Embed Size (px)
Citation preview
Applied Mathematics and Computation 217 (2010) 2184–2190
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate /amc
Exceptional family of elements for general ordercomplementarity problems
S.Z. NémethThe University of Birmingham, School of Mathematics, The Watson Building, Edgbaston, B15 2TT Birmingham, United Kingdom
a r t i c l e i n f o
Keywords:General order complementarity problemOrdered Banach spaceVector latticeLeray–Schauder alternativeExceptional family of elementsIntegral operator
0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.07.018
E-mail address: [email protected]
a b s t r a c t
The notion of exceptional family of elements for general order complementarity problemsin Banach spaces will be introduced. It will be shown that for general order complementar-ity problems defined by completely continuous fields the problem has either a solution oran exceptional family of elements. Finite dimensional examples and an application to inte-gral operators will be given.
� 2010 Elsevier Inc. All rights reserved.
1. Introduction
Several problems in optimization, economics, mechanics elasticity and engineering can be modelled by complementarityproblems [5,8,10,14].
The notion of exceptional family of elements (EFE) is a topological method introduced via the topological degree in [16]for completely continuous fields (see also [4]). By using the Leray–Schauder type alternatives, this notion was extended toother classes of mappings and was successfully applied to prove existence theorems for complementarity problems[13,18,21,15]. In [17] a nice practical application of EFE to oligopoly models was presented.
Since 1997 a large number of papers have been published in this topic. Several notions of EFE for other types of comple-mentarity problems have been also developed. A quite exhausting list of these papers can be found among the references of[15,28]. The notion of EFE was extended to variational inequalities in Euclidean [34], Hilbert [29] and Banach spaces [1,2].
The nonexistence of EFE is a very general coercivity notion which implies the existence of a solution for the correspondingcomplementarity problem (or variational inequality problem). Therefore, it is very important to find conditions for the non-existence of an EFE. In the above cited papers there are several sufficient conditions for the nonexistence of an EFE. Necessaryand sufficient conditions for the nonexistence of EFE were first given in [26,25]. In [27] the notion of exceptional family ofelements was related to the isotonicity of the metric projection for the first time.
A particular class of complementarity problems deals with order complementarity. There are several types of ordercomplementarity problems. The linear order complementarity problem was systematically studied in [3]. The latterproblem was extended to the general linear order complementarity problem and several results were presented in[6,19,32]. Another extension of the linear order complementarity problem is the nonlinear order complementarity prob-lem studied in [9]. The notion of the general order complementarity problem considered in this paper is taken from[19,15].
The order complementarity problems can be applied in lubrication theory [31] and economics [14], and is related to thefold complementarity problem [12,24]. The general order complementarity problem when one of the mappings is the iden-tity mapping was systematically studied in [22], and for set valued mappings in [23,7]. Numerical methods for the ordercomplementarity problems were obtained in [11,20].
. All rights reserved.
S.Z. Németh / Applied Mathematics and Computation 217 (2010) 2184–2190 2185
The order complementarity is only briefly discussed in [15], because no notion of EFE has been developed which is nat-urally related to these type of problems. The only attempt to relate EFE to order complementarity can be found in [16]. How-ever, the corresponding results are rather particular. Therefore, Isac in [15] rightly claims that the problem of finding a notionof EFE for order complementarity problems is open.
In this paper we shall give a notion of EFE well-fitted for general order complementarity problems (OCPs) called orderexceptional family of elements (OEFE) and provide a rather general condition for the nonexistence of an OEFE which implythat the corresponding OCP has a solution. This is the first notion of EFE which is naturally related to an order complemen-tarity problem. In finite dimension several rather general examples are given which emphasize the strength of our results.Moreover, a rather general result is obtained for integral operators in L2 spaces too.
We note that in finite dimension when there is only one mapping defining the general order complementarity, the prob-lem reduces to the existence of a solution for a nonlinear system of equations.
Consider a general order complementarity problem on a Hilbert space which has a vector lattice structure. Suppose thatthe positive cone of the vector lattice is self-dual (e.g., Rn
þ, or more generally the positive cone of a Hilbert lattice). If the prob-lem is defined by two mappings and the positive cone of the vector lattice, then the problem reduces to an implicit comple-mentarity problem. If one of these mappings is the identity mapping then the problem further reduces to a nonlinearcomplementarity problem.
The structure of the paper is as follows. First we present some preliminary notions. Next we state the Leray–Schauderalternative. We continue by defining the general order complementarity problem. Then, we present our main resultswhich are based on the Leray–Schauder alternative. At the end we draw some conclusions and present some openquestions.
2. Preliminaries
Let (E,k�k) be a Banach space and K � E a closed set. K is called a wedge, if for any k P 0 and x, y 2 K, kx 2 K and x + y 2 K. Awedge K is called a cone if K \ (�K) = {0}.
A relation � on E is called reflexive if x � x for all x 2 E. A relation� on E is called transitive if x � y and y � z implies x � z. Arelation � is called antisymmetric if x � y and y � x imply x = y. A relation � on E is called an order if it is reflexive, transitiveand antisymmetric. A relation � on E is called translation invariant if x � y implies x + z � y + z for any z 2 E. A relation � on Eis called scale invariant if x � y implies kx � ky for any k > 0. A relation � on E is called continuous if for any two convergentsequences fxngn2N and fyngn2N in E with xn � yn, for all n 2 N we have x* � y*, where x* and y* are the limits of fxngn2N andfyngn2N, respectively.
The relation6 on E is a continuous, translation and scale invariant order if and only if it is induced by a cone K � E; that is,x 6 y if and only if y � x 2 K. The cone K can be written as K = {x 2 E: 0 6 x} and it is called the positive cone of the order 6.The triplet (E,k�k,K) is called an ordered Banach space.
The ordered Banach space (E,k�k,K) is called a vector lattice if for every x, y 2 E there exists x ^ y :¼ inf{x,y} with respect tothe order induced by K. In this case we say that the cone K is latticial and for each x 2 E we denote x+ = �0 ^ (�x). By standardarguments, it can be shown that
ðxþ zÞ ^ ðyþ zÞ ¼ x ^ yþ z ð1Þ
and
x ^ y ¼ y� ðy� xÞþ; ð2Þ
for all x,y,z 2 E.A continuous mapping f: E ? E is called completely continuous mapping if for every bounded set M � E the set f(M) is rel-
atively compact. We denote by I the identity mapping of E, defined by I(x) = x. The mapping f is called completely continuousfield if I � f is a completely continuous mapping. If f, g: E ? E are completely continuous mappings and h: E ? E is a contin-uous mapping, then h � f and g � f are completely continuous mappings.
Recall the following Leray–Schauder alternative [30] (see also [15]).
Theorem 2.1. (Leray–Schauder alternative) Let (E,k�k) be a Banach space, C � E a closed convex set, and U � C a bounded andopen set (with respect to the topology of C) such that 0 2 U. If f : U ! C is a completely continuous mapping, then either f has afixed point in U, or there exist an element x* 2 @CU and a real number k*2]0,1[ such that x* = k*f(x*).
Recall the following definition of general order complementarity problems (see [19,15]).
Definition 2.1. Let (E,k�k) be a Banach space ordered by the latticial cone K � E and D � E a nonempty closed convex set.Consider m mappings f1, . . . , fm: E ? E. The General Order Complementarity Problem defined by the family of mappings ffigm
i¼1and the set D is
OCP ffigmi¼1;D
� �:
find x� 2 D such thatf1ðx�Þ ^ � � � ^ fmðx�Þ ¼ 0:
�
2186 S.Z. Németh / Applied Mathematics and Computation 217 (2010) 2184–2190
3. Main results
The following definition is the first notion of exceptional family of elements well-fitted to an order complementarityproblem.
Definition 3.1. Let (E,k�k) be a Banach space ordered by the latticial cone K � E and D � E a nonempty closed convex set.Consider m mappings f1, . . . , fm: E ? E. We say that a family of elements {xr}r>0 � D is an order exceptional family of elements forthe family of mappings ffigm
i¼1 with respect to D (OEFE) if the following conditions are satisfied:
(1) kxrk? +1 as r ? +1,(2) For every real number r > 0, there exists a real number lr > 0 such that u1
r ^ � � � ^ umr ¼ 0, with ui
r ¼ lrxr þ fiðxrÞ, fori = 1, . . . ,m.
Then, we have as follows:
Theorem 3.1. Let (E,k�k) be a Banach space ordered by the latticial cone K � E and D � E a nonempty closed convex set. Considerm completely continuous fields f1, . . . , fm: E ? E such that Sm(D) + K � D, where Sm = I � fm. If the family of mappings ffigm
i¼1 iswithout OEFE with respect to D, then the problem OCPðffigm
i¼1; DÞ has a solution.
Proof. Consider the mapping U: E ? E defined by
UðxÞ ¼ x� f1ðxÞ ^ � � � ^ fmðxÞ:
Then, OCPðffigmi¼1; DÞ has a solution if and only if U has a fixed point in D. Let
UkðxÞ :¼ x� f1ðxÞ ^ � � � ^ fkðxÞ;
for all x 2 E and k = 1, . . . ,m. We shall prove by induction that Uk is completely continuous mappings for all k = 1, . . . ,m. De-note Sj = I � fj, j = 1, . . . ,m. By assumption Sj are completely continuous mappings. Since U1 = S1, it is a completely continuousmapping. Suppose that Ui is a completely continuous mapping where i 2 {1, . . . ,m � 1}. By (1) and (2) we have
Uiþ1ðxÞ ¼ x� ðx�UiðxÞÞ ^ fiþ1ðxÞ ¼ Siþ1ðxÞ þ ðUiðxÞ � Siþ1ðxÞÞþ: ð3Þ
Hence, Uk are completely continuous mappings, for all k = 1, . . . ,m. It follows that U = Um is a completely continuous map-ping. Moreover, by using (3) it follows that U(D) � D. If the mapping U has a fixed point in D, then the problemOCPðffigm
i¼1; DÞ has a solution and the proof is complete.Let us suppose that U has no fixed points in D. Let r > 0 be arbitrary. Denote Dr = {x 2 D: kxk < r}. Then, U has no fixed
points in Dr ¼ fx 2 D : kxk 6 rg. The Leray–Schauder alternative for the sets C = D, U = Dr and the mapping f ¼ UjU : U ! Cimplies the existence of xr 2 @Dr = {x 2 D: kxk = r} and kr 2]0,1[ such that
xr ¼ krUðxrÞ ¼ krðxr � f1ðxrÞ ^ � � � ^ fmðxrÞÞ:
We have kxrk = r ? +1 as r ? +1. Denote lr ¼ k�1r � 1 > 0 and uir ¼ lrxr þ fiðxrÞ, i = 1, . . . ,m. Then,
lrxr ¼ �f1ðxrÞ ^ � � � ^ fmðxrÞ:
Hence, by using (2) successively, we haveu1r ^ � � � ^ um
r ¼ ðlrxr þ f1ðxrÞÞ ^ � � � ^ ðlrxr þ fmðxrÞÞ ¼ lrxr þ f1ðxrÞ ^ . . . ^ fmðxrÞ ¼ 0:
Therefore, {xr}r>0 is an OEFE for the family of mappings ffigmi¼1 with respect to D. h
Since the nonexistence of OEFE implies the existence of a solution for the corresponding general order complementarityproblem, it is important to find conditions which imply that a family of mappings is without OEFE. Theorem 3.2 gives arather general condition for the nonexistence of OEFE, based on the following definition.
Definition 3.2. Let (E,k�k) be a Banach space ordered by the latticial cone K � E, D � E a wedge and f1, . . . , fm: E ? E mappings.We say that the family of mappings ffigm
i¼1 satisfy the K condition with respect to D if there is a q > 0 such thatf1(x) ^ � � � ^ fm(x) R �Dn{0}, for all x 2 D with kxk > q.
Theorem 3.2. Let (E,k�k) be a Banach space ordered by the latticial cone K � E and D � E a wedge and f1, . . . , fm: E ? E mappings.If the family of mappings ffigm
i¼1 satisfy the K condition with respect to D, then it is without OEFE with respect to D.
Proof. Suppose that {xr}r>0 is an OEFE for the family of mappings ffigmi¼1 with respect to D. We shall use the notations from
the definition of an OEFE. There is a sufficiently large r > 0 such that kxrk > q. Thus, we have u1r ^ � � � ^ um
r ¼ 0. Hence, as in theproof of Theorem 3.1, we have f1(xr) ^ � � � ^ fm(xr) = �lrxr 2 �Dn{0}, which is a contradiction. Hence, the family of mappingsffigm
i¼1 is without OEFE with respect to D. h
S.Z. Németh / Applied Mathematics and Computation 217 (2010) 2184–2190 2187
Theorems 3.1 and 3.2 imply as follows.
Corollary 3.1. Let (E,k�k) be a Banach space ordered by the latticial cone K � E, D � E a wedge and f1, . . . , fm: E ? E completelycontinuous fields such that Sm(D) + K � D, where Sm = I � fm. If the family of mappings ffigm
i¼1 satisfy the K condition with respect toD, then the problem OCP ffigm
i¼1; D� �
has a solution.
4. Finite dimensional examples
Example 4.1 is given for the sake of completeness only. It shows that the descriptive conditions of Example 4.2 for themappings fi, i = 1, . . . ,m � 1 can be satisfied for a very large class of mappings.
Example 4.1. Let E ¼ ðRn; k � kÞ, where k�k is an arbitrary norm in Rn; K ¼ D ¼ Rnþ; aij > 0; bij > 0; cij P 0 real constants;
uij : Rn ! R;wij : Rn ! R;qj : Rn !� �1;1� arbitrary continuous functions with
limkxk!þ1
uijðxÞP aij;
limkxk!þ1
wijðxÞP bij;
fiðxÞ ¼ ui1ðxÞkxkwi1ðxÞ � ci1; . . . ;uinðxÞkxk
winðxÞ � cin
� �;
i = 1, . . . ,m � 1, j = 1, . . . ,n; and
fmðxÞ ¼ ðq1ðxÞjx1j; . . . ;qnðxÞjxnjÞ;
for all x 2 Rn. If kxk is large enough the components of fk(x), k = 1, . . . ,m are nonnegative. Hence, the components off1(x) ^ � � � ^ fm(x) are nonnegative too. Therefore, the family of mappings ffigm
i¼1 satisfy the K condition with respect to D.It is a technicality to check that Sm(D) + K � D, where Sm = I � fm. The details are left to the reader. Thus, all conditions of Cor-ollary 3.1 are satisfied. Therefore, the problem OCP ffigm
i¼1; Rnþ
� �has a solution.
Example 4.2. Let E ¼ ðRn; k � kÞ, where k�k is an arbitrary norm in Rn and K ¼ D ¼ Rnþ. Suppose that the components of each fi,
i = 1, . . . ,m � 1 are arbitrary continuous functions from Rn to R whose graphs in Rnþ1 are in the half-space xn+1 P 0, for allx 2 Rn with sufficiently large norm and let fm be of the form given in the Example 4.1. Similarly to Example 4.1, it can bechecked that the conditions of Corollary 3.1 are satisfied. Therefore, the problem OCPðffigm
i¼1; RnþÞ has a solution.
5. Application to integral operators
Let X � R be a bounded open set, L2(X) the set of functions on X whose square is integrable on X, and
L2þðXÞ ¼ fu 2 L2ðXÞjuðtÞP 0 for almost all t 2 Xg:
L2(X) is a Hilbert space with respect to the scalar product
hu;vi ¼Z
XuðsÞvðsÞds
and L2þðXÞ is a latticial cone of L2(X). If u, v 2 L2(X), then
ðu ^ vÞðtÞ ¼minðuðtÞ; vðtÞÞ;ðu _ vÞðtÞ ¼maxðuðtÞ;vðtÞÞ
and
uþðtÞ ¼maxðuðtÞ;0Þ;
for almost all t 2X. Let L : XX R! R, K : XX! R, and Fj : X R! R. We recall the following definition [33] andresult (see Theorems 1.5, 1.12 and 1.13 of [33]).
Definition 5.1. We say that L is a Caratheodory function if Lðs; t; uÞ is continuous with respect to u for almost allðs; tÞ 2 XX and is measurable in (s, t) for each u 2 R.
Theorem 5.1. If the following conditions are satisfied,
(1) L is a Caratheodory function;(2) jLðs; t; uÞj 6 Rðs; tÞðaþ bjujÞ for almost all s, t 2 X, 8u 2 R, where a, b > 0 and R 2 L2ðXXÞ;
2188 S.Z. Németh / Applied Mathematics and Computation 217 (2010) 2184–2190
(3) For any a > 0 the function Raðs; tÞ ¼maxjuj6ajLðs; t;uÞj is summable with respect to t for almost all s 2X;(4) For any a > 0,
limmesðDÞ!0
supu 2 L2ðXÞjuðtÞj 6 a;8t 2 X
PD
ZXLðs; t; uðtÞÞdt
��������
L2ðXÞ¼ 0;
where mes (D) is the Lebesgue measure of D and PD is the operator of multiplication by the characteristic function of the set D �X;(5) For some b > 0,
limmesðDÞ!0
supkuk
L2ðXÞ6b
ZDLðs; t;uðtÞÞdt
��������
L2ðXÞ¼ 0;
then the operator A : L2ðXÞ ! L2ðXÞ defined by
AðuÞðsÞ ¼Z
XLðs; t;uðtÞÞdt;
is completely continuous.The integral of an almost everywhere nonnegative function is nonnegative, therefore by Theorem 5.1 we have the
following:
Corollary 5.1. If conditions 1–5 of Theorem 5.1 and condition
6. Lðs; t;uÞP 0 for all u P 0, for all s 2X, and for almost all t 2X
are satisfied, then the operator A : L2ðXÞ ! L2ðXÞ defined by
AðuÞðsÞ ¼Z
XLðs; t;uðtÞÞdt;
is completely continuous and A L2þðXÞ
� �� L2
þðXÞ.Consider m operators Ai : L2ðXÞ ! L2ðXÞ, i = 1, . . . ,m defined by
AiðuÞðsÞ ¼Z
XLiðs; t; uðtÞÞdt;
where Li : XX R! R are measurable in (s, t) for each u 2 R. Then, the general order complementarity problem definedby the family of mappings fAigm
i¼1 and the set L2þðXÞ is
OCP fAigmi¼1; L
2þðXÞ
� �:
find u� 2 L2þðXÞ such that
minm
i¼1
RX Liðs; t;u�ðtÞÞdt ¼ 0;
for almost all s 2 X:
8>>><>>>:
ð4Þ
Theorem 5.2. Let Li : XX R! R be m functions measurable in (s, t) for each u 2 R.
Suppose that conditions 1–5 of Theorem 5.1 with all
1mesðXÞu� Liðs; t;uÞ;
i = 1, . . . ,m in place of Lðs; t;uÞ are satisfied; condition 6 of Corollary 5.1 for
1mesðXÞu� Lmðs; t;uÞ;
in place of Lðs; t;uÞ is satisfied; and Liðs; t;uÞP 0, i = 1, . . . ,m, for some C # X with mes (C) > 0, for all s 2C, for almost all t 2 Xand for all u P 0. If we denote
AiðuÞðsÞ ¼Z
XLiðs; t; uðtÞÞdt;
then the problem OCP fAigmi¼1; L2
þðXÞ� �
has a solution.
Proof. Let E = L2(X) and K ¼ D ¼ L2þðXÞ. Since condition 6 of Corollary 5.1 for
S.Z. Németh / Applied Mathematics and Computation 217 (2010) 2184–2190 2189
1mesðXÞu� Lmðs; t;uÞ;
in place of Lðs; t;uÞ is satisfied, we have ðI �AmÞðKÞ# K. Since conditions 1–5 of Theorem 5.1 with all
1mesðXÞu� Liðs; t;uÞ;
i = 1, . . . ,m in place of Lðs; t;uÞ are satisfied, it follows that Ai, i = 1, . . . ,m are completely continuous fields. SinceLiðs; t;uÞP 0, i = 1, . . . ,m, for some C # X with mes (C) > 0, for all s 2C, for almost all t 2X and for all u P 0, it follows that
AiðuÞðsÞ ¼Z
XLiðs; t;uðtÞÞdt P 0;
for all s 2 C and all u 2 L2þðXÞ. Thus,
minðA1ðuÞðsÞ; . . . ;AmðuÞðsÞÞP 0;
for all s 2 C. Hence, mes (C) > 0 yields A1ðuÞ ^ � � � ^ AmðuÞ R �D n f0g for all u 2 D. Hence, we can apply Corollary 3.1 for anarbitrary q > 0 and fi ¼ Ai, i = 1, . . . ,m to conclude that the problem OCP fAigm
i¼1; L2þðXÞ
� �has a solution. h
6. Conclusions
In this paper we introduced for the first time a notion of exceptional family of elements naturally related to an order com-plementarity problem. We called this notion order exceptional family of elements (OEFE). We showed that there is either anexceptional family of elements, or the corresponding order complementarity problem has a solution. We gave two rathergeneral examples and presented an application for integral operators. The following open questions can be considered forfuture research:
(1) Give other (possibly more explicit) conditions for the nonexistence of order exceptional family of elements.(2) Find an algorithm which can decide whether an order complementarity problem has an order exceptional family of
elements or not.(3) Give a more general theorem for integral operators where the condition
A1ðuÞ ^ � � � ^ AmðuÞ R �D;
is true for all u 2 D with large enough norm, but is not true for all u 2 D.
(4) Find necessary and sufficient conditions for the nonexistence of order exceptional family of elements.(5) Find classes of order complementarity problems where the nonexistence of an order exceptional family of elements isequivalent to the existence of a solution for the corresponding order complementarity problem.(6) Find practical consequences of the method presented in this paper for economics and the lubrication problem pre-
sented in [31].
Acknowledgments
The author was supported by the Hungarian Research Grant OTKA 60480. The author is grateful to the reviewer for her orhis helpful comments which contributed to a substantial improvement of the paper.
References
[1] M. Bianchi, N. Hadjisavvas, S. Schaible, Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities, J.Optimiz. Theory Appl. 122 (1) (2004) 1–17.
[2] M. Bianchi, N. Hadjisavvas, S. Schaible, Exceptional families of elements for variational inequalities in Banach spaces, J. Optimiz. Theory Appl. 129 (1)(2006) 23–31.
[3] J.M. Borwein, M.A.H. Dempster, The linear order complementarity problem, Math. Oper. Res. 14 (3) (1989) 534–558.[4] V.A. Bulavsky, G. Isac, V.V. Kalashnikov, in: A. Migdalas, P. Werbrant (Eds.), Application of Topological Degree to Complementarity Problems, Kluwer
Academic Publishers, Dordrecht, Netherlands, 1998, pp. 333–358.[5] R. Cottle, J. Pang, R. Stone, The Linear Complementarity Problem, Academic Press, Boston, Massachusetts, 1992.[6] M.S. Gowda, R. Sznajder, The generalized order linear complementarity problem, SIAM J. Matrix Anal. A 15 (3) (1994) 779–795.[7] N.J. Huang, Y.P. Fang, Fixed point theorems and a new system of multivalued generalized order complementarity problems, Positivity 7 (3) (2003) 257–
265.[8] D. Hyers, G. Isac, T.M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific, Singapore, Republic of Singapore, 1997.[9] G. Isac, Complementarity problem and coincidence equations on convex cones, Bollettino Unione Mat. Ital. 6 (5) (1986) 925–943.
[10] G. Isac, Complementarity Problems, Lect. Notes Math., vol. 1528, Springer-Verlag, Berlin, Germany, 1992.[11] G. Isac, in: S.P. Singh (Ed.), Iterative Methods for the General Order Complementarity Problem, Kluwer Academic Publisher, 1992, pp. 365–380.[12] G. Isac, The fold complementarity problem, Topol. Methods Nonlinear Anal. 8 (2) (1996) 343–358.
2190 S.Z. Németh / Applied Mathematics and Computation 217 (2010) 2184–2190
[13] G. Isac, Exceptional family of, elements, feasibility, solvability and continuous path of e-solutions for nonlinear complementarity problems, in: P.Pardalos (Ed.), Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, Kluwer Academic, Boston, 2000, pp.323–337.
[14] G. Isac, Topological Methods in Complementarity Theory, Kluwer Academic Publishers, Dordrecht, Netherlands, 2000.[15] G. Isac, Leray-Schauder Type Alternatives, Complementarity Problems and Variational Inequalities, Springer-Verlag, New York, 2006.[16] G. Isac, V.A. Bulavski, V.V. Kalashnikov, Exceptional families, topological degree and complementarity problems, J. Global Optim. 10 (2) (1997) 207–
225.[17] G. Isac, V.A. Bulavsky, V.V. Kalashnikov, Complementarity, Equilibrium, Efficiency and Economics. Nonconvex Optimization and its Applications, vol.
63, Kluwer Academic Publishers, Dordrecht, Netherlands, 2002.[18] G. Isac, A. Carbone, Exceptional families of elements for continuous functions: some applications to complementarity theory, J. Global Optim. 15 (2)
(1999) 181–196.[19] G. Isac, D. Goeleven, Existence theorems for the implicit complementarity problems, Int. J. Math., Math. Sci. 16 (1) (1993) 67–74.[20] G. Isac, D. Goeleven, The implicit general order complementarity problem, models and iterative methods, Ann. Oper. Res. 44 (1-4) (1993) 63–92.[21] G. Isac, V.V. Kalashnikov, Exceptional families of elements, Leray–Schauder alternative, pseudomonotone operators and complementarity, J. Optimiz.
Theory Appl. 109 (1) (2001) 69–83.[22] G. Isac, M. Kostreva, The generalized order complementarity problem, J. Optimiz. Theory Appl. 71 (3) (1991) 517–534.[23] G. Isac, M. Kostreva, Kneser’s theorem and the multivalued general order complementarity problem, Appl. Math. Lett. 4 (6) (1991) 81–85.[24] G. Isac, M. Kostreva, The implicit general order complementarity problem and Leontief’s input–output model, Appl. Math. 24 (2) (1996) 113–125.[25] G. Isac, S.Z. Németh, The nonexistence of a regular exceptional family of elements. A necessary and sufficient condition. Applications to
complementarity theory, J. Global Optim. 42 (3) (2008) 359–368.[26] G. Isac, S.Z. Németh, REFE-acceptable mappings: a necessary and sufficient condition for the nonexistence of a regular exceptional family of elements, J.
Optimiz. Theory Appl. 137 (3) (2008) 507–520.[27] G. Isac, S.Z. Németh, Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems,
Optim. Lett. 2 (4) (2008) 567–576.[28] G. Isac, S.Z. Németh, Scalar and Asymptotic Scalar Derivatives. Theory and Applications, Springer Optimization and its Applications, vol. 13, Springer,
New York, 2008.[29] G. Isac, Y.B. Zhao, Exceptional families of elements and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert
spaces, J. Math. Anal. Appl. 246 (2) (2000) 544–556.[30] J. Leray, J. Schauder, Topologie et equations fonctionnelles, Ann. Sci. École Norm. S. 51 (1934) 45–78.[31] K.P. Oh, The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem, J. Tribol. 108 (1986) 598–604.[32] R. Sznajder, Degree theoretic analysis of the vertical and horizontal linear complementarity problem. Ph.D. Thesis, University of Maryland, USA, 1994.[33] P.P. Zabreiko, A.I. Koshelev, M.A. Krasnoselskii, S.G. Mikhlin, L.S. Rakovshchik, V.Y. Stetsenko, Integral Equations. A Reference Text, Nordhoff
International, 1975.[34] Y.B. Zhao, Exceptional families and finite-dimensional variational inequalities over polyhedral convex sets, Appl. Math. Comput. 87 (2-3) (1997) 111–
126.
