Topics in
NONLINEAR ANALYSIS APPLICATIONS
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Topics in
NONPAR ANAlksiS APPLIOJIONS Donald H Hyers Department of Mathematics University of Southern California, USA
George Isac Department of Mathematics and Computer Science Royal Military College of Canada
Themistocles M Rassias Department of Mathematics National Technical University of Athens, Greece
World Scientific Singapore • New Jersey • London • Hong Kong
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TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd. A11 rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-2534-2
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Printed in Singapore by UtoPrint
CONTENTS Preface CHAPTER 1
IX 1
Cones and Complementarity Problems 1. INTRODUCTION 2. CONVEX CONES • Normal cones • Regular and completely regular cones • Well based cones • Isotone projection cones • Galerkin cones 3. COMPLEMENTARITY PROBLEMS • The explicit complementarity problem • The implicit complementarity problem • The generalized order complementarity problem 4. EXISTENCE THEOREMS • Galerkin cones and the generalized Karamardian condition • Galerkin cones and conically coercive functions • Variational inequalities and explicit complementarity problems • Isotone projection cones and complementarity problems • Comment • Complementarity problems and condition (S) S-variational inequalities and the implicit complementarity problem • Heterotonic operators and the generalized order complementarity problem • Topological degree and complementarity
1 2 4 8 10 16 23 27 30 39 42 47 47 54 59 69 81 82
•
95 104 117
CONTENTS 5. SOME SPECIAL PROBLEMS IN COMPLEMENTARITY THEORY • Boundedness of the solution-set • Solution which is the least element of the feasible set • The cardinality of the solution-set • Nonexistence of solution • Sensitivity analysis • Nonlinear complementarity and quasi-equilibria 6. COMPLEMENTARITY AND FIXED POINTS 7. REFERENCES
132 132 135 139 139 139 139 140 148
CHAPTER 2
167
Metrics on Convex Cones 1. 2. 3. 4. 5. 6.
INTRODUCTION HILBERT'S PROJECTIVE METRIC THOMPSON'S METRIC WORKING WITH TWO CONES MONOTONE SEMIGROUPS AND METRICS ON CONES REFERENCES
CHAPTER 3
167 170 208 220 228 233 241
Zero-Epi Mappings 1. 2. 3. 4. 5. 6.
INTRODUCTION ZERO-EPI MAPPINGS ON BOUNDED SETS ZERO-EPI MAPPINGS ON THE WHOLE SPACE ZERO-EPI MAPPINGS ON CONES ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION ZERO-EPI MAPPINGS AND COMPLEMENTARITY PROBLEMS 7. ZERO-EPI MAPPINGS AND k-SET CONTRACTION 8. REFERENCES
VI
241 243 255 271 284 304 315 318
CONTENTS
CHAPTER
4
325
Variational Principles 1. 2. 3. 4. 5. 6.
INTRODUCTION PRELIMINARIES CRITICAL POINTS FOR DYNAMICAL SYSTEMS VARIANTS OF EKJELAND'S VARIATIONAL PRINCIPLE THE DROP THEOREM STRONG FORMS AND GENERALIZATIONS OF EKELAND'S PRINCIPLE 7. EQUIVALENCIES 8. EKELAND'S VARIATIONAL PRINCIPLE FOR VECTOR VALUED FUNCTIONS 9. APPLICATIONS • Existence of solutions for minimizing problems • Coercivity condition • A global variational principle on cones • Density result • Mountain pass lemma • The Bishop-Phelps theorem • Clarke's fixed point theorem • Borwein's c-principle • R (the field of real numbers), satisfying the separation axioms: 1) (x0,y) = 0 for ally =E implies xQ=0, 2) (^^o) = 0 for all x e E impliesyo = 0. Let £ be a vector space over R which is endowed with an order structure defined by a reflexive, transitive and anti-symmetric binary relation " < ". We say that £ is an ordered vector space over R if the following axioms are satisfied: i) x\f > O], is not normal with respect to the Schwartz topology onZ>. 6. Let Q c R" be a bounded domain. For every k e Nandpe [1, + oof, let PF*(Q) be the vector space 0j(fl) = {/ e Lp(Q)| Daf e Lp(n) for all aeN"
such that \a\ < k\,
where Daf is the derivative of / in the sense of distributions. The space Wp(Ci) is a Banach space with respect to the norm
"/ C ■■
nliip
ii J lip
r
=- l 7y.lz^/l = (r,\ ■pp) ^ iIW JaJft"h.(a)
Vp -UP
i
HP
«—i II
= y \D f\ i—i
Li«is* z * ii
J
J
Hi (0) 7
CONES AND COMPLEMENTARITY PROBLEMS The convex cone K = I / e W* (fi) | / > 0 on Qj is not normal with respect to the topology defined by the norm | | . • Regular and completely regular cones Let E{x) be a locally convex space and JT c E a closed pointed convex cone. Definition 2.2 We say that K is regular [resp. sequentially regular] if every net [resp. every sequence] monotone increasing and order bounded of elements ofK, is convergent. Definition 2.3 We say that K is completely regular [resp. sequentially completely regular] if every net [resp. every sequence] monotone increasing and topologically bounded of elements ofK, is convergent. From definition, we obtain immediately the following result. Theorem 2.5 In a locally convex space, every pointed normal completely regular convex cone is regular. Examples 1. If / is an arbitrary non-empty set, we consider on the vector space R1 the product topology. The convex cone K = R+' is a regular cone. 2. If £ = I00, the convex cone K = /+°° is not regular. Theorem 2.6 Let E{x) be a locally convex space and K c E a pointed convex cone. IfK is normal and weakly complete {resp. weakly sequentially complete), then it is completely regular {resp. sequentially completely regular). Proof Since K is weakly complete, it is weakly closed, and hence it is closed with respect to the topology x. The normality of K implies also, that E* = K* -K*. Let {xt}.eI be a net of elements of K. Suppose that {*,}. is
8
CONVEX CONES monotone increasing and topological bounded. For every/ e it, the net of real numbers {/(■*,)}. ., is a Cauchy net (since it is monotone increasing and bounded). Hence, the net {;c,}. , is a weakly Cauchy net and because K is weakly complete, it is weakly convergent to an element x* e K. From Theorem 2.4, we deduce that {*,}. is x-convergent to the same limit x* and therefore the theorem is proved. ■ Corollary 2.7 In a semireflexive locally convex space, every pointed, closed normal convex cone is sequentially completely regular. Proof Since every semireflexive locally convex space is x(E,E ^quasicomplete, we have that every semireflexive locally convex space is weakly sequentially complete, and the Theorem 2.5 is applicable. ■ Corollary 2.8 In the space Lp(Q,/j), with 1 < p < oo, every pointed closed normal convex cone is completely regular. We recall that a locally convex space E is nuclear if and only if there exist an equicontinuous sequence {/„}„eA, 0 and b e B.
10
CONVEX CONES Remark 2.2 Since a base B for a convex cone must be a convex set, it is clear that 0 g B. If B is a non-empty convex subset of E and iST c E is a convex cone, we say that K is generated by B, ifK = (J XB. ASO
Proposition 2.11 Let E(z) be a locally convex space, B a E a convex subset and K 1 and c\, Ci e B such that X0cl - (/l0 - l)c 2 . Since we obtain a contradiction of the uniqueness property for base representations, we must have 0 ^ . To show the converse, we suppose that B satisfies the stated conditions. If Xybx = A2b2 for bh bi e B and positive real numbers X\, A^, then we have 0=
1 / t i — A*2
(Xfa - X~2A b22)b2e) Ae JLif X\ {A>\b\ \fX\* *X2. XQ. .
It follows that X\ = Xq. and hence b\ = 62 ■ Consequently, B is a base for K.
m
Proposition 2.12 Let E(x) be a locally convex space and K 1, we can show that lim/?,. = 0, which implies ie/
that l i m a , = 0 .
16/
Since B is bounded, for every circled neighborhood
iel
V e 1/ (0), (where f(0) is the filter of neighborhood of zero in E) we have that there exists a > 0 with the property that X B cz V for all -a < X < a. Because there exists z'o e / such that, for all i > i0, we have 0 < a, < a, we deduce that for every circled neighborhood F e ^ ( 0 ) we have or, bt e Vfor all i > z'o, which implies that lim:c( = 0 and the theorem is proved. ■ iel
We note that it is easy to show that Theorem 2.16 is a consequence of Theorem 2.15 (which was proved several years after Theorem 2.16). Corollary 2.17 In a Hausdorff locally convex space, every locally compact cone is normal. ■ Corollary 2.18 In Rn every pointed closed convex cone is normal. ■ The reader can find several examples of well based cones in (Isac, G. [1]) as well as the proof of the following Theorem: Theorem 2.19 Let E(x) be a locally convex space and K c E a pointed convex cone. IfK is well based and complete, then it is completely regular.
15
CONES AND COMPLEMENTARITY PROBLEMS Proof Since K is complete it is closed and, being well based, it has a closed bounded base B. Let {*,},e/ be a monotone increasing and topologically bounded net of elements of A". Let U be a circled neighborhood of zero and e > 0 such that e B c U. The set |x, | i > /„ j , where r0 is a fixed arbitrary element of/, is bounded. Consider the elements yi = xt -x^,(i
> ; 0 ). For
every i > I'O, there exist bt e 5 and /i, > 0 such that >>, = A,- bt. Since | v, 11 > i0) is a bounded set there exists P > 0 such that |v,|/>/0]c/fl
(J>£5
which implies that |A,|/>z' 0 | is a bounded set.
^-(KASl
We put n = sup! A,,|/>i 0 }, and we consider an index I'I such that X, >/j-e.
For every i > iu we have yi -y, =/?,c,, where c, 6 B and
B{ = Aj - Xt < e. The last relation is a consequence of the fact that B is a base for K. Hence, xt -x(i eU,for all i > iu which implies that {*,}iel is a Cauchy net in K, and, because K is complete, {x,}.
is convergent and in
conclusion, K is completely regular. ■ • Isotone projection cones Let (//,, K) be an ordered Hilbert space. We say that (H,, K) is a Hilbert lattice if and only if the following properties are satisfied: i) H is a vector lattice, ii) J|jc|I = ||x| for allx e H{i.e. the norm \\ \\is absolute), Hi) 0 <x K (i.e. K is super-adjoint). Theorem 2.20 Let (//,, K) be an ordered Hilbert space. If H is a vector lattice, the following statements are equivalent: 1) H is a Hilbert lattice, 2) for all x,y e Kwe have <x,y> > 0 (i.e.KczK*), K c K) and is K-local inner product. Proof (1) =>(2). Indeed, for all x,y e K we have \x - y\ = (x - v) + (x - v) = 0 v (x - y) + 0 v (v - x) < x + y, which implies (since H is a Hilbert lattice) that ||x - v|| < \\x + y\\ and by the formula .•a ||x + v| - | x - v | | =4{x,y), we obtain (x,y) > 0 for all x,y e K. For all x,y e H, we have x A y = 0 if and only if jx\ -\y\\- \x\ + \y\. If x,y e K are such that x A y = 0, then |JC - y\ = x + y. Since H is a Hilbert lattice, we deduce that \\x + y\\ = \\x - y\\, which implies ii?
(x,y} = -^\x + yf-\\x-yf\ = 0. (2)=>(1).
Indeed, if 0 < x < y then ||vf -\xf in
iii2
\\1
I +
=(y-x,y
+ x)> 0 [by
\
assumption (2)] and |belli -|bc| =4{x+,x )-0= 0 (since the inner product assumption (2)] and ||x|| -|pc| =4{x ,x~)-0 (since the inner product is JT-local). ■ Theorem 2.21 a) If(H, K) is a Hilbert lattice, then K = K'.
17
CONES AND COMPLEMENTARITY PROBLEMS b) If{H,, K) is an ordered Hilbert space which is a vector lattice and is a K-local inner product, then H is a Hilbert lattice, if and only if, K = K'. Proof a) If H is a Hilbert lattice, then by Theorem 2.20 we have that K a Jf*. The inclusion K* c K is also true. Indeed, x eZ* and x g K imply x > 0 and (x,x~\ = (x+ - x~ ,x~\ = (x+,x~)-(x~
-II2
,x~) = -lx~j (1). Indeed, if z e H is an arbitrary element, we set x = P^z) and y = z-x. We have (z-x,u-x) = (z-x,u)-(z-x,x) - x,u) - (z - 3x,x) =(2 =z-x,u) (z-x,u) 0) we obtain that (/I - l)(.y,x) ^ 0. Since A - 1 can be positive or negative we obtain (y, x) = 0, and, because we showed that (y, w - x) < 0 for all M e A, we deduce that (y, u) < 0 for all u e K. Hence, y € A0. Applying Theorem 2.22 the theorem is complete. ■ Corollary 2.24 PK is a positively homogeneous operator, that is, PK(ax) = aPK(x) for all a eR+ and all x e H. Proof
Indeed, from Moreau's Theorem, we have x = PK(x)
where (PK(X),
P^x))
+PAx),
= 0.
If a e R, then we have ax = aPK(x)+aPKo(x),
where aPKaP (xK(x)eK, =*,
ceP^x) e K° and (aPK(x),aPKo(x)j = 0. Since the decomposition given by Moreau 's Theorem is unique, we deduce in particular that aP ccPKK{x) {x)==PPKK(ox). {ax) ■■ ■ Let (H,) be an arbitrary Hilbert space and KCZHSL fa closed convex cone. The order monotonicity of the projection operator onto a closed convex cone has been studied in (Isac, G. [10]), (Isac, G. and A. B. Nemeth [1-5]). (Isac, G and L. E. Persson [1]) and (Bernau, S. J [1]).
19
CONES AND COMPLEMENTARITY PROBLEMS Definition 2.6 We say that K is an isotone projection cone if and only if, y-x X &K implies PK(y) - PK(x) eK for every x, y e H. The main properties of isotone projection cones are given in the following: Theorem 2.25 If (H,) i) every isotone projection ii) every isotone projection Hi) every isotone projection
is a Hilbert space then: cone KczH is sub-adjoint (i. e. K a K). cone KaH is normal, cone KaH is regular and completely regular.
Proof i) We suppose that K is isotone projection. From Theorem 2.2 S we have ^ ' ( 0 ) = -K*{= K°). If x e K, then -x < 0 and since PK is monotone increasing with respect to the order defined by K it follows that 0 < PEKTX) < Pi(0) = 0, that is, -x e ^ ' ( 0 ) = -K',
i.e. x e K*
ii) Suppose that K is isotone projection. If 0 < x < y we have that 0 < (y - x,y) and 0 < (y -x,x) (since y-xeKcK).KcK). From that we have I*! < (x,y) < \\y\\ , that is
||JC||
u) = sup(x,u)>.
Theorem 2.27 Let K a H be an isotone projection cone. If K0 is an exposed face of K, then PK projects spK0 onto KQ and K0 is an isotone projection cone in this subspace. Proof A proof can be found in (Isac, G. [10]). ■ Examples of isotone projection cones are given in the papers (Isac, G. and A B. Nemeth [2],[5]). Theorem 2.28 1) If [H,,K) is a Hilbert lattice, then K is isotone projection and moreover, P^x) = x+. 2) A Hilbert space (//,) ordered by a closed self-adjoint cone K is a Hilbert lattice if and only ifK is isotone projection. Proof A proof is given in (Isac, G. and A. B. Nemeth [3]). ■ An important and difficult problem in the study of isotone projection cones was to find a characterization of isotone projection cones in a general Hilbert space. For the Euclidean space (R",\, this problem is now completely solved by the following result. Definition 2.9 We say that a convex cone K c R" is correct if and only if, for every face FcK,we have PspF[K) c F. Theorem 2.29 [Isac-Nemeth] Let KczR" be a generating convex cone. 21
CONES AND COMPLEMENTARITY PROBLEMS The following assertions are equivalent: i) K is isotone projection, ii) K is latticial and correct, iii)K is polyhedral and correct, iv) there exists a set of vectors |w, | i si) with the property that (",,«,) ^ 0 for all i *j , ij e I and K = Uut | / e / ] j , (where ( )° denotes the polar cone), v) K is latticial and PK(x) < x+ for every x e R. Proof The evidence of this result can be found in (Isac, G. and A. B. Nemeth [5]). It is a long proof with several technical details. ■ Thus, it would be useful to present a simple approach of this result. We now consider that (H,) is a general infinite dimensional Hilbert space. In this case, the following definition is fundamental. Definition 2.10 We say that a cone K^His weakly correct if and only if for every face FczK, we have Pyj(K) c K. Theorem 2.30 (Isac, G. and A. B. Nemeth [2]) Every generating isotone projection cone in an arbitrary Hilbert space is latticial and weakly correct. ■ The open problem posed in (Isac, G. and A. B. Nemeth [2]) takes the following form. Open problem It is true that, in an arbitrary Hilbert space, every latticial and weakly correct cone is isotone projection ? This problem has been recently studied by S. J. Bemau [1]. He considered the following stronger form of Definition 2.10.
22
CONVEX CONES Definition 2.11 We say that the cone KaH is strongly correct if and only if for each face FcK,we have P^yi^) C F ■ The strong correctness is exactly the correctness which we used in the Euclidean space [Rn,\, since in R" every subspace is closed Using several technical results, S. J. Bernau [1] proved the following result. Theorem 2.31 In an arbitrary Hilbert space (H, ) a generating cone is isotone projection if and only if it is latticial and strongly correct. ■ It is interesting to know if a weakly correct cone is strongly correct. • Galerkin cones Let (E,\\ ||) be a Banach space and KczEa closed con vex cone. Definition 2.12 The convex cone K c E is a Galerkin cone if and only if there exists a countable family of convex subcones {JSTn}ngAf of K such that: i) K„ is locally compact for every n e N, ii) ifn < m then Kn c Km
iii)K=XJKn. nzN
A Galerkin cone will be denoted by K(Kn)n
form x- E 0 for each n eW. Iff*.} It an>0 for each n eJV. n If {x„}nsN is a Schauder basis
r
•i
for every n e N, for JT, then Kn = cone\xx,....,xn) - * ; for K, then Kn = cone{xl,....,x„) = Q> for every n e N,
A*.l^°
23
CONES AND COMPLEMENTARITY PROBLEMS defines a Galerkin approximation of K. In this case K(Kn)neN is a n \x\ < \y\ for all x, y e E . Theorem 2.32 Let [E,\ |,JSTJ be an infinite-dimensional separable Banach lattice with the KMP. If K has a closed bounded base, then it has a Schauder basis. Proof Let B be a bounded convex base for K. We recall that
K= [x&E\ x>0}. Since £ is a Banach lattice, K is closed. Since 0 £ B, pick a > 0, such that a < \\x\\ < b for each x e B.
b>0 (2.3) (2-3)
Let x be an extreme point of B. For each y e K such that 0 < y < x, we claim
24
CONVEX CONES that y = Ax for some k e ] 0,1]. Indeed, x = y + z with z e K, and we may find A, > 0, A2 > 0, bx,b2 in 5 such thaty = A\b\, z = A^bi. We have _Al^ — iUI + .— ^A2 — 6L2 . x = (A,+A (/Ij + A22)) — *1 /li + A>2
Aj + A2
By virtue of the uniqueness of the decomposition of x, we necessarily get that x is a convex combination of b\ and b2. Hence, since xe ext{B), x = bx=b2 and the claim is proved. In particular, for x, y e ext{E) with x A y 5* 0, the relations 0 < x A j / < x and 0 < x A y < y and the preceding observation yield x A y = /be = juy for some A e ] 0 , l ] j y u e ] 0 , 1]. Hence, x = — y and because x, y e B, we must have x = y. Thus, the extreme points of B are pairwise disjoint and according to the relations x + y = x V y+jc A y and 2(x A y) = x + y - \x -y\ (which are valid in any vector lattice) for x,y eext(B), we deduce 0<x<x+y = \x-y\. (2.4) This forces the set of extreme points of B to be discrete since, by virtue of (2.3), (2.4) yields: a < \x\ < \x - y\\ for all x,y e ext(B). (2.5) Since E is separable and normed, it satisfies the second axiom of countability as does its subset ext(B). Being discrete, ext{B) is then necessarily at most countable. Thus, we may assume that there exists {b„}n^N such that ext{E) - {bn}ni_N. For each x e B, since 0 < x A bn < b„ and b„eext(B), there exists x(n) such that x A b„ = x(n)b„. Since x (n)b„. < bm we get JC(H)||6„|,.< £xA6,= Xx(06, w A v, we *derive x i=e / (lim By continuity of the operator (w,v) -> w A v, we derive x =jt-»0O lim (x f\xk)< x
jt-»oo
+00
+00 di)b and the proof is complete. and < t, and and therefore therefore x=x x=x = = ^! * (x(/)6,, the proof is complete. ■
i1==i1
Let (E,j I) be a general Banach space and C c £ a closed convex set. We say that a continuous operator (not necessarily linear) P : E —>E is a projection onto C if P(E) = C and /'(x) = x for every x e C. Given a Galerkin cone K(K„)noo
To show this, we use the following classical result. Theorem 2.33 [Krasnoselskii-Zabreiko] Let D be a closed convex subset of a Banach space \E,\ |). For every positive a, there exists a projection Pa onto D which satisfies \x-Pa(x)\ 0> and B = \i\y* > OJ, (where y* = ly*\ _ ) are complementarity subsets of N„ (that is, A-Nn\B). ,5 The problem LCP can be generalized for nonlinear mappings. If f:R"^>R" Rn is an arbitrary mapping, the Nonlinear Complementarity Problem associated to / and to the convex cone R" is: {find find xJC*t e Rl R" such that JK f(x,) f(x,)J e R" NCP:V NCP-.V 7 and(xt,f{xt))-0. 0 The study of economic equilibria (Ann, B. H. [1]), (Manne, A. S. [1]), (Nagurney, A. [1]), the study of some problems in elasticity (Isac, G. and M. Thera [1]), (Isac, G. and M. Thera [2]), (Goeleven, D., V. H. Nguyen and M. Thera [1]) or in lubrication (Kostreva, M. [1]), (Oh, K. P. [1], [2]) have been important motivations for the development of the study of nonlinear complementarity problems. An interesting collection of nonlinear complementarity problems are also presented in (Dirkse, S. P. and M. C. Ferris [1]). The Complementarity Theory is now a wide domain of Mathematics. If we consider, in particular, only the Linear Complementarity Theory, we remark that it is a rich mathematical theory formed by a variety of algorithms and a variety of applications in applied sciences and in technology. On this subject alone, there are probably more than one thousand papers published (see for instance the references published in (Cottle, R. W., J. S. Pang and R. E. Stone[l]), (Isac, G. [12]), (Murty, K. G. [1]). Nevertheless, Nonlinear Complementarity Theory is now in developing. Our principal aim in this chapter is to present Nonlinear Complementarity Theory as a new and fascinating field for applications of Nonlinear Analysis. For this aim, we selected only some representative results. We will show that Complementarity Theory can be considered as an interesting source of new problems for Nonlinear Analysis, and, also, Complementarity Theory can be used to obtain new results and new concepts in Nonlinear Analysis. Finally, since the Complementarity Theory has many applications in Economics, Applied Sciences, Engineering, Optimization, 29
CONES AND COMPLEMENTARITY PROBLEMS Game Theory, etc, this domain can be considered as a new opened door for new applications of Nonlinear Analysis. The relations between Complementarity Theory and Nonlinear Analysis are deep. Not only Nonlinear Analysis is used to study the Nonlinear Complementarity Problems, but also to study and solve the Linear Complementarity Problem. We consider now the most important classes of Complementarity Problems. • The explicit complementarity problem Let \E,E)
be a dual system
of locally convex spaces and K cz E a closed convex cone. Given a mapp i n g / : K-^E , the explicit complementarity problem associated to / andK is: r find xxn0 eK find e K such such that that ECP(f,K):\ r and {x (x00,f(x == 0. 0. 0. f(x(x0)0) eeK' K and ,f(x00)) )) = Models 1) Let c = (c,) e K", b = (6,) e Rm be arbitrary vectors and A = {ay) an (m, «)-matrix of real numbers. Consider the primal linear program minimize (c, A x) xe^ I F 5 ' xe9 (PLP): (PLpy.l • ! . ?9,f = \x sR" eR" |IxxeR? eR? and and Ax-b Ax-b € Je eR?) /H
(
and its dual (DLP): (DLpy.l
maximize (y,b) y^7i y ^{ ?7zZ = jx [x eR sRm | x eR? eR™ and A'y-c
eeR^V /^j.
A fundamental result in Linear Programming Theory is the following: Theorem 3.1 If there exist x0 e 7\ and y0 e ? 2 such that (c,x0) = (y0,b), then x0 is a solution of the problem PLP andy0 is a solution of the problem DLP. By this result, we can associate to the couple of problems (PLP, DLP) a By this result, we can associate to the couple of problems (PLP, DLP) a
30
COMPLEMENTARITY PROBLEMS linear complementarity problem. Indeed, considering the slack variables u eR" and v eRm such that Ax - v = b and A'y + u = c and denoting -A'' xx~\ u [u] [c1 0 -A' z= ; wM = ; q = z= ; w = V ; q = -b and M - A 0 }, 'we obtain the following y. -6 A 5
_>>J
|_vj
L J
\_ °
linear complementarity problem
,n+m find zeR"+n such that
+m +m LCP[A,q;Rn+m+m eRl+m ,w=Mz Mz + q qeR" eR^+m LCP(A,q;RT ):):- zzeR" ,w=Mz
and and (z, {z, w) w) = = 0. 0. The main contribution of Complementarity Theory to Linear Programming is the fact that the complementarity problem transforms an optimization problem in an equation. 2) Consider the convex program minimize minimize f{x) fix) ? = {xeR:\gi(x)0;j j{x,u)>0;j 7 == l,2,...,n
= l,2,-,m . - ^du- W h* (x,u)>0;i . ) *««-!*".» n+i
t
(3.4)
x > 0, u > 0 x>0,u>0 n
m
U, ==Ii
i=\
U ^Xjh,(x,u) a«J X "MAA++,/(*' £ X ; 7 J , ( * > " ) = 0 and ( * > " ) = 0. 0
x
If we set z == '
u
and h(z) [^(z),...,A„(z),An+1 (z),...,^n+m h(z) = [fy(z),...,/z„(z),/z n+1(z),...,A +m (z)]', then the si
system (3.4) can be stated as the following complementarity problem: n+m find zeR zeRn+m such that that n+m such +m m +m n +m ECP(h,R"+m ECP(h,R" ):-2 ): • zeRl z e ^+m+,h(z)^R , /J(Z) + e J^
and{z,h(z)) and(z,h(z)) = Q. 0. 3) In the books (Isac, G. [12]), (Cottle, R. W., J. S. Pang and R. E. Stone [1]), some other mathematical programming problems which can be transformed in complementarity problems are also presented. We cite for example the Quadratic Programming Problem, some Saddle Point Problems (some max-min problems) etc. (see (Isac, G. [12])). 4) Consider a two-persons game (a bimatrix game), that consists of two players. The first player solves the program maximize maximize ((xx, Axx2 )
G (G.
i &-\
x *A
f
H
(*W) -
n
w—\
1
j=y
and the second player solves the program
32
1
i x isx2 given ■: £z*j ^ =. 0, 2) no commodity is in excess demand, b + Ay, - d(pt) > 0, 3) no prices or activity levels are negative,/?. > 0, y* > 0, 4) an activity earning a deficit is not used and an operated activity has no loss, [c- A'p*} -yt = 0 , 5) commodity in excess supply has zero price and a positive price implies market clearance, p[{b + Ay, -d[pt)) = 0. We note that the vector c of operating costs represents factors of production that are exogenous to the economy or sector under certain considerations. If (1) - (5) describe a general equilibrium problem of a closed economy,
34
COMPLEMENTARITY PROBLEMS then the cost vector c = 0, because all prices will be determined simultaneously and no single price will be exogeneously given. In this case, the demands d,{p) for i=l, 2, ....,m are functions of all prices in the economy, i. e. both product and factor prices. Furthermore, these demand functions will usually be specified in a manner consistent with individual household utility maximization, that is dt(p) = T)x* where x\ is h
the Kth household's utility maximizing demand of the commodity i. Households excess demand are given by dip) -b. We remark that if the demands satisfy each individual household's budget and there is nonsatiation, then p'd(p) = p'b, and the demand function d(p) are homogeneous of degree 0 in all prices. We observe that, when c = 0, conditions (1) - (5) determine only relative prices, that is, if the vector pt represents equilibrium prices, so does Xp* for any scalar X > 0. Hence, we are free to normalize the prices. Now, we observe that we can associate to this model the following complementarity problem: find
z&Rn++m such that
\F(z)eR:+mand(z,F(z)) *); = 0 U"l
ECP(F,R^m):\ iwh \>here z =
"Ud
F(z)
\c-A'p
and D
b + Ay-d(p)
Remark 3.1 The problem ECPi F,R?m i is a nonlinear complementarity problem since d(p) is generally a nonlinear mapping. 6) Variational inequalities form another source of nonlinear explicit complementarity problems. The study of variational inequalities is an important domain in Nonlinear Analysis with many and important
35
CONES AND COMPLEMENTARITY PROBLEMS applications. Given a locally convex space E(x), a mapping f:E—>E E* and a closed convex subset D of E, the variational inequality associated to / and D is the following problem:
(
Vl{f,D)
find x, x, £e D D such such that that ( x - xx.,. ,f(x / ( xt)). ) ) > 00 for all x xeD. eft (x
When the set D is a closed convex cone, we have the following result. Theorem 3.2 Let E(T) be a locally convex space, K c £ a closed convex cone and f:E—>E a mapping. An element x, e K is a solution of the problem VI(f,K), if and only if, x, is a solution of the problem ECP(fK).
r
Proof Suppose that x. e K\s a solution of the problem VI(f,K). We have (x - x.,f(x.))
> 0 for ally e K
(3.5)
and if y e K is an arbitrary element and x= y + x,, then, from (3.5), we obtain (y,f(xt))> 0 for ai\yeK, (3.6) which implies t h a t / * , ) e K\ If we consider* = 2xt in (3.5), we deduce (*.,/(*.)) >0 and if we put in (3.5) x = 0, we have (*♦,/(*.)) 0 for every x e K. We obtain immediately ( x - x , , / ( x » ) ) > 0 for all x e K, that is x, is a solution of the problem VI{f,K). ■ From Theorem 3.2, it follows that the problem ECP(fK) is essentially, a 36
COMPLEMENTARITY PROBLEMS variational inequality but with respect to a convex cone. Because the most important results obtained about variational inequalities are obtained assuming that the set D is compact or bounded, not all results about variational inequalities are applicable to explicit complementarity problems. Hence, for complementarity problems a new theory must be developed. A more concrete example is the following. We consider a differential equation of the form
M-
' ^ - ■== Ax(t); Ax{t);t>0 t>0 W ■ dt dt x(0) = x0,x0 &K
(3.7)
where K cR" is a closed convex cone and A : R" -> R" is a mapping. In some practical problems, as for example the study of economical systems, we are interested to find x(t) such that x(f) e K and we are interested to study the problem find x such that find such that
• x(t) x{t) e K for all t > 0, x(0] x(0) = xn0 and
(3.8) (3.8)
ldx(t] x(t))\>0, > 0,for for allt>0 allt>0 and and all all yeK. yeK. / — ^ -- Ax(t), y - x(t)
A*
By (3.8), we obtain an explicit complementarity problem. 7) An interesting application, of Nonlinear Complementarity Problem in infinite dimensional vector spaces, is the study of the post-critical equi librium state of a thin elastic plate. Consider the mathematical model constructed by means of the classical nonlinear description proposed by Von Karman for plates undergoing large deflections relative to their thickness. Suppose that Q is a thin elastic plate (the thickness is supposed to be constant) resting without friction on a flat rigid support. The material is supposed to be homogeneous and isotropic. From the mathematical point of view, Q is identified to a bounded open connected subset of R2. The •37 37
CONES AND COMPLEMENTARITY PROBLEMS plate Q is assumed to be clamped on yi c y and simply supported on Y2 = Y \ Yi, where y is the boundary of Q which is supposed to be sufficiently regular. Suppose that a lateral variable load XLa,{a = 1,2, as it is indicated in (Cimetiere, A. [1]) where X is positive and increasing representing the magnitude of lateral loading, is applied to the boundary of Q. Because Q is a thin elastic plate, we observe that if X exceeds a critical value, termed the critical load (specific for each plate Q), then the plate deflects out of its plane, and we say that it buckles. Consider the Sobolev space, A*,,
A,.
2 2 22 (Q),v;,y i,2 equipped with the H2(Q) = \\uueL e L(Q)\ (Q) | -— >; _ ^ _ -€L Vij = 1,21 ec L (C1), dx{ dxtdx} norm |*|La/oy anc* l e t E De m e closed subspace of H2(Q) defined by
dz
=o, J
2 2 0,a.e, B-\, E = \zeH 0, a. e. on Q j . The cone K represents the vertical admissible displacements. For a fixed A the post-critical equilibrium of the plate Q is governed by the following complementarity problem find z eK e.K such that 0 is the intensity of the lateral load (XLa), L : E -> E is a selfadjoint linear compact operator and C : E -> E is a nonlinear continuous compact operator connected with the expansive properties of the plate. The operator C is positively homogeneous of order p = 3, it is the Frechet
38
COMPLEMENTARITY PROBLEMS derivative of the mapping z -> — (z,C{zfj and it satisfies also the following properties: i) (z, C{z)) > 0 for each z e E, ii) z e E and (z, C(z)} = 0 imply z = 0. The operator T(z) = z - X L{z) + C(z) is the Von Karman operator, which is very important in elasticity. For further details see (Cimitiere, A. [1]), (Isac, G. and M. Thera [2]), (Goeleven, D., V. H. Nguyen and M. Thera [1]). 8) The Explicit Complementarity Problem is the mathematical model used in the study of some contact problems in Mechanics, of some free boundary problems, of some problems in lubrication, of equilibrium of traffic flows and to maximize the oil production. (Isac, G. [12]). Also, the Explicit Complementarity Problem has interesting applications in Structural Engineering, in Fluid Mechanics, in Electrical Circuit Simulation, etc. (Isac, G. [12]), (Cottle, R. W., J. S. Pang and R. E. Stone [1]). 9) Finally, we remark that any nonlinear perturbation of a linear complementarity problem is a nonlinear complementarity problem. • The implicit complementarity problem Let (E,E*) be a dual system of locally convex spaces, K a E a closed convex cone and D c E a nonempty subset. Given the mappings /:£>—»£* and g:D^> E, the implicit complementarity problem associated to fg and K is find x0 e€ D such that ICP(f,g,Kh ICP(f,g,K):< Ag(x g(x00)eK,f{x ) eJT,/(xo) 0)(=K e f and ,(g(*o),/(*o)) {{g[x = 0. O. o),J{x0)) = This problem was considered for the first time by A. Bensoussan and J-L. Lions [1], A. Bensoussan, M. Goursat and J.-L. Lions [1] and studied by I.
39
CONES AND COMPLEMENTARITY PROBLEMS Capuzzo Dolcetta and U. Mosco [1], by U. Mosco [1] and other authors, namely J. S. Pang [1], [2], D. Chan and J. S. Pang [1], M. A. Noor [2], I. Capuzzo Dolcetta, M. Lorenzani and F. Spizzichino [1], G. Isac [4], [9], [12], G. Isac and D. Goeleven [2], etc. Models The implicit complementarity problem was introduced by A. Bensoussan and J-L. Lions as the mathematical model of some stochastic impulse control problems (Bensoussan, A. and J-L. Lions [1]). Generally, the implicit complementarity problem is related to the theory of QuasiVariational Inequalities (Capuzzo-Dolcetta, I. and U. Mosco [1]) which are currently used in the study of some stochastic optimal control problems or in the study of various free boundary problems associated to some differential operators. 1) Quasi-variational inequalities and the implicit complementarity problem In the study of some practical problems, we have the following situation. We have two spaces H and E such that E0, 1 R" as the columns of the following matrix
43
CONES AND COMPLEMENTARITY PROBLEMS
//to /ato-zAto- « Z.'to /22W-« «
- « - «
//•(*) Z/to z/"to //w
z*to /iW - «
where a is an arbitrary real number such that a > 0, then the problem (3.13) is exactly the problem EGOCP{Tx, Tt,...,Tm, R"). The problem (3.13) has been studied in (Cottle, R. W. and G. B. Dantzig [1]), (Ebiefung, A. and M. Kostreva [1]) and (Szanc, B. P. [1]). 2) The Mixed Lubrication Consider the case of mixed lubrication in the context of a journal bearing with elastic support. In this case, we have two operators T}(X) = H0 + a + L(X) and
0H\ +-VI T2(X) =-R{X) 0, a.e. o n Q j , the problem to compute the pressure is to solve the problem EGOCP{T\, T2, K). This problem was considered in (Kostreva, M [1]), (Oh, K. P. [1], [2]) and (Isac, G. and M Kostreva [1]). 3) The Discrete Dynamic Complementarity Problem. This problem was defined by Harrison and Reiman in 1981 [see the references of (Mandelbaum, A. [1])], and, it is important since this model is a unifying framework for fluid and diffusion approximation of stochastic flow network. Consider E = Rm endowed with the Euclidean structure and the ordering defined by K= R+ . Suppose given x = |x(0),x(l),...,x(«),...] a sequence in Rm with x(0) > 0 and & an (m, /M)-matrix. The Discrete Dynamic Complementarity Problem is, find the sequence y- {y(0),y(l),..,y(n,..)\ •)} such that i) z(n) = x(n) + TRy{n) > 0, for all n = 0,1,2,.. ii) ^(0) = 0, Ay(rt) = y(n) - y(n -1) > 0 and
(3.14)
Hi) (z(n), Ay(ra)) = 0, for all n = 0,1,2,... (we consider by conventiony{-\) = 0). From (ii) we have that each coordinate y}, = |v y (w)|« = 0,1,2,..j of y is nondecreasing and from Hi) we have that yy (« - 1) strictly increases to^,{M) only when zfri) - 0 for j = 1,2, ,m. Consider the vector space S = (x|x:N-> Rm) ordered by K = [x GS\ x(n) > 0, for all n eiVJ where A^={0,1,2,....,n,....}. 5" is a Frechet space with respect to the topology defined by the family of seminorms {pr}ri.N,
where pr(x)=
sup ||^(")|. 00,...,Fm(x)-y°>0\ -J> - v °°> : >) is non-empty. For this >uj '+ I model, the problem is to show that given v° > 0 the problem %,.. IGOCPi T\,T 2,---,T +\ has a solution x° > 0 which is the least element of Tm,R K)
46
EXISTENCE THEOREMS it is Sy0, where 7j(x) = Fi(x)-y°,...,T Fmm(> (x)-y°. X -y * - y°, m(x) [* = F production x° is realizing v° with a minimal social cost.
0
In this case, the
The Complementarity Theory studies several kinds of problems associated to a particular complementarity problem. The most important problems are the following: 1. the study of solvability, 2. the study of existence of a solution which is the least element of the feasible set. For example for the problem ECP(f,K) the feasible set is ?={xeK\f{x)eK*}, K 3. 4. 5. 6.
the study of the solution-set from the topological point of view, the study of the cardinality of the solution-set, important to study when the solution-set is a bounded set, the study of a complementarity problem must be finished by a sensitivity analysis.
4. Existence theorems In this section we present several existence theorems based on some special ideas. Every idea can be considered as the starting-point of new results in Complementarity Theory. • Galerkin cones and the generalized Karamardian condition Let (E,F) be a duality between two locally convex spaces, K F a mapping. We denote by if* the dual of K with respect to the duality < - , - > . Theorem 4.1 [Karamardian] The problem ECP(f,K) has a solution if: 1) the function (x,y)->(xj(y)) is continuous onKxK, 2) there exists a compact convex set DczK such that for each x e K\D there exists z e Z> such that (x - z,f(x)) > 0.
47
CONES AND COMPLEMENTARITY PROBLEMS Proof A proof is in (Karamardian, S. [1]). ■ Karamardian 's Theorem was generalized by M. S. Bazaraa, J. J. Goode and M. Z. Nashed, and by G. Luna (see the references of (Isac, G. [12]). Another generalization is in (Isac. G. [5]). An analysis of Karamardian's theorem proves that assumption (2), in the case of infinite-dimensional spaces is a strong one, because it implies a compact localization of a solution of the problem ECP(fK). Using the concept of Galerkin cone we will introduce a generalization of Karamardian's condition, which is more flexible. Let (//,) be a Hilbert space and K{K^)n&N a Galerkin cone in H. Definition 4.1 We say that a mapping f: K —» H satisfies the condition (GK) (the generalized Karamardian's condition) with respect to the Galerkin cone K(Kn)n&N, if there exists a countable family {D„\ [A, neJVN of subsets ofK such that: 1) for every n e. N, Dn is a convex compact subset ofKn 2) for every x e K„\D„ there exists y e D„ such that (x - y,f(x)} > 0. The next result assumes that condition (GK) is satisfied with an equibounded family {Dn} neNN ■ This assumption is naturally satisfied in several situations. Examples : 1) We assume t h a t / : K -> H has the following form (used in the study of some problems in elasticity theory), f(x) = x - AL(x) + T(x), for every i • , , 1 (x,L(x)) xeK,K where Lr is an operator such that — = supV-^>0 is well P x*K ||x| defined (for example if L is a linear self-adjoint compact operator, as in the study of elastic plates), X > p and T is a JT-coercive operator, that is,
48
EXISTENCE THEOREMS lim
(x,T(x)) X
M_u* xsK
\ " = +oo.
wr
In this case, condition (GK) is satisfied with an equibounded family {Dn}neN with respect to every finite rank Galerkin approximation (JSTn)neAr of K, (i.e., for every n l
51
CONES AND COMPLEMENTARITY PROBLEMS
H*JIMklD-
for every k e N such that r-*» n tii MI my IKIh-" ' neiV is bounded last relation is impossible. In conclusion, we have that {*„}neAf
Kl
and then it admits a weakly convergent subsequence j x„t | keN whose limit xt belongs to K, since K is weakly closed. The sequence \xn\
IteJV
is
strongly convergent. Indeed, since/is a (ws)-field we find a subsequence {yj}j.jeNN
of
W*keN 6*
s u c h t h a t A:=
HZ^j) j-t+oo
c»M n») ^ well defined ({?{v,)}.^ '\y, jeN
is strongly convergent). Using the weak lower semicontinuity of the norm
2 Nir= M*)> M^)}
>i\yj || || and the relation ||yy||2 = \Vj,liyj\\,
wr
for every j e Nv/e get L2
II II 2 / \ ||^|| 2 +a>'VJW
(4.2) (4.2)
For every j e N we set, xXjJ == Pj(x.) PJ(x.) ■ Then, by the convexity of Kj, we have 111 1 _- -f f(\\-- 1) ll }/j 6eKj. Zj 1 ^xt,f(xt)) ==00..0. Indeed, let z e K be x../"(x. an arbitrary element and denote Pfz) by z, for everyj e JV. We observe that Izj - y •} is strongly convergent to z - *♦, and we have ieN
fa - yj y,,f ,f(yj))\Vi > every; A , for every/ €T.JV. (zji■■~ypf(yjf) *>0,^ 0for every j e€ N.
(4.5) (4.5)
Taking the limit asj tends to +oo in (4.5) we obtain, (z-x (z-x*,f(x,)) > -xt,f(x,))>0, >(0, for t,t[x, each zZ e€ K, = K, which is equivalent to f(x*) (x.,f(x.)) = 00 (see / ( * . ) ee K* * and /x«,/(x»)) fix, =C Theorem 3.2). ■ Remark 4.2 It is evident that, if in Theorem 4.2 assumption (2) is replaced by the following assumption: 2b) condition (GK) is satisfied with a family [Dn }nsN such that (3rt > 0XV« Opn e N)(Vx eD e£>„)(||x| > r,) (3n r,) n)(jx\\ > then the problem ECP(f,K) has a solution JC, and x»5* 0. Theorem 4.2 is ap plicable to the model (3.9) which is used in the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions (Isac, G. and M. Thera [2]). Other results, similar to Theorem 4.2 are also proved in the papers (Isac, G. and M. Thera [2]) and (Goeleven, D., V. H. Nguyen and M. Thera [1]). An interesting generalization of Theorem 4.2 for arbitrary closed pointed cones (not necessarily Galerkin) is in (Zhou, Y. and Y. Huang [1]). We
53
CONES AND COMPLEMENTARITY PROBLEMS remark also a generalization of Theorem 4.2 for reflexive Banach space; proved in (Thera, M. [1]), with a proof which, modulo some details basec on duality mapping, is similar to the proof of Theorem 4.2. • Galerkin cones and conically coercive functions The concept of conically compact coercive function which we introduce now, can be used to study the explicit complementarity problem with respect to a Galerkin cone. The conically compact coercivity is based on the concept of conical bornology (Isac, G. [8], [17]). A bornology on a non-empty set X is a collection g" of subsets of X such that i) S covers X, i.e.,X= i.e.,X ( J 5 , i) SCOVQXSX, Beff
n
ii) if 5, ,B2 ,..,B„ e S(n e AO then B,. eff, ii) if5, ,B2,..,B„ e S(n e AOthen (J#, e^» =i iii) if B e g"and C c B then C € g". iii) if B e g"and C c B then C e f. Let E(T) be a locally convex space. For every x e Ewe denote: £B) & b(x)==\B (?c(x) ==[B #.(*) [5 cc £E || 5B isis convex convexcompact compact and and xx g€ B\. B\. £(0)] we set K0 (B) = \J AB. KB. If x * 0 and For every 5B € &(0) [resp. 5B e &(0)] Aao (x) [resp. [resp.5B ee &(*)] &(*)]we wedenote denote 5B e€ (? gbb(x) KK,xX(B) (B) B}. A> }. (5) == {Ab {^A ++ (1 - A)x | A A> >0,0,bb e€e55} Definition 4.4 We say that a subset AR K -> R HI a function. The level set of order X (X (X eR) eR) of the function/is by definition LW(f){xeK\f{x)<x). (f) =: [xxeK € K\ f(x) ix(f) fix <X\. <X We recall that the function/is coercive over K (in the classical sense), if and only if, the level sets Lrff) of/for every X are /,:= inf f{x) f[x) < +oo. bounded sets. We suppose that -oo < /*:= JCSA XEK
Definition 4.6 We say that f is conically compact coercive, if and only if, for each X > f. the level set Lx(f) is a conically compact set. X>f. The next result was proved by G. Isac and M. Thera in [2] and it is a conically compact coercivity test.
NII)
Theorem 4.3 Let K be a convex cone in a Banach space \E,\ E. ||J and {fi\^Li>\Sj\ ._ two families of mappings from K into R. Suppose that the WZA*j}% i=V
>i
following assumptions are satisfied: 1) f is lower-semicontinuous and positively homogeneous of degree pt >0 for x}, for every i e{\,2,...,m e{\,2,-,m\), {\,2,...,m\}, 2) f{x) > Ofor each x e K\ {0} and each i e {\,2,...,m\}, 3) for each] e {l,2,...,m2}, there exists r, rj > > 0 such that
55
CONES AND COMPLEMENTARITY PROBLEMS *^ /M * < 00,, {L2,..,™,}} and lim \^J^-< rrjj < vaax[p max{/?,.t |11i ee {L2,..,/«,}} so, 0 0 Mh-* xeK xeK
lljcfl HP
II II
«2
_ isa « ^ then, for any at > 0 and $ > 0 the function hix) h[x) == Z «,.^(x) -YjPjgj Z A/#7 i=i
;=i 7=1 7=1
.=1
conically compact coercive function. Proof A proof is in (Isac, G. and M. Thera [2]). ■
The next result extends the classical Weierstrass variational principle to conically compact coercive functions. This result was proved in (Isac, G. and M. Thera [2]). Theorem 4.4 Let (E,\\ ||J be a Banach space and K a E a closed convex
w;, '"2
cone. Let {/a;, {/•}"!, and IgA _ be lower-semicontinuous mappings from K into R that satisfy assumptions (1) (2) and (3) of Theorem 4.3. Iff: K -» i? tf->J? is lower-semicontinuous such that f(x) > Z ^.« 0 and p) > 0,for each i andj, then f 'achieves its minimum on every locally compact convex pointed subcone K0 c K. Moreover, the set of minimal points off is compact. Proof Let h be the function from K into R defined by A(*) =
I '"I
i=\
/=!
-z W*) tn->
hfi\X)
7=1
i=i
By Theorem 4.3 we have that h is conically compact coercive and since f(x) > h(x) for each x e K, we have t h a t / i s conically compact coercive too. Hence, for each locally compact convex subcone K0 of A" we have that x mf{h(x)\xeKQ} is finite, which implies that inf{f(x)\xeK {/(*)|xe* F "0) o0} is also
M
56
EXISTENCE THEOREMS
[-
M
finite. Moreover, for each £>0 £ > 0 the set L f(x) + e6 \ LE£ = \x eiKKQ0 fix) < inf fix) eA0 0 )J [ ' ' ixi=K is non-empty and compact, since it is a closed and bounded subset of KQ , which is locally compact. Therefore f ] Le is non-empty and compact and e>0 £>0
each element x0 of this intersection satisfies f{x = mmf(x). f(x). 0) /(*b)= xeK
■
0
We apply now the concept of conical compact coercivity to the study of the solvability of the explicit complementarity problem. Theorem 4.5 Let H() be a Hilbert space, K(K )ni_N a Galerkin cone V n »/n R two Junctions. If If, the the following following assumptions assumptions K^>H and O : H^R are satisfied: are satisfied: 1) fis a (ws)-field {wherefix) T(x)), (w/zereX^) == *x -- T(x)), 2) fis the gradient of®, 3) , +co, (x,f(x)) x Jx|-»+oo xsD xeD
"
then the problem ECP(fK) admits a solution x*e K, which is a cluster point of a sequence {x„} .N, where for each nn eN, solution of.ofECP(fK ECP(f,Knn). eN, x„ isIs aasolution * . .nineN' Proof By virtue of assumption (3), we have that O achieves its minimum on every cone K„. That is, for every nn eN x„ e K„ such that eN there exists x„ O(x„) = ®(x,) - min min0>(x). (;c). xeK„ xeKn
I I \ In particular, since (/(*„)>*-*«) In particular, since (fix\x„ ),x-x,x-x, n)x
every xx (f(xAx-x n)>0 0 (f(x „),x-x„)> i for for every n),x-x n)>
'
X X n) X ®(xxn+t(x-x ))-®(x n+t ®(XX-x n n( ~ n))~®{ +n -w\xnn) we get we get /lim— -*u+ t
/-->o+
t
which implies implies (by ee JST„ K„ which (by Theorem Theorem 3.2) 3.2) that that
57
CONES AND COMPLEMENTARITY PROBLEMS (/(*„ ),*,,) = 0. 0. This means that x„ solves ECP(f,Kn) an< /f{x„) ( * . ) eK* « < n and (/(*„),*„} hence we have .2 \\xn(={T{xn),xn). (4.6; (4.6; kf=(%,)>*»)■ By assumption (4), the sequence {*„}neN neJV is bounded and therefore admits i weakly convergent subsequence |JC„ | izN
whose limit x* belongs to K
Since/is (w.s)-field, we may find a subsequence \yt)te/V ( „ of \xn }
ietf
sucl
that A = lim T(xn)exists, and, by using the weak lower-semicontinuity o: l'-f-W0 I'-f-WO
the norm, (4.6) yields 2
W 2t< \\x f 0 ;/(y,.),z,.-y,.)>0 from which we derive
>*,^ n ^ p
((y,,*,) y,*,) > ( % ) , * , . ) .
M^U
(4.8; (4.s;
ftL*
Since [T{yi)}ieN and {*,},l eA , norm converge respectively to A and*., w< etf ieAf obtain (from (4.8)), II 22, =(x,,x.)>(/i,x,). ||x.| \\x,f=(x„x.)>(A,x ( * , , * . ) > (i4,JC») t).
{ML,
(4.9; (4.9
N.
Finally, from (4.7) and (4.9), we have that {||y,|}ieA J-JV tends to |x.|, whil
oo
'fr) = /(*) •
z e K be an arbitrary element. Then z = limf,., where f.Si^PM* = PK (z) and since i-»oo l'->00
'
v,- solves ECP(f,K,) we have ft-*./(*)) (£,. ->',,/(>',■))*o > 0 for eachieN. /' sN. Thus, taking the limit as i tends to +00 in the last inequality, we get (z-x.,f(x (z- x,,f(x,)) t))>0 > 0 x*,J[x,)>U for each z e K which implies (by Theorem 3.2) that x solves ECP(f,K). ■ Other results similar to Theorem 4.5 are proved in (Isac, G. and M. Thera [2]), (Isac, G. and M. Thera [1]). • Variational inequalities and explicit complementarity problems Considering Theorem 3.2, we remark that another way to obtain new existence theorems for explicit complementarity problems, is to find new existence theorems for variational inequalities, with respect to unbounded closed convex sets. Several authors have recently worked in this sense, and some interesting general results were obtained by J. S. Guo and J. C. Yao in [1]. We present now some of their results. Let (E,l I) be a Banach space K c E a closed non-empty set a n d / : K -»£* a mapping. We recall that the variational inequality associated with/and K is Ind xxtt eK find e K such such that that
(
Vl{f,K):
x-xtt,/(*.)) ,/(**)) ^ 00 for for each each x xeK eK )) > \ (x-x
We say that / is continuous on finite-dimensional subspaces, iff is conti nuous onK f)F for every finite-dimensional subspace F ofE. We consider on F the restriction of the topology of E. Theorem 4.6 [Ky Fan] Let X be a compact convex set in a topological vector space. Let A be a closed subset of X x X with the following properties: 1) (x, x) e A for every x e X, 59
CONES AND COMPLEMENTARITY PROBLEMS 2) for anyfixedy e X, the set Ix eeX\{x,y) X\ (x,y)0 (y --xnk x„ (y-x ni,f(x k ,f(x„ t)) > 0 for each k e N. nk))>OforeachkeN.
(4.10)
From the assumption and (4.10) we deduce that (y-x,f{x))>0 (y-x,f(x))>0. y-x,f(x)} Therefore, x e L\y) and hence L\y) is weakly countable compact. By Eberlein 's Theorem, L\y) is weakly compact and hence weakly closed for every y e K. We put S = {(x,y) e K x K\\(y-x,f(x))>0\ (y --x,f(x)>0 x,f(x)) > o}. Since, for each
60
EXISTENCE THEOREMS v e K, we have D{y) L\y) = = ixeK\(x,y)eS\ |x e{xeK\{x,y)eS} K\ (JC, V) GSJ we obtain that for eachy e tf, the \xeK\(x,y) set \x eK\(x,y) G^J iyyeK\(x,y)eS) 0 for all -Xt,f(x \xKczS t,t[Xttt))>0 2 U 7 E JC and the proof is complete. ■ Theorem 4.8 Let K be a non-empty closed convex set in a Banach space [E\ I) andf: K K^E* -»£* an arbitrary mapping. Then x, e K is a solution of the problem VI(f,K), if and only if, there exists a set D a E such that int(D) is non-empty and the set K f]D satisfies the following properties.: 1) K f]D is bounded closed and convex, 2) xt eKC\'mt(D), x eEKHD K C\D. f)D. (x-x.,f(x (x-x„f(x )) allxsK X . , t/))>0 I X . I ) ^>0for t Proof Suppose x* e K is a solution of VUf,K). Let p > 0 be a positive real number such that |x,|0 t,f(x t,f(x*)}>Q t))>0 t,f(xt))>0 ^ u for all (x. >U and hence (x-x x e K, that is, xt is a solution of VI(fK). ■ Theorem 4.9 Let (E,\\ : * :.. l II)||j be a reflexive Banach space, K c:E a non-empty
closed convex set andf: K—> K-+E E i a mapping. Suppose that there exists a set D a E such that int(Z)) is non-empty and the following assumptions are satisfied: 1) Kf]D is non-empty bounded, closed and convex,
61
CONES AND COMPLEMENTARITY PROBLEMS
kL*U,
such that
then VI(f,K) has a solution. Proof Let L\K) f]D. Since Theorem 4.7 is applicable for/and L\K), EKK)==KKC\D. there exists JC. G D(K) such that (x-x,,f(x.)) (x-x.,f(x.)) > >0for 0 for allxVtorallx
(4.11)
If x. e KC\int(D) we have that x» is a solution of VI(fK) by Theorem 4.8. Suppose now that x. eKf)dD Then, by hypothesis there exists eKHdD ( X ,-- Kw,/(x„)) , / ( X . ) ) > 00 . From (4.11) and the last ine u eKf)int(D) such that (x, -u,j u , > ^u quality we have (x [xt♦t -- ww,/(x,)) u,f(x.)) K be an arbitrary element. ,/lx.l) = = U0 . Let xx ee K There exists X e]0,l[ such that Ax + X)u e L\K) (since + (l - (l-A)ueD(K) UGK flint(D)). We have 0 < (A(x-u) + u-x.,f(x.))x,)) = Xix 0< A.(x-u,f(x.)) A(x- u,f(xt)) = A(x-xt,f(xt)), = MX-X..
+(w-x.,/(x.)) +(«-x.,f(x.))
nx.
which implies that x. is a solution of VI(f,K). ■ Theorem 4.8 and 4.9 have interesting applications relating to the solvability of the explicit complementarity problem.
N I)
Theorem 4.10 Let (E,\ fl) be a Banach space, KczE KczE a closed convex cone and ff:K^E : K -> E an arbitrary mapping Then x. e KK is a solution oj X. G —* ECP(f,K) if and only if there exists a set D c E such that int(£>) is nonDczE empty and the set Kf]D satisfies the following properties: rni 1) KCiD K H D is bounded closed and convex, 62
EXISTENCE THEOREMS eKClint(D), 2) xx,t eJSTI eKf\int(D), 3) (x-xt,f(xt))>0forallxe ))>Oforallxe Kf)D. tor all x € Kf]D Proof The theorem is an immediate consequence of Theorem 4.8. ■
NII)
Theorem 4.11 Let \E,\ E* a mapping. We say that/is : i) weakly coercive if lim \x,f(xj) +<x>; x,f(x)) = +co; ||x|->+oo xsX
(*./W == +°o +oo;; ii) coercive if lim -—^-r.—W-MMl xeK M xeK
63
CONES AND COMPLEMENTARITY PROBLEMS Hi) a-copositive if there exists an increasing function a : [0,+M4#I) ||x||a(||x|) (*,/(*)-/()) r—>-KO r-»+co
for all x e K. If cif) = kr for some k > 0, then/is said to be strongly copositive. Corollary 4.13 Let \E,\ ||l be a reflexive Banach space, K a E a closed convex cone andf: K—>Ea completely continuous operator. If one of the following conditions is satisfied: 1) lim lim inf/x,fix)) infix,f(x))>0, (*,/(*)) >0, w->-**> xeK
2) fis weakly coercive, 3) fis coercive, 4) fis a -copositive, 5) fis strongly copositive, then the problem ECP(fK) has a solution. Proof If, condition (1) is satisfied, then there exists an r > 0 such that > (x,f(x)\ > 0 for all x e K with \\x\\ \\x\\ >irr . In this case the corollary follows x 2 T\x))*v ;*./(*)> *o from Corollary 4.12. Since, about conditions (1) - (5) we have the implications (5) => (4) =>(3) =>(2) =>(1), the corollary follows. ■ Other interesting results about the solvability of the problem ECP(f,K) can be obtained supposing that the mapping/ is a pseudo-monotone operator. In this sense R. W. Cottle and I. C. Yao [1] worked recently. We present some of their results. Let (H, ) be a Hilbert space, K c Ha non-empty subset a n df-.K^H. / : K^>> K-> H. We say that f is pseudo-monotone, if and only if, for any x and y in K, we have (xx-y,f(y))>0 - y, > 0 implies (x --y>j\*) y,f(x)) *> 0. This concept was defined (x-y,f(x)}>0 v. f(y)) nv 2 and studied by S. Karamardian in [2] (see also (Karamardian, S. and S. Schaible [1]).
64
EXISTENCE THEOREMS We say that / is continuous on finite-dimensional subspaces if it is continuous on KC\F for every finite-dimensional subspace F of H with A"f]F s* ■ It is well known that a monotone mapping (in the sense of Browder and Minty) is pseudo-monotone, but not conversely. Pseudo-monotone maps are also studied in (Karamardian, S. and S. Schaible [1]). Proposition 4.14 Let (H, ) be a Hilbert space, KczH a closed convex subset. Iff: K —» H is a pseudo-monotone mapping continuous on finitedimensional subspaces, then x* e K is a solution of the variational inequa lity (u --x,f{x))>0,forallu&K x,f(x)) > 0,for allueK (u-x,f(x))>0,foraUueK (4.11) (4.H) if and only if (u-x [u-xtt,f(u))>0,forallueK , f{u)) >0,forallueK
(4.12)
Proof Suppose that*, satisfies (4.12). Consider an arbitrary element u e K A, 0. > 0. Computing the limit as X \->0 (4.12) we deduce A(u-x„f(x A(u-xt,f(xxx)) —» 0 and using the continuity of / on finite-dimensional subspaces we obtain (u-x Therefore x* is a solution of (4.11). Conversely, ' « - Xt,f{x . , / (t^>0. J C . ) ) >> 0 . supposing that x* e K is a solution of (4.11) and using the fact that / i s ,f(u)}>0K « for all u e K, that is, x« is a pseudomonotone we have (u-xt,f(ufj>0 solution of (4.12). ■ Theorem 4.15 [Cottle-Yao ] Let (H, ) be a Hilbert space andK a H a closed bounded and convex subset. Iff :K -> H is a pseudo-monotone mapping continuous on finite-dimensional subspaces, then there exists xt e K such that {x-x,, [x-x.,f(xf(x 0 for all x e K. t))>0 t)) > >U Proof Let "P be the family of all finite-dimensional subspaces F of H with Kf]F * 0. For every F e 7 let PF be the orthogonal projection of Honto F. KC\F*. Let ffFF == PPFF°f°f be the composition of Pp and / By Hartman-Stampac65
CONES AND COMPLEMENTARITY PROBLEMS chia's theorem [Hartman, G. J. and G. Stampacchia [1], [Lemma 3.1]] or [(Isac, G. [12]), Theorem 3.1], there exists for every F e ? a n element xF eKf]F such that )}>0, ueKC\F. (u-xFFx,f(x ueKftF. F))>0, F F,f(x F))>0, u, for all ueKftF.
(4.13)
We denote by KF = |x G \ F a G e "p\ for every F e ?. For F e ?, we denote by KF the weak closure of KF . The family JJT/| F e ? } has the finite intersection property. Indeed, for F, G e "P, let M e "? be such that FU G a M. Then . Let x. e H ^ F - Consider an FFe? e?
Fe?
arbitrary element u e K, and F e ? such that u s F. Since K° if£F is weakly compact and x. x. e e f]K which K nneN„ CczJTp n *F,; , there exists a sequence {*„} iMnetf F Fe? Fe?
converges weakly to x. (We used also Eberlein's theorem ). By Proposition 4.14 and formula (4.13), we have (u\u-xnx,f(u))> Q, 0, for forallaWnzN. n e N. n,f(u))>
(4.14) ;4.i4)
The function («-*,/(«)) (u - *x,f(u)} , / (u is weakly continuous in x . Computing the limit in (4.14) as n -> +oo we have, (u-x.,f(u)) (u - x,,f(u)} > 0, 0 for all u e If and applying again Proposition 4.14 we have («-x„/(x,))>0, (u-x,,f(x,Jt))>0, for X, ) ^ t all u e K and the theorem is proved. ■ The importance of the next result is the fact that it gives some necessary and sufficient conditions for the existence of solutions to the variational inequality problem for unbounded sets, when the mapping / is pseudomonotone. 66
EXISTENCE THEOREMS Theorem 4.16 [Cottle-Yao ] Let (H, ) be a Hilbert space and K a closed convex subset in H and f:f: K -> —> HH a pseudo-monotone mapping which is continuous on finite-dimensional subspaces. Then the following statements are equivalent: 1) there exists x* e K such that (x - x*,f(x.)j x, ,/(*.)) > 0, for all x e K,
(4.15)
2) there exist u e K and a constant r> r > \\u\\ \\u\\ such that(x-u,f(x))>0 (x - u,f{x)} > 0, for all x € Kwith \\x\\ IWI ==r,r, 3) there exists r > 0 such that the set Ix ejK"||jc| 0,
\\x\\ = r, there exists u e Kwith \\u\\ < r
4) there exists a closed, convex set E in H such that int(is) i±ty, Kf)E is nonempty and bounded and, for each x e K f] d(E), there exists usKf] int(£) such that (x - u,f(x)) > 0, 5) there exists a non-empty closed, bounded convex subset B of K with \X&K(B) * | such that the following condition is satisfied: for each x e dK{B), there exists u e int^i?) such that (x - u,f{xf> > 0. Proof (1) implies (2). Indeed, let*. eJTbea solution of (4.15). Choosing r > 0 such that |x*| < r and considering u = x*, we remark that (2) follows from the pseudo-monotonicity of the mapping / (2) implies (3). This implication follows immediately. &H\\x\0. .-w,/(x.))>0.
As an immediate consequence we have the following result for explicit complementarity problems. Theorem 4.17 Let (H, < >) be a Hilbert space, K c H a closed convex cone and f: K —> H a pseudo-monotone mapping which is continuous on finite-dimensional subspaces. Then the following statements are equivalent: 1) the problem ECP(f,K) has a solution, 2) there exist u e K and a constant r > \\u\\ (x - u,f(x)\ > 0 for \\u\\ such that {x-u,f(x))>0 all x € Kwith M=r, ||x|| = r, 3) there exists r > 0 such that, for each x e K with \\x\\ Ml ==r,r, there exists \u\\ 0, u eK with \\u\\ ( x - w,/(x)\ > 0,
68
EXISTENCE THEOREMS 4) there exists a closed convex setEcH such that int(E) * (j), Kf]E is nonempty and bounded and for each xxeKfl^E), eKf]o[E), there exists u e Kfixnt(E) such that ((x* - «u,f(x)) ,/(*)) > >00, ueKr\'mt(E) 5) there exists a non-empty closed, bounded, convex set B c:K with intjy(5) BdK non-empty and satisfying the following condition: for each x e dg(B), dniB} there exists u e intx(2?) such that (x - u,f(x)) > 0. ■ 'x-u,f(x))>0. The next result, which is a particular case of Theorem 3.1 proved in (Guo, J. Sh. and J. Ch. Yao [1]) can be considered as a variant of Karamardian's Theorem [Theorem 4.1] for pseudo-monotone mappings which are continuous on finite-dimensional subspaces. Theorem 4.18 Let \E,\ ||j be a reflexive Banach space andK a E a closed convex cone. Let Letf: f :K — ->» EE be a pseudo-monotone operator which is continuous on finite-dimensional subspaces. If there exists a bounded subset D ofK such that for each x e K\D there is u e D with (x-u,f(x))>0, (x-u,f(x))> )> 0, then, the problem ECP(fK) has a solution. ■ Some other existence results for variational inequalities with respect to unbounded sets can be used to obtain new existence theorems for explicit complementarity problems. In this sense we cite the papers (Guo, J. Sh. and J. Ch. Yao [1]), (Yao, J. C. [1]) and (Hadjisawas, N., D. Kravvaritis, G. Pantelidis and I. Polyrakis [1]). Results on complementarity problems, based on variational inequalities are also proved in the papers (Noor, M. A. [1] [7]), (Noor, M. A. and Th. M. Rassias [1]) and (Noor, M. A., K. I. Noor and Th. M. Rassias [1]). • Isotone projection cones and complementarity problems The con cept of isotone projection cone was introduced in 1985 by G. Isac, with the aim to use the geometry of cones in Hilbert spaces, in the study of comple mentarity problems. This interesting class of cones was studied by G. Isac and A. B. Nemeth [1] - [5] and recently by S. J. Bernau [1]. Using isotone
69
CONES AND COMPLEMENTARITY PROBLEMS projection cones we will study the complementarity problem by some special iterative methods. Let (H,) (#,>) be a Hilbert space, K a H a closed convex cone and f, g : K —» H two arbitrary mappings. We consider in this section the following complementarity problems: ECP(f,K)
((
find ee K K such such that that find xx0n G )sK'and(x f(x0o)) f(x00)eK'and(x )eK'and(x0> ,f{x )) 00,f(x 0)) ( f(x
W)
and
=0
find Ind x0 eeKK such that r(x0Q)eK,f(x )eK,f(xQ0))eK'and ICP{f,g,K): ICP(f,g,K):< g(x eK'and eK • "n+l n ++1, ^nl+ )i ))) -- ^d 2 == ^ rPK{Xn l ()* -„ /+ i.+)l)-fl{PK{Xn*l))~fz{PK(y„+l))-d W *,(*„)£*. :X, !*/»,(?„) i have [using also (1) and (2)] P K(xn) < x, < PK(y„) which implies 0< x, PKn)
*»+i f{xn)\ + hH? \)>nh n\ r ^ - -^ % v j ) - .-/ /{ynW )\ +] /z^„ ■yn+ yn+i i ==^ Ps[y» +%) for every n - 0,1,2 If the following assumptions are satisfied : i) xa<xxandy\ -0 ii) if dimH = +oo, the mapping 0{x) (x) ==h(x) h(x)++ PJx-hlx)-f PK\xPK-[x-h{x)-f{x)] h[x) - /(*)] hlx) X) is nonexpansive or condensing, then the problem ICP(f,I-h,K) has a solution x* E K such that xn<x,oo n-»oo
n->oo n->oo
Proof The proof is similar to that of Theorem 4.23, however, we are using instead the mapping O(JC) 0>(JC) -= *(x) h(x) -PKh[x) - f(x)], f(x)] defined for every Hx) ++ PPKK[x[x-h(x)\x-Hx)-flx) xeK.K. xeK. n ny« for Since x0,y0 E E K, £ then the fixed point JC. of O satisfying xnn<x <xt 1. l(px)
^ \{\{o)o) T2T(u jUoaTl(uQ),
ftll
T2{HQU0)>
a Mo ^ T2{u0),
which imply x
x \ * *Mo"[T > >"o / / o Ca[[K K _ 1'"o] " o ] = /V Mo\M o = = *o» o> ^ " [ ^ ( l"{u o )0 )+ - 1 - rT2("o)] 2(U0)] 2:
yi Tx{yQ)) ++ 7 T 7i = = 71(^0 22^(x0Q)•).
Also from assumptions,
?i(«)^K("o). ^(/"o^oj^K^^o)T2(/J.Q\) 0 ^(y"o T
2{MOUO)^T2[MOUO),
whence xxl *;
(A(u),vv -u)>(f,v-u), -») for all v e D (A(u),
(4.28)
where A : H -+H (H is a Hilbert space), / e H and D c H is closed and convex (Shi, P. [1]). In (Shi, P. [1]) it is proved that the solvability of variational inequality (4.28) is equivalent to the solvability of the following equation A(PD{x)) + x-PD{x) = f. (4.29) This equation is named in (Shi, P. [1]) the Wiener-Hopf equation. A similar operator as the normal operator it seems, was used by M. Kojima in [1] and by M. Kojima and R. Hirabayashi in [1] in the study of some problems in nonlinear programming. ■ • Complementarity problems and condition (S) + Let ( £ , | |) be a Banach space and K a E a pointed closed convex cone. Given a mapping / : K -> -» E*, we consider the problem: ECP{f,K\
((
[find find x, eJT e K such that
' [f(x.) f(x.)eK'and(x.,f(x.)) e K' and (*.,/(*.)) = = 0.
In this section, we consider the case when E is a reflexive Banach space and f(x) = = 7j(x) - 7^(x), T2(x), with T\ and T2 satisfying special conditions. Such a case seems to be frequently used in many practical problems. We denote by (w) - lim the limit with respect to the weak topology and by (w*) - lim the limit with respect to the weak star topology. Definition 4.8 Let D be a subset ofE. We say that a mapping T : D —>£ satisfies condition (S) [S)++, if for any sequence {x {x„} n } BgAf ngAf c D with (w)- lim xn=xt, (w*)- lim T(xn)= u e £ * and limsup(xn,7T(jc„)) R+ such that O(O) 0(0) = 0 and lim O(r) = +oo. Given Given J:E-> J:E->
/->+oo
aa weight weight O, O, aa duality duality mapping mapping on on E E associated associated to to O O is is aa mapping mapping E E J(x) = - lx* sE* (x,x*\ = = |x|||x*| \\x\\ix*land be* tI = 0(|x||)> 22 '' such a«
A duality mapping is a monotone operator and it is strictly monotone if E is strictly convex. If (E,j [E\ ||) is a reflexive Banach space with IE*,|| \E*\ ||J strictly convex, then a duality mapping associated to a weight function O is a demi-continuous point-to-point mapping (see Cioranescu,I. [1]). If (E,j I) is a Banach space, then a duality mapping on E is a point-to-point [E\
83
CONES AND COMPLEMENTARITY PROBLEMS mapping and norm continuous, if and only if, the norm of E is Frechet differentiable. Definition 4.9 A mapping T:E^>E E' is said to satisfy condition {S)+ if for any any sequence sequence \x \xnn ]] NN^E ^E which which converges converges weakly weakly to to x, x, in in E E and andfor for which which x limsup(x„ - x.,T(xn)) < 0, we have the norm convergence °f{ „}neN to x,. n-*oo
The condition (S)+ is currently used in nonlinear analysis and it was introduced by F. E. Browder [2] - [5]. The importance of condition (5)+ is the fact that one can verify this property, under suitable concrete hypotheses, for some maps of a Sobolev space W"'P(Q) into its conjugate P W^m'p\Q) (where P' p'=^—), of the form = p-V 7>' T{u)= £ ( - il ) H DaTa(x,u,...,Dmu)
"")
a<m
(Browder, F. E. [5]).
Proposition 4.28 If a mapping T ; E -> E' satisfies condition (5)+, then it satisfies (S) . Proof Let {*„} {x„}nn6jv ^N be a weakly convergent sequence to x, in E such that [T(x„)} {T(x„)}nnt^_NN is weakly-star convergent to u e E* and l i m c i i n / -v T% -v i \ -rf* ■/•mi - . \ Yimsup(x R+ such that (x - y,f(x) - /(v)) f(y)) > p(\x pQx - y§), yfj, for all x,y e E. p(0) = 0 and (x Proposition 4.29 Any strongly p-monotone mapping T : E —> ■E'E satisfies condition (S)+. Proof Let {x„}neN be a sequence weakly convergent to x. in E and such that limsup(x„ -x*,T(xn))
< 0. Since we have
fl-»oo
4k _ X* II) - (Xn ~ X'. r (*« ) - ^ ( ^ )) = (Xn ~ X*. r (*« )) - (*« - X* > ?t*« )) _ r X _TX X TX X X 4k * * II) (*» " **' ( « ) i * )) = (*« * > ( n )) ( n ~ *»?t*« )) we obtain fl-»oo
we obtain
0 < liminf/o{|xn - x t | ) < limsup/?(||xn - x,|) 0 < liminf /J(|JC„ - x t |) < limsup/?(||xn - x»|) < lim sup(x„ - x,, r(x„)) - lim(x„ - x., T{x,))
] «- 0(|x.|)][h| - N ||x.|] J(x.)) > [0(K|) - 0(|M)][|k|| | ] > 0, which implies
(4.30)
0(||,.||)][||,„||-|k.||] 0(||,.||)][||,„||-|k.||] (K||)-0(|,.||)][|K||-|h|] n->oo
< lim sup(x„, J(xn)) - lim (x,, j(xn)) - lim (x„ - x,, J(x,))
limO(Jjc1.|) = 0(|jc.| limO(||x 0(|x.| + c) and O(|;c,|| E* satisfies Altman's
condition with respect to TX:K -> E* for r > 0, if for every x e K with |;c| = r we have (x,T2(x)) < (je,7j(;c)y. |x| Remark 4.5 If E is a Hilbert space and Tl(x) = x for every x e E, then we obtain from Definition 4.10 the classical Altman's condition for T2 . This condition is an essential assumption in several known fixed point theorems (Altaian, M. [1], (Shinbrot, M. [1]). In the next result, we use Altman's condition to obtain an existence theorem for the problem ECP(f,K) with respect to a locally compact convex cone. The following generalization of the classical Hartman-Stampacchia theorem will be used in the next result.
87
CONES AND COMPLEMENTARITY PROBLEMS Theorem 4.32 [Hartman-Stampacchia] Let yE,\\ \E,\ |) ||j be a Banach space, E its topological dual and let C be a compact convex set in E. If E' f\C^>E*E' is a continuous mapping, then there exists x* e C such that (x --x,,f(x.)) x,,f(x.)) > 0, for every x e C. Proof This variant of the classical Hartman-Stampacchia theorem is a particular case of Theorem 3.1 presented in (Isac, G. [12]) and proved in (Holmes, R. B. [1]) and (Kinderlerer, D. and G. Stampacchia [1]). ■ Theorem 4.33 Let (E,\\ [E,\\ ||j |j be a Banach space, K azEE a locally compact convex cone and TX,T2:K—» :K —•E' > E'i two mappings. If the following assumptions are satisfied: 1) T\ and T2 are continuous, 2) T2 satisfies Altman's Altman 's condition condition with with respect respect to to T\for T\for some some rr > > 0, 0, then the problem ECP{T ECP(T\T , K) h has a solution x. with |x.|| < r. { 2 M||x»|| < r. Proof Since K is locally compact, the set Kr = jx ix e K\ \X\ |X| 0, 0, that is , (x.,7](x.)- r2(x»)) < 0 or (x.,7;(x,)) (*,2j(x.)) < (x. - T2(xt)), and using
88
EXISTENCE THEOREMS assumption (2) we obtain (x*,Tx(xt)-T22(x(jc.)) = 00 (x.,7;(x,)-r = t))
(4.33) (4.33)
The proof will be finished if we show that Tx(xt)-T2{xt)
e K*. K. Indeed, from (4.32) and (4.33) we have (X,7J(JC,)- T2(xt)) > 0 for all Kr hence all
x ezK,i K, that is, we have 7j(x*) - T2 (x*) e^T'a e K and the theorem is proved. ■ 7j(x t )-^(x.) Corollary 4.34 Let \E,\ |) \\j be a Banach space, K a E a locally compact convex cone. If T:K —> E is a continuous mapping andfor some r> Owe have (x,T(x)j > 0 for all x e K with \x\ = r. then the problem ECP(T,K) has a solution x* with p»| < r. ■ The first main result of this section is the following theorem. Theorem 4.35 [Isac-Gowda ] Let (E,\\ \E,\ I) |) be a reflexive Banach space and let K{K } be a Galerkin cone in E. Suppose given two continuous V nn/neiV mappings TX,T2:K->E* :K —> » E . If the following assumptions are satisfied: 1) T\ is bounding (i. e. maps bounded subsets into bounded subsets) and satisfies condition (S) with respect to K, 2) T2 is a (ws)-compact operator, 3) T2 satisfies Altman's Altman 's condition with respect to T\ for some r > 0 with respect to K, \\x4 such that |x*| < r. Proof Let xn be a solution of the complementarity problem ECP{TX - T2, Kn) obtained by Theorem 4.33, with \\x„\\ < r. We have ft x-- Tr22)(x \xJxn)) (T )(x„) < and and (x(xa,{T 0. n) eEKl 22 a,(Txx-- TT n)) ==0.
(4.34)
89
CONES AND COMPLEMENTARITY PROBLEMS Since {xn}neN and (7l(x„)l J7j(jc„)|l n e # are bounded, there exists a subsequence ne/i I xs(n) I
n*N
($ '■ N—> N, strictly monotone) such that KT
^-J-^-** t t n ) - ^ *
md T ^
u i( «n))-^» W'))~^~*
(4(4.35) 35)
-
T x
for some x. eG K A and u e E*. £*. By the (ws)-compactness of T2 there exists a further subsequence of j x^ x^B\ J
nzN
I ^2\xMn))}
n
, for which
'*ss n o r m convergent to some v e E*
(4.36)
Then x, eeJf l f and (7J - T2 \x^ Moreover,we wecan canshow showthat that )(^^(n))] —' '' > aw-- v.v. Moreover, K'. In fact, let x e JA^ f ^B ) and m > n. Then /x,(7J »u-- v e A*. /x,(?; - T2ix^m))\ > 0 holds since (7J - ^^)(^( ^ mj ))e e ^ m) ) 0, W>2 (x - S(u0),T(x,)) K?. ,7t*.)) > 0, for all x e K?.
(4.43) (4.43)
If x e K is an arbitrary element, then there is a sufficiently small A. e ]0,1[ such that v = Ax + (l - X)S(un) e K?. Now , if we put x = v in (4.43) we
98
EXISTENCE THEOREMS obtain (x - S(u0),T(xt)) > 0, 0, for all x e K.
(4.44) (4.44)
Since H^wo)!! < r we can put x = S(u0) in (4.40) and we deduce (s(u0Q)-S(x )-S(x,),T(x.))>0 t),T(xt))>0 >>0
(4.45)
From (4.44) and (4.45) we obtain (JC S(x,), T(x,)) T(x,)) >> 0, 0, for for all all xx ee K. K. (x -- S(x,),
(4.46)
Since S(x-) e K, from (4.46) and Proposition 4.40 we obtain that x* is a solution of the problem ICP(T, S, K) and the proof is finished. ■ We extend now Theorem 4.43 to Galerkin cones. Definition 4.12 Let K(Kn) be a Galerkin cone in E. We say that S :K -► ^ EE is subordinate to the approximation (K„)n£N of the cone K if there exists n0 e JV such thatfor every n>n0we have S(Kn) c Kn. Some examples of subordinate mappings are given in (Isac, G. and D. Goeleven [1]) Definition 4.13 We say that S : K -> E is r-subordinate to the approximaDefinition tion (K„)neN if there exist r > 0 and «0 sN such that for every n >n0we have s(K^.)czKn,
where K% = {x sK„ s K„ |\\x\\ | \\x\\£is subordinate to the approximation1 (tf„) neJV °f (*.U > r-subordinate for any r > 0.
99
CONES AND COMPLEMENTARITY PROBLEMS Remark 4.6 If S : K -» -> E is continuous and r-subordinate to the approxi mation {Kn)neN,
then s(Kf)^K. SIK^CK.
Indeed, if x * e Kf, then we have two
cases: a) ||*|| 11*11 > nnuu which which n > max\n ,n }, we have 0 x maxfo,^),
implies implies ||x*nn||oo limx*n. We have that xt e K and IbtJ < r , since K? is closed. Hence S(xn->oo t) e K. Let x e K be an arbitrary element. For every closed. Hence S(x e Jf. Let x e A" be an arbitrary element. For every n > max{«0,/n}, wet)have n > max{«0,/n}, we have (Pn{x)-S(xn),Tix:))>0, >0, (4.47) (Pn{x)-S(x:),T{x:))>0, (4.47) where {P„}nc.N is a sequence of projection (P„ is a projection onto Kn). Since 5 and T are strongly continuous, computing the limit in (4.47), we obtain (x - S(x,), T(x,)) > 0, for all x <E z K.K. (4.48) The proof is finished since from (4.48) by Proposition 4.40 we have that x* xt is a solution of the problem ICP(T, S, K). ■ We consider now the particular case when S(K) c K. Theorem 4.45 Let \E,\ ||j ||) be a Banach space, K a E a pointed locally compact convex cone and S :K^> K; T : K -> E continuous mappings. If the following assumptions are satisfied: 1) {S{x), T(x)) < (x, T(x)), for all x e K, 2) there exists r > 0 such that for every x e K with r < \\x\\ there exists an 0,0, elementvxx €G Ksuch that ||vj| \\vx\\ || x*xt ||\\< r, then there is a natural number n such that r < & < * * < « .
For this x*n, by
assumption (2) there is an element v . e K such that v . < r and x
II xn II
n
(S(x'nn)-)-v (s(x Vx.,r{x n))>0. x:,r{x:))>o.
(4.50)
But, since v^. < r < |JC*| < n, from (4.49) we have (s(x*\ - v^., jix*)\ < 0, which is a contradiction of (4.50). Hence, lx'\
:}."IneN
\
K is locally compact the sequence |x*J \
subsequence \x* \ keN
"IneN
is bounded and because
has a norm convergent
x'„ .\ now show that x. is a solution Let x, x. == lim x' n .We k^yoo "'
of the problem ICP{T, S, K). Indeed, if v € K is an arbitrary element, then there ism e JV such that for every n > m we have, v e Dn and for every nk>m,v £ £>„t and ^(x* t ) - v,l(x^)^ < 0. Using the continuity of 5and Twe obtain (Sfc,)- v.T^x,)) < 0 for all v e K, that is x. is a solution of the problem S-VI(T,S,K), which by Proposition 4.40, is equivalent to the
102
EXISTENCE THEOREMS problem ICP(T, S, K). Obviously, by assumption (2) we must have |JC»| < r || and the theorem is proved. ■ Corollary 4.46 Let (E,\ |) be a Banach space, K a E a pointed locally compact cone and S : K -> K ; T : K -» E continuous mappings. If the following assumptions are satisfied: 1) (S{x), T{xj) < (x, T(x)}, for all x e K, 2) there is a number r > 0 such that for every x e K with r < ||JC|[ we have
(S(x),T{x))>0, then the problem ICP(T, S, K) has a solution x* such that \\ x* || < r. Proof We apply Theorem 4.45 with vx = 0 for every x e K satisfying \x\>r. U Corollary 4.47 Let (E,\\ ||j be a Banach space, KaE
a pointed locally
compact cone and S : K -» K, T : K —> E continuos mappings. If the following assumptions are satisfied: 1) (S(x), T(x)) < (x, T(x)), for all x e K, 2) there exist a number ro > 0 and u0 e K such that for every x e K with r0 < \\x\\ we have (S(x) - u0,T(x)/ > 0, then, the problem ICP{T,S,K) has a solution x* such that ||JC*|| < 1 + max{r0,|w0|}. Proof If we denote r = max(r0,|w0|| + 1 , we have r > r0 and r > ||u0||. Now, we can apply Theorem 4.45 since assumption 2) of this theorem is satisfied with Vf = UQ for every x e K with ||x|| > r. ■ Remark 4.7 Condition 2) of Corollary 4.47 is satisfied if T is semicoercive with respect to S in the following sense:
103
CONES AND COMPLEMENTARITY PROBLEMS / (S(x)-u0,T(x)) ) N W (3«o eK) ti. i lim urn -t-ou . . (3u0*» \H-*»
IH| |H|
JJ
Theorem 4.48 Le? (i?,|| |) be a reflexive Banach space and K(Kn)n&N a Galerkin cone in E. Let S : K -> K and T: K -> E be strongly continuous mappings. If the following assumptions are satisfied: 1) S is subordinate to the approximation (Kn))n 0 such that for every n>n0 and every x e K„ with r < ||;c|| there is an element vx ei5Tn such that \\vx\\ < r and (S(x)-vx,T{x))>0, )>0, then the problem ICP(T, S, K) has a solution x» such that \\x, || < r. Proof Since, for every n > «o we have S(Kn) c Kn and all the assumptions of Theorem 4.45 are satisfied, we have that, for every n > n0, the problem ICP(T, S, K„) has a solution x*n. Because of the fact for every n > no, we have pc* V' ' ){A(T {x ),T {x,),...,T (x,)) l t 2 m \' J[A(Tl{xt),T2{xt),...,Tm(xt)) = = 0. 0. In many practical problems, we have D = K. In this case, we denote our problem by IGOCPUT^ ,K\.
(w:>4
Given the problem IGOCPUT^^K.D),, * , £ > ) , we define the operators: H[x) - V H[x) v^x-T ( X - 7l(x),x-T J ( X ) , X2-(x),...,x-T 2 ^ ( J C ) , . .m.(x)y, , X - 7 ^ ( X ) ) ; for for all all x e E, E, G(x) = A(X + TX(X),X + TT22(X),...,X (x),...,x + +T Tmm(x)); (x)); for for all all JJCC Ge E. E. Proposition 4.49 IGOCPUTA^KfDh
The element xt e D is a solution of the problem if and only if, x, is a fixed point ofH or, if and only
if xt is a fixed point ofG.
105
CONES AND COMPLEMENTARITY PROBLEMS Proof The proof is in terms of an elementary calculus based on the proper ties of the latticial operators "A" and "v". ■ From Proposition 4.49, we conclude that it is important to study the operators H and G and to establish some useful properties with respect to the ordering. In this sense we consider the following general case. Let F\, F2l...,Fmbem operators from E into E. We denote F {x),...,F allxx e E, F.{x) (x)) for all A{x) = A(Fl]{x),F22{x),... >Fm m{x)) Fv(x) {x) = v(Fx(x),F2{x),...,Fm{x))
for allxe£.
Definition 4.14 We say that T : E —» E is a heterotonic operator on a set DcE, if an only if, there exists an operator T:E x E —> E such that: i) f(x,x) = T(x) ,for allx e D, ii) t(x,y) is monotone increasing on D with respect to xfor anyy, Hi) f[x,y) is monotone decreasing on D with respect to yfor any x. The concept of heterotonic operator was introduced and studied by V. I. Opoitsev [1]. When we say that T is heterotonic we suppose that f is selected. Remarks 4.8 1) A monotone increasing (resp. decreasing) operator Tis heterotonic. Indeed, if T is monotone increasing (resp. decreasing) we take f{x,y) = T\x) + x-y (resp. f{x,y)=T(y) +x-y ). 2) If T is heterotonic the choice of T is not unique. 3) The sum and the composition of two heterotonic operators is a heterotonic operator. Proposition 4.50 If Ft = /?,. + St;i = 1,2,..,m where i?, is increasing andSt is decreasing, then FA andFy are heterotonic operators. Proof If we take 106 106
EXISTENCE THEOREMS t{x,y) = A(Rl{x) + S,{y),...,Rm{x) + Sm{y)); and Fv(x,y) = V(_RJ(JC) + S1(y),...,Rn(x) + Sn(y)), for all x,y e E, we remark that Definition 4.14 is satisfied for FA, respectively for F v . ■ A more general result, is the following. Proposition 4.51 IfFifor every i = 1,2,..m is heterotonic, then FA and F v are heterotonic, also. Proof In this case we take K{*>y) = ^(Fi{x,y),...,Fm(x,y)); and Fv(x,y) = vJF^x,y),...,F m (x,yj}, for all x,y e E. ■ Suppose now that E is a Banach space. Let || || be the norm defined on E. Definition 4.15 Let a - o~(t) be an increasing real-valued continuous at 0 function from R+ into R+ such that a(0) = 0. Let DaE be a non-empty set. We say that T: D—> E is a-Holder continuous if \T(X) - T(y)j < cr(||x - yfj, for all x,ye.D. We say that Tis of Holder type if there is an increasing continuous function a: R+ -> R+ withCT(0)= 0 and such that T is a-H6lder continuous. Suppose that (E,\ ||,jn is an ordered Banach space such that with respect to the ordering defined by K, the space E is a vector lattice.
'(*.! I.')
Definition 4.16 -/JM£,| | , ^ j is an ordered Banach space which is a vector lattice we say that the norm \\\\is a Riesz norm if: i) I |JC| I = ||jc||,_/br allx e E (the norm is absolute),
107
CONES AND COMPLEMENTARITY PROBLEMS ii) 0 < x < y implies \x\ < \\y\\, for all x, y e E (the norm is increasing). Remark 4.9 It is easy to show that a norm || || on E is Riesz if and only if |JC| < \y\ implies jjcf < \y\, for all x, y e E. Proposition 4.52 If (E,\\ \\,K) is an ordered Banach space which is a vector lattice and the norm \\ \\ is Riesz, then for every set ofm operators of Holder type Fi,F2,...,Fm:E -> E we have that FA andFv are operators of Holder type too. Proof It is sufficient to prove the theorem for m = 2. Suppose that F\ (resp. F2) is cr, (resp. ar2)-H6lder continuous. We have F,(*) VF2(x) = F,(x) Fl(x) + [F2(x) - F,(x)] F,(*)]+ and Fi{y) VF2(y) = F,(y) Ffo) Ft(y) + [F2{y) (y) - F,(v)] F(y)] + which implies Hx)vF2{x)-F,{y)vF2{y)\ v^W| = Fl(x) +
•fcM-[F (x)-F (x)] -F (y)-[F (y)-F,(y)] 2
l
+
l
2
+
++ + = [m -m] +fcM--FH F W] -Fl(y)} (y)Y - m]+teM ~-[F te2W(y)- *i
Ewe say that (x*,y*) is a coupledfixed point ofT if f(x* ,y,) = x, and f(yt, x*) = yt. This concept was introduced by V. Lakshmikantham and studied by D. Guo and V. Lakshmikantham [1] and also by Y. Z. Chen [1]. Every fixed point is a coupled fixed point. The set of coupled fixed points localizes the set of fixed points. Definition 4.18 We say that a coupled fixed points (x;y*) of a heterotonic operator T is minimal and maximal on D if for every coupled fixed point {x,y) e D ofTwe have x,<x\ and x,y e£>, 2') f(x,fiy)
> fTaf{x,y);
for all // > 1 andx,y x,yeD.
When (E(r),K) (E(T), k) is an ordered locally convex space with a topology x de-
111
CONES AND COMPLEMENTARITY PROBLEMS fined by a family of seminorms {pa}a^, and K is normal, we can suppose «£/#' that every seminorm/?ahas the following property: 0<xpa(x)<pa(y), 00, for every x, y e K. We suppose that the family of seminorms {Pa} {pa}aSjt *s sufficient (Marinescu, aejt\A Gh. [1]). Theorem 4.55 Let
w.[p
\E(T),
a
}
J be a locally convex space ordered by a
regular normal, pointed closed convex cone K. Suppose, given m heterotonoic operators Tx,T2,...,Tm:E -> E and consider the problem IGOCPi{T^^Kj. :,K).
Denote by T the heterotonic operator H or G associa-
ted ted with with this this problem. problem. Suppose Suppose that that T T is is a-(concave, a-(concave, convex) convex) and and T t is is continuous. If there exist Mo > 1 and UQ > 0 such that computing x0 = fax U0 , y0 = fa H0, Xj = T(xQ,y0) and v, = T(y0,x0) we have that y\^y x0<xloo
\x T(x n == x,yn_x) for [** T{nx_n-\^n-\) M allall neN, neN, [yn = T{yn-\>xn-\) far for all and, for any a e/t, we have, Pa{x*-x )<M0 Pa{x*-xnn)<Mo
ff
\
1 v V
neN,
1 \ ^ r Pa{«o) for all n eN
Mo
)
Proof Consider the sequences {*„}n(EA, and i-^n/neAf {yn}n€N defined above. We •neN have xn-\ -x <x ^xnn 0, when T(xn) -xn 0 for all i = \,2,...,m that is xn is feasible. We recall that the feasible set of the problem IGOCPUT,}",K,D\K,D\ is the set F = [x BD\ T,(X) eK,for
alii = L2,...,m}.
From Theorem 4.56, we obtain the following result. Corollary 4.57 If all the assumptions of Theorem 4.56 are satisfied, ifK is regular and for every n e N, x„ is feasible, then {xnfneN is convergent to a n} solution of the problem IGOCPl IGOCPUT^^K). We finish this section with the remark that other results about the problem IGOCP({i;}™,K,D) are proved in the papers (Isac, G. and D. Goeleven [1], [2]) and (Isac, G. [4]). • Topological degree and complementarity The topological degree considered as one of the most important mathematical instrument in nonlinear analysis can be used in Complementarity Theory. With the topological degree we can study both the Linear Complementarity Problem and the Nonlinear Complementarity Problem. In relation to the Linear Complementarity Problem, the topological degree has been effectively used to study the existence of solutions and the cardinality of the solution-set (Kojima, M. and R. Saigal [1]), (Howe, R. and R. Stone [1]), (Howe, R. [1]) and (Garcia, C. D. [1]) and also to study the stability of solutions (Ha, C. D. [1]). The Nonlinear Complementarity Problem was studied by the topological degree in papers (Goeleven, D., V. H. Nguyen and M. Thera [1]), (Pang, J. S. and J. C. Yao [1]) and (Isac, G. , V. Bulavski and V. Kalashnikov [1]). For the discussion in this section we make use of some basic results from degree theory. Those readers who are not familiar with this theory can consult the books (Lloyd, N. G. [1]) and (Rothe, E. H. [1]).
117
CONES AND COMPLEMENTARITY PROBLEMS We now recall some properties of topological degree. Let D be a bounded open subset of R" andy a point of R" . The closure of D is written D and its boundary 3D . We denote by &{p) the linear space of continuous functions from D into R" . If F E