SOME RECENT ADVANCES IN VARIATIONAL

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the Wiener-Hopf equations technique and ideas of Dafermos [24] to develop a sen sitivity analysis framework for variational inequalities. Using this technique ...

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 26 (1997), 229-255

SOME RECENT ADVANCES IN VARIATIONAL INEQUALITIES PART II: OTHER CONCEPTS M

uham mad

A

slam

N oor

(Received February 1994)

Abstract. Variational inequality theory has become an effective and powerful technique for studying a wide class of problems arising in various branches of mathematical and engineering sciences in a unified and general framework. This theory has been developed in several directions using new and novel meth­ ods that have led to the solutions of basic and fundamental problems thought to be inaccessible previously. Some of these developments have made mutually enriching contacts with other area of pure and applied sciences. In this paper, we focus mainly on the recent iterative algorithms for solving various classes of variational inequalities. We also suggest some open problems with sufficient information and references. For notations and basic concepts, see Noor [145].

7. Fuzzy Variational Inequalities In recent years, fuzzy set theory introduced by Zadeh [138] in 1965 has emerged as an interesting and fascinating branch of pure and applied sciences. The applica­ tions of fuzzy set theory can be found in many branches of regional, physical, math­ ematical and engineering sciences including artificial intelligence, computer science, control engineering, management science, economics, transportation problems and operations research, see [13, 138, 141] and the references therein. Motivated and inspired by the recent research work going on in these two different fields, Chang and Zhu [13], and Noor [99] introduced the concept of variational inequalities and complementarity problems for fuzzy mappings. Noor [99] has shown that the vari­ ational inequalities for fuzzy mappings are equivalent to fuzzy fixed problems. This equivalence was used to suggest an iterative algorithm for solving variational in­ equalities. Here, 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 map­ pings. By an appropriate rearrangement of the Wiener-Hopf equations for fuzzy mappings, we suggest and analyze a number of iterative algorithms. We denote the collection of all fuzzy sets on H by F (H ) = {/i : 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 of v in T(u).

1991 A M S Mathematics Subject Classification: 49J40, 49K40, 90C20, 90C30, 35J85. Key words and phrases: Variational inequalities, Wiener-Hopf equations, projection algorithm, auxiliary principle, invex, preinvex, variational-like inequalities, parallel algorithms, complemen­ tarity problems.

230

MUHAMM AD ASLAM NOOR

Let A E F (H ), v

G

(0,1], then the set (A )u =

{u E H : A(u) > i/}

is said to be an u-cnt set (i^-level set) of A. For a given fuzzy mapping T : K —►F (H ), we consider the problem of finding u G K such that w G (T(u)) and (w, v —u) > 0,

for all v

(7.1)

G K.

The inequality of the type (7.1) is known as the variational inequality for fuzzy mappings, which is mainly due to Noor [99], where the projection technique was used to suggest an iterative algorithm. It has been shown in Noor [98] that the auxiliary principle technique can be used to study the existence of a solution of (7.1) and to suggest a general iterative algorithm. Let P k be the projection of H into K and Q k = I — P k » where I is the identity operator. Consider the problem of finding z G H , u G K such that w G (T (u ))u and w + p~lQ K z -

0,

(7.2)

where p > 0 is a constant. The equations of the type (7.2) are called the WienerHopf equations for fuzzy mappings. Definition 7.1. For all x, y £ H , a fuzzy mapping T : H —> F (H ) is said to be: (i)

F - s t r o n g m o n o to n e , if there exists a constant a G (0,1) such that ( u - v , x - y ) > a\\x - y\\2,

(ii)

for all u

G (T (x ))u, v G (T {y ))v.

F - L ipschitz c o n t in u o u s , if there exists a constant (3 G (0,1) such

that D (T (* ))„,

(T (j,))„) 0 is a constant and P k is the projection of H into K .

From Lemma 7.2, 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 7.3. For a given uo the iterative schemes

G K

such that wq

G (T(u o))^,

wn G {T(un))^ : ||wn - iun+i|| < D d T ^ n ) ) ^ un-i-i = P k [wn

where p > 0 is a constant.

P^n\t

compute {un} by

(T(un+ i))„)

7i = 0 ,1 ,2 ,... ,

PART II:

231

OTHER CONCEPTS

We now prove that the variational inequality (7.1) is equivalent to the WienerHopf equations (7.2) for fuzzy mappings using essentially Lemma 2.1 and technique of Shi [123]. Theorem 7.4. The variational inequality for fuzzy mappings (7.1) has a solution u G K , such that w G (T(it))i/ if and only if the W iener-H opf equations for fuzzy mappings (7.2) has a solution z G H , u G K such that w G (T {u ))v, where

P roof. Let u G K such that w 2.1 and 7.2, we have

u

=

PK z ,

(7.3)

z

=

u — pw.

(7.4)

G {T (u ))v

be a solution of (7.1). Then by Lemmas

u = P k [u — pw\.

(7.5)

Using the fact Q k — I — P k and (7.3) —(7.5), we obtain Q k [u — pw] = u — pw — P k [u — pw] = ~pw ,

from which and (7.5), it follows that w 4- p ~ 1Q k z = 0.

Conversely, let z e H , u e K such that w

G (T(u))^

be a solution of (7.2), then

pw = - Q k z = P r z - z.

Now from Lemma 2.1 and (7.6), for all v

G K,

(7.6)

we have

0 < (P k z — z, v — P k z ) = p(w, v — P k z ). Thus (u ,w ), where u = P k z is a solution of (7.1).



Theorem 7.4 establishes the equivalence between the variational inequality (7.1) and the Wiener-Hopf equations (7.2) for fuzzy mappings. This alternate formu­ lation is very important from numerical and approximation points of view. For a suitable and appropriate rearrangement of the Wiener-Hopf equations (7.2), we can suggest a number of iterative algorithms for solving the variational inequality for fuzzy mappings (7.1) and related problems. I. The equations (7.2) can be written as Q

kz

=

-p w ,

from which it follows that z = P k z - pw = u — pw,

using (7.3).

(7.7)

which is a fuzzy fixed point problem. This fixed point formulation enables us to suggest the following iterative schemes. Algorithm 7.5. For given zq G H , uq G K such that wq {zn}, {u n} and {w n} by the iterative schemes.

G ( T ( uo ) ) u,

compute (7-8)

un



PK zn,

wn

G

(T(un))v : ||tUn —u?n+i|| < D ((T(wn))^,

Zn+1

=

un - p w n,

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

(T{un.i-i))^)

(7-9) (7.10)

232

M UHAMM AD ASLAM NOOR

II. The equations (7.2) may be written as Qk z =

- w + (I - p ~ l )Q K z,

=

PK z - w

+ (I - p ~ l )Q K z

=

u - w + (I - p ~ 1)Q k z ,

from it follows that

z

using (7.3).

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

Algorithm 7.6. For given zq

G H , uo G K such that wo G

(T(uo))i/, compute

{zn}, {u n} and {w n} by the iterative scheme. w?i



P k Zni

wn

G

(T{un))v : ||w;n - w n+i|| < D ((T (u n ))v, (T (un+ 1 )) J ,

Zn + l

=

'U'n

“1“ ( 7

p

) Q k Z wi

^ — 0, 1 , 2 , . . . .

We now study those conditions under which the approximate solution obtained from Algorithm 7.5 converges to the exact solution of the Wiener-Hopf equations (7.2).

Theorem 7.7. Let K be a closed convex set in H .

Let the fuzzy mapping

T : H —> F (H ) be a F-strongly monotone with constant a G ( 0 ,1 ) and F-Lipschitz continuous with constant /3 G (0 ,1 ). If

(7.11)

0 < p < 2 ^ ,

then there exist z G H , u G K such that w G (T {u))^, which satisfy the W ienerHopf equations (7.2) and the sequences {z n}, {u n} and {« ;n} generated by Algo­ rithm 7.5 converge to z, u, and w strongly in H respectively.

P roof. From Algorithm 7.5, we have ll^ ri+ l

z n\\

^ n —1

P i^ n

=

H^n


u G H , as n —* oo. Using the continuity of the fuzzy operator T , P k and Algorithm 7.5, we have z = u — pw =

— pw e H.

Using Theorem 7.4, we see that z e H , u € K such that w G (T(w)) are the solution of (7.2) and consequently z n+1 —>2 , wn+i —* u and u>n+i —►u> strongly in H . This completes the proof. □ 8. Random Variational Inequalities It is well known that the study of the random equations involving the random operators in view of their need in dealing with probabilistic models in applied sci­ ences is very important. It is well known that the introduction of randomness leads to several questions including the measurability and the probabilistic aspects of the solution, see, for example, Bharucha-Reid [9]. Motivated and inspired by the recent research work going in these fascinating areas, we introduce and consider random variational inequalities. Using the projection technique, we show that the random variational inequalities are equivalent to the random equations, which are called the random Wiener-Hopf equations. This equivalence is then used to sug­ gest and analyze some new iterative algorithms for solving the random variational inequalities. Let (Q ,F ,n ) be a complete probability measure space and H be a separable Hilbert space with norm and inner product denoted by || • ||and (•, •) respectively. Let A' be a closed convex set in H . We denote by B h , the Borel field of if. X (Cl, H ) is the space of all mappings / : —» H such that f ~ 1(B ) G F , whenever B G B h Let X (C l,K ) be the subset of X ( Q ,H ) , made up of the maps of / with range in K.

An operator T : x H —> H is called a random operator,if T ( •,u ) G X (C l,H ) for all u G H . The operator T is said to be continuous, if for u> outside some null set, the operators T(uj, •) are continuous on H . The operator T is called measurable, if it is measurable with respect to the u-algebra F x B h , that is, if for all B G B h , {( uj,u);T(uj,u ) e B } e F x B h - It is also well known that 1. A measurable operator is necessarily a random operator. 2. T

is a random operator if and only if the real-valued map w

—> (T(uj,u),v)

is

measurable for all u, v G H .

3. A continuous random operator is measurable. Let T : 0 . x H —> H be a random operator. We shall write Twu(u;) = T(u,u(u>)), for all u G H and uj G fi. Problem 8.1. We consider the problem of finding u almost all u G f2, (TtL)u(u>),v — u(uj)) > 0,

for all v

G X (C l,K )

G K.

such that for (8.1)

The inequality of the type (8.1) is called a random variational inequality. First of all, we prove the following result.

234

MUHAMM AD ASLAM NOOR

Lemma 8.2. If T is a measurable random operator and (Au)(cj) = Tuu(u), then Au G X {C l,H ) whenever u G X (C l,H ). P roof. Let J : Cl —>Cl x H be defined by J(uj) = (u>, u(w)). Then J is measurable when Cl x H is given the product a - algebra F x B h - In fact for all B G F, E G B h , J ~ 1{B x F ) = B CIu ~ 1(E ) which belongs to F as u G X(f2, H ). Now Au = T o J , so that for B G B h , (Aw)-1 (£') = J ~ 1{T ~ 1(E )). Since T is measurable, T ~ 1(E ) G F x B h and hence J ~ 1(T ~ l (E )) is in F . We consider the problem of finding 2 € X(Cl, H ) such that for almost all a; G Cl, TwP k z {uj) + p ~ l {uj)QK z(u) = 0,

(8.2)

where p{u) > 0 is a real-valued random variable. Here P k is the projection of H onto K and Q k = I — P k , where I is the identity operator. The equations of the type (8.2) are called the random Wiener-Hopf equations. □ Definition 8.3. A random operator T : Cl x H —>H is said to be: (i)

r a n d o m ly s t r o n g m o n o to n e , if there exists a real-valued random variable a(w) > 0 such that for all u, v G X(Cl, H )

(Twu(u>) —Tuv(u),u{ a(uj) | |u(u>) — v(u;) ||2,

(ii)

a.e.

(8.3)

r a n d o m ly L ip s c h itz c o n tin u o u s , if there exists a real-valued ran­ dom variable (5{oj) > 0 such that for all u , u G X (Q, H ),

||Twu (uj) - Twv (uj) ||< /3(uj) ||u(u) -

v (uj)

||,

a.e.

(8.4)

Note if T is assumed to be continuous, randomly strong monotone and Lipschitz continuous, then T is measurable and outside a universal null set, (8.3) and (8.4) are satisfied by all oj. Note also that then a (a;) < /3(u>). We now prove that the random variational inequality problem (8.1) is equivalent to solving the random Wiener-Hopf equations (8.2). Theorem 8.4. The random variational inequality (8.1) has a solution u(uj) G X (Cl, K ) if and only if the random W iener-H opf equation (8.2) has a solution z(w ) G X (C l,H ), where z( u>) =

u (uj)

— piu^TuU^)

(8.5)

P k z (u>).

(8.6)

and u(u) =

Here p(co) > 0 is a real-valued random variable and P k is the projection o f H into K.

From Theorem 8.4, it is clear that the random variational inequality (8.1) is equivalent to solving the random Wiener-Hopf equations (8.2). This equivalence is very important from numerical and approximation point of views. We use this equivalence to suggest and analyze a number of iterative algorithms for solving random variational inequalities. We now discuss the following cases.

PART II:

235

OTHER CONCEPTS

I. The random Wiener-Hopf equations (8.2) can be written as Q k z (uj) =

- p (uj)Tu P k z (uj)

from which it follows that z(u)

=

P k z {u ) - p{uj)TwPK z(uj)

=

u(uj) - p(w)Twu(uj),

(8.7)

by (8.6).

This formulation enables to suggest the following iterative algorithm for the random variational inequality (8.1). Algorithm 8.5. For a given zq[u) G X ( Q ,H ) , compute zn+i(u;) by the iterative schemes un(uj) = PK zn(u)

(8.8)

Zn+i(u ) = Un(hj) ~ pi^TuUniuj),

(8.9)

and

where p(u>) > 0 is a real-valued random variable. II. We can rewrite the random Wiener-Hopf equations (8.2) as 0

= - p ~ l (u )Q K z { u j ) - T u P k z ()-

Using this formulation, we suggest the following iterative algorithm. Algorithm 8.6. For a given zq(u;) G X ( Q ,H ) , compute zn+i(u >) by the iterative schemes UH(UJ) = PK Zn{u)

and Zn+li^y = 'U'niyj)

rPujUn{uj)

+ (7

p

(w))Q/^Zn(ct>),

Tl

— 0, 1, 2, . . . .

III. If for almost all u> the operator Tw is linear and T ~ l exists, then the random Wiener-Hopf equations (8.2) become: z (uj)

=

(7 - p ~ l (u )T ~ l ) Q K z{w).

This random fixed point formulation may be used to suggest the following algo­ rithm. Algorithm 8.7. For a given zq() is the exact solution of the random Wiener-Hopf equations (8.2).

236

MUHAMM AD ASLAM NOOR

Theorem 8.8. Let zn+i(uj) satisfy (8.9). If the random operator T is randomly strong monotone and randomly Lipschitz continuous, and 0 < p(uj)
0 such that

(T(u, A) - T (v, X),u —v) > a||u —i?||2,

238

M UHAMM AD ASLAM NOOR

(b) L o c a l l y L ip sch itz c o n tin u o u s if there exists a constant /3 > 0 such that

l|r(u,A) —T(v, A)|| < 0\\u —t>||.

We consider the case, when the solutions of the parametric Wiener-Hopf equa­ tion (9.2) lie in the interior of X . Following the ideas of Dafermos [24], we consider the map F(z, A) where

(u, A)

=

PKxn x (z , A) - pTPKxn x { z , A)

for all (z, A) € X x M

=

( u ,\ ) -p T ( u ,\ ) ,

(9.5)

=

PKxn x (z ,X ).

(9.6)

We have to show that the map F(z, A) has a fixed point, which is a solution of the Wiener-Hopf equation (9.2). First of all, we prove that the map F (z , A) is a contraction map with respect to z, uniformly in A e M . Lemma 9.3. For all z\, z2 € X , and A € M , we have | | F ( * i ,A ) - F ( z 2,A)|| < 0\\zi - z2||, where 9 = y / l - 2pa + p2/32 < 1

for

0< p
(x ))

=

0,

—u( 0) = u'( 0)

=

1/ ( 1 )

0< x < 1

(

10.6 )

where

f

1,

for t > 0,

/

7(0 = { 0,

for i < 0,

(1°-7)

is a discontinuous function and is known as the penalty function. Here ip is the given obstacle function defined by (10.2). Thus from (10.2), (10.6) and (10.7), we obtain the following system of differential equations 0, u"'(x) =

for 0 < x
H , we consider the problem of finding u € H such that (Tu, v — u) + ip(v) — ip(u) > 0,

for all v € H.

(11.1)

The problem of the type (11.1) is known as the mixed variational inequality or variatonal inequality of the second type. It has been shown in [4, 14, 28, 33, 34, 49] that a wide class of nonlinear and linear problems arising in pure and applied sciences can be studied via the variational inequality of type (11.1). Glowinski, Lions and Tremolieres [34] and Noor [83, 85] has developed the auxiliary principle technique for solving problem (11.1) and suggested some iterative methods. We note that if is the indicator function of K , that is, if u € K , otherwise, then problem (11.1) is equivalent to the problem (2.1). We need the following well known result, which plays an important part in obtaining our main results.

244

M UHAMM AD ASLAM NOOR

Lemma 11.1 ([11]). For a given u 6 H , z G H satisfies the inequality (u —z ,v —u) + p 0,

for all v € H,

which can be written as, for p > 0, (u — (u —pTu ), v — u) + p^p(v) —p(p(u) > 0,

which is equivalent to u = J