Analysis And Topology

104

H.

Begehr and A. Dzhuraev

min{/2,v}

akl...kMll...IV = 1
p=o

1Ii<

that the orders of the indices of the a's are interchangable. Here the last sum is understood to be taken over all ordered sets {Ci,.. }' pairwise disjoint and being empty if the highest {k1,... , assuming

.

. . ,

index i.e. p, p u(z)

—

p, ii — p,

respectively is zero. Thus d(k

=

k=1

min{'2,v}

/2,L'l

/L+V
1k_zkI2 n

/2,L/1 p/2+Vfl

p=o

fl
'DL71

lDLTp

xf

[flog

I

(CTAZCTA

2

A=1 '2—p

A=1 (kA — ZkA

H(1 =

1A

—

(56) ZIA

1 —

which more clearly expressed as 1

/2,L/1

/2+Vfl

1

II I

JDkM 12)ii

lli<. •
Schwarz Problem for Cauchy—Riemann Systems

i-ri/

105 /2

LI

x

1 IA

1A

H c1

+

—

n

p=l

/2,V=1

/2+V
fl
p=/2+V—fl

lDk1

'DIV_p

p

2

p

H log A=1

Z(TA

(CiA

I

1Z(TA((TAI

dl/(TA

'2—p

LI—p

x

+ IA

ailkA

A

)=1

A

kA

kA

Simplification of condition (52) can easily be done by using (37) and (38) which

lead to (24). Denote for 1
1k 11k

f

= x

H

(k+1 — Zk+1

-

L1k+2 1 —

f

II

k

'2k

Zk+1

v=1

—

VZLI (['(V =

ñ we write, see (49), U0 =

TIM

i)

Zk+1

(k+1 — Zk+1

H. Begehr and A. Dzhuraev

106

with, see (47), F11

•

(k2

•

+

1
I F——

(Ilk + I2k) /1=2

l1(:2(:A(:A+1(:,4

)=1

1D11

JD',4

Zi

x 1 —

1 —

1 — z1A+1 (1A+1

1—

Z1,4

(1,.

Similarly as for the derivatives of u0 here

=

+

•

1
and hence,

= Thus the solvability condition (52) becomes

I/k L \v=1

_i)

—

Zk+ 1

(k+1 — Zk+1

Zk+1

— 1) (k÷1 — Zk+1]

—

xfi v=k+2

(

(I, — ZI,

+

Z1,

— Z1/

v=1

n /2—1

/22

'=1

it/I

I II •••

'01LI

Schwarz Problem for Cauchy—Riemann Systems

xf

—

—

107

ZkMV

Zk1

1 — ZkM_v (kM_V

1

x

Zi

Z11

—

l—Z11(11

(57)

Summarizing the following result is proved.

Theorem 4. Let fkl, 1 < k, 1 n, have mixed derivatives of first order with respect to all variables (zi,. and the complex conjugate variables E . . ,

. belonging to L1 and satisfy the compatibility conditions , (46) and (48). Let -y E be complex-valued. If (57) is satisfied then the The solution Dirichlet problem (50) for system (44) is uniquely solvable in is given by (56). .

5. Riernann-Hjlbert Problem in the Unit Ball of V 1. Let lB = {z E < 1} be the unit ball of (JJfl + = v"IziI2 + and ÔJB = {( 'JJ = 1} be the unit sphere, the boundary of lB. Let A(() and be functions given on ÔJB, satisfying there a Holder condition — p, E — with respect to the Euclidean metric and 0 < < 1. The Riemann—Hilbert problem in the ball lB is to find a function p(z), holomorphic in JB, continuous up to ÔJB, satisfying the condition

on ÔJB.

Re

(58)

Since for n > 1 the ball lB is not biholomorphically reducible to a polydisc, then this problem has special features. The Schwarz problem = 1 is a special case of the Riemann—Hilbert problem.

Lemma 12. The Schwarz problem in lB is solvable if and only if

f holds for any (

E

(Re

(1-

- i)

=0

(59)

Ô1B, where ((,q)

Proof. From (58) and the Cauchy formula for the unit ball it follows that a solution of the Schwarz problem is given by

H.

108

=f

Begehr and A. Dzhuraev

- i)

((i_

+ ic

(60)

with an arbitrary real constant c. Using the Plemelj formula for the boundary values of the Cauchy integral (see [10]) as z E lB tends to ( E Ô1B we conclude

that (60) satisfies (58) if and only if (59) holds for any (

Ô1B.

Let us

denote by H(p, q) the vector space of all harmonic homogeneous polynomials in that have total degree p in the variables z1,. , zr-, and total degree q in , (see [12], p. 255). The following lemma gives another the variables description of the condition for the right-hand side of the Schwarz problem to . .

.

be solvable.

Lemma 13. The Schwarz problem is solvable if and only if -y(() is orthogonal to the subspace H(p, q) of the Hilbert space L2(ÔJB) for p 1, q 1. Proof. If the problem is solvable, then it follows that -y(() extends to lB as the real part of a holomorphic function in JB, i.e. as a pluriharmonic function and therefore it satisfies the equations (see [3])

= where —

0,

1
(61)

-o

a

are the Cauchy—Riemann operators, which are tangential to OJB and this is q) with p 1, q 1. In case n = 2 equations (61) reduce equivalent to to the single equation

E2Ly=0. where

L

(i

a

- (2

a

—

L

(62)

—a -—a (2

Moreover, the operator L becomes a one-to-one mapping of the space H(p, q)

with q 1 onto the space H(p + 1, q — 1) and the operator L becomes a oneto-one mapping of the space H(p, q) with p 1 onto the space H(p — 1, q + 1). Hence for any P E H(q, p) with p 1, q 1 there is a Q E H(p + 1, q + 1) such that P = LL2Q. Therefore if satisfies equation (62), then

f

=

=

=

= 0.

Schviarz Problem for Cauchy-Riemann Systems Because

-______

109

of the Green's formula it is easy to see that

[Lf,g] = —[f,tg], i.e. L* =

q) with p 1, q 1.

Let us now assume that

In order to prove the solvability of the Schwarz problem we just show that the holomorphic function ço(z) represented by (60) actually gives the solution of the Schwarz problem. Let Kpq(z, () be the reproducing kernel for the space H(p, q) that z) E H(p, q) and the integral operator (f, z))L2(äJB) with the is kernel Kpq(z, () is the unitary invariant orthogonal projection from L2(ÔJB) onto H(p,q) C L2(ÔJB), i.e.

=f

(f,

f

for example [1],

z

p. 118)

_2) (q+n _2) p±q-

Kp,q(z,() =

-q, n

xF

2)

-1,1-

(63)

)

where F(a, b, c; x) is the Gauss hypergeometric function and the obvious expansion

(1- (z for IzI

=

po

(p

=

<1 to obtain from (60)

=f

(Re

- i)

(1-

((1-

+

(1-

- i)

H. Begehr and A. Dzhuraev

= 818

(64)

= 818

z), and the the Hermitian symmetry of the kernel, () = () following from (63) is used. Since particular symmetry z) = Kqp(z, () H(p, q), then according to our assumption we have

Here

=

z)dcr(() 818

0

ÔJB

for p 1,q 1. That is, if we take into account that Koo(z,()

1, then the

right-hand side of (64) is just the Poisson integral dcr(()

as the series

converges to the Poisson Kernel (1

when z E lB tends to ( tends to the Dirichlet problem for the ball.

Hence, the function Re ço(z)

ÔJB, because the Poisson integral solves

which satisfies conCorollary 3. Every real-valued function -y(() E dition (59) or is orthogonal to the spaces H(p, q) with p 1, q 1 can be

represented as the boundary values of the real part of a function holornorphic in lB. Remark. The condition (59) as well as the condition -y(C,)..LH(p, q), p 1, q 1 automatically holds for any function E Ca(ôlB) in case n = 1. Indeed in = 1, Re — 1 = 0 and H(p, q) is empty — 1 = Re this case (I =

for p> 0,q >0. Turning to the Riemann—Hilbert problem (58) with arbitrary nonvanishing coefficient

)t E

on Ô1B we note that in case n > 1 the function

Potential Theory on Ordered Sets

177

it follows that the relation B'4s =

s

is equivalent with the relation BAGY

y E supp

=

therefore, by Proposition 2.4, with the relation

and

yE Let now put =

—

= be two positive measures on X with s = A ,i2 i E {1, 2) we get j.4 A = 0 and therefore,

If

we

(supp /4) n (supp i4) = 0.

From G'4 =

we deduce

= and

B5uPP

(G'4) = B5uPP

=

therefore, using the first part of the proof supp /4 = supp /4 = 4,,

= /4 = 0.

Hence

=

=

A

Proposition 3.5. For any finite subset F of X and any s E S(x) we have BF5 E Sr(X). The set Sr(X) is a convex subcone of S(X) solid with respect to the specific order and increasingly dense in S(X) with respect to the natural then we have order. Ifs E Sr(X), s =

s E P(X)

=0 = 0.

s E 11(X)

Proof. Let s E Sr(X), s =

and let t E S(X) be such that t

s.

From

yEX

and since S(X) is a complete lattice with respect to the specific order ([2]) then there exists a family (/4)yEx of positive real number, ,4 < such that t

>,t4Gy yEX

H. Begehr and A. Dzhuraev

112

I

(Re

- i)

2

(1-

=0

+Re

=

holds for any (E ÔJB, where F(z) = —

TF

. .

( — ij)dzi(ij)

[

- n J (118

-

,

and 67

Proof. The conditions

afk

of1

—0

necessary and sufficient in order for the system (66) to be solvable in lB and then all the solutions of this system in lB are given by are

=

+J

-

if

where is an arbitrary holomorphic function in lB. Hence, we get the Riemann—Hilbert boundary condition

=

Re for the holomorphic function

I'(() =

with the right-hand side — Re

and can apply Theorem 5.

3. As an immediate application we give a solution to the Dirichlet problem

u(c)=y(c), (EOJB,

(68)

for pluriharmonic functions in the ball lB. Since any pluriharmonic function is a sum of a holomorphic and an antiholomorphic function in 1B,

u(z) =

Schwarz Problem for Cauchy—Riemann Systems

113

condition (68) is equivalent to the Schwarz conditions

Re [cp(() +

= Re

Re {i[cp(()

= Re

—

on

ÔJB

i(p(z) — i,1'(z)) which are holomorphic in JB, so for the functions p(z) + that an application of Lemma 12 gives the following result.

Theorem 7. The Dirichlet problem for plurihacrmonic functions in the unit ball is solvable if and only if

I

- i)

+ (1

=

0

(69)

holds for any ( E ÔJB or 'y(c)±H(p,q),p 2 1,q 1. In this case the unique solution of the Dirichlet problem is given by u(z)

1

1

=J

-

1

1

+

(1-

-1)

References 1.

2. 3.

4. 5.

A. Alexandrov, Function theory in the ball. Several Complex Variables, Encyclopedia Math. Sci. 8, Springer—Verlag, Berlin etc., 1991. M.B. Balk, Polyanalytic functions. Akad. Verlag, Berlin, 1991. E. Bedford, The Dir-ichlet problem for some overdetermined systems in the unit ball in iT1t. Pacific J. Math. 51, 19—25, (1974). H. Begehr, Complex analytic methods for partial differential equations. An introductory text. World Scientific, Singapore etc., 1994.

H. Begehr and R.P. Gilbert, Transformations, transmutations and kernel

functions, I, II. Longman, Harlow, 1992, 1993. 6. H. Begehr and A. Kumar, Bi—analytic functions of several variables. Complex Variables, Theory Appl. 24 (1993), 89—106. 7. A. Dzhuraev, Methods of singular integral equations. Longman, Harlow, 1992. 8. A. Dzhuraev, Degenerate and other problems. Longman, Harlow, 1992. 9. R.P. Gilbert and J.L. Buchanan, First order elliptic systems: A function theoretic approach. Acad. Press, New York, 1983. 10. N. Kerzman, Singular integrals in complex analysis, Proceedings of Symposia in Pure Math. A.M.S., Vol. 35, Part 2 (1979), p. 3—41. 11. M.Z. Li, Generalized Riemann—Hilbert problem for a system of first-order quasilinear elliptic equations of several complex variables. Complex Variables, Theory Appl. 7 (1987), 383—394.

H. Begehr and A. Dzhuraev

114

12. W. Rudin, Function theory in the unit ball of (V 13. 14.

Springer—Verlag, New York

etc., 1980. I.N. Vekua, Generalized analytic functions. Pergamon Press, Oxford, 1962.

G.C. Wen and H. Begehr, Boundary value problems for elliptic equations and systems. Longman, Harlow, 1990. 15. W. Wendland, Elliptic systems in the plane. Pitman, London, 1979.

Heinrich Begehr Freie Universität Berlin Fachbe reich Mathematik und Inforrnatik I. Mathematisches Institut 14195 Berlin, Germany E-mail address: begehr©math .fu-berlin.de

Abduhamid Dzhuraev Academia Nauk Tadjik SSR Institute of Mathematics 734025 Dushanbe Tadjikistan

ANALYSIS AND TOPOLOGY (pp. 115-142) eds. C. Andreian Cazacu, 0. Lehto and Th. M. Rassias © 1998 World Scientific Publishing Company

GENERALIZED MULTIVALUED VARIATIONAL INEQUALITIES H. BEN-EL-MECHAIEKH AND G. ISAC

Abstract Using recent fixed point theorems for approachable multifunctions, we prove some very general multivalued quasi-variational inequalities involving such multifunctions. We deduce that both the generalized multivalued variational-like and complementarity problems are solvable for large classes of upper semicontinuous multifunctions with non-convex values. The theorems presented here unify numerous known results. Various applications to existence problems in optimization are presented.

1. Introduction Variational inequalities and in particular the complementarity problem for multifunctions have been intensively studied and applied to numerous nonlinear existence problems (see for instance the books of Baiocchi and Capelo [4], Cottle, Pang and Stone [14], Isac [26], and Murty [32]). The purpose of this paper is to study the existence problem for generalized multivalued variational inequalities involving upper semicontinuous multifunctions that are approximable — in the sense of the graph — by continuous single-valued functions. The class of approachable multifunctions is very broad; it contains upper semicontinuous multifunctions with convex values (Cellina

[11]), or contractible values (Mas Colell [30]), or oo-proximally connected values (Górniewicz, Granas and Kryszewski [20], Ben-El-Mechaiekh [5], The authors acknowledge supports from the Natural Sciences and Engineering Council of Canada and the Academic Research Program of the Department of the National Defence of Canada respectively. 115

H. Ben-El-Mechaiekh and G.Isac

Ben-El-Mechaiekh and Deguire [6], Ben-El-Mechaiekh, Oudadess and Tounkara [7]), or decomposable values in V (Bressan and Colombo [10]). Hence, an existence theory for generalized variational inequalities and complementarity problems for approchable multifunctions will very likely find numerous applications in various areas of nonlinear analysis (e.g. differential inclusions and control theory, smooth and non-smooth optimization, game theory, etc.. .). The results presented here unify existence results for multifunctions with convex values (see for instance Chan and Pang [12], Dien [16], Gowda and Pang [22], Itoh, Takahashi and Yanagi [29], Panda and Sen [33], Panda, Sen and Kumar [35]), contractible values (Yao [40]), and provides new results for multifunctions with non-contractible values (see Problem 8.8.B in [26]). It also provides some typical applications of these results to optimization problems involving non-convex multifunctions. The paper is organized as follows. We start by recalling in Section 2 the basic facts about approachable multifunctions, particularly, stability features and fixed point properties needed in the proofs of the main results. Section 3 contains our main existence results. We essentially show (Theorem 3.2 below) that given a compact absolute retract X, an absolute neighborhood retract Y, a continuous self-multifunction of X, an upper semicontinuous multifunction 4 from X into Y, and a continuous extended proper real function p on X x Y x X, the generalized multivalued variational inequality

with p(x0,y0,x0)

p(xo,yo,x),Vx

E

(1)

has a solution provided both 4 and the marginal multifunction := {u E

= inf p(x,y,z)},(x,y) E X x Y (2) zEW(x)

approachable. For example, 4 could be a composition of multifunctions having R6 values, and could be continuous with non-empty compact

are

convex values (in which case the multifunction given by (2) is approachable). Non-compact versions of particular instances of this result are also

presented. Applications of the abstract results of Section 3 to minimization of cost functions along trajectories of differential inclusions is provided in Section 4. Based on recent results on qualitative properties of solutions sets of differential inclusions of Bressan [9], Górniewicz, Ricceri and Slosarski [21], and Plaskacz [36], problem (1) is applied to solve a minimization problem of a cost function y, z) along the trajectories of the Cauchy problem

Generalized Multivalued Variational Inequalities

y'(t)

F(t,y(t))

117

a.e. t E [O,T]

(3)

I

F is of upper semicontinuous or lower semicontinuous type, x E X, a set of being the solutions set Poincaré operator of (3) and allowed return points). Section 5 is devoted to quasi-variational inequalities and complementarity problems involving subgradients with applications to

where

minimization problems for non-smooth real functions of pseudoconvex type.

2. Preliminaries In order to make this paper self-contained, we briefly set forth below some basic properties of the spaces and multifunctions which we study here. For more details concerning the notions in this section, the reader is referred to [5], [6] and [7]. For an elementary discussion on absolute (neighborhood) retracts (A(N)R for short), we refer to the books of Hu [25] and Van Mill [39]. All topological spaces are assumed to be Hausdorif. Recall that a topological space X is said to be contractible (in itself) if there exist a fixed element X and a continuous mapping h X x [0,1] —÷ X such that h(x, 0) = x xo and h(x, 1) = x0, Vx X. Clearly, every convex and more generally, every starshaped subset of a topological vector space is contractible. Some important solution sets of differential equations and differential inclusions satisfy a more general contractibility property (see Section 4 for examples); roughly speaking, they are contractible in each of their open neighborhoods; (recall that a subset A of a topological space X is contractible in its neighborhood U if there exists X x [0,1] —p U such that h(x, 0) = x and h(x, 1) = a continuous mapping X, where xU is a given fixed point in U). This leads us to formulate XU, Vx the following definition.

Definition 2.1. (Dugundji [17]) A subspace Z of a topological space Y is said to be x-proximally connected in Y if for each open neighborhood U of Z in Y, there exists an open neighborhood V of Z in Y contained and contractible

mU. For example, the set {(t, sin(

0

< t

1} U ({0} x [—1, 1]) is not con-

tractible in itself, but it is contractible in each of its open neighborhoods in R2.

H.

118

Ben-El-Mechaiekh and G. Isac

Example 2.1. (i) If a subspace Z of an ANR Y has trivial shape in Y (that is Z is contractible in each of its neighborhoods in Y) then Z is connected in Y (see [39]). be decreasing sequence of compact spaces having trivial (ii) Let shape in an ANR Y. Then Z = Z2 is tx-proximally connected in Y (see connected in [17]). In particular, every R5 set in an ANR Y is Y; (recall that an R5 set is the intersection of a countable decreasing sequence of compact contractible metric spaces).

Definition 2.2.

Let (X,U) and (Y, V) be two uniform spaces. A multifunction 4) : X —÷ P(Y) is said to be approachable if and only if VU E U, VV E V, 4) admits a continuous (U, V)-approximative selection, that is a single-valued function s : X —÷ Y verifying s(x) V[4)(U[x])], Vx X. ([7])

One readily verifies that the multifunction 4) is approachable if and only if for each member W of the product uniformity U x V on X x Y, there exists a continuous single-valued function s X —p Y satisfying the inclusion: graph(s) C W[graph(4))]. We list some examples of approachable multifunctions.

Example 2.2. (i) (Convex case, [11]). Let X be a paracompact topological space equipped with a compatible uniformity U and let Y be a convex subset of a locally convex topological vector space F. Let 4) : X —÷ P(Y) be an upper semicontinuous (u.s.c. for short) multifunction with non-empty convex values. Then 4) is approachable. (ii) (Contractible case, [31], [6]). Let X and Y be two ANRs with X compact. Then every u.s.c. multifunction 4) : X —+ P(Y) with compact contractible (hence non-empty) values is approachable.

Example 2.3. (Non-contractible case, [5]). Let X be an approximative absolute neighborhood ertension space for compact spaces, and let Y be a uniform space. Then every u.s.c. multifunction 4) X —p P(Y) with non-empty compact x-proximally connected values in Y is approachable; (if X and Y are ANRs with X compact, this result can be found in [20]). The preceding example can be refined as follows:

Generalized Multivalued Variational Inequalities

119

Example 2.4. Let X be an ANR, let (Y, V) be a uniform space, and let 4) : X P(Y) be a u.s.c. multifunction with non-empty values such that the restriction of 4) to any finite polyhedron P C X is approachable. Then the restriction 4)1K of 4) to any compact subset K of X is approachable.

Proof. We only provide a sketch of the proof. Given an open subset U of a normed space and a compact subset K of U, by a result of Girolo [19] there exists C, a compact ANR, such that K C C C U. One readily verifies that if X is an ANR and K is a compact subset of X, then there exists C, a compact ANR such that K C C C X. Since compact ANRs are dominated by finite polyhedra, a result in [6] implies that the restriction 4)IC of 4) to C is also approachable, and by a result of [5] on restrictions of approachable multifunctions to compact subsets, the restriction 4)1K of 4) to K is approachable. 0

We now formulate two stability properties together with some fixed point properties for approachable multifunctions crucial for our purposes.

Proposition 2.1. ([7]) Given three uniform spaces (X,U), (Y, V) and (Z, W), let 4) : X —÷ P(Y) be a u.s.c. approachable multifunction with non-empty compact values, and let 4' Y —+ P(Z) be a u.s.c. multifunction with non-empty values such that the restriction WI4)(X) is approachable. Then the composition product W4) : X —÷ P(Z) is u.s.c. and approchable provided the space X is compact.

Definition 2.3.

(i) A decomposition , 4) of sets (X0 = X, X1, .. , = Y) is a sequence of multifunctions X2_i —+ such that 4)(x) = o ... o )(x), Vx E X. A decomposition , 4) is said to be an upper semicontinuous decomposition if all spaces X2 are topological, and all multifunctions are u.s.c. with non-empty compact . .

.

.

values.

(ii) An upper semicontinuous decomposition , a sequence (Xo = X, Xi,.. . , = Y) is said to be a V-decomposition if all spaces Xi,... , are ANRs, and if for each i = 1,.. , n, 4)2(x2_1) is x-proximally connected in X2, Vx2_1 E X2_1. . .

4)

.

The next example of approachable multifunctions follows from the Definition 2.3, Proposition 2.1, and Example 2.3.

H. Ben-El-Mechaiekh and G. Isac

Example 2.5. Let X, Y be ANRs with X compact, and let 4) : X —p P(Y) be a multifunction that admits a V-decomposition. Then 4) is approachable.

Proposition 2.2. Let {4), : (X2,U2) —÷

be a family of u.s.c. approachable multifunctions between uniform spaces. Then the multifunction 4) E defined by 4)(x) = JJjEIXi —+ HiEIXZ, is also u.s.c. and approachable. :

The next result generalizes the fixed point theorems of Fan [18] and Himmelberg [24] to approachable multifunctions.

Theorem 2.3. ([6]) Let X be non-empty convex subset of a Hausdorff locally convex space E, and let 4) X —÷ P(X) be a u.s.c. multifunction with nonempty closed values. Assume that 4) is compact, that is there exists a compact subset Y of X such that 4)(X) Y. If one of the following conditions is satisfied: (i) 4) is approachable; or

(ii) for each finite subset N of X, the multifunction 4)N

conv{N} P(Y) defined by 4)N(X) = 4)(x),x E conv{N}, is approachable. Then has a fixed point, that is, E X with x0 E 4)(xo).

The following purely topological fixed point property is an immediate consequence of Definition 2.2 together with a simple compactness argument:

Proposition 2.4.

Let X be a topological space and let 4) : X —÷ P(X) be a u.s.c. approachable multifunction with non-empty compact values. If 4) is compact and X has the fixed point property for continuous single-valued mappings, then 4) has a fixed point. Remark. Proposition 2.4 holds true when X is an acyclic compact ANR since it has the fixed point property for continuous single-valued mappings.

3. Generalized Variational Inequalities We start this section with our first main result.

Theorem 3.1.

Let C be a non-empty compact convex subset in a locally convex space E, and let Y be a non-empty complete convex subset of a locally

Generalized Muitivalued Variational Inequalities

121

—

P(Y) be a u.s.c. multifunction with non-empty compact values such that one of the following condition is satisfied: convex space F. Let 4) : C —÷

(i) 4) is approachable, or (i)2 for each finite subset N of C, the restriction

4)Iconv{N} to the convex hull conv{N} is approachable. Let W C —p P(C) be a continuous multifunction with non-empty be a compact convex values, and let p : C x Y x C —÷ R U

continuous extended proper real function satisfying: (ii) V(x, y) E C x Y, the function ço(x, y,.) is quasiconvex on C: Then the problem

E 4)(xo), such that 1 so(xo,yo,x) co(xo,yo,xo),Vx E W(xo)

f

E

(3)

has a solution. : C x Y —p P(C) by putting:

Proof. Define the marginal multifunction :={u E

W(x);ço(x,y,u) =

ml ço(x,y,z)},(x,y) E Cx

Y. (4)

zEW(x)

The compactness of the values of 'I', together with the continuity of ço,

implies that has non-empty compact values. The convexity of the values of W, together with (ii), implies that has convex values. We verify that is u.s.c.. To do this, observe that = where := {u E C;ço(x,y,u) = infZEw(X)so(x,y,z)}. Since W is u.s.c. and has compact values, it suffices to verify that the graph of is closed. To be a net in do this, let (Xa, converging to (x, y, u) E C x Y x C. Then, ço(x, y,u) =

lim

a

= limsup where

inf

inf

z)

inf ço(x,y,z),

zEW(x)

the inequality above follows from the upper semicontinuity of the

marginal function

ço(., ., z)

(this follows from the facts that p is lower

semicontinuous as a real function and that W is lower semicontinuous as a multifunction, see Theorem 1.4.16 in [3]). Hence, (x, y, u) Now, since Y is convex and complete, canv4)(C) is a convex compact subset

of Y. By Example 2.2 (i), since the product C x conv4)(C) is compact, the restriction of the multifunction to C x is approachable.

122

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Ben-E1-Mechaiekh and C. Isac

Define a multifunction F : C x ccinv4(C) —f P(C x conv4(C)) by putting:

T(x,y) :=

x

x

By Proposition 2.2, the multifunction F is u.s.c., approachable and has non-empty compact values. All conditions of Theorem 2.3 are thus satisfied. Therefore, F has a fixed point (x0, Yo) E r(x0, yo), that is, xo W(xo), Yo E and ço(xo,yo,xo) <ço(xo,yo,x),Vx Remarks.

0

(1) If in addition, V x E X with x E W(x),Vy e

one has

so(x,y,x) 0, then co(xo,yo,x) 0, Vx E W(xo). (2) If

=

C, Vx

C, the continuity assumptions on p can be slightly

relaxed to is

l.s.c. and ço(., ., u) is u.s.c.;

in which case, our theorem generalizes Theorem 1 in [33] to infinite dimensional spaces and to composites of convex as well as non-convex multifunctions.

We can provide a purely topological formulation of Theorem 3.1 that generalizes the main abstract existence result in [40].

Theorem 3.2. Let X be an acyclic compact ANR, and let Y be an ANR. Let 1 : X —÷ P(Y) be a u.s.c. approachable multifunction with non-empty compact values. Let : X —+ P(X) be a continuous multifunction with nonempty compact values, and let cc: X x Y x X —+ R U {±cx} be a continuous extended proper real function. Assume that for any finite polyhedron P contained in X x Y, the restriction of the marginal multifunction defined by (2) to P is approachable. Then problem (1) has a solution. Proof.

Since 4(X) is a compact subspace of Y, by Girolo's result (see the proof of Example 2.4) there exists C, a compact ANR such that ç C C Y. The restriction of to X x C is approachable by the stability of approachability under the passage {finite {compact ANRs} [6]. Moreover, as in the proof of Theorem 3.1, one easily verify that is upper semicontinuous with non-empty compact values. By Proposition 2.1, the

Generalized Multivalued Variational Inequalities

123

P(X) is compact-valued and approachable. XxC It has a fixed point by Proposition 2.4 and this ends the proof. EJ multifunction X

Our result generalizes Theorem 3.1 of [40] in several directions. Indeed, observe first that an ANR is contractible if and only if it is an AR. A compact AR is certainly acyclic and has the fixed point property for singlevalued continuous functions. Furthermore, condition (i) of Theorem 3.1 in [40] can be removed due to Girolo's result. Moreover, if the values of the Remark.

multifunction 4) and that of the marginal multifunction

are cio-proximally

connected (in particular if they are contractible), then, by Example 2.3, our hypotheses are satisfied.

As a particular case of the Theorem 3.2, we have in view of Definition 2.3 and Example 2.5:

Corollary 3.3. Let X, Y, W, p,

be as in Theorem 3.2, and let 4) : X P(Y) be a multifunction that admits a V-decomposition. Then problem (1) has a solution.

It suffices to observe that since X is a compact ANR, then by Proof. is approachable. The conclusion follows immediately from Example 2.5, Theorem 3.2. 0 One can compensate the full compactness of the set X in Theorem 3.1 by a Karamardian type coercivity condition due to a classical argument used for instance in [29] for solving variational inequalities. In order to do this, given two subsets X and C of a topological space E, we shall denote by Bdx(C) = Cn X \ C the boundary of C relative to X, and by = C fl (E \ Bdx(C)) the interior of C relative to X.

Theorem 3.4. Let X be a non-empty subset in a locally convex space E, and let Y be a non-empty complete convex subset of a locally convex space. Let X —+ P(Y) be u.s.c. multifunction with non-empty compact values, and let '1' : X —+ P(X) be a multifunction. Assume that there is a non-empty compact convex subset C in X such that:

: C —f P(C) defined by (i) the multifunction := W(x) n C, x E C, is continuous with non-empty compact convex values;

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Ben-E1-Mechaiekh and C. Isac

(u)1 41C is approachable; or (u)2 for each finite subset N of C, the restriction 4lconv{N} to the convex hull conv{N} is approachable. Let p : C x Y x X —p R U {±cx} be an extended proper real function satisfying:

(iii) p is continuous on C x Y x C; (iv) V(x, y) E C x Y, the function ço(x, y,.) is convex on (v) Vx E C such that x E E

with

ço(x,y,u) ço(x,y,x), Vy E 4(x). Then problem (1) has a solution with xo E C.

Proof. By co(xo,

Theorem 3.1,

such that ço(xo,yo,x)

E

Given

Yo, xo), Vx E

x E W(xo) \

One

Case 1: x0 E so that z

= Ax +

(1 —

.),

it

z

By (v),

<

possible. 1

small enough

Aço(xo, Yo, x) + (1 —

with

E

co(xo, Yo, xo). One can choose 0 < A

(1 —

are

ço(xo,yo,x).

2: x0 E

= ,\x +

can choose 0 < A

follows that co(xo, Yo, xo)

)t)co(xo,yo,xo). Thus, co(xo,yo,xo)

ço(xo, Yo, uo)

cases

,\)xo E WC(xo). Hence ço(xo,yo,xo) ço(xo,yo,z), and

by convexity of co(xo, Yo,

Case

C, two

<

1

small enough so that

E WC(xo). Hence co(xo,yo,xo)

A)ço(xo, Yo, uo) < Aço(xo, Yo, x) + (1 —

xo)

follows from (v). Thus, p(xo,

Aco(xo,yo,x) + (1 —

xo) where the last inequality

ço(xo, x) and the proof is complete.

0 Remarks. (1) can

Again,

if

= X, Vx E

be slightly relaxed to: p is l.s.c.

and

C,

the

ço(.,

.,

continuity assumptions on

u)

is

u.s.c.

(2) One may formulate a purely topological version of the preceding theorem based on Theorem 3.2. (3) Theorem 3.4 generalizes Theorem 1 in [33] in several directions.

4.

Applications to Control Problems A striking result of Aronszajn [1] asserts that the solution set of the

Cauchy problem with proximally connected

continuous right hand side is an R5 set, in the space of continuous functions.

hence x-

This qualita-

tive property of solution sets was extended by many authors to differential inclusions.

Generalized Multivalued Variatzonal Inequalities

125

and F : [0, T] x K —p

be a multifunction with non-empty compact values. Denote by SF(x; K) (SF(X) for SF(x; Rn)) the sets of Carathéodory solutions viable in K (i.e. y E S(x; K) if and only if y(t) e K, Vt e [0, T]) of the Cauchy problem with initial value x: Let K be a non-empty subset of

a.e. in [0, T],

(5)

{

Let C be a non-empty subset of K. Assume that for any given x E C there C C of return points. Starting at a point x E C, we corresponds a subset travel along the trajectory y of problem (5). We then follow a return path to a point z e Assume that a cost ço(x, y, z) is associated to this journey (for instance, ço(x,y,z) could be the sum of an attack cost coi(x,y) and a retreat cost ço2(y(T), z)). We are interested in the problem:

Find x0 e C,x0 E 'I'(xo),yo E SF(xo;K),such that ço(xo,yo,xo) = in ZEW(xo)co(xO,yO,z). ( on recent results on qualitative properties of solutions sets of differential inclusions, Theorem 3.1 readily implies the solvability of (6) whenever F is a Carathéodory multifunction or F is lower semicontinuous. Before stating our results, let us recall some important notions. Based

Definition 4.1. F is a Carathéodory multifunction if the following conditions are satisfied:

(i) F has convex values; (ii) y '—÷ F(t, y) is u.s.c. a.e. t E [0, T]; (iii) Vy e K,t i—÷ F(t,y) is measurable; (iv) e F(t,y),y e K} < p(t) +cx) is an integrable function. where IL: [0, T] —+ [0,

Definition 4.2. ([36]) A non-empty closed subset K of is said to be a proximate retract if there exists an open neighborhood U of K in and a continuous mapping r : U —p K(called neighborhood retraction) such that the following two conditions are satisfied: (i) r(x) = x,Vx e K; (ii) IIr(x)—xII = dist(x,K) = infUEK IIx—uII,Vx E U.

Note that any closed convex subset of

and any C2-submanifold of

is a proximate retract ([36]).

Theorem 4.1. Assume that C is convex compact, W is continuous with nonempty convex compact values, p is continuous on C x C([0, T], x C, and p is

H.

126

Ben-El-Mechaiekh andGlsac

quasiconvex with respect to the return variable z. Then problem (6) has a solution provided K is a proximate retract and F is a Carathéodory multifunction satisfying the tangency condition

F(t,y) flTK(y) where TK(y) := {v e cone to K at y.

O,V(t,y) e [O,T] x K, V,K)

(7)

= O} is the Bouligand contingent

: K —p P(C Proof. In view of Theorem 1.1 in [36], the multifunction := SF(x; K), x e K, has R5 values. Moreover, one ([0, T], K)) defined by can show that is u.s.c.. Indeed, the multifunction F extends to a multifunction F: [0, T] x

—p

in such a way that Sp(x; K) =

(see [36]

for details). One then invokes well-known results to obtain that the solution set is u.s.c. (see [2]). Being a closed—+ P(C([O, T], multifunction Sp : K —+ P(C([O, T], K)) given the multifunction graph multiselection of by

:= Sp(x;K) =

e K, is also u.s.c.. By Example 2.3,

approachable. The conclusion immediately follows from Theorem 3.1.

is

0

Remark. Observe that in case K is an open subset of then TK(y) = the inwardness condition (7) being automatically satisfied. When F is l.s.c., the situation is quite different. Generally, the multifunction SF (.) is neither u.s.c. or l.s .c., nor are its values always closed. A remarkable result of Bressan [9] asserts that the multifunction SF(.) admits a u.s.c. multiselection with compact values. This leads to a lower semicontinuous version of the preceding theorem.

Theorem 4.2. Assume that C is a closed disk D(uo, b) and that K is an open subset of containing the closed disk D(uo, b + LT), where b, L > 0. Assume also that 'I' and p are as in Theorem 4.1. If F is l.s.c. with values in the open ball B(0, L), then problem (6) has a solution.

Proof. By Theorem 6.1 of [9], the multifunction SF : D(uo, b) —+ P(C([0, T], R71)) admits a u.s.c. multiselection with compact (connected) values. In fact, for x e D(uo, b), is the set 51(x) of Carthéodory solutions of a singlevalued Cauchy problem y'(t) = f(t, y(t)), y(O) = x, where f is a directionally continuous selection of F on [0, T] x K. By Corollary 5.2 of [9], for every > 0,

Generalized Multivalued Variationallne qualities

127

there exists a single-valued continuous mapping

:

D(uo, b) —÷ C([O, T],

such that the graph of is contained in the f-ball around the graph of that is, is approachable. The conclusion readily follows from Theorem 3.1. D

[21] on Remark. Based on a recent result of Górniewicz, Ricceri and the solution set of differential inclusions of order k in a Banach space E, we obtain a version of Theorem 4.1 with the Cauchy problem (5) replaced by the differential inclusion

y(k)(t) E F(t, y(t), y'(t),

5'

y(O) = xo,y'(O) = where

F:

,

y(k_1)(t))

xi,. ,y(k_l)(o) =

—+ P(E) has non-empty compact convex values, is

[0, T] x

measurable in t, completely continuous in x, integrably bounded, and sends compact sets on compact sets.

5. Generalized Quasi-Variational Inequalities and Applications The general results of Section 3 find immediate applications in the theory of variational inequalities and complementarity problems. We refer to [261 for a comprehensive discussion on the various aspects as well as the many applications of these problems.

5.1. Quasi-variational inequalitites Let X be a subset of a topological vector space E, let Y be an arbitrary set and let F be a locally convex space with topological dual F* and duality P(X), X —+ P(Y) be two product (.,.) F* x F —+ R. Let W : X F*,1l multifunctions, 0 X x Y —+ X x X —+ F be two mappings, and 4: X —+ R be a real function. The generalized quasi-variational inequality problem (GQVIP for short) :

:

associated to (X, Y, W, I

0,

is defined as:

such that

find x0 e W(xo),yo E (O(xo,yo),ii(x,xo))

—

çb(x)

for all

x E 'F(xo).

(8)

128

I-f.

Ben-El-Mechaiekh and G. Isac

Particular instances of problem (8) were studied in [12], [16], [22], [29], [33], [35], [40], and many other papers. The following theorem is an immediate consequence of Theorem 3.2. It generalizes Theorem 3.2 of [40] to a broader class of multifunctions.

Theorem 5.1. ANR,

Assume that X is an acyclic compact ANR in E, Y is an : X —+ P(Y) admits a V-decomposition, and that W : X —p P(X)

continuous with non-empty compact values. Assume also that 0 and continuous and satisfy: is

are

(i) (0(x,y),ii(x,x)) 0, V(x,y) E (ii) V(x,y) E X x Y, the marginal set := {u E W(x); (0(x,y),ri(u,x)) = inf (0(x,y),ii(z,x))} zEW(x)

is oo-proximally connected in X.

Then problem (8) has a solution. Proof. It suffices to apply Theorem 3.2 to the function

y, u) = (0(x, y),

0

ii(u,x)).

Assume now that E is a normed space. Given a real p> 0, let

be the closed disk of radius p centered at the origin in E, and let be the set Denote by W,, the compression of W to := given by x E Xi,. Then we have:

Corollary 5.2. Assume that X is an AR in E, Y is an ANR and P(Y) admits a V-decomposition. Assume also that 0 and

:X are continuous and

verify:

(i) (0(x,y),ii(x,x)) 0, V(x,y) E Assume that apo > 0 such that Vp p0: (ii) the marginal set

= {u is

(iii)

E

(0(x,y),'q(u,x)) =

inf

(0(x,y),ii(z,x))}

x Y. oo-proximally connected in V(x, y) E is compact and W,, is continuous with non-empty compact values.

Generalized Multivalued Variational Inequalities Then: (1)

a solution

Po has a cluster point, then problem (8) has a

Y, W,,,

(2) if the set

129

0,

has

Vp

solution.

Proof. For each p Po, being a retract of X, the set

is a compact AR. Conclusion (1) readily follows from Theorem 5.1. Assume now that the set a subsequence and for some converging to x0 E X. For each n, E E { is Since for any large p, 0, Vx E ij(x, u.s.c. with closed values, it follows that x0 E W(xo). Furthermore, the sequence

of solutions to the problems

Y, 'I',,,

0,

has

{yn} being contained in the compact set } U {xo}) has a cluster point The continuity of 0 and implies that (0(xo, yo), ii(x, x0)) Yo E

0,VxEW(xo).

12

Remark. Corollary 5.2 generalizes Theorem 3.8 of [40] in several directions. An immediate application of Theorem 3.4 is the following:

Theorem 5.3. Assume that Y is a complete convex subset of a locally convex space and let C be a non-empty compact convex subset of X. Assume that admits is u.s.c. with non-empty compact values such that the restriction := a V-decomposition, the multifunction fl C, x E C, is continuous on C and has non-empty compact convex values. Suppose that 0 and are continuous and satisfy:

(i) (0(x,y),ii(x,x)) 0, V(x,y) E graph(4); (ii) V(x,

X x F*, the function

x)) is convex on

(iii) çb is convex and its restriction to C is continuous;

(iv) Vx E C such that x E

E

cb(x)

—

with

q5(u),Vy E

Then problem (8) has a solution. Proof. It suffices to apply Theorem 3.4 to ço(x,y,u) =

Remarks. (1) If E = F and ij(x, u) = x — u, then hypotheses (i)-(ii) are obviously satisfied. If in addition X, then problem (4) has a solution.

In the case where in addition 0(x, y) = y, we obtain a generalization of the main theorem in [29] to non-convex multifunctions.

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Ben-El-Mechaiekh and G. Isac

(2) If ii(x, x) = 0, Vx E X, then (i) is obviously satisfied. However, it may happen that is not identically zero on the diagonal of X x X and yet problem (8) has a solution (see e.g. [16]). In normed spaces, a simple coercivity condition easy to verify is sufficient for hypothesis (iv) in Theorem 5.3 to hold true. More precisely, we have:

Corollary 5.4. Assume that E is a norrned space, that X is a non-empty convex subset of E, that Y is a non-empty complete convex subset of a locally convex space, and that both 0 and are continuous. Assume also that condi> 0 such that tions (i)-(iii) of the preceding theorem are satisfied and that VP

p0:

(iv) Vx

lxii = p,

E

E W(x),

max

cb(x)

is compact and the multifunction

(v)

non-empty compact convex values on (vi) the restriction of the multifunction 4 to Then problem (8) has a solution.

Proof.

Take

Remarks.

(1)

C = X fl Let cb

in

Theorem 5.3.

—

is

continuous and has

admits a V-decomposition.

0

0. If there exists x0 E flXEX 'P(x) such that lim

max (0(x, y), ii(xo, x)) <0,

then hypothesis (iv) is satisfied. We thus obtain a generalization of a result in [40].

(2) It is routine to verify that an alternative coercivity condition to (iv) is: (iv)' there exists a non-empty compact convex subset C of X such that Vx E

X with max (0(x,y),ii(u,x)) <çb(x)—çb(u).

With the appropriate changes to hypotheses (v)-(vi) we obtain a generalization of a result of [16].

Generalized Multivalued Variational Inequalities

131

Existence results for so-called monotone multifunctions are other immediate consequences of Theorem 5.3. Let us recall that given three sets X, Y, Z, a rpultifunction 4) : X —p P(Y), and a mapping 0 graph(4)) —p Z, :X P(Z) is defined by 4)o(x) = the parametrized multifunction {O(x,y);y E 4)(x)},x E X. :

Definition 5.1. Let E, F be two norrned spaces. Let X be a non-empty subset of E, and let Y be an arbitrary non-empty set. Let : X —÷ P(Y), 'I' : X —p P(X) be two multifunctions and let 0: X x Y —+ F*, 11: X x X —÷ F be two mappings. (i)

4)

is said to be (0,q)-monotone with respect to 'I' on X if Vx1 E

'I'(x1),Vx2 E W(x2),Vzi E 4)o(xi),V z2 E 4)o(x2),

(z2,q(x2,xi)) + (zl,q(xi,x2)) 0 (ii) 4) is said to be i — (0, ij)-monotone with respect to 'I' on X if there with p.(O) = 0 and exists an increasing function p.: [0, [0, +cx as r +cx such that Vx1 E W(x1),Vx2 E 'I'(x2),Vzi E

4)o(xi),Vz2 E 4)o(x2),

(z2,rl(x2,xl)) + (z1,ij(x1,x2)) p. —

(0,11)-monotone

with respect to 'I' on X,

then it is (0,11)-monotone with respect to W on X.

(2) If there exists x0 E

such that 4)(xo) is non-empty, then the on

(0,

4)

with respect to 'P at x0 in the sense that: E 4)O(XO),VX E 'I'(x) Vz E

(zo,q(xo,x)) + (z,ii(x,xo)) 0. (Similarly, the p. — (0,

implies a corresponding notion

of p. — (0,

u, and if p.(r) = where > 0 is a fixed constant in (ii), we obtain concepts of strong (0, and which, in a finite dimensional setting with strong (0, and 11(x, u) = x — u, reduce to the concepts of strong 0(x, y) = y monotonicity and strong copositivity introduced in [12].

(3) If 11(x, u)

0 whenever x

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132

Ben-El-Mechaiekh and G. Isac

reduces to the concept of (4) When O(x, y) = y, our (0, — monotonicity used in [33]. If ii(x, u) = x u, all of the concepts above and x0 = 0, the X= were introduced in [40]. If in addition concepts of copositivity reduce to the classical ones (see Saigal [37]).

Proposition 5.5. Let E be a normed space, X a non-empty convex subset of E, and Y a non-empty complete convex subset of a locally convex space. Assume that 0 and are continuous and verify the following conditions:

(i) ii(x,u) = —ii(u,x), Vx,u E X; X x F*, the function (ii) V(x,

x)) is convex on (iii) for each u E X,II11(x,u)IIF —+ whenever IIxIIE (iv) 4) is ij)-monotone with respect to 'I' on X. Assume that > 0 such that Vp p0: is continuous and has non(v) is compact and the multifunction empty compact convex values on (vi) Vx E W(x) with lxii = p, E W(v) with livil < p such that the line segment [v, x] intersects '11(x); (vii) the restriction of the multifunction 4) to admits a V-decomposition. Then problem (8) (with 0) has a solution.

Proof. By (iv), there exists an increasing function i : [0, [0, —4 +oo as r —+ +cx such that Vx E W(x),Vv E W(v),V with = 0 and z E 4)o(x),Vz' E 4)o(v),

(z, —ij(x,v)) < (z','q(v,x)) Thus, (z,'q(v,x)) whenever lixil —4 x.

=

—

(z, —'q(x,v)) <

—

One can choose p> 0 large enough so that, Vx E X, E W(v) with ilvil
(0(x,y),ii(u,x))

+ (1 —

= A(0(x,y),ii(v,x))

A)(0(x,y),ii(x,x))

0.

The conclusion follows at once from Corollary 5.4 with çb

0.

U

Generalized Multivalued Variational Inequalities

133

(1) As the following example suggests, condition (vi) in the preceding proposition is essential for the solvability of problem (8). Let X the rotation be a closed subset of R2 \ {O} and Y = R2. Denote by Remarks.

of angle t in R2, and let 4(x) :=

with

to E (0, i.),

and W(x) u — x. Clearly,

A < 1}. Let O(x,y) y,ii(u,x) is u.s.c. with non-empty compact convex values. Furthermore, for p > 0 is non-empty compact and convex Vx E X. It is large enough, W(x) fl routine to verify that W,, is continuous (lower semicontinuity follows from a direct elementary argument; upper semicontinuity follows from the fact the has closed graph). Furthermore, Vx1, x2 E R2, {x + (1 —

(O(x1,Rt0(xi)),r'(xi,x2)) +

=

— x1) =

— x2) +

=

I1x2 — x1 112 cost0

=

,u(IIxi — x211)11x2 —

— x2),x1

—

x2)

II

— (0,11)-monotone with = r cos t0, r 0. That is, 4) is respect to W on X. However, problem (8) has no solution. Indeed, given with any x E X, any x' E 'I'(x)\{x} is of the form x' = x + (1 —

with

0 < A < 1, and

(0(x,y),11(x',x)) =

—

=

(1

—

A)cos(to

+

11x112 <0.

A simple geometric argument shows that condition (vi) of Proposition 5.5 fails.

(2) The conclusion of Proposition 5.5 holds true if hypotheses (iv) and (vi) W(x) such that 4) is — (0, ii)-copositive with are replaced by E respect to W at x0 on X. In this case, we generalize (and correct) Theorem 3.6 of [40] to non-contratible multifunctions. (3) Simple counter-examples show that the hypothesis of 11)-monotonicity cannot be replaced by mere (0, ij)-monotonicity.

5.2. Complementarity problems Complementarity problems are particular cases of problem (8). Recall that given a closed convex cone X in a locally convex space E with dual cone = P(E*) E E*; 0 for all x E X}, let 4) X —p be a multifunction, let {y (y, x)

H.

134

f : X x E*

Ben-El-Mechaiekh and C. Isac

be a mapping, and let 0: X —p R be a functional. The

generalized multivalued complementarity problem (associated to (X, 4), f, 0))

X*,f(xo,yo) E J find x0 E X,yo E 4)(xo)fl

such that

(f(xo,yo),xo) = cb(xo).

1

If f(x, y) =

y,

(9) reduces to the classical generalized multivalued

complementarity problem. We formulate now some typical existence properties for problem (9) that generalize to non-convex multifunctions results in [29]. Their proofs are similar to those presented there for convex-valued 4) and are left to the reader.

Theorem 5.6.

Assume that 4) is u.s.c. with non-empty compact cc-proximally connected values and that 0 : X —÷ (—cc, 0] is an l.s.c. convex functional. Assume also that there exists a compact convex subset C of X with non-empty interior relative to X such that for each x E Bdx(C) there exists with — u) 0(u) — uE Then: (1) Problem (9) has a solution provided 0(0) = 0 and Ø(Ax) = x X. V(A,x) e (2) with 0 (Yo, x0) <—0(x0) provided 0(0) = 0 E C, E

and çb(x+y) <0(x),

XxX.

By way of illustration, we have the following generalization of well-known results that could be applied to finding Kuhn—Tucker type stationary points for nonsmooth programming problems with very general objective functions; (specific situations will be discussed elsewhere). Let E = and X= 0: X —p 0] be an l.s.c. convex functional with 0(0) = 0 and Ø(Ax) = AØ(x), V(A, x) E [1, +oo) x X. Let g be a locally Lipschitz real function on X, and let us assume that := h(ôf(x)) is a homeomorphic image, lying in X, of the Clarke generalized gradient [12] of g at x (such a mapping 4) is of course u.s.c. and has non-empty compact contractible values). If there exist a constant > 0 and a vector d e X such that Vx

{x

X;(d,x) =

i3},

E {x E X;(d,x)
then, by putting C := {x E X; (d, x) 13} (which is compact), one immediately obtains the solvability of problem (9). Observe here that our coercivity

Generalized Multivalued Variational Inequalities

135

condition above is independent of the mapping f (which could be of the form f(x, y) := Mx + y + r, M E as in [35], or of any other form for rE

that matter).

Theorem 5.7. 4) X —p

Assume that X is a closed convex cone in

be such that for any compact convex subset C of X, is compact-valued u.s.c. and approachable. Assume that

the restriction

f(x,y)

= y,

V(x,y) Then 5.3.

and let

that 0

= 0, and

> 0 such

that

that (y —

v,x) >

graph(4)), Vv E problem (9) has a solution.

Minimization and saddle points for pseudoconvex functions

The preceding result applies to minimization problems of functions of pseudoconvex type. They extend to nonsmooth functions and to more general classes of multifunctions results of [40]. Before starting, let us recall some important concepts of generalized differentiability; the reader is referred to [2] and [12] for details. Let i/ be a locally Lipschitz real function on an open subset U of a normed space E. The Clarke directional derivative at x in the direction u is (x;u) :=

lim sup

t

x'—+x,tjO

The Clarke generalized gradient of :=

E

at x

E*;

E

E},

a non-empty, weak-* compact convex set; moreover, the multifunction is upper semicontinuous. The lower and upper Dini derivatives at x in the direction u are respectively given by is

- (x; u) := lim inf 'ih'(x + tu) .

—

+ ,

t

Obviously, pseudo-regular

.

(x; u) := lim sup t—40+

+ tu) — t

pointwise. It is said that at

x if

=

function is pseudo-regular on its domain.

.).

is

Note that a continuous convex

136

H.

Ben-E1-Mechaiekh and C. Isac

Let X be a non-empty subset of a Banach space E and let f be a real function on X.

Definition 5.2. We say that the function f is of pseudoconvex type on X if there exist a continuous mapping T from X into a finite dimensional Banach space F, a locally Lipschitz function g F —p R, and a continuous mapping that:

(i) f=goT; (ii) Vx0,x

X, g°(T(xo);ijo(T(x),T(xo))) 0

f(x) f(xo).

Remarks. Let us observe that this notion of pseudoconvexity encompasses notions studied elsewhere. (1) If g is pseudo-regular, T is a diffeomorphism of X=F= and ijo(z, zo) = z — z0, our definition recovers the notion of an essentially pseudoconvex function of Tanaka et al. [38]. It was shown in [38] that an essentially pseudoconvex function f is pseudoconvex along smooth arcs in its domain and that arcwise pseudoconvexity implies invexity. Assuming for

simplicity that X = F = respect to a mapping ij:

and T be the identity mapping, if f is invex with x in the sense that, —p

f(x) — f(xo) f°(xo;ij(x,xo)), Vx,x0 E X,

(see Hanson [23] for the continuously differentiable case) then f is of pseudoconvex type. (2) Consider now the minimization problem

minf(x).

(10)

sEX

If f is of pseudoconvex type and x0 E X is a solution of the inequality g°(T(xo);ijo(T(x),T(xo)))O, Vx

X,

(11)

then x0 solves problem (10). Assume that f : X = E -L F R where F is a finite dimensional Hilbert space with inner product (., .), T is strictly differentiable (see [12]), and g is locally Lipschitz. Then f is also locally Lipschitz and the chain rule holds: ôf(x) C ôg(T(x)) o D3T(x) (where D3 is the strict derivative operator). Assume that g is pseudo-regular or that T maps every neighborhood of a given point x into a dense subset of a neighborhood of T(x) (e.g. T is onto); then equality holds in the chain rule above (see [12]). In this

Generalized Multivalued Variational Inequalities

137

case, a slight modification of an argument of Ben—Israel and Mond [8] shows that if every point x0 with 0 e ôf(xo) is a global minimum for f, then f is of pseudoconvex type. Indeed, if ôf(xo) actually contains 0 it suffices to take ijo(T(x),T(xo)) = 0. If 0 ôf(xo) define

e F,

Tio(z,zo) :=

where arg eEag(zo)

:=

g(z)—g(zo)>O, g(z) — g(zo) = 0, g(z)—g(zo)
if

any nonzero vector

if if

— g(T(xo)) and = g(T(x))

Clearly,

g°(T(xo); ijo(T(x),

T(xo))) =

ijo(T(x), T(xo)))

max

0,

Vx E R71.

As an immediate consequence of Corollary 5.4 we obtain

Theorem 5.8. Assume that X is closed and convex in E and let f : X

R

be a function of pseudoconvex type. Assume that: (i)

E F x F*,

0;

(ii) V(x, X x F*, the function > 0 such that Vp P0: Assume that (iii) Vx e X with lxii = p, E X with hull g°(T(x); i10(T(u) ,

T(x))) is convex on X.