Existence of generalized variational inequalities

r~

Operations Research Letters 15 (1994) 35 40

ELSEVIER

Existence of generalized variational inequalities* Jen-Chih

Yao

Department of Applied Mathematics. National Sun Yat-sen University. Kaohsiung. 80424. Taiwan. ROC

(Revised 29 October 1992; revised 10 May 1993)

Abstract In this paper we investigate certain generalized variational inequalities in Hilbert spaces which contain general mildly nonlinear variational inequalities and the classical variational inequalities as special cases. We employ the fixed point technique used by Glowinski, Lions and Stampacchia to obtain some existence results for these nonlinear inequalities. In particular, we obtain some existence results for both general mildly nonlinear variational inequalities and implicit complementarity problems. Key words." General mildly nonlinear variational inequalities; Implicit complementarity problems; Fixed point; Hilbert space

1. Introduction Let H be a real Hilbert space with n o r m II " II and inner product (.,.), respectively. Let T, A, G be m a p p i n g s from H into itself and f : H ~ ( - ~ , + ~ ] . We shall study in this paper the following generalized variational inequality: Find x 6 H such that (Gv - Gx, Tx - Ax) >~f ( x ) - f ( v )

for all v e H.

(1)

The m o t i v a t i o n of studying the P r o b l e m (1) is that it has deep connections with nonlinear analysis and it also has i m p o r t a n t applications in optimization theory, engineering, structural mechanics, elasticity theory, lubrication theory, economics, variational calculus, transportation, network equilibrium, etc. [2, 3, 6-8, 10]. B e f o r e we proceed any further we m a k e the following observations. (i) If f is the indicator function of some closed convex subset K of H, i.e.,f ( v ) = 0 for v e K and f(v) = + ~ otherwise, then inequality (1) is equivalent to finding x e K such that (Gv-Gx,

Tx-Ax)>~O

for a l l v e K .

*The work of this paper was supported by the National Science Council grant NSC 83-0208-M110-020. 0167-6377/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD! 0 1 6 7 - 6 3 7 7 ( 9 3 ) E 0 0 8 0 - B

(2)

J.-C. Yao / Operations Research Letters 15 (1994) 35 40

36

If we further assume that G K = K, then (2) is equivalent to finding x ~ H such that Gx • K and (Gv - Gx, Tx - Ax) >~ O

for all Gv • K .

(3)

The inequality (3) is now known as the general mildly nonlinear variational inequality studied by Noor [11] where an iterative algorithm is suggested using a projection method to find the approximate solution of (3). (ii) If A -- 0, then (1) reduces to finding x • H such that (Gv - Gx, Tx) >~f ( x ) - f ( v )

for all v • H.

(4)

The problem (4) is similar to the mixed variational inequality problem studied by Noor [12]. (iii) If G is the identity mapping on H a n d f i s the indicator function of some closed convex subset K of H, then (1) is equivalent to finding x • K such that (v-x,

Tx-Ax)>lO

for a l l v e K .

(5)

Inequalities like (5) are known as the mildly nonlinear variational inequalities which were introduced by Noor [13] in the theory of constrained mildly nonlinear partial differential equations. Some existence results of (5) for nonmonotone operators in Banach space setting are obtained by Yao [15]. (iv) If G K = K, A - 0 a n d f i s the indicator function of some closed convex subset K of H, then (1) is equivalent to finding x • H such that Gx • K and (Gv - Gx, Tx) >>.0

for all Gv • K .

(6)

The inequality (6) was studied by Noor [14] whose approximate solutions can be obtained by an iterative method. We note that Problem (6) has potential applications to the boundary value problems [14]. (v) If G is the identity mapping on H, A = 0 a n d f i s the indicator function of some closed convex subset K of H, then (1) is equivalent to finding x • K such that (v-x,

Tx)>~O

for a l l v • K .

(7)

Inequalities like (7) are known as the classical variational inequalities which have been extensively studied in the literature both in finite and infinite dimensional spaces, See, e.g., Allen [1], Browder [2], Harker and Pang [7], Kinderlehrer and Stampacchia [8], Lions and Stampacchia [9], Nagurney [10] and the references therein. (vi) If G is the identity mapping on H, A = 0 a n d f = 0 then (1) is equivalent to finding x • H such that Tx = 0,

(8)

and we have the theory of operator equations. Therefore, Problem (1) is more general than Problems (2)-(8). In Section 2, we shall give some preliminaries that will be used throughout this paper. In the final section, we shall state and prove the main results of this paper by employing the fixed point technique used by Glowinski et al. [4]. In particular, we obtain some existence results for both general mildly nonlinear variational inequalities and implicit complementarity problems.

2. Preliminaries

We first recall the following definitions. Definition 2.1. Let H be a real Hilbert space and T: H ~ H.

J.-C. Yao / Operations Research Letters 15 (1994) 35 40

37

(i) The m a p p i n g T i s strongly m o n o t o n e with constant ct > 0 if for each pair of distinct elements x, y ~ H, (x-y,

Tx-

Ty) >>.c t N x - yH 2.

(ii) The m a p p i n g T is Lipschitz continuous with constant fl > 0 if for each pair of distinct elements x, y~H, [I Tx - Ty H <~ fl ll x - Y l[.

We note that if the m a p p i n g T is both strongly m o n o t o n e with constant ~z > 0 and Lipschitz continuous with constant fl > 0 then ct ~< ft. Definition 2.2. Let H be a real Hilbert space and f : H - - , ( - oc, + oc]. (i) The functional f i s convex if for each pair of distinct elements x, y • H and for each 0 ~< ct ~< 1, f(ctx + (1 - coy ) ~< ctf(x) + (1 - oOf(y).

(ii) The functional f is lower semicontinuous at x • H /f lim i n f f ( x , ) >~f(x), Xn~X

and f is lower semicontinuous on H if it is lower semicontinuous at each x ~ H. Recall that the epigraph (epi(f)) of a functional f is the set {(x, ~) e H x ~: f ( x ) ~< ~}. Then f is convex if and only if e p i ( f ) is a convex set a n d f i s lower semicontinuous if and only if e p i ( f ) is a closed set. A subset K of H is said to be a cone if 0~x e K for every x e K and e v e r y , / > 0. If a cone K is also convex then it is easy to see that ~x + fly • K for all ~,/~/> 0 and for all x, y e K. I f K is a cone then the dual cone K* of K is the following set K*={y~H:(x,y)>~O

for all x • K}.

3. Main results We now state and prove the main result of this paper. Theorem 3.1. Let H be a real Hilbert space, T, G, A : H --+ H and f : H --+ ( - oo, + oc]. Suppose that the following conditions are satisfied: (i) T is both strongly monotone with constant ~ > 0 and Lipschitz continuous with constant fl, (ii) G is both strongly monotone with constant 7 > 0 and Lipschitz continuous with constant 6, (iii) A is Lipschitz continuous with constant 2 >1 O, (iv) f is convex and lower semicontinuous, (v) the functional x F-+(y, Gx) is convex for each y • H, (vi) [ ( x , x ) + f ( x ) ] / l l x l l ~ + o o , as Ilxll + + ~ , (vii) ct > 62 + (1 - 27 + 62) 1/2 • Then there exists an x • H such that (Gv - Gx, Tx - Ax) >>.f ( x ) - f ( v )

for all v • H.

Proof. Essentially we follow the fixed point technique used by [4, Appendix 1, Section 2, p. 545, L e m m a 2.1]. F o r each u • H and for each 0 < p ~< 1, we consider the auxiliary p r o b l e m of finding w • H satisfying the

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J.-C. Yao / Operations Research Letters 15 (1994) 35-40

following variational inequality problem: P ( p , u):(v -

w, w) + p f ( v ) - p f ( w ) >~ (v - w, u) - p ( G v -

Gw, Tu -

Au)

for all v • H.

F o r fixed 0 < p ~< 1 and fixed u • H the p r o b l e m P ( p , u) has a unique solution. T o see this, let T1 : H --. H and fl : H --* ( - oo, + oo] be defined as follows: Tlx = (x -

u)/p,

x • H,

and fl(x) = f ( x ) + ( G x , T u - A u ) ,

x • n.

Then T1 is strongly m o n o t o n e and continuous and f l is convex and lower semicontinuous by (iv) and (v). Since G is Lipschitz continuous, there is a constant, say, C such that (x, T l x ) + f l ( x ) Ilxll . f ( x ) Ilxll >1

p

(x, x) + f ( x ) + C >1

+ ~

+ C

IIxN

from which it follows by (vi) that (x, T , x ) + f l ( x )

Ilxll

--, + oo,

as Ilxll --, + 00.

Hence by Browder [3, T h e o r e m 1], there exists a unique w • H such that (v - w, Tlw) ~>fl(w) - f l ( v )

for all v • H,

from which it follows that w is the unique solution of P ( p , u). F o r fixed 0 < p ~< 1, define Sp : H --, H by S p u = w where w is the unique solution of P(p, u) for each u • H. It is easy to see that any fixed point of the m a p p i n g Sp for some 0 < p ~< 1 is a solution of(I). Consequently, T h e o r e m 3.1 will be proved if we can show that for some 0 < p ~< 1, Sp is a contraction, i.e., there exists 0<0< 1 such that IlSpx - SpYll ~ OIIx - Yil

for all x, y • H. T o this end, let Ul, u2 • H and wl = S o U l , W 2 = S o u 2 . Then we have (wl - w2,w2) + pf(wl)

- p f ( w 2 ) >1 ( w l - w2, u l ) - p ( G W l -

Gw2, Tul - Aul),

(9)

and (Wz -

Wl, Wl) + pf(wx) - pf(wl)

>i (WE -

w l , u2) - p ( G w 2 -

GWl, Tuz -

Au2).

(10)

Adding (9) and (10), we then have Ilwl - w2112 ~< (w2

-

Wl,/./1

-

p(Gwl

u2) +

-- Gw2,

= (WE -- W l , U l - - U2 -- p ( T u I -- T u 2 ) + p(GWl -

Gw 2 -

TUl -

Aul

p(Gwl

-

--

Tu 2 +

Au2)

GWE, A U l - A u 2 )

w I + WE, T u 1 - - Tu2).

(11)

pz/~Z)llul u2ll IIw, w211,

(12)

By conditions (i)-(iii), we have Ilul - u2 - p ( T u l - - Tu2)II z <~ (1 -- 2pc~ + }(Gwl

-

IIGw1 -

G w 2 , A U l - - Au2)l ~< a21tu, aw2 -

-

w l + w2112 ~< (1 - 2~ + a2)llwl - w2112

N o w combining (11)-(14), we obtain flSpx -

-

uzll z,

Spyll ~ Oflx - Yll

(13) (14)

J.-C. Yao / Operations Research Letters 15 (1994) 35 40

39

where 0 = (1 - 2pet + p2f12)l/2 + p[t~)) + (1 - 27 + ~2)1/2]. It follows that 0 < 0 < 1 for any p satisfying 0
2 f 1 2 _ z 2 , ~,1 ,

(15)

where z = 6y + (1 - 27 + 62) 1/2. Note that fl ~> ~ > r by condition (vii). Consequently, S o is a contraction for any p satisfying (15) and thus has a fixed point from which the result follows. Remark 3.1. Condition (v) of Theorem 3.1 will be satisfied if for example G is affine or in particular linear. If G is the identity mapping then conditions (ii) and (v) are automatically satisfied. We note that for the existence of a solution to problem (7) Theorem 3.1 requires T to be both strongly monotone and Lipschitz. These conditions are stronger than those in the classical result which requires strong monotonicity only [7]. Theorem 3.1 can be employed to obtain existence results for various kinds of variational inequalities. For example, we have the following result on the unification of the calculus of variations. Corollary 3.1. Let H be a real Hilbert space, T: H ~ H and f : H ~ ( - oc, + oc]. Suppose that the following conditions are satisfied: (i) T is both strongly monotone with constant ~ > 0 and Lipschitz continuous, (ii) f is convex and lower semicontinuous, (iii) [ ( x , x ) + f ( x ) ] / l l x l l - - " + ~ , as Ilxll ~ + ~ , Then there exists x E H such that (v - x, Tx) >~f(x) - f ( v )

f o r all v e H.

Remark 3.2. If ct >~ 1 then Corollary 3.1 is a direct consequence of [3, Theorem 1] because in this case we have [(x, T x ) + f ( x ) ] / l l x l l - - , + ~ as Ilxll--' + ~ . We note that conditions (iv) and (vi) of Theorem 3.1 will be satisfied if f is the indicator function of some closed convex subset of H. With this observation, we have the following existence result of the general mildly nonlinear variational inequality (c.f. [9, Theorem 3.1]). Corollary 3.2. Let H be a real Hilbert space, K a closed convex subset o f H and T, G, A : H --, H. Suppose that the following conditions are satisfied: (i) T is both strongly monotone with constant ot > 0 and Lipschitz continuous with constant fl, (ii) G is both strongly monotone with constant 7 > 0 and Lipschitz continuous with constant 3, (iii) A is Lipschitz continuous with constant 2 >>.O, (iv) the functional x~-~(y, Gx) is convex f o r each y ~ H, (v) G K = K,

(vi) ct > 62 + (1 - 2;; + ~2)1/2. Then there exists an x e H such that Gx ~ K and (Gv - Gx, T x - A x ) >~ O f o r all Gv s K.

Corollary 3.3. I f in addition to the hypotheses o f Corollary 3.2, we further assume that K is a cone, then there exists an x ~ H such that G x ~ K and T x - A x ~ K*,

(Gx, T x - A x ) = O.

Proof. By Corollary 3.2, there exists an x e H such that Gx (Gv - Gx, T x - A x ) >~ O

for a l l G v e K .

~ K

and (16)

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J.-C. Yao / Operations Research Letters 15 (1994) 35 40

For each u • K, u = Gv for some v • K. Since K is a convex cone, u + Gx • K. By substituting u + Gx into (16), we get (u, T x - A x ) >1 0 for all u • K. Hence T x - A x • K*. On the other hand, by substituting 0 and 2Gx into (16), respectively, we get (Gx, T x - A x ) <~ 0 and (Gx, T x - A x ) >~ O. Therefore, (Gx, T x - Ax) = 0 and the result follows. Remark 3.3. Under the assumptions of Corollary 3.3, ifA = 0, then Corollary 3.3 says that there exists x e H such that Gx • K , T x • K * and (Gx, Tx) = 0 and we have an existence result for the implicit complementarity problem studied by Isac (c.f. [5, Theorem 1]).

Acknowledgement The author thanks the referees for their comments and suggestions that improved this paper substantially.

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