not in a T-boundary, with T # +co, then they belong to the +oo-boundary of S+[z,J, hence they are inseparable from each other.

Case 2. &, E S’[z,] (or C#Q E S’[z,]). Let T be the parameter such that y,(T) E &,. Since & $ S’[z,J, the (H,)-orbit passing through y,(t) intersects C, for small positive values of t, while this does not happen for t 2 T. Hence Ci contains points of Z’[z], that have to belong to a r-boundary, for T 5 T. By remark 2.1, in this case there also exist inseparable orbits. W

THEOREM 2.1. Assume that (Hf) is parallelizable. Then H is injective.

Proof. By [ll, theorem 51, (Hf) has no separatrices, hence it has no inseparable orbits. By lemma 2.2, this implies that for any a E R, the set f -‘(a) has at most one connected component, consisting of a single orbit 4. If there exist z0 # z1 such that H(z,) = H(z,) = (a, b), then zO, z1 E f -‘(a). Since g is decreasing along the solutions of (H,), we have g(zO) = g(d(t,)) #

g(@(tA) = gkJ. n

Hadamard global inverse function theorem 1073

THEOREM 2.2. Assume that, for any solution y(t) of (Vf), one has

Then:

lim If(r(0) I = 00 t+*CC

(i) every level set off is nonempty; (ii) (Hf) is parallelizable;

(iii) H is injective.

Proof. (i) Obvious. (ii) In order to prove the parallelizability of (H,) it is sufficient to prove that (H_) has no inseparable orbits. By contradiction, let us assume that there exist two inseparable orbits &, , 41. By continuity, f assumes the same value on both orbits. Let C, z0 and z, be as in lemma 2.2. The orthogonal strips Sl[zi] are nonintersecting neighbourhoods of Zi, (i = 0, l), so that, for any z* E I%,[z,,] n C # 0, one has z* # zl. By lemma 2.1, S,[zJ is the union of two semi-strips S:[zJ, S;[zO] where f is, respectively, greater or less than f (z,,).

Assume that z* E JS;[zJ. By the invariance of S;[zO] with respect to (V’), yZ* is contained in X~;[Z,,]. Then, for any w E yZ*, f(w) I f(z,,), contradicting the unboundedness off on yZ*. A similar argument works if z* E 13s,‘[zJ. (iii) It comes from point (ii) and [l 1, theorem 51 as in the proof of theorem 2.1. N

In theorem 2.2 it would be sufficient to request the unboundedness both from above and from below off (y,(t)). Since f (y,(t)) is monotone, this is equivalent to the chosen hypothesis. A simple sufficient condition for f to be unbounded on the solutions of (Vr) is given in the next corollary.

COROLLARY 2.1. Let H = (f, g) E C?~,,,(Z?, R2) be nonsingular. If

then: i

+CC inf IVf(z)] dr = +oo

fJ IzI=r

(i) for any solution y(t) of (Vf), the function f(y(t)) assumes every real value; (ii) H is injective.

(0

Proof. By theorem 2.2, it is sufficient to prove point (i). Without loss of generality, we can assume that y(t) does not pass through the origin. Denoting by &y(s)) the angle determined by the vectors y(s) and Vf(y(s)), we have

If(W) -fWo))I =

The orbit v(t) is positively and negatively unbounded, hence f (y(t)) cannot be upper or lower bounded. H

1074 M. SABATINI

Remark 2.2. A different proof of the above corollary can be given following the approach introduced in [3], and developed in [13] (see also [2, Section 2, P4 * FP; 141).

Theorem 2.1 and corollary 2.1 admit natural extensions to maps not defined on all of R’, or not verifying the unboundedness condition on the whole plane. We report them without proof.

THEOREM 2.2’. Let N be a connected component of (z, E R2: a <f(z) < b). If H = (f, g) E ei,,(N, R2) is nonsingular, and

lim If(~(0)l = 00, t+*CC

for any y c N, then H is injective.

COROLLARY 2.1’. Let N be a connected component of (z E R2: a <f(z) < bJ. If Z-Z = (f, g) E e:,,,(N, R2) is nonsingular, and

s

+CO

o Nnt;71f=d]V_Z(~)I dr = +a

then: (i) for any solution y(t) c N of (V’), the functionf(y(t)) assumes every real value;

(ii) H is injective.

The integral condition (I) does not imply (1.9) in [13]. In fact, the map H(x, y) := (y - tanh(x) + 1, -y) satisfies (I), and

s

+m inf IH( dr 5

0 IzI=r i +=]ZZ(r, O)l dr = +O” [l - tanh(r)] dr < +co. 0 .i 0

The Jacobian matrix of H has eigenvalues with negative real part, but there are no points z E R2 such that H(z) = 0, so that the arguments used in [2, Section 21 to prove the equivalence of the various hypotheses considered in relation to the Jacobian conjecture do not apply.

Remark 2.3. Condition (Z) can be verified after composing H with a function of the type (o(x), P(y)), where a’(x), /Y(y) # 0. This leads to check whether the more general condition

.i’ +* inf (c~‘(f(z)) . IVf(z)l) dr = +m (I,) 0 IzI=r

holds, for some (Y E e’(R, R).

Condition (I,) is not necessary for a nonsingular map H to be injective. The map H(x, y) := (y2 - e’, (y - 1)2 - e”) is nonsingular, and its components do not satisfy (I,). In fact, the level set y2 - eX = 0 has two connected components, the graphs of the functions y = *eX’2. Analogously, the second component of H has a level set with two connected components. However, H is injective, since the system

(

y2 - ex = a

(y - 1)2 - eX = b

has a unique solution, given by (x = ln[(a - b + 1)2/4 - a], y = (a - b + 1)/2).

Hadamard global inverse function theorem 1075

Moreover, even if bothf and g satisfy (I), so that they are unbounded, H is not necessarily surjective. An example is given by the map H(x,y) := (y - eX,y + e”). There are no points (x, y) such that H(x, y) = (0, - 1). A stronger condition has to be imposed on the components of H to also get the surjectivity.

In the next theorem we denote by v(t) a solution of (H,).

THEOREM 2.3. H is a diffeomorphism if and only iff(W(R)) = R for any solution v/ of (H,), and g(q5(R)) = R for any solution 4 of (Hf).

Proof. (-) By absurd. Assume that there exists an (HJ-orbit I,U such that f(ty(t)) is bounded from above. The other cases can be easily reduced to this one. Let us set

M := sup(f(y(t)): t E R). (*)

Since g is constant on y, there exists b E R such that g(ty(t)) = b. By the surjectivity of H, there exists z E R2 such that H(z) = (M, b). Let y,(t) be the (H,)-solution starting at z. Since f(ty,(t))

is monotone, we have f(w,(t)) > A4 for t < 0, f(w,(t)) < A4 for t > 0, (or vice versa). This contradicts the injectivity of H, unless z E I,Y, that contradicts (*).

(=) Surjectivity. Without loss of generality, we may assume that H(O,O) = (0,O). Let (a, ZJ) E R2, and $J,, be the orbit corresponding to the level set f -l(O). Since g(&(t)) assumes every real value, there exists a tb E R such that g(&(tb)) = b. Let vb be the (H,)-orbit passing through &(tb). The function f(y&)) also assumes every real value, so that there exists an s, such that f(v&,)) = a. Since g is constant along the solution of (H,), we have

H(V/&)) = (a, b). Injectivity. It comes from the same argument developed in lemmas 2.1 and 2.2 and theorems

2.1 and 2.2, replacing the solutions of (Vf) with the solutions of (H,). n

COROLLARY 2.2. If H is a diffeomorphism, then (Hf) and (H,) are parallelizable.

Proof. If not, (Hf) (or (H,)) has a couple of inseparable orbits 4r, &. By continuity, there exists a E R such that f(&) = a = f(&). By theorem 2.3, g(4,(R)) = R = g(&(R)). This implies that, given b E R, there exist zi E &, z2 E q52, such that g(zJ = b = g(z2). This contradicts the injectivity of H. W

COROLLARY 2.3. Let H satisfy

‘+- i

inf ldetJdz)I dr = +a,

0 lzI=r lVg(z)I (J)

Then H is a diffeomorphism of R2 onto R2.

Proof. In order to prove the injectivity it is sufficient to show that (J) implies (I). This comes from the inequality (det JN(z)(/IVf 1 5 lVg/.

In order to prove the surjectivity, we proceed as in corollary 2.1

1076 M. SABATINI

As in the proof of corollary 2.1, this leads to

The orbit 4(t) is positively and negatively unbounded, hence g($(t)) cannot be bounded above or below. The same holds for the functionf(w(t)), where v(t) is a solution of (H,). Then the thesis comes from theorem 2.3. n

3. TWO APPLICATIONS

Our first application is to the asymptotic stability Jacobian conjecture. We recall that it is concerned with the global asymptotic stability of a critical point 0 of a plane differential system

i = H(z) zeR2, H E C1(R2, R’), (S)

such that, for any z E R2, det J&) > 0, tr .ZH(z) < 0. In the next theorem we impose a weaker condition on the trace, but we request more regularity to the partial derivatives of H.

THEOREM 3.1. Let H E e:,, . If there exists (Y E C?‘(R, R) such that condition (I,) holds, and: (i) N(0) = 0;

(ii) det JH(z) > 0, v z E R2; (iii) tr J&O) < 0 and tr J,(z) 5 0, V z E R2.

Then the origin is a globally asymptotically stable equilibrium point of (S).

Proof It is a straightforward consequence of theorem 5 in [3] and corollary 2.1. n

It is clear from the proofs of Section 2, that the local Lipschitzianity of the derivatives of H can be replaced by any hypothesis ensuring existence and uniqueness of solutions of the systems

(Vf) and (Hf). The next result allows us to deduce the global invertibility of a plane polynomial map from

an algebraic condition. We emphasize that we do not request the Jacobian determinant to be constant, as in the case of the map H(x, y) := (x + x3, y).

THEOREM 3.2. Let H = (P, Q): R2 + R2 be a nonsingular polynomial map. If one of the

following holds: (i) there exists a polynomial Q1 such that the map H, = (P, Q1) is nonsingular and injective;

(ii) there exist constants h, k > 0 such that the polynomial lvP12(x2 + y2 + h) - k has no real roots; then H is a diffeomorphism of R2 onto R2.

Proof. (i) If H, is injective, it is a diffeomorphism [5, 61. By corollary 2.2, (HP) is paralleliz- able. By theorem 2.1, H is invertible.

(ii) As in point (i), it is sufficient to prove that H is injective. M(x,y) := IVP12(x2 + y2 + h) is a positive polynomial, hence it is not bounded from above. If M(x, y) - k has no real roots, then:

[VP1 > k J x2 + y2 + h ’

on all of R2, so that condition (I) of corollary 2.1 is satisfied. q

Hadamard global inverse function theorem 1077

Remark 3.1. We emphasize that condition (i) does not make assumptions on the degrees of Q and Q, . This could help to avoid some of the difficulties involved in “reduction of degree” techniques [ 11.

Remark 3.2. Condition (ii) seems to be a natural one to impose on polynomial maps. In fact, on any line ((x, y) = (t cos(f3), I sin(@)) A4 reduces to

MB(t) := IVP(t cos(@, t sin(8))12(t2 + h),

that is a one-variable nonvanishing polynomial. Hence there exists a constant K@ > 0 such that MB(t) > K@. This does not imply immediately the existence of a positive constant k such that M(x,y) > k. In fact, K, varies with 19, and it could tend to zero. Proving that this cannot happen for singular polynomial maps would give a positive answer to Keller’s Jacobian conjecture for two-variable real polynomials.

Acknowledgements-The author wishes to thank Professors R. Conti, P. Habets, C. Olech and Dr L. Mazzi for useful remarks on this paper.

Note added in proof-After this paper was completed, Professor Druikowski of the University of Krakow gave me an example of a polynomial with nonvanishing gradient, not satisfying condition (I). The polynomial is p(x, y) = x + X’J. It is easy to verify that p(x, y) has a level set with three connected components. In fact, p-‘(O) is the union of the y-axis and the hyperbola xy + 1 = 0. This shows that there are no polynomials q(x, y) such that (x, y) + (p(x, y). q(x, y)) is an invertible map.

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