Variational inequalities for fuzzy mappings (II)

Variational inequalities for fuzzy mappings (II)

ZZ¥ sets and systems ELSEVIER Fuzzy Sets and Systems97 (1998) 101-107 Variational inequalities for fuzzy mappings (II) Muhammad Aslam Noor Mathemat...

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sets and systems ELSEVIER

Fuzzy Sets and Systems97 (1998) 101-107

Variational inequalities for fuzzy mappings (II) Muhammad Aslam Noor Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received September 1995; revisedOctober 1996

Abstract

In this paper, we establish the equivalence between the variational inequalities for fuzzy mappings and the Wiener-Hopf equations for fuzzy mappings. We use this equivalence to suggest a number of new iterative algorithms for solving the variational inequalities for fuzzy mappings. We also study the convergence criteria of these iterative algorithms. The auxiliary principle technique is used to study the existence of a solution of the variational inequality for fuzzy mappings and to suggest a novel and innovative iterative algorithm for computing the approximate solution. The results proved in this paper represent an improvement of previously known results. (~) 1998 Elsevier Science B.V. All rights reserved. Keywords: Variational inequalities; Fuzzy mappings; Wiener-Hopf equations; Auxiliary principle technique; Fuzzy numbers;

Mathematical programming; Operators; Optimization

1. Introduction

In recent years, the fuzzy set theory introduced by Zadeh [22] in 1965 has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of the fuzzy set theory can be found in many branches of regional, physical, mathematical and engineering sciences including artificial intelligence, computer science, control engineering, management science, economics, transportation problems and operations research, see [1, 7, 22, 23] and the references therein. Equally important is the concept of variational inequality theory, which constitutes a significant and important extension of the variational principles. Variational inequality theory provides us with a simple, natural, novel and innovative framework for study a wide class of unrelated linear and nonlinear problems arising in pure and applied sciences, see, for example [1, 2, 6, 8-21]. Motivated and

inspired by the recent research work going on in these two different fields, Chang [3], Chang and Zhu [5], Chang and Huang [4] and Noor [ 11, 12] introduced the concept of variational inequalities and complementarity problems for fuzzy mappings in different contexts. Noor [11] has shown that the variational inequalities for fuzzy mappings are equivalent to the fuzzy fixed point problems. This equivalence was used to suggest an iterative algorithm for solving variational inequalities. In this paper, we introduce the concept of the Wiener-Hopf equations for fuzzy mappings. Using essentially the projection technique, we establish the equivalence between the variational inequalities and the Wiener-Hopf equations for fuzzy mappings. This equivalence is more general and flexible. By an appropriate rearrangement of the Wiener-Hopf equations for fuzzy mappings, we suggest and analyze a number of iterative algorithms. We also study those conditions under which the approximate solution obtained from

0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PII S01 65-0 114(96)00323-5

M. Aslam Noor/Fuzzy Sets and Systems 97 (1998) 101-107

102

the iterative algorithms converges to the exact solution. The Wiener-Hopf equations technique is used to give a new formulation of the complementarity problems. It is well known that the problem of finding the projection of the space onto the convex set is itself a difficult problem except in some simple cases. To overcome this problem, we use another technique which does not depend upon the projection. This technique is called the auxiliary principle technique, which has been developed by Glowinski et al. [8] and Noor [12]. This technique is used to study the existence of a solution of variational inequality for fuzzy mappings as well as to suggest a general iterative algorithm. This technique enables us to develop a number of efficient and implementable numerical methods, see e.g., [12, 13] and the references therein. In Section 2, we introduce the basic problems and review some relevant results. The main results are considered and discussed in Section 3.

2. Formulations Let H be a real Hilbert space, whose norm and inner product are denoted by I]"II and (.,-), respectively. Let K be a closed convex set in H. We denote the collection of all fuzzy sets on H by F ( H ) = {ia : H --. I = [0, 1]}. A mapping T from H into F ( H ) is called a fuzzy mapping. If T : H --* F ( H ) is a fuzzy mapping, then the set T(u), for u E H, is a fuzzy set in F ( H ) and T(u, v), for all v E H is the degree of membership ofv in T(u). For more details and fundamental concepts, see [3, 7, 22, 23]. Let A E F ( H ) , v E (0, 1], then the set (A)v = {u EH: A(u)>~v} is said to be an v-cut set (v-level set) of A. For a given fuzzy mapping T : K --~ F ( H ) , we consider the problem of finding u E K such that w E (T(u))v and

(w,v - u) >.0

for all v EK.

(2.1)

The inequality of the type (2.1) is known as the variational inequality for fuzzy mappings, which is mainly due to Noor [ 11 ]. For the applications and existence results, see [1, 3, 4, 11-14] and the references therein.

I f K = H, then problem (2.1) is equivalent to finding u E H such that w E (T(u))v and

(w,v) = 0

forall v E H ,

(2.2)

which is the Riesz-Frrchet representation theorem for fuzzy mappings and appears to be a new one. We note that if K* = {u E H: (u, v) ~>0 for all v E K} is a polar cone of the convex cone K in H, then problem (2.1) is equivalent to finding u E K such that

wE(T(u)LNK*

and

(w,u)=0,

(2.3)

which is known as the complementarity problem for fuzzy mappings. Let PK be the projection of H into K and QK = I PK, where I is the identity operator. Consider the problem of finding z E H, u E K such that w E (Tu)~ and

w + p-lQKz = 0 ,

(2.4)

where p > 0 is a constant. The equations of the type (2.4) are called the Wiener-Hopf equations for fuzzy mappings. For general discussion and applications of the Wiener-Hopf equations, see [12, 13, 15, 16, 19-21]. Lemma 2.1 (Baiocchi and Capelo [2]). Let K be a closed convex set in H. Then, 9iven z E H, u E K satisfies the inequality

(u-z,v-u)>~O

JorallvEK,

if and only if U

PKZ,

=

where PK is the projection o f H onto K. Furthermore, Px is non-expansive. Definition 2.1. For all x, y E H , a fuzzy mapping T :H ---+F ( H ) is said to be: (i) F-strong monotone, if there exists a constant E (0, 1 ) such that
v,x -

y> ~ > ~ l l x -

yll 2

for all u E (T(x))~, v E (T(y))~. (ii) F-Lipschitz continuous, if there exists a constant/~ E (0, 1 ) such that

D((T(x))v, ( T ( y ) ) ~ ) ~/~llx - yll, where D(., .) is the Hausdorff metric on F(H).

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M. Aslam Noorl Fuzzy Sets and Systems 97 (1998) 101-107

In particular, from (i) and (ii), it follows that 0~[Ix -- yll 2 <~ (U -- V,X -- y) ~< IlU -- vii

IIx -

yll

<~ O((Z(x))~, ( T ( y ) L ) I I x - yll flllx

- YI] 2,

which implies that e ~
3. Main results

In this section, we first establish the equivalence between the problems (2.1) and (2.2) using the projection technique. Invoking Lemma 2.1 and following the technique of Noor [11], one can easily prove the following result. Lemma 3.1. L e t K be a closed convex set in H. Then (u, w ) is a solution 03" the variational inequality f o r f u z z y mappings (2.1) i f and only i f u = P r [ u - pw], where p > 0 is a constant and PK is the projection o f H onto K.

From Lemma 3.1, we conclude that the variational inequality is equivalent to the fuzzy fixed point problem. This equivalence formulation enables us to suggest the following iterative scheme. Algorithm 3.1. For a given uo E K such that w0 E ( T(uo ) )v, let ul = ( 1 - 2)u0 + 2Px[uo - pwo]. Since w0 E (T(uo))~ there exists Wl E (T(ul))~ such that

Un+l = (1 - 2)un + 2 P r [ u , - pw,],

n = 0, 1,2,...

where p > 0 is a constant and 0 < 2 < 1 is a parameter. We now prove that the variational inequality (2.1) is equivalent to the Wiener-Hopf equations (2.4) for fuzzy mappings using the technique of Noor et al. [15, 16] and Shi [19]. Theorem 3.1. The variational inequality f o r f u z z y mappinos (2.1) has a solution u E K , w E (T(u))~ i f and only i f the W i e n e r - H o p f equations f o r f u z z y mappings (2.4) has a solution z E H, u E K such that w E (T(u))v, where u = Px,z,

(3.1)

Z = U -- pW,

(3.2)

and p > 0 is a constant.

Proof. Let u E K such that w E (T(u))v be a solution of (2.1). Then by Lemmas 2.1 and 3.1, we have u = Px[u - pw].

(3.3)

Using the fact QK = I - PK and (3.1), (3.3) we obtain Qx[u - pw] = u - p w - PK[u - pw] = --pw,

from which and (3.3), it follows that w + p-lQxz=O.

Conversely, let z E H, u E K such that w E (T(u))v be a solution of (2.4), then pw = - Q r z = P K z -- z.

(3.4)

llwo - w~ II <~D((T(uo))v, (T(ul)),,),

Now from Lemma 2.1 and (3.4), for all v E K, we have

where D(.,-) is the Hausdorff metric on F ( H ) . Let

0<<.(PKz - z, v - PKz) = p(w, v - PKz).

u2 = ( 1 - 2)Ul + J.PK[Ul -- pWl].

Thus ( u , w ) , where u = P x z , is a solution of(2.1).

Continuing in this way, we can compute {u,} and {w, } by the iterative schemes Wn E ( T ( U n ) ) v : ]]Wn -- Wn--1 II

<~ D((T(un))v, (T(un+l)),~)

[]

Theorem 3.1 establishes the equivalence between the variational inequality (2.1) and the Wiener-Hopf equations (2.4) for the fuzzy mappings. This alternate formulation is very important from numerical and approximation point of views. For suitable and

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M. Aslam Noor/Fuzzy Sets and Systems 97 (1998) 101-107

appropriate rearrangement of the Wiener-Hopf equations (2.4), we can suggest a number of iterative algorithms for solving the variational inequality for fuzzy mappings (2.1) and related problems. I. Eqs. (2.4) can be written as QKZ = - pw,

converges to the exact solution of the Wiener-Hopf equations (2.4). Theorem 3.2. L e t K be a closed c o n v e x set & H. L e t the f u z z y mapping T : H ~ F ( H ) be F - s t r o n g l y m o n o t o n e with constant ~ E (0, 1) and F - L i p s c h i t z continuous with constant fl E (0, 1 ). I f

from which it follows that z=Pxz

- pw=u

- pw

c(

using (3.1)

(3.5)

which is a fuzzy fixed point problem. This fixed point formulation enables us to suggest the following iterative schemes. Algorithm 3.2. For given z0 E H, u0 E K such that w0 E (T(u0))~, compute {z,}, {u,} and {Wn} by the iterative schemes. (3.6)

u, = P K z , , w. ~ (T(u.))v

:llwn

-

0 < p < 2~-2,

(3.9)

then there e x i s t z E H , u E K such that w E ( T ( u ) ) v , which satisfy the W i e n e r - H o p f equations (2.4) and the sequences {Zn}, {u,} and {w,} 9enerated by A l g o r i t h m 3.2 converge to z, u, and w stronoly in H , respectively.

Proof. From Algorithm 3.2 and using the strongly monotonicity D-Lipschitz of the operator T :H F ( H ) , we have

Wn÷l]l IIz.+, -z~ll z (3.7)

<~ O ( ( T ( u n ) ) v , (T(Un+l))v) Zn+ 1 = u n - pWn,

n = 0 , 1,2 .....

(3.8)

= IlUn - Un-I - p(wn - Wn-1)112 <~ IlUn - u , - x II2 - 2 p ( w , - w n - l , u , - u , - 1 )

+ p=llw. - w . - i II2

II. Eqs. (2.2) may be written as QKz = - w + (I - p - 1) QKz,

Ilu. - Un-, II2 -- 2 p ( w .

from which it follows that

q- p 2 { O ( ( Z ( u n ))v, ( T ( u n - 1 ) ) v ) } 2

~<(1 - 2 p ~ + p 2 f l 2 ) l l u ,

z = P x z - w + ( I - p - 1) Q~z = u-w+(I-p-1)Q1,;z

using(3.1).

-- W n - I . U n

-

-

Un--l)

- u,_l]l 2.

From (3.1) and (3.6), we have U._l II ~ IIPKZn

II

Using this fuzzy fixed point formulation, we can suggest the following iterative scheme.

IlUn

Algorithm 3.3. For given z0 C H, u0 E K such that wo E (T(uo))v, compute {z,}, {u,} and {w,} by the iterative scheme.

since PK is a nonexpansive operator. Combining (3.10) and (3.11 ), we obtain

-

Pgzn-1

IIz. - Zn-1 II,

<~ D ( ( T ( u n ) ) v , (T(un+1))v),

n = 0 , 1 , 2 .....

We now study those conditions under which the approximate solution obtained from Algorithm 3.2

(3.11)

-- 2 p a + p2f12 }IIZ" _ Z.--111

= OIIz. -- Z.--1 II,

w . ~ ( T ( u . ) ) v l l w . - w.+~ II

Zn+l=Un--Wn+(I--p-1)QKZn,

-

IlZn--~ -- Z. II ~ { ¢ 1

Un = PKZn,

(3.10)

(3.12)

where 0 = v/1 - 2p~ + p2f12. From (3.9), we see that 0 < 1 and consequently, from (3.12), it follows that {z, } is a Cauchy sequence in H, i.e., z,+l - - ~ z E H as n---* oc. From (3.11), we know that {u,} is also a Cauchy sequence in H, i.e.

M. Aslam Noorl Fuzzy Sets and Systems 97 (1998) 101-107 Un+ 1 - - ~ u E H ,

as n--* oo. Also, from (3.7), we have

Ilwn - w . + l II ~<

D((T(un))v,

~
(T(un+l))~)

u . + l II,

which implies that the sequence {Wn} is also a Cauchy sequence in H , so that there exists w E H such that Wn+ 1 -"4 W.

Using the continuity of the fuzzy operator T, Px and Algorithm 3.2, we have z = u - pw = Pxz - pw E H. Now we shall prove that wE (T(u))v. In fact,

d(w,(T(u))v) ~ IIw - w.II + d(wn,(T(u))v)

IIw- w. II "bO((T(un))v,(T(u))v) <~ Ilwn

-

wll + / ~ l l u .

- utl --' 0

as n ---*oo,

105

which appears to be a new one. We remark that this new formulation may be useful in developing some efficient numerical techniques for solving complementarity problems for fuzzy mappings. We remark that in order to use Algorithms 3.1-3.3, one has to find the projection of the space onto the convex set, which is itself a difficult problem. In addition, the projection method and its variant forms cannot be applied to suggest and analyze iterative algorithms for computing the approximate solution of some classes of variational inequalities involving nondifferentiable forms, see e.g. [12]. These facts motivated Glowinski et al. [8] and Noor [12] to develop another technique, which does not involve projection. This technique is known as the auxiliary principle technique. For the applications of this technique, see [12, 13] and the references therein. We use this technique to study the existence of a solution of the variational inequality for fuzzy mappings (2.1) and this is the main motivation of our next result.

where d(w,(T(u))v) = inf{Jlw- vii; vE(T(u))~}; we have d(w, (T(u))~) = 0. This implies that w E (T(u))~ since (T(u))~ E F ( H ) . Using Theorem 3.1, we see that z E H , u E K such that w E (T(u))~ are the solution of (2.4) and consequently Z,+l ---~z, u~+l ~ u and Wn+l--~w strongly in H. This completes the proof. []

Theorem 3.3. Let the f u z z y mappin9 T : H ~ F ( H ) be F-strongly monotone with constant ~ E (0, 1 ) and F-Lipschitz continuous with constant fl E (0, 1). I f (3.9) holds, then there exists a solution u E K such that ~ E (T(u))v satisfyin9 ( 2 . 1 ) f o r all v E K .

R e m a r k 3.1. Theorem 3.1 can be used to obtain a new formulation of the complementarity problem (2.3). It is well known that any arbitrary z E H can be written as

Proof. We use the auxiliary principle technique, as developed by Noor [12], to prove the existence of a solution of (2.1). For a given u E K, we consider the problem of finding q E K such that 09 E (T(u))v satisfying the auxiliary variational inequality for fuzzy mappings

z : P x z + P-K*z : P K z + P K * ( - z ) ,

(3.13) (q,v - q) >>. (u,v - q) - p(w,v - q)

where K* is the polar cone of the convex cone K in H. The W i e n e r - H o p f equation (2.4) for p = 1 can be written as

for all v E K ,

(3.15)

Hence using (3.1), (3.14), the complementarity problem (2.3) can be formulated as follows: Find z E H , u E K such that

where p > 0 is a constant. The relation (3.15) defines a mapping u ~ q. In order to prove the existence of a solution u E K satisfying (2.1), it is enough to show that the mapping u ---+q defined by the relation (3.15) has a fixed point belonging to K satisfying (2.1). Let ql,q2 E K be two solutions (3.15) related to ul,u2 E K such that 091 E (T(Ul))v, (o2 E (T(u2))v, respectively, i.e.

Px(z)EK,

( q l , v - ql) >~ ( u l , v - ql) - p(~1,v - ql)

w =

-

Qxz

:

-

z + PKz =

-

PK*(-z),

(3.14)

where we have used (3.13).

-PK.(-z)E((T(u))v)MK*,

(t'x(z),Px. ( - z ) ) = o,

for all v E K ,

(3.16)

M. Aslam NoorlFuzzy Sets and Systems 97 (1998) 101-107

106

and

(iii) For given e > 0, if wise repeat (ii).

Ilq.+l -

qn ]1 ~< g, stop. Other-

(qz, v - q2) >1 (uz, v - q2) - p(a)2, v - q2)

for all v C K .

(3.17)

Taking v = q 2 in (3.16) and v = q l in (3.17), and adding the resultant inequalities, we have (ql - q 2 , q l - q2) <~ (ul - u e , q l - q2) - p(021 - 092,ql - q2) =

(Ul -- U2 -- P((D1

--

~2),ql -- q2)

which implies that ]]ql - q21] ~< ]]Ul - -

U2 -- fi((D1

~<{V/1 - 2p~ +

Remark 3.3. In recent years, it has been shown that many computational techniques including projection, linear approximation, relaxation, decomposition, descent and Newton's methods for solving variational inequalities for fuzzy mappings can be derived from the auxiliary principle technique by a suitable and appropriate choice of the fuzzy mappings T, M and the convex set K. Furthermore, the auxiliary principle technique can be used to find a number of equivalent differentiable optimization problems. For recent stateof-the art, see [12] and the references cited therein.

- - (2)2)11

p2fl2}llu,

- u2ll

= Ollu, - u2]] using (3.10),

where 0 = v/1 - 2p~ + p 2 f 1 2 . From (3.9), it follows that 0 < 1, so that the mapping v --+ q defined by (3.15) has a unique fixed point belonging to K, a closed convex set in H satisfying (2.1), the result. [] Remark 3.2. Following the technique of Noor [12], one can suggest a more general auxiliary variational inequality problem for fuzzy mapping than (3.15). For a given u E K, we consider the problem of finding q E K such that coC(T(u))v and (M(q), v - q) >_, ( M ( u ) , v - u) - p(co, v - q) for all v c K ,

(3.18)

where M : H - - + F ( H ) is a nonlinear fuzzy mapping. Clearly, if M_= I, the identity operator, then the auxiliary problem (3.1 8) is exactly the problem (3.15). We remark that if q = u, then q is a solution of the problem (2.1). On the basis of this observation, we can now suggest and analyze an iterative algorithm for (2.1). This is a novel way to compute the approximate solution of problem (2.1) as long as the problem (3.18) is easier to compute than (2.1). Algorithm 3.5. (i) At n = 0, start with initial q0. (ii) At step n, solve the auxiliary problem (3.18) with u = qn+l. Let qn denote the solution of problem (3.18).

Remark 3.4. It is worth mentioning that in many important applications in economics and equilibrium problems, the convex set K also depends on the solution u explicitly or implicitly. In these cases, the variational inequality for fuzzy mappings (2.1) is called the quasi-variational inequality. To be more specific, given a point-to-set mapping K : u - - , K ( u ) , which associates a closed convex-valued set K ( u ) with any element u o f / / , we consider the problem of finding u C K ( u ) such that o2 E (T(u)v) and (to, v - u) ~>0 for all v E K ( u )

(3.19)

which is called a quasi-variational inequality for fuzzy mappings. We note that i f K ( u ) - - K , then problem (3.19) collapses to the problem (2.1). The problem (3.19) has been studied by Noor and A1-Said [18] using the projection method and the Wiener-Hopf equation technique. An extension of the auxiliary principle technique for quasi-variational inequality for fuzzy mappings is still an open problem and this is another direction for further research.

4. Conclusions

In this paper, we have introduced the concept of Wiener-Hopf equations for fuzzy mappings. Using essentially the projection technique, we have established the equivalence between the variational inequalities for fuzzy mappings and the WienerHopf equations for fuzzy mappings. This alternate

M. Aslam Noor/Fuzzy Sets and Systems 97 (1998) 101-107

formulation is used to suggest and analyze a number o f new iterative algorithms for solving variational inequalities. W e have also used the auxiliary principle technique to study the existence o f a solution o f the variational inequality for fuzzy mappings and to suggest a general iterative algorithm. It is worth mentioning that the fuzzy set theory provides a strict mathematical framework in which vague conceptual phenomena can be studied precisely and rigorously. Fuzziness has so far not been defined uniquely semantically, and probably never will. It will mean different things, depending on the application area and the way it is measured. Consequently, it is clear that the concept o f variational inequalities and the W i e n e r - H o p f equations for fuzzy mappings can be defined differently depending upon the area in which one seeks to study these concepts. The study o f this area is a fruitful and growing field o f intellectual endeavor. Much work is needed to develop this interesting subject.

References [1] J.P. Aubin, Mathematical Methods of Game and Economic Theory (North-Holland, Amsterdam, 1979). [2] C. Baiocchi and A. Capelo, Variational and QuasiVariational Inequalities (Wiley, New York, 1984). [3] S.S. Chang, Variational Inequality and Complementarity Problems Theory and Applications (Shanghai Scientific and Technological Literature Publishing House, Shanghai, China, 1991). [4] S.S. Chang and N.J. Huang, Generalized complementarity problems for fuzzy mappings, Fuzzy Sets and Systems 55 (1993) 227-234. [5] S.S. Chang and Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems 32 (1989) 359-367. [6] J. Crank, Free and Moving Boundary Problems (Clarendon Press, Oxford, UK, 1984).

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[7] D. Dubois and H. Prade, Fuzzy Sets and Systems, Theory and Applications (Academic Press, London, 1980). [8] R. Glowinski, J.L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities (North-Holland, Amsterdam, 1981). [9] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity (SIAM, Philadelphia, 1988). [10] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Applications (Academic Press, New York, 1980). [11] M. Aslam Noor, Variational inequalities for fuzzy mappings (I), Fuzzy Sets and Systems 55 (1993) 309-312. [12] M. Aslam Noor, Theory of Variational Inequalities, Lecture Notes, Mathematics Department, King Saud University, Riyadh, Saudi Arabia, 1996. [13] M. Aslam Noor, Some recent advances in variational inequalities (I, II), New Zealand J. Math. 26 (1997). [14] M. Aslam Noor, General variational inequalities for fuzzy mappings, J. Fuzzy Math. (1997). [15] M. Aslam Noor, K. lnayat Noor and Th.M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993) 285-312. [16] M. Aslam Noor, K. lnayat Noor and Th.M. Rassias, Invitation to variational inequalities, in: H.M. Srivastava and Th.M. Rassias, Ed., Analysis, Geometry and Groups: A Riemann LegacT Volume (Hadronic Press, Florida, 1993), 373-448. [17] M. Aslam Noor and W. Oettli, On general non-linear complementarity problems and quasi-equilibria, Le Matematiche 49 (1994) 313-331. [18] M. Aslam Noor and E.A. AI-Said, Quasi variational inequalities for fuzzy mappings, Preprint, 1996. [19] P. Shi, Equivalence of variational inequalities with WienerHopf equations, Proc. Amer. Math. Soc. 111 (1991) 339-346. [20] F.O. Speck, General Wiener--Hopf Factorization Methods (Pitman, London, 1985). [21] S.M. Robinson, Normal maps induced by linear transformations, Math. Oper. Res. 17 (1992) 691-714. [22] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [23] H.J. Zimmermann, Fuzzy Set Theory and its Applications (Kluwer Academic Publishers, Boston, 1988).