Integral representation of solution for \bar{\partial}u=f and its uniform estimate on ellipsoids

Author(s): Shouchuan Hu and Nikolaos S. Papageorgiou

Source: Transactions of the American Mathematical Society, Vol. 347, No. 1 (Jan., 1995), pp. 233-259

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Volume 347, Number 1, January 1995

GENERALIZATIONS OF BROWDER'S DEGREE THEORY

SHOUCHUAN HU AND NIKOLAOS S. PAPAGEORGIOU

ABSTRACT. The starting point of this paper is the recent important work of F. E. Browder, who extended degree theory to operators of monotone type. The degree function of Browder is generalized to maps of the form T+f+G, where T is maximal monotone, f is of class (S)+ bounded, and G(*) is an u.s.c. compact multifunction. It is also generalized to maps of the form f + NG , with f of class (S)+ and NG the Nemitsky operator of a multifunction G(x, r) satisfying various types of sign conditions. Some examples are also included to illustrate the abstract results.

1. INTRODUCTION

The resolution of a large variety of problems in nonlinear analysis depends on the study of equations of the form Tx = y, where T is an operator de- fined on an appropriate space X and y E X. The Leray-Schauder degree has proven to be a very powerful tool in such investigations. The most important property of this degree is, of course, the homotopy invariance property, which forms the basis for the continuation method, which was originally developed by Poincare and which consists of embedding the problem in a parametrized family of problems and considering its solvability as the parameter varies. Ever since the introduction of the Leray-Schauder degree theory in 1934 (which is an infinite-dimensional extension of Brouwer's degree theory), there have been various extensions and generalizations in different directions. By far the most important of these generalizations is due to F. E. Browder. In a series of impor- tant papers [7-12], Browder developed a degree theory, which is a generalization of the Leray-Schauder degree theory, for maps from a bounded open subset of a reflexive Banach space X into its dual X*. Browder's breakthrough work paved the way for the application of degree-theoretic techniques to large classes of nonlinear partial differential equations.

Browder's degree theory is defined primarily for (S)+ mappings (see ?2) and (S)+ mappings with maximal monotone perturbations, which cover a substan- tially large class of nonlinear partial differential operators. Browder demon- strated that the (S)+ maps are the right class to consider and he proved the

Received by the editors February 2, 1994.

1991 Mathematics Subject Classification. Primary 35J60, 47H05, 47H1 1.

Key words and phrases. Degree function, monotone operator, operator of class (S)+ , Nemitsky operator, sign condition, multifunction, approximate selector, normalization, additivity on domain, homotopy invariance, compact embedding.

The paper was written while the first author was on sabbatical at Florida Tech. The hospitality in the Department of Applied Mathematics there is acknowledged.

? 1995 American Mathematical Society

existence of his degree through ingenious arguments involving Galerkin approx- imations. Specifically he proved

Theorem A [11]. Let X be a reflexive Banach space. Then there exists one and only one degree function on the class of maps T + f, with T maximal monotone and f bounded and of class (S)+, which satisfies the additivity on domain property, is normalized by J, the duality map of X into X* corre- sponding to an equivalent norm on X with respect to which both X and X* are locally uniformly convex, and is invariant under affine homotopies of the form (1 - t)(T + f) + tf1 with T maximal monotone, f and fi of class (S)+

Remark. In fact, Browder showed that the unique degree function is invariant under a much broader class of homotopies, namely homotopies of the form

Tt

f,

f

When applied to partial differential operators, we can have X = _WOmP(Z) and X* = W-m q(Z) with 1 < p, q < oo and p + i = 1, T: D C X

2x*\{z} and f: U -* X* where U is a bounded open set of X. As will

be clear from the definitions (cf. ?2), a compact perturbation of (S)+ maps is still an (S)+ map. Therefore compact maps from X into X* are harmless, in the sense that they can alwavs be absorbed in the original (S)+ operator. But if such a compact map originates from a Nemitsky (superposition) operator Ng(u)(x) = g(x, u(x)), x e Z, u E WomYP(Z) with g: Z x R - R, it is clear that g(x, r) has to satisfy certain growth conditions. To avoid such restrictions which are not always satisfied in applications, Browder [ 12] proved the following theorem. Let Z C Rn be a domain in Rn (bounded or unbounded), X =

Wom P(Z), and U C X open and bounded. We will say that g: Z x IR -*R

satisfies the sign condition provided it has the following two properties: (i) for each fixed r E ]R, x -- _{g(x, r) is measurable, for each fixed x E Z,}

r -- _{g(x, r) is continuous, and for each integer s > 0, there exists a} function h, E

L c(Z)

Then the theorem of Browder [12, Theorem 7] reads as follows:

Theorem B [12]. Assume that f is a bounded mapping of class (S)+ of U into X*, and Ng : X -- _{X* is the Nemitsky operator corresponding to a function} g(z, r) satisfying the sign condition. Assume that yo E X* is a target point such that yo ? (f + Ng)(OU). Then the degree d(f

+

In this paper we present the following generalizations of Browder's degree theory, contained in Theorems A and B above. First we prove that the degree function stipulated by Theorem A can be extended uniquely to the case where f is allowed to have a multivalued compact perturbation (i.e. for operators of the form T + f + G with G( - ) being the multivalued compact perturbation). Second, we establish the existence of a unique degree function for maps of the form f + NG, where NG is the multivalued Nemitsky operator corresponding to a multifunction G(x, r) satisfying a sign condition. This extends Theorem

B, since the function g: Z x R -1 IR is replaced by a multifunction G: Z x R --

2R\{0}. These two extensions are presented in ??3 and 4. In ?5, we present some examples illustrating the applicability of our abstract results.

2. PRELIMINARIES

Let X be a Banach space and consider the family F of all continuous map- pings f: U -- _{X, with U a bounded open subset of X (we consider all} possible such sets U) and with (I - f

)

for all t E [0, 1] and K C X is compact. Then the Leray-Schauder degree

theory states that there is an integer-valued degree function d(*, *,

*)

triples (f, U, yo), with f E F, yo E X\f(a U) such that

(a) Normalization: If yo E U, then d(I, U, yo) = 1.

(b) Additivity on domain: If U1 and U2 are disjoint open subsets of U such that yo S f(U\(U1 U U2)), then

d(f , U, yo) = d(f , U1, yo) + d(f , U2, yo)-

(c) Homotopy invariance: If the homotopy {ft : t E [0, 1]} belongs in H,

y : [0, 1] -- _{X is continuous, and y(t) S f (&U) for any t E [0, 1],} then d(f , U, y(t)) is independent of t E [O, 1].

By a result proved independently by Fuhrer [16] and Amann-Weiss [1], prop- erties (a), (b), and (c) above determine uniquely the Leray-Schauder degree function.

In order to discuss a degree theory for maps from X into X*, where X is a reflexive Banach space, we need to introduce the type of mappings we will be dealing with.

Definition 1. (i) A map T: D C X -, _2x \{0} is said to be "monotone" if (x* -*, x-y) > 0

for all [x, x*], [y, y*] E GrT. Here GrT denotes the graph of T(-) and

(

(ii) We say that T( - ) is "maximal monotone" if it is monotone and for any [u, u*] E X x X* for which (u* - x*, u - x) > 0 for all [x, x*] E GrT we

have [u, u*] E Gr T.

Following Browder, we will be concentrating on maps of type (S)+ as the primary class to define a degree function. The class (S)+ of operators was first introduced by Browder [3, 4].

Definition 2. Let B C X and f:

B

and (ii) if {x,n},> 1 C B and x, w x for some x E X and lim(f(x ), x, -x) < 0, then x -+x in X.

Finally let us introduce the kind of multivalued perturbations that we will be considering:

Definition 3. A multifunction G: B C X -- _{2x* \{0} is said to belong to class}

(P) if it maps bounded sets to relatively compact sets, for every x E

B

C

X* G-(C) = {x

B: G(x)

#z

By a well-known renorming theorem due to Troyanski [20], given a reflexive Banach space, we can always renorm it equivalently so that both X and X* are locally uniformly convex. Thus without loss of generality we may assume from the beginning that both X and X* are locally uniformly convex. Recall that a locally uniformly convex Banach space has the Kadec-property; i.e. if x, w- x

and llxnll -llxll, then xn -x in X.

Define J: X -* X*, the duality map, by

J(x)

{x*

X*:

Then from Browder [1 1, Proposition 8], we have the following result: Lemma

4

J(-) is a well-defined,

single-valued

map from X onto X* ,

which is a homeomorphism

and is also monotone

_{and of class (S)+.}

Remark.

J(-)

bounded subset of X.

Using the duality map

J(.),

Lemma 5. A monotone operator T : D C X -- _{2x \{0} is maximal monotone}

if and only iffor every

(equivalently

for 3ome

R(T +

The following approximate selection theorem due to Cellina [13] will be im- portant in our extension of Browder's degree theory to a multivalued context. Lemma 6 [13]. If Y,

V

an u.s.c. multifunction

with closed and convex values, then given

> 0, there

exists a continuous map g, : B -- _{V such that} ge(y) E G((y+Be)fnB)+Be

for all y E B and

convG(B), with BE = {Y E Y:

IIYIIY

< e} and

BE = {v E V: IVIlv <E}-

Remark.

3. THE DEGREE FOR THE MAPPINGS OF THE FORM T + f + G

Let X be a reflexive Banach space, equivalently renormed so that both X and X* are locally, uniformly convex and let J(*) be the duality map corre- sponding to this locally uniformly convex norm. Assume that U is a bounded open set in X, T: D C X -- _{2x*\{0} is maximal monotone, f: U} -- _{X* is}

a map of class (S)+, and G : U -- _{2X* \{0} is a multifunction of class (P).}

In this section we will define a degree function

d(T

f

Yo

X*\(T

f

among all possible degree functions satisfying the three characteristic proper- ties of normalization, additivity on domain, and homotopy invariance, to be defined precisely in the present context in the sequel.

Recall that homotopy invariance is with respect to a certain class of permis- sible homotopies. We will now introduce those permissible homotopies for the maps T, f, and G. The permissible homotopies for T and f (see Definitions 6 and 7) are due to Browder [1 1, 12], while the permissible homotopies for G (see Definition 8) are a natural extension to multifunctions of the permissible homotopies for compact maps used in the Leray-Schauder degree theory. Definition 7 [1 1]. A family of maximal monotone maps {Tt: t E [0, 1]} is said to be a "pseudomonotone homotopy" of maximal monotone maps if it satisfies the mutually equivalent conditions:

(i) Suppose that t, -n t in [0, 1], [xn, x,*] E _{GrTt, with x,} w x in X, xn* I _{x* in X*, and lim(xn*, xn) < (x*, x). Then [x, x*] E GrTt}

and (xn*, xn) -- _{(x*, x).}

(ii) ip(t

x*)

(iii) For each x* E X*, t -- _{(o(t, x*)} = _{(Tt + J)-I(x*)} is continuous from [0, 1] into X endowed with the norm topology.

(iv) Given [x, x*] E GrTt and tn -+ t in [0, 1], then there exists a se- quence [xn, xn*] E GrTt, such that xn -- _{x in X and xn*} - _{x* in}

X* (i.e. Gr Tt C lim Gr Tt, which is of course equivalent to saying that t -+ _{GrTt is l.s.c. from [0, 1] into} 2xxx* \{0}).

The admissible homotopies for f are given in the next definition.

Definition 8 [12]. Let

{ft

{ft}

such that tn -- _{t for which}

lim(ftn(xn) Xn -X) < 0

we have that xn - x in X and

ftn

Finally we introduce the family of admissible homotopies for the multivalued perturbation G(

*

) .

Definition 9. A one-parameter family of multifunctions Gt : U - _2x*\}

t E [0, 1], is said to be a "homotopy class (P)" if (t, x) -- _{Gt(x) is u.s.c. from} [0, 1] x U into 2x*\{0}, for every [t, x] E T x U Gt(x) is a closed and convex subset of X*, and

{U

The next proposition paves the way for the eventual definition of the degree function on maps of the form T + f + G by producing a crucial approximation to it on which Browder's degree function can be defined. From Lemma 6, we know that if G: U - 2x \{0} is a multifunction of class (P) and e > 0, then we can find gE U) X* a continuous function such that _{g, (U) C} conv G(U)

and for all x E U ge(X) E G((x +BE) n U) +B* where BE = {X E X :IIXII <E}

and _BE = {X E X* : IIX*II* < E}. In what follows _gE(.) will denote this

approximate selector of G( ).

Proposition 10. Let U be a bounded open set in X, T: D C X -- _{2x \{z} a} maximal monotone map with 0 E T(O), f: U -+ X* a bounded map of class

(S)+, G: U - 2x*\,' J a multifunction of class (P). Let yo* E X* such that

y* 0 (T+f+G)(QU). Then

(i) y* 0 (T+f+ge)(&U) for alle>0 small;

(ii) f + g, is a mapping of class (S)+ and so by Theorem A, the Browder degree d(T + f + ge, U, y*) is defined for all e > 0 small;

(iii) there is el > 0 such that for all 0 < _{e < el} and all approximations

g(-), d(T + f + g6, U, y*) has the same value (that is, {d(T +

f +

}

Proof. From the remark following Lemma 6, we know that g,

(*)

f

could find a sequence en 1 0 and {un}n>i C AU with un -- u in X such that y* E (T+f+gen)(un), n > 1.

Let _{v* = yo} - _(f(un) +

gan(un)) * Then v,* E T(un). By passing to a sub- sequence if necessary, we may assume that g6, (un) --. _{g*, f(un)} w

f

* +

lim(v* + f(un), un - u) = lim(y* - _{gA (un), Un} - u) = 0

-

Since by hypothesis

f

lim(vn*, un - _u) < 0

?* v E _{T(u) and (vn*, un)} (v*, u) (since T is maximal monotone). Therefore (f(un), un - u) -p 0 as n -- ox and so un - u in X and f(u) =

f**

f

To prove (iii), we proceed again by contradiction. So suppose that there exists 0 < en < an -+ 0 such that

d(T+f

Then from the homotopy invariance property of Browder's degree function (cf. Theorem A), we get tn -+ _{t in [0, 1] and un} E AU such that

y*

+ f

Note that (tng6n(X) + (1- tn5)g6 (X)) E G((x + B5") n U) + B* for all x E U

and so arguing as in the proof of part (i), we get y* E (T + f + G)Q( U), a contradiction. So the proof is complete. Q.E.D.

In the light of this proposition, the following definition makes sense: Definition 11. We define d(T + f + G, U, y*) to be the common value for E > 0 sufficiently small of d(T + f + ge, U, y*) (this last degree being the

Browder degree).

The next theorem shows that the degree function just defined has the three characteristic properties of normalization (with normalizing map the duality map J), of additivity on domain, and of homotopy invariance (with admissible homotopies being given by Definitions 7, 8, and 9).

Theorem 12. The degree function defined by Definition 10 has the following prop- erties:

(i) Normalization: d(J, U, y*) = 1 for all y* E J(U).

(ii) Additivity on domain: If U1, U2 are disjoint open subsets of U and

y* 0 (T+ f + G)(U\(U1 u U2)), then

d(T+ f + G, U, y)= d(T+ f + G, Ui, y*) +d(T+ f + G, U2, _y*). (iii) Homotopy invariance: Let {Tt}tE[o, ] be a pseudomonotone homotopy

of maximal monotone maps from X into 2X with 0 E Tt (0) for all te[0, 1], {ft}tE[o,l] isahomotopy of class (S)+ of mapsfrom U into

a bounded subset of X*, and {Gt}tE[o, 1] is a homotopy of class (P) of multifunctions from U into the nonempty, closed, and convex subsets of X*. Let y* : [0, 1] -+ _{X* be a continuous map such that y*(t) 0}

(Tt + f + Gt)(aU) for all t e [0,1]. Then d(Tt + f + Gt, U, y(t)) is independent of t E [0, 1].

Proof. (i) This property follows immediately from Theorem A.

(ii) This property too follows from Theorem A, since by Definition 10, d(T

+ f

+ f +

f

a (S)+ map.

(iii) Let G(t, x) = Gt(x). Recalling (cf. Definition 8) that (t, x) -) G(t, x) is u.s.c., we can apply Lemma 6 with B = [0, 1] x U and, for any e > 0, get a continuous function ge(t, x) from B into conv G([O, 1], U) such that

g,(t, x) E G(([t - , t + E], X + BE) n B) + BEfor all (t, x) E B . We claim that for e > 0 small enough, y*(t) ? (Tt + ft + gt,e)(aU) for all t E [0, 1], with gt,e(x) = ge(t, x). Assume the contrary. We then have

tn-+ t in [0, 1], En I 0, {un},n> C aU with un w u in X, and y*(tn) E

(Ttn

+

for all n > _{1. By passing to a subsequence if necessary, we may assume that} y*(tn) - y*(t), vn* * v*, ftn(un) wf*, _{and gt,,(un) -} g*. Hence

lim(vn* + A, (un) X Un -U) = O-

Also since

_{ft}

So we get

lim(vn*, un - _{u) < 0}

and this by Definition 7 implies that v* E Tt(u) and (v, un) - _{(v*, u). Thus}

lim(ft,,(Un) , Un -U) = O

and so we have un -) u in X and _ft(un) w f (u) in X* (cf. Definition 8). Also it is easy to check that g* E Gt(u). All these facts combined tell us that

y*

(

with u E C9 U , which is a contradiction. So indeed for E > O small enough, we

haeta y. .t 0 ,T

.

It is routine to verify that {ft+g9,C}1E(O, 1] is a homotopy of class (S)+. So by Theorem A, we have that d (Tt+ ft +gt, , U, y*(t)) is independent of t e [0, 1] for e > 0 small enough. It only remains to show that for e > 0 sufficiently small and for each t E [0, 1] fixed, we have d(Tt +ft +

gt

d(Tt + ft +Gt s

Fix t E [0, 1]. Then by hypothesis Gt: U --* 2X' \{} is a multifunction of class (P). Apply Lemma 5 _{to Gt(*) with B} = _{U, to get g:} U -- _{X* a}

compact map which satisfies

ge(x)

E

for all x E U. Consider the affine homotopy sg, + (1 - s)gt,, with variable s E _{[0, 1]. The same arguments used before show that}

y*(t) 0 (T1 + ft + sge + ( - _s)gte)(aU)

for s E [0, 1] and e > 0 small enough. Then Theorem A tells us that

d(Tt + ft +gt^,c U,~y*(t)) = d(Tt + ft + ge IU, y*(t))

for e > 0 sufficiently small. But for a > 0 sufficiently small the last degree equals d(

Tt

ft

d(Tt + ft + Gt,~

U,~

y*(t)) = d'Tt + ft +

I U

for small e > 0. Since t e [0,

1]

Next we establish the uniqueness of the degree defined above with respect to the three properties of Theorem 12.

Theorem

13. There exists exactly one degree

function on the class of maps T +

f + G, with T maximal monotone, f bounded

and of class (S)+, and G a

multifunction

of class (P), which satisfies the normalization

and additivity

prop-

erties of Theorem

12 and is also invariant

under

all affine homotopies

of the form

(I

- t) (T ? f + G) + tf1 with t

[0, 1

T maximal monotone, fI, f

and of class (S)+, and G a multifunction of class (P).

Proof. Let d1 be such a degree function. By setting G 0, from Theorem A, we have that d1 coincides with Browder's degree function, which is uniquely defined on maps of the form T + f . Using the above affine homotopy we will show that this unique identification carries on to the broader class (T + f + G). Suppose yO* (T + f + G) (& U) . Consider the affine homotopy

(1 -t)(T+ f +G)+t(T+ f +ge)

with Te = (T-1 +eJ'1)-I and ge( - ) is as always the compact selector of G( ) guaranteed by Lemma 6 such that g,(x) E G((x + Be) n U) + Be, for all x E U. Using Definition 7 we can easily check that (T, + f ) is of class (S+); hence (T, + f + g6) is of class (S)+ (recall the class (S)+ is closed under compact perturbations). Since we have

then d(T+

f

?

f +G,

y )

u, and

y0 E (1-tn)T(Un) + _tnTn(Un) + (1 -tn)G(Un) + tnge (Un) + f(un).

Assume that v*

T(un) and gn*

G(un) are such that

= (1 - _{tn)v* + tnTen(Un) + (1} -

tn)gn* + tng,,(Un) + f(Un),

by passing to a subsequence

if necessary,

we may assume that

w f

and

(1 - _{tn)gn* + tngen(un)}

- h* in

In what follows, we use the arguments

of

Browder [11] (see the proof of Theorem 12). Let w

*

T(un -enJ(wn*)). This and v *

T(un) imply that en

Iw

un) and

0 <

_(vn*,

un). Thus we obtain

tn gn IIWn*112 < _{(Yo -( 1-tn)gn* -tngzn (Un)- f (un)} X Un) < M M > O -

Hence {tnCen wn* I2}n>l is bounded and so tnen,IIw*II -+

0

[x, x*I

Gr T. Then

(W*- X*, Un -EnJ1 (W*)-X) > 0, (V*- X*, Un -X) > 0

Thus (w* - x*, un - x) > (w* - _{x*, enJ-1(w*)) > -,en,Iw*II*IIx*II*} * Conse-

quently, we have

(yO - (1 - tn)gn- tngen(un) -f(un) - X%, un - x) _{-tn6n||Wn*|I*|X*} II* 0. Hence if we let _zn = -[(1- _tn)gn* + _{tngen(un) + f(un)],} then Zn w _{z* with} z*=yo*-(h*+f*) and

lim(z*- x*, Un -X) > 0

and the latter means that lim(zn,

(x*,

x)

.

On the other hand, since

(1

Also

un

u)

0, since

f(.)

is of class (S)+. Therefore

we get

lim(zn*, Un -U) < O =E- lim(zn*, Un) < (Z U) -

Thus (z*, u)

_{lim(zn, Un) ?}

(zn, un) > (x*, u-x) + (z*, x) . It follows

that 0

x) for all [x, x*]

Because of the maximal

monotonicity

of T, z*

T(u). Then, by replacing [x, x*] by [u, z*], we get

(zn, Un) -- (z*, u). Hence

lim(f(un), un-u) =0-

So we conclude that un -- _{u in X; hence u E 9U and f(un)} w- _f(u) = _f

in X* (since f(.) is demicontinuous

being of class (S)+; cf. Definition 2).

Also it is straightforward

to show that h* E G(u). Thus y*

(T + f + G)(u),

with u

a contradiction.

Therefore

the two degrees coincide and so we

have established

the uniqueness of the degree function on maps of the form

(T+f+G). Q.E.D.

Remarks. (1) It is clear that the degree function of Definition 11 can be ex-

tended to the broader class of maps of the form T + f + G, with f being

pseudomonotone and bounded by defining

Note that for any e > 0, f+eJ is of class (S)+. This extended degree function

has properties similar to the degree function for pseudomonotone maps defined by Browder [ 1 1 ].

(2) The condition 0 E T(O) can always be satisfied by appropriately trans- lating the domain and the operator. More precisely if [xo, x0*] E Gr T, define T1: (D -xo) 5 X -* 2x \{0} by T1(x) = T(x +xo) -xo*. Clearly T1(.)

is still maximal monotone (if T(-) is) and 0 E T(O). For the permissible maximal monotone homotopies the condition 0 E _{Tt(O) can be replaced by the} requirement that [xo, x *] E Gr Tt for every t E [0, 1].

4. DEGREE FOR MAPS OF THE FORM f + NG

As we already pointed out in the introduction, the condition that G: U 2x* \{o} is compact translates into some growth condition on G when applied to partial differential inclusions. In Theorem B, this restriction was replaced by a sign condition. In this section, we pursue this idea and achieve a two-fold extension of Theorem B. On the one hand, we allow a multivalued function G(x, r) in place of g(x, r) and on the other hand, we relax the sign condition. Let ZC Rn beanopensetandfor m > 1, 1 <p < o,let X= W- l(Z). Then its dual is X* = W-m,q(Z) with I + I = _{1. Let G: Z x} Rt 2R'\{o} be a multifunction with compact, convex values such that (x, r) - G(x, r) is measurable and r -- _{G(x, r) is u.s.c. It is well-known that un-} der these assumptions we can write G(x, r) = [(o(x, r), ,v(x, r)] = {h E R: Vo(x, r)

<

while r -k -(p(x, r), V(x, r) are both u.s.c. We want to impose sign conditions

on G(x, r) and so we make the following definition:

Definition 14. A multifunction G(x, r) is said to satisfy the "sign condition" if the following properties hold:

(i) G(x, r) = [g(x, r), ,v(x, r)] is measurable in (x, r) and u.s.c. in r and for each s > 0, there exists hs (* ) E _{L1 (Z) such that for}

_{In <}

jo(x, r)j, jj(x, r)j < hs(x) a.e. on Z;

(ii) for all x E Z, (o(x, r)r > 0 for r < 0 and qi(x, r)r > 0 for r > 0. If G(x, r) is single valued, this definition coincides with the sign condition of Browder [12] (cf. Definition 5). As we already indicated earlier, we want to relax this condition. So we introduce

Definition 15. A multifunction G(x, r) is said to satisfy the "generalized sign condition" if the following properties hold:

(i) G(x, r) = [(p(x, r), u(x, r)] is measurable in (x, r), u.s.c. in r, and for each s > _{0 there exists hs E L1(Z) such that for Irj < s}

ko(x, r)I IV1(x, r)l < h5(x) a.e. on Z; (ii) there is an ro > 0 such that for all x E Z

Let G: Z x R -- 2R\{z} be a multifunction satisfying Definition 14 or 15.

We formally define the Nemitsky operator NG from D C X = Wo P(Z) into

2x*\{0} by

NG(U) = IV E W-m q(Z) n 4lc(Z) :V(X) E G(x, u(x)) a.e. on Z}, with D

=

A multifunction G(x, r) which is only measurable in x and u.s.c. in r is not in general jointly measurable (cf. [18]). That is why we need to assume joint measurability of G(x, r). Note that if G(x, r) = g(x, r) is single-valued, then this joint measurability is automatically satisfied by the Caratheodory con- ditions; i.e. g(x, r) is measurable in x, continuous in r. This is the case in Browder [12]. Also note that our joint measurability hypothesis implies that for every u Z -- _{R measurable, x} -- _{G(x, u(x)) is measurable and so it has a}

measurable selector.

Let U be a bounded open set in X = W0m'P(Z),

f:

(f

The following proposition which will be needed in the sequel is due to Brezis and Browder [2].

Proposition 16. Let u be an element of W0m'P(Z), T an element of W-m,q(Z) n _L4lr(Z) such that T(x)u(x) > h(x) for some h summable function on Z. Let (T, u) denote the distribution action of T on u (i.e. the duality brackets for [T, u] E X* x X). Then T(.)u(.) is summable on Z and

(T, u) = T(x)u(x) dx.

A critical step in defining d(f + NG, U, y*) is to approximate G(x, r) by single-valued, Caratheodory functions g, (x, r) which satisfy the corresponding sign conditions. This is done in the next proposition.

Proposition 17. If G: Z x R -- _{2R\{1} is a multifunction which satisfies the} generalized sign condition and e > O, then there exists g: Z x JR - _{R., a}

Caratheodory function satisfying:

(i) g,(x, r) E G(x, r + B) + Be for all (x, r) E Z x R and with Be - (-X, e);

(ii)

for each s

O, there exists hs

LI (Z) such that for Irn

and hs(*) can be chosen independent of e > 0; (iii) for all x E Z and all Irl > rO + 1, g6(x, r)r > 0.

Proof. Let ,u: Z -- _{R be a continuous function such that 0 < u(x) < 1 for all} x E Z and fzu(x) dx < xo.

Step 1: Define (0*, y,*: Z x R -- R by

and

Yl*(x, r) ={min[0,

y/(x,

and G*: Z x R -- _{2R\{z} by G*(x, r)} = _[p*(x, r), y/ *(x, r)]. It is clear that G* (x, r) C G(x, r) and that G* (x, r) is measurable in (x, r) and u.s.c. in r.

Fix x E Z and apply Lemma 6 on G* (x, *) with 3 > 0 to get iq6: R- *R R a continuous map such that

tq5(r) E G*(x, r+B)+Ba

for all r ER (recall B' = (-3,3)). Take 3 < min[I, eu(x)]. We then have

max{y E R:y

E G*(x,

<O

min{y ER:y E

G*(x,

Define 1* : Rt

R

min[0, ?qa

(r)],

i7j* (r) = g 7 (r) , - (rO + I2) < r < (rO +2- max[O, ?tj5(r)], r > ro + 1,

and on the intervals -(ro + 1) <

r

r <

+

?1*

(r)

(x, r +

+

Step 2. Define a multifunction F, Z 2C(R, R) by

J76(X) = {ti E

C(R, R):

0

$

Let G(x, r) =G(x,

r +

Let G(IF) be the graph of (). We have

G(J7) = {[x, ?J] E Z x C(R, R): d(?I(r), G(x, r)) =

0

for all rER, and ?(r)r >

O

an enumeration of the rationals in irl > ro + 1. Note that since G(x, *) is u.s.c., r -+ d (i(r), G(x, r)) is l.s.c. for any ? E C(R, R). So we can write

G()=

n

n

{[x,

1E

Z

x

C(R,

For each n > 1, (x, t1) -*

d(ti(rn),

from Z x C(R, R) into

(recall C (R, R) is a Frechet space). So it is jointly

measurable

and therefore {(x,

Z x C(R,

: d(77(rn),

G(x, rn))

0}

2'(Z) x B(C(R, R)) for every n > 1 with Y(Z) being the Lebesgue a-field

of Z and B(C(R, R)) the Borel a-field of C(R, R). Consequently,

G(I,)

Y (Z) x B(C(R, R)). Applying Aumann's

selection theorem (cf. Wagner [21,

Theorem 5.10]) we get y,: Z

C(R, R) a Lebesgue measurable

map such

that YB(X) E Fe(X) for all x E Z.

Set ge(x, r) = (ye(x))(r) . Then g: Z x R -*

R

which satisfies conditions (i) and (iii) of the proposition. It is easy to see that

condition (ii) is also satisfied and in fact h,(.) can be chosen independent

of

E > 0

small, since

Q.E.D.

Let gg(x, r) be the Caratheodory

approximate

selector

obtained

in the above

proposition. We need yet another approximation

method, namely the trunca-

tion procedure

on g(x, r).

Definition 18. Let

be an increasing

sequence of relatively compact,

open subsets of Z such that Z = U=I1 Zk . Let &k(*) be the characteristic

function of

Consider

the truncation

of g, at level k; i.e.

g,k(x r) g= I E x r) if 1ge(x, r)l < k,

ksign(g8(x, r)) if Ige(x, r) I> k.

We define the kth-approximant

of the Nemitsky

operator

as a map from X into X*, by

Nk (U)(X) - =kgk(X, U(X)).

It is then clear that each Nk is a compact map of X into X* n L*(Z).

In what follows, g8(x, r) will be the Caratheodory

approximate

selector of

G guaranteed

by Proposition 17 and gk(x, r) the corresponding

truncation

and Nk its Nemitsky operator. Also by A(.) we denote the Lebesgue

measure

on the set Z.

Theorem 19. If U is a bounded

open set of X

f a bounded

map of class (S)+ from U into X*

(Z), G: Z x

is a

multifunction

satisfying the generalized

sign condition,

and y* E X* such that

Yo

0 (f +NG)(a U), then

(i) y

₀

_{+ Nk )( U) for k}

_{> 1}

_sufficiently

_{large and E sufficiently}

_small.

Since (f

is of class (S)?, the Browder

degree d(f + Nk, U, y*)

is well defined;

(ii) for k > 1 sufficiently

large and

> 0 sufficiently

small, d(f+NNk, U, y*)

is independent

of k,

and the selector g (.) from Proposition 17.

So we can define this ultimate common value to be the degree

d(f +NG, U, yO).

Proof. (i) Suppose that the conclusion was false. Then we can find

a

U and

l 0 such that

As before, by passing to a subsequence if necessary, we may assume that Uk W u in X, f(uk) w f*, _Nkk(Uk) and. `* V in X*. For any bounded subdomain Z' of Z, by Sobolev's embedding theorem we know that W? 'P(Z') embeds compactly in LI(Z') . Hence we may assume that Uk(X) --+ u(x) a.e. on Z.

Contrary to the single-valued situation considered by Browder [12] (cf. Theo- rem B), where Nk (Uk) - Ng(u) in LI (Z'), in the present multivalued context,

Nk (Uk)(*) is not pointwise convergent in general. It is only weakly convergent in L '(Z) as we will show next. Nevertheless, using this weak convergence of {Nk (Uk)( )}k>l in L1(Z) we will arrive at a contradiction, establishing part (i) of our theorem.

First we will show that

SUp IN k (Uk)(x)l dx < 00.

k>I Z e

To this end, let Z(k) = {x E Z: IUk(Z)I

<

M> jN _kek (Uk)(X)Uk(x)

dx

|

Lk

> -(ro?

1)L

1)L

Z(k) Z\Z(k)

JZ\Z(k) Nek (Uk)(x)l dx < rM + j hro+I (x) dx IN (Uk)(x)l dx <

M

(x)

JZk r0 +1 J +lJ

Now let Z C Z be measurable with A (Z) < o and I > ro + 1. Let Z' - {X E Z: lUk(X)I?< } and Zk+ = X E Z: IUkI(Z)IL > 1 We have

ek (Uk) (X)

I

dx

dx

+

dx

<fhi(x)dx+ (M +(ro+1)jhro+I(x)dx).

Since I > ro + 1 was arbitrary and h1(*) E L1 (Z), it is immediate from the above inequality that SUPk>I fZ, INkk(uk)(x)I dx -- 0 as A(Z') -- 0, and for every E > O there is a Z C Z, A(Z) < 0 such that fz,z, _{INekk(uk)(x)ldx} <

E. So finally invoking the Dunford-Pettis theorem (see Dunford-Schwartz [15, p. 347]), we get that _{Nkk(Uk)( )}k>I is relatively sequentially weakly compact

in LI(Z). Hence we may assume that Nkk (Uk) w Vl in L1(Z).

Now we will show that v u E L1 (Z) and that the following inequality holds:

L

dx

<

limL

dx.

Define Z_ = {x E Z: ju(x)I < rO + i} and Z+ = {x E Z: Iu(x)I >

v*(x)u(x) > 0 a.e. on Z+. In fact for any Z1 C Z+ with A(Z1) < xc and any 3 > 0 by Egorov's and Lusin's theorems we can find a closed set Zj 6 _Z, with A(Z1\Z,) < 3 such that Uk(Z) ` _{u(z) uniformly on Zj and u(.) is}

continuous on Zj. Hence Uk ` _{u in L??(Z,) and so since Ne(uk)} - vl in ' L1 (Z,) we get that for any B C Zj measurable

jv*(x)u(x)dx = lim _{Nek(uk)(X)Uk(x)} dx > 0

? O<v (x)u(x) a.e. onZ,.

Since Xz6 (x)v (x)u(x) converges to Xz, (x)v (x)u(x) in A-measure we get that 0 < v (x)u(x) a.e. on Z1 . Finally recall that Z1 c Z+ with A(Z1) < oo was arbitrary and Z+ is a-finite to conclude that 0 < v* (x)u(x) a.e. on Z+.

Next we prove that

_fz

dx

<

6 > 0, there exists Z1 C Z+ such that A(Z1) < x and SUp IN |k(Uk) (X) I dx < xo .

k>1 Z\Z le

If Zj C Z1 is as above, we define

Z(J) ={x EZ+ n nZ6: IUk(X)I <ro+ 1} and

Z+k(3) = {X _{E Z+} n _{Z,: IUk(X)I > rO+ 1}.}

Then _{IN(ekk(Uk)(x)uk(x)I < (ro+1)hro+1(x) on ZL (3) and Nekk(uk)(X)uk(x)} > 0 on Z+k(3). Clearly A(Z1 n Zk (j)) < A(Z1 n _{Zf) <} 3. _{Since hro+ (*)} E L1 (Z), we have

sup [(ro + 1) _hro+(x)dx ] = p(3) 0

k> 1 Z Z

as 3 10. Thus

lim

+Nkk

dx

=lim

[IN~(k)()k

d?_

Nk

Uk(x)

dx

_ | ek (Uk) (X) Uk (X)

d x

+ j _{Nek(Uk)(X)Uk(x)}

_dx]

dx

1)

hro+I

(x) dx

lgk~ ~~~~~~~Z nZk (j)

- SUp J I ek(kk)(X)I dx k> 1 Z\Zj

>

dx -

-

.

Since J > 0 was arbitrary, vj*(x)u(x) > 0 a.e. on Z+ and p(3) , 0 as I O, we get that v u E LI(Z+) and

Now we will show that the same inequality holds also over Z_. For any 3 > 0, find a measurable set Z1 C Z_ with A(Z1) < oo such that

J lv*(x)u(x)l dx < 3 Z_nZc

and

(ro+1)j hro+I(x)dx<36.

Recall that v u E Ll (Z_) and hro+i(*) E L' (Z) and so the above choice is

possible. Let Z C Z1 with A(Z1\Z6) < 3 such that Uk(x) - u(x) uniformly

on Z and u(-) is continuous on Z . Define

ZL(3) ={X E Z- n Zg: IUk(X) <?ro+ 1} and

Z+k() ={x E ZfnZ": IUk(xX)I > ro+ }.

Hence since fz v (x)u(x) dx = lim _{fi N,kk} (Uk) (X)Uk(X) dx, we have

jv*(x)u(x) dx < lim f N(Uk) (X)Uk)(x) dx +

j

z

~

~

~

< lim Nk (Uk) (X)Uk (x) dx + _p(3) + |v*(x)u(x) |dx where p(3) = _{supk>1 [(ro + 1)}

fik (6) hro+I (x) dx]. Recalling our initial choice of

Z1 C Z, we see that p(3) - O and fz nz lv*(x)u(x)l

dx

j

_{v*(x)u(x)dx <}

Therefore, we finally have that v u E L' (Z) and

L

vj*(x)u(x)dx

<

Recapitulating, we have that _Ngkk(Uk) w- _{v* in L'(Z)} and _Ngkk(Uk) w v in X* = W-m q(Z). Since both modes of convergence imply weak convergence in the space of distributions O(Z)', we get that v* = v* = y* - f* and so

V1*E W-m,q()

Define

h: Z

,

h(x)

=

h _{v*(x)u(x) if lu(x)l < rO + 1,}

Then h E L'(Z) and v*(x)u(x) > h(x) a.e. on Z. So by Proposition 16, we have that

Hence, since

we have

lim(f(uk), Uk-u) = (yo-f*, u) -limj Nk(Uk)(X)Uk(x)dx

< (YO* -f* u)_ -|vl*(x)u(x)

dx

Since f is a map of class (S)+, we get that

in

and

so u E OU. Furthermore, f(uk) -y f(u) in X* and so

f*

it is straightforward

to check that

Thus we have shown that

y*

(f +

with u

U, a contradiction.

This completes the proof of

part (i) of the theorem.

(ii) Again we proceed by contradiction.

Then we can find sequences

ek > 0, 3k > 0

such that

cc e

and

l

as

k

and

furthermore

d(f +N , U, y*) 0 d(f

By the homotopy

invariance

property

of the degree function for affine homo-

topies of class (S)+, we know that we can find

[0, 1] and

U such

that

f(Uk) + ( -sk)N k (Uk) + skNkJ(Uk) = y0.

Without loss of generality,

we may assume that the above equation

holds for

all k > l. Let V* = _{(1 -sk)Nk (Uk)} + SkN kk _{(Uk) .} We may assume that Sk -- S

in [0,

in X and

in X*. Wehave

Jv*(x)uk(x) dx = (yO-f(uk), Uk) < M

Let 4k(x, r) = (1 - sk)k(x)g k (x, r) + skXk(x)g nk (x, r). Then vZ*(x) = g k(x, Uk (x)). It is easy to see that g (x, r) satisfies (i)-(iii) of Proposition 16,

with

replaced

by

and

replaced

by max[ek,

With the same argument as in part (i), we can show that vk w- v*

in

L1(Z), v*(x)u(x) 0> a.e. on the set

{x

Z:

and

L vj*(x)u(x)

dx < lim

(recall that because of the compact embedding

of

P

into

(Z') for

any Z' C Z bounded, we may assume that Uk(X) -- _{u(x) a.e. on Z)}

_.

and

Thus we get yo*

with

AU,

which is a contradiction.

Therefore,

we have proved part (ii) and the proof of the theorem is com-

plete. Q.E.D.

For the degree function established

with the previous theorem,

we will prove

the three characteristic

properties.

For this we need to introduce

the permissible

Definition 20. Let {Gt(x, r)}tE[O, 1] be a family of multifunctions from Z x R into 2R\{0}. Such a family is said to be a "permissible homotopy of multi- functions" satisfying the generalized sign condition, if the following conditions are satisfied:

(i) Gt(x, r) = [(o(x, r, t), V(x, r, t)] is measurable in (x, r, t) and u.s.c. in (r, t); for each s > 0, there exists hs E LI (Z) such that for Iri ? s and all t E [0, 1]

max[l(px . r, t)J, I (x, r, t)J] <

hs(x).

(ii) There is an ro > 0 such that for all (x, t) E Z x [0, 1] we have (o(x, r, t)r > 0 for r < -ro, vt(x, r, t)r > 0 for r > ro.

Having defined the permissible homotopies for the multifunction G(x, r) we can now introduce the permissible homotopies for the degree function defined by Theorem 19.

Definition 21. The class H of permissible homotopies of maps of the form f + NG consists of all homotopies ht =

_(f

Theorem 22. The degree function defined in Theorem 18 has the following prop- erties:

(i) Normalization: d(J, U, y*) = 1 if yo e J(U).

(ii) Additivity on domain: If Ul, U2 are disjoint open subsets of U such that _{y* 0 (f} + NG)(U\(UI U U2)), then

d(f +NG

U, y5) = d(f

(iii) Homotopy invariance: Let {ht = ft + NG,}tE[o, _I] be a homotopy in the class H and let y* : [0, 1] -* _{X* a continuous map such that yt} *

(ft + NG,)(9 U) for all t E [0, 1]. Then d (ft + NG, U, y*) is indepen- dent of te[0, 1].

Proof. Properties (i) and (ii) are obvious. To establish property (iii), first we obtain a single-valued approximate selector g6(x, r, t) of Gt(x, r) which is measurable in x and continuous in (r, t) (cf. Proposition 17) and satisfies all conditions of Proposition 17 uniformly in t E [0, 1], and then repeat the arguments employed in the proof of Theorem 19, using the fact that Browder's degree function on maps of class (S)+ is homotopy invariant. Q.E.D.

Remark. The degree function defined by Theorem 19 on triples (f? NG, U, y*) is not unique in general, since not every approximate continuous selector g, (x, r) of G(x, r) necessarily satisfies the same sign condition as G.

A careful reading of the proof of Theorem 19 shows that in the definition of the generalized sign condition, we had to assume that the control function hs(-) E L1(Z). If G(x, r) satisfies the sign condition of Definition 14, then we only need to assume that hs(*) E Lll c(Z) (see also Browder [12]).

Definition 23. The class HI of permissible homotopies of maps of the form

of all homotopies

_{ff

(i) for every s > 0, there is

hs

Lll

n

max [o(x , r , t), y/ (x , r,. t)] < hs

(x) ,

(ii) q(x, r, t)r > O for all r

<

>

Theorem 24. The same approach as in Theorem 19 will define a degree function d(f + NG, U, y0) with f of class (S)+, G a multifunction which satisfies the sign condition (cf: Definition 14). In addition, this degree function has the three characteristic properties of normalization, additivity on domain, and invariance under homotopies of class H1.

Proof. As in Proposition 17, we can obtain a Caratheodory approximate selector g6(x, r) of G(x, r) satisfying the sign condition; i.e. g6(x, r)r

>

To have uniqueness of the degree function, we need to restrict the class of multifunctions G(x, r).

Definition 25. G: Z x IR -* 2R\{z} is a multifunction satisfying the "strict sign

condition" if the following hold:

(i) G(x, r) = [o(x, r), yi(x, r)] and is measurable in (x, r) and u.s.c. in r;

(ii) y/(x, r)r > 0 for r < 0 and (x, r)r > 0 for r

> O;

(iii) for any s > 0, there exists h, E LI (Z) such that for Irl < s max[j(o(x, r)j, kI (x, r)jJ < h,(x).

As in Definition 23, we can define the class H2 of all permissible homotopies of maps of the form

f

Proof. Suppose that there were another degree function d1 different from d obtained in Theorem 24; i.e. dl(f

+

(f+

d(f +NG, U, yO) = d(f + NEk, U, y*)

for k > 1 large enough and e > 0 small enough. Recalling that Browder's degree function is unique on maps of class (S)+ (cf. Browder [11, Proposi- tion 14]), we have d(f + Nk , U, y*) = d1

(f