Generalized quasi-variational-like inclusions with nonconvex functionals

Applied Mathematics and Computation 122 (2001) 267±282 www.elsevier.com/locate/amc

Generalized quasi-variational-like inclusions with nonconvex functionals q Xie Ping Ding Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, People's Republic of China

Abstract In this paper, two new concepts of g-subdi€erential and g-proximal mappings of a proper functional is introduced in Hilbert spaces. The existence and Lipschitz continuity of g-proximal mapping of a proper functional are proved. By applying these concepts, we introduce and study a class of generalized quasi-variational-like inclusions with a nonconvex functional. A new iterative algorithms for ®nding the approximate solutions and the convergence criteria of the iterative sequences generated by the algorithm are also given. These algorithm and existence result generalize many known results in the literature. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Generalized quasi-variational-like inclusion; g-Proximal mapping; Iterative algorithm

1. Introduction In 1994, Hassouni and Mouda® [1] introduced and studied a class of variational inclusions and developed a perturbed algorithm for ®nding approximate solutions of the variational inclusions. Adly [2], Huang [3], Kazmi [4] and Ding [5,6] have obtained some important extensions of the results in [1] in various di€erent assumptions. We observe that all authors, in [1±6], assume that the functionals included in variational inclusions or generalized quasivariational inclusions are proper convex and lower semicontinuous. q

This project was suppoted by the National Natural Science Foundation of China (19871059). E-mail address: [email protected] (X.P. Ding).

0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 0 2 7 - 8

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In this paper, we ®rst introduce two new concepts of g-subdi€erential and gproximal mappings of a proper functional on Hilbert spaces and show the existence and Lipschitz continuity of g-proximal mappings of a proper functional. By using the new concepts, we introduce and study a new class of generalized quasi-variational-like inclusions with a nonconvex functional. An innovative iterative algorithm for ®nding approximate solutions is suggested and analyzed. The convergence criteria of the algorithms is also given. These algorithm and existence result generalize many known results of generalized quasi-variational inclusions and generalized quasi-variational inequalities in literature. 2. Prelimilaries Let H be a real Hilbert space endowed with a norm k
k and a inner product h
;
i. Let CB…H † be the family of all nonempty bounded closed subsets of H . De®nition 2.1. A mapping g : H ! H is said to be (i) c-strongly monotone if there exixts a constant c > 0 such that hg…x†

g…y†; x

yi P ckx

yk

2

8x; y 2 H :

(ii) r-Lipschitz continuous if there exists a constant r P 0 such that kg…x†

g…y†k 6 rkx

yk 8x; y 2 H :

De®nition 2.2. Let E : H ! CB…H † be a set-valued mapping. A mapping N : H  H ! H is said to be (i) a-strongly monotone with respect to E in the ®rst argument if there exists a constant a > 0 such that hN …u;
†

N …v;
†; x

yi P akx

yk

2

8x; y 2 H ; u 2 E…x†; v 2 E…y†:

(ii) b-Lipschitz continuous in the ®rst argument if there exists a constant b > 0 such that kN …u;
†

N …v;
†k 6 bku

vk

8u; v 2 H :

(iii) A mapping T : H ! H is said to be a-strongly monotone with respect to E if there exists a constant a > 0 such that hT …u†

T …v†; x

yi P akx

yk

2

8x; y 2 H ; u 2 E…x†; v 2 E…y†:

(iv) E is said to be -Lipschitz continuous if there exists a constant  > 0 such that H …E…x†; E…y†† 6 kx

yk 8x; y 2 H ;

where H …
;
† is the Hausdor€ metric on CB…H †.

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269

In a similar way, we can de®ne the n-Lipschitz continuity of N …
;
† in the second argument. De®nition 2.3 (Zhou and Chen [7]). A functional f : H  H ! R [ f‡1g is said to be 0-diagonally quasi-concave (in P short, 0-DQCV) in x if for P any ®nite set fx1 ; . . . ; xn g  H and for any y ˆ niˆ1 ki xi with ki P 0 and niˆ1 ki ˆ 1, min f …xi ; y† 6 0:

16i6n

De®nition 2.4 (Noor [8]). A mapping g : H  H ! H is said to be (i) d-strongly monotone if there exists a constant d > 0 such that hg…x; y†; x

yi P dkx

yk

2

8x; y 2 H ;

(ii) s-Lipschitz continuous if there exists a constant s > 0 such that kg…x; y†k 6 skx

yk

8x; y 2 H :

De®nition 2.5. Let g : H  H ! H be a single-valued mapping. A proper functional / : H ! R [ f‡1g is said to be g-subdi€erentiable at a point x 2 H if there exists a point f  2 H such that /…y†

/…x† P hf  ; g…y; x†i 8y 2 H ;

where f  is called a g-subgradient of / at x. The set of all g-subgradients of / at x is denoted by D/…x†. The mapping D/ : H ! 2H de®ned by D/…x† ˆ ff  2 H : /…y†

/…x† P hf  ; g…y; x†i 8y 2 H g

…2:1†

is said to be g-subdi€erential of / at x. Remark 2.1. If g…y; x† ˆ y x 8y; x 2 H and / is a proper convex lower semicontinuous functional on H , then De®nition 2.5 reduces to the usual de®nitions of subdi€erential of a functional /. If / is di€erentiable at x 2 H and satis®es /…x ‡ kg…y; x†† 6 k/…y† ‡ …1

k†/…x†

8y 2 H ; k 2 ‰0; 1Š;

then / is g-subdi€erentiable at x 2 H , see [9, p. 424]. De®nition 2.6. Let g : H  H ! H be a mapping and / : H ! R [ f‡1g be a proper functional such that / is a g-subdi€erentiable on H . If for each x 2 H and for any q > 0, there exists a unique point u 2 H such that

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hu

x; g…y; u†i P q/…u†

q/…y† 8y 2 H ;

…2:2†

then the mapping x 7! u, denoted by JqD/ …x†, is said to be g-proximal mapping of /. By (2.1) and the de®nition of JqD/ …x†, we have x that

u 2 qD/…u†. It follows

1

JqD/ …x† ˆ …I ‡ qD/† …x†; where I is the identity mapping on H . Remark 2.2. If g…y; x† ˆ y x 8y, x 2 H and / is a proper convex lower semicontinuous functional on H , then De®nition 2.6 reduces to the usual 1 de®nition of the proximal mapping of / and JqD/ …x† ˆ Jqo/ …x† ˆ …I ‡ qo/† …x†: Let E; F ; G : H ! CB…H † be set-valued mappings, N ; g : H  H ! H and g : H ! H be single-valued mappings. Let / : H  H ! R [ f‡1g be a proper functional such that for each ®xed y 2 H , /…
; y† : H ! R [ f‡1g is a lower semicontinuous and g-subdi€erentiable on H and g…H † \ dom D/…
; y† 6ˆ ;: We will consider the following generalized quasivariational-like inclusion problem:  Find x 2 H ; u 2 E…x†; v 2 F …x† and z 2 G…x† such that …2:3† hN …u; v†; g…y; g…x††i P /…g…x†; z† /…y; z† 8y 2 H : Special cases (I) If N …u; v† ˆ u v 8u; v 2 H , E; F : H ! H are both single-valued mappings and G is the identity mapping, then problem (2.3) reduces to the following general quasi-variational-like problem:  Find x 2 H such that g…x† 2 dom D…
; x† and …2:4† hE…x† F …x†; g…y; g…x††i P /…g…x†; x† /…y; x† 8y 2 H : Problem (2.4) was introduced and studied by Ding and Lou [10]. (II) If g…y; x† ˆ y x 8y; x 2 H and for each y 2 H , /…
; y† is a proper convex lower semicontinuous functional, then problem (2.3) reduces to the following generalized implicit quasi-variational inclusion problem:  Find x 2 H ; u 2 E…x†; v 2 F …x† and z 2 G…x† such that …2:5† hN …u; v†; y g…x†i P /…g…x†; z† /…y; z† 8y 2 H : Problem (2.5) includes many classes of variational inclusions and generalized quasi-variational inclusions considered by Hassouni and Mouda® [1], Kazmi [4] and Ding [5,6] as special cases.

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271

(III) If /…x; y† ˆ /…x† 8x; y 2 H , G is the identity mapping on H and N …u; v† ˆ f …u† p…v† 8u; v 2 H ; where f ; p : H ! H are single-valued mappings, then problem (2.5) reduces to the following problem:  Find x 2 H ; u 2 E…x† and v 2 F …x† such that …2:6† hf …u† p…v†; y g…x†i P /…g…x†† /…y† 8y 2 H : Problem (2.6) is called set-valued nonlinear generalized variational inclusion problem which was introduced by Huang [3] and, in turn, includes the variational inclusion problems studied by Hassouni and Mouda® [1] and Kazmi [4] as special cases. (IV) If K : H ! 2H is a given set-valued mapping such that each K…x† is a closed convex subset of H (or K…x† ˆ m…x† ‡ K where m : H ! H and K is a closed convex subset of H ) and if g…x; y† ˆ x y for all x; y 2 H , / : H  H ! H is de®ned by /…x; z† ˆ IK…z† …x† 8x; z 2 H ; where IK…z† …x† is the indicator function of K…z†, i.e.,  0; if x 2 K…z†; IK…z† …x† ˆ ‡1; otherwise; then the problem (2.5) reduces to the following generalized nonlinear quasivariational inequality problem: 

Find x 2 H ; u 2 E…x†; v 2 F …x† and z 2 G…x† such that g…x† 2 K…z† and hN …u; v†; y g…x†i P 0 8y 2 K…z†: …2:7†

(V) If N …u; v† ˆ S…u† ‡ T …v† 8u; v 2 H , where S; T : H ! H are given singlevalued mappings, then problem (2.7) reduces to the following completely generalized strongly nonlinear implicit quasi-variational inequality problem considered by Huang [11]:  Find x 2 H ; u 2 E…x†; v 2 F …x† and z 2 G…x† such that …2:8† g…x† 2 K…z†; and hS…u† ‡ T …v†; y g…x†i P 0 8y 2 K…z†: Problems (2.5) and (2.8) include a number classes of variational and quasivariational inequalities, and complementarity and quasi-complementarity problems, studied previously by many authors as special cases, see [11±22]. The proof of the following result can be found in [23]: Lemma 2.1. Let D be a nonempty convex subset of a topological vector space and f : D  D ! ‰ 1; ‡1Š be such that (i) for each x 2 D, y 7! f …x; y† is lower semicontinuous on each compact subset of D,

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Pm (ii) for each Pm finite set fx1 ; . . . ; xm g  D and for each y ˆ iˆ1 ki xi with ki P 0 and iˆ1 ki ˆ 1, min1 6 i 6 n f …xi ; y† 6 0, (iii) there exist a nonempty compact convex subset D0 of D and a nonempty compact subset K of D such that for each y 2 D n K, there is an x 2 co…D0 [ fyg† satisfying f …x; y† > 0: Then there exists y^ 2 D such that f …x; y^† 6 0 for all x 2 D. Now we give some sucient conditions which guarantee the existence and Lipschitz continuity of the g-proximal mapping of a proper functional on a Hilbert space. Theorem 2.1. Let g : H  H ! H be continuous and d-strongly monotone such that g…x; y† ˆ g…y; x† for all x; y 2 H and for any given x 2 H , the function h…y; u† ˆ hx u; g…y; u†i is 0-DQCV in y. Let / : H ! R [ f‡1g be a lower semicontinuous g-subdifferentiable proper functional on H . Then for any given q > 0 and x 2 H , there exist a unique u 2 H such that hu i.e., u ˆ

x; g…y; u†i P q/…u†

q/…y† 8y 2 H ;

…2:9†

JqD/ …x†:

Proof. For any given q > 0 and x 2 H , de®ne a functional f : H  H ! R [ f‡1g by f …y; u† ˆ hx

u; g…y; u†i ‡ q/…u†

q/…y† 8y; u 2 H :

Since g…y; u† is continuous and / is lower semicontinuous, we have that for each y 2 H , u 7! f …y; u† is lower semicontinuous on H . We claim that f …y; u† satis®es the condition (ii) ofP Lemma 2.1. If it is false, then Pm there exist a ®nite set m fy1 ; . . . ; ym g  H and u0 ˆ iˆ1 ki yi with ki P 0 and iˆ1 ki ˆ 1 such that u0 ; g…yi ; u0 †i ‡ q/…u0 †

hx

q/…yi † > 0

8i ˆ 1; . . . ; m:

Since / is g-subdi€erentiable at u0 , there exists a point fu0 2 H such that /…y†

/…u0 † P hfu0 ; g…y; u0 †i 8y 2 H :

It follows that hx

u0 ; g…yi ; u0 †i > q/…yi †

q/…u0 †  P hqfu0 ; g…yi ; u0 †i

8i ˆ 1; . . . ; m:

Hence we must have m X iˆ1

ki hx

qfu0

u0 ; g…yi ; u0 †i > 0:

On the other hand, by the assumption, h…y; u† ˆ hx 0-DQCV in y, we have

…2:10† fu0

u; g…y; u†i is

X.P. Ding / Appl. Math. Comput. 122 (2001) 267±282 m X

ki hx

iˆ1

fu0

273

u0 ; g…yi ; u0 †i 6 0

which contradicts the inequality (2.10). Hence f …y; u† satis®es the condition (ii) of Lemma 2.1. Now take a ®xed y 2 dom /: Since / is g-subdi€erentiable at y, there exists a point fy 2 H such that /…u†

/… y † P hfy ; g…u; y†i

8u 2 H :

Hence we have f … y ; u† ˆ hx P h y P dk y P k y

u; g… y ; u†i ‡ q/…u† q/… y† u; g… y ; u†i ‡ hx y; g… y ; u†i ‡ qhfy ; g…u; y†i uk

2

ykk y

skx

uk‰dk y

uk

s…kx

uk

sqkfy kk y

yk ‡

uk

qkfy k†Š:

y uk 6 rg: Then D0 ˆ f yg Let r ˆ s=d…kx yk ‡ qkfy k† and K ˆ fu 2 H : k and K are both weakly compact convex subset of H and for each u 2 H n K, there is a y 2 co…D0 [ f y g† such that f … y ; u† > 0. Hence all conditions of Lemma 2.1 are satis®ed. By Lemma 2.1, there exists an u 2 H such that f …y; u† 6 0 8y 2 H ; i.e., h u

x; g…y; u†i P q/… u†

q/…y† 8y 2 H :

Now, we show that u is a unique solution of problem (2.9). Suppose that u1 ; u2 2 H are arbitrary two solutions of problem (2.9). Then we have hu1 hu2

x; g…y; u1 †i P q/…u1 † x; g…y; u2 †i P q/…u2 †

q/…y† 8y 2 H ; q/…y† 8y 2 H :

…2:11† …2:12†

Taking y ˆ u2 in (2.11) and y ˆ u1 in (2.12), and adding these inequalities, we obtain hu1

x; g…u2 ; u1 †i ‡ hu2

Since g…x; y† ˆ (2.13) that dku1

x; g…u1 ; u2 †i P 0:

…2:13†

g…y; x† 8x; y 2 H and g is d-strongly monotone, it follows from 2

u2 k 6 hg…u1 ; u2 †; u1

u2 i 6 0

and hence we must have u1 ˆ u2 : This completes the proof.



Remark 2.3. Theorem 2.1 shows that the existence of the g-proximal mapping JqD/ : H ! H of a lower semicontinuous g-subdi€erntiable proper functional /. We emphasize that the functional / may not be convex in Theorem 2.1. The following example shows that the existence of the mapping g : H  H ! H satisfying all conditions in Theorem 2.1.

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Example. Let H ˆ R be real 8
line and g : R  R ! R be de®ned by if jxyj < 1; if 1 6 jxyj < 2; if 2 6 jxyj:

Then it is easy to see that 2 1. hg…x; y†; x yi P jx yj for all x; y 2 H , i.e., g is 1-strongly monotone; 2. g…x; y† ˆ g…y; x† for all x; y 2 R; 3. jg…x; y†j 6 2jx yj for all x; y 2 R, i.e., g is 2-Lipschitz continuous; 4. for any given x 2 H , the function h…y; u† ˆ hx u; g…y; u†i ˆ …x u† g…y; u† is 0-DQCV Pn in y. If it is false, then Pn there exist a ®nite set fy1 ; . . . ; yn g and u0 ˆ iˆ1 ki yi with ki P 0 and iˆ1 ki ˆ 1 such that for each i ˆ 1; . . . n, 8 if jyi u0 j < 1; < …x u0 †…yi u0 † 0 < h…yi ; u0 † ˆ …x u0 †jyi u0 j…yi u0 † if 1 6 jyi u0 j < 2; : 2…x u0 †…yi u0 † if 2 6 jyi u0 j: It follows that …x 0<

n X

ki …x

u0 †…yi

u0 † > 0 for each i ˆ 1; . . . ; n and hence we have

u0 †…yi

u0 † ˆ …x

u0 †…u0

u0 † ˆ 0

iˆ1

which is impossible. This proves that for any given x 2 R the function h…y; u† is 0-DQCV in y. Therefore g satis®es all assumptions in Theorem 2.1. Theorem 2.2. Let g : H  H ! H be d-strongly monotone and s-Lipschitz continuous such that g…x; y† ˆ g…y; x† for all x; y 2 H and for any given x 2 H , the functional h…y; u† ˆ hx u; g…y; u†i is 0-DQCV in y. Let / : H ! R be a lower semicontinuous g-subdifferentiable proper functional on H and q > 0 be a arbitrary constant. Then the g-proximal mapping JqD/ of / is s=d-Lipschitz continuous. Proof. By Theorem 2.1, the g-proximal mapping JqD/ of / is well-de®ned. For any given x1 ; x2 2 H , we have that u1 ˆ JqD/ …x1 † and u2 ˆ JqD/ …x2 † are such that hu1 hu2

x1 ; g…y; u1 †i P q/…u1 † x2 ; g…y; u2 †i P q/…u2 †

q/…y† 8y 2 H ; q/…y† 8y 2 H :

…2:14† …2:15†

Taking y ˆ u2 in (2.14) and y ˆ u1 in (2.15), and adding these inequalities, we obtain hu1

x1 ; g…u2 ; u1 †i ‡ hu2

x2 ; g…u1 ; u2 †i P 0:

Since g is d-strongly monotone and s-Lipschitz continuous, and g…u1 ; u2 † ˆ g…u2 ; u1 † , we have

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dku1

u2 k2 6 hg…u1 ; u2 †; u1

u2 i

6 hg…u1 ; u2 †; x1 x2 i 6 kg…u1 ; u2 †kkx1 6 skx1 x2 kku1 u2 k: It follows that

D/

Jq …x1 †

JqD/ …x2 † ˆ ku1

275

u2 k 6

s kx1 d

x2 k

x2 k;

i.e., JqD/ is s=d-Lipschitz continuous.  Remark 2.4. If g…x; y† ˆ x Lemma 2.2 in [1].

y for all x; y 2 H , then Theorem 2.2 reduces to

3. Existence and algorithms of solutions We ®rst transfer the generalized quasi-variational-like inclusion problem (2.3) into a ®xed point problem. Theorem 3.1. …x; u; v; z† is a solution of problem (2.3) if and only if …x; u; v; z† satisfies the following relation: g…x† ˆ JqD/…
;z† …g…x†

qN …u; v††;

…3:1† 1

where u 2 E…x†, v 2 F …x†, z 2 G…x†, JqD/…
;z† ˆ …I ‡ qD/…
; z†† is the g-proximal mapping of /…
; z†, I is the indentity mapping on H and q > 0 is a constant. Proof. Assume that …x; u; v; z† satis®es relation (3.1), i.e., g…x† ˆ JqD/…
;z† …g…x†

qN …u; v††: 1

Since JqD/…
;z† ˆ …I ‡ qD/…
; z†† ; the above equality holds if and only if N …u; v† 2 D/…g…x†; z†: By the de®nition of g-subdi€erential of /…
; z†, the above relation holds if and only if /…y; x†

/…g…x†; z† P

hN …u; v†; g…y; g…x††i

8y 2 H ;

and hence hN …u; v†; g…y; g…x††i P /…g…x†; z†

/…y; z†

8y 2 H ;

i.e., …x; u; v; z† is a solution of the generalized quasi-variational-like inclusion problem (2.3).  Remark 3.1. Eq. (3.1) can be rewritten as

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x ˆ …1

h k†x ‡ k x

g…x† ‡ JqD/…
;z† …g…x†

i qN …u; v†† :

This ®xed point formulation enables us to suggest the following iterative algorithms: Algorithm 3.1. Let E; F ; G : H ! CB…H † be set-valued mappings. Let N : H  H ! H , g : H ! H and g : H  H ! H be single-valued mappings and / : H  H ! R be such that for each given z 2 H , the mapping /…
; z† : H ! R is lower semicontinuous g-subdi€erentiable proper functional on H satisfying g…H † \ dom D/…
; z† 6ˆ ;. For given x0 2 H , u0 2 E…x0 †, v0 2 F …x0 † and z0 2 G…x0 †, let i h x1 ˆ …1 k†x0 ‡ k x0 g…x0 † ‡ JqD/…
;z0 † …g…x0 † qN …u0 ; v0 †† ‡ e0 : By Nadler [24], there exist u1 2 E…x1 †, v1 2 F …x1 † and z1 2 G…x1 † such that ku0 kv0 kz0

u1 k 6 …1 ‡ 1†H …E…x0 †; E…x1 ††; v1 k 6 …1 ‡ 1†H …F …x0 †; F …x1 ††; z1 k 6 …1 ‡ 1†H …G…x0 †; G…x1 ††:

Let x2 ˆ …1

k†x1 ‡ k‰x1

g…x1 † ‡ JqD/…
;z1 † …g…x1 †

qN …u1 ; v1 ††Š ‡ e1 :

By induction, we can de®ne sequences fxn g, fun g, fvn g and fzn g satisfying 8 xn‡1 ˆ …1 k†xn ‡ k‰xn g…xn † ‡ JqD/…
;xn † …g…xn † qN …un ;vn †Š ‡ en ; > > < 1 un 2 E…xn †; kun un‡1 k 6 …1 ‡ …n ‡ 1† †H …E…xn †; E…xn‡1 ††; 1 > > vn 2 F …xn †; kvn vn‡1 k 6 …1 ‡ …n ‡ 1† †H …F …xn †;F …xn‡1 ††; : zn 2 G…xn †; kzn zn‡1 k6 …1 ‡ …n ‡ 1† 1 †H …G…xn †;G…xn‡1 ††; n ˆ 0;1;.. .; where q > 0 and k 2 …0; 1Š are both constant and en 2 H is an error to take into account a possible inexact computation of the g-proximal point. Remark 3.2. If g…x; y† ˆ x y 8x; y 2 H and k ˆ 1, then Algorithm 3.1 reduces to Algorithm 3.1 of Ding [6] and Algorithm 3.2 of Ding [5]. For an suitable choice of the mappings E; F ; G; g; N ; g and /; Algorithm 3.1 includes many known iterative algorithms for generalized variational and quasi-variational inclusions, and generalized variational and quasi-variational inequalities as special cases, for example see [1±6,8±22]. Algorithm 3.2. Let E; F ; G : H ! CB…H † and S; T ; g : H ! H . Let K : H ! 2H be such that for each x 2 H , K…x† is nonempty closed and convex. For any given x0 2 H , u0 2 E…x0 †, v0 2 F …x0 † and z0 2 G…x0 †, by using similar

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277

method in Algorithm 3.1, we can construct the following iterative sequences fxn g, fun g, fvn g and fzn g satisfying 8 xn‡1 ˆ …1 k†xn ‡ k‰xn g…xn † ‡ PK…xn † …g…xn † q…S…un † ‡ T …vn †††Š ‡ en ; > > > > < un 2 E…xn †; kun un‡1 k 6 …1 ‡ …n ‡ 1† 1 †H …E…xn †; E…xn‡1 ††; 1 > > vn 2 F …xn †; kvn vn‡1 k 6 …1 ‡ …n ‡ 1† †H…F …xn †; F …xn‡1 ††; > > : zn 2 G…xn †; kzn zn‡1 k 6 …1 ‡ …n ‡ 1† 1 †H…G…xn †; G…xn‡1 ††; n ˆ 0; 1; . . . ; where q > 0, k 2 …0; 1Š and en is the error. Now we prove the existence of a solution of problem (2.3) and the convergence of Algorithm 3.1. Theorem 3.2. Let E; F ; G : H ! CB…H † be -Lipschitz continuous, j-Lipschitz continuous and f-Lipschitz continuous respectively. Let g : H ! H be c-strongly monotone and r-Lipschitz continuous. Let g : H  H ! H be d-strongly and s-Lipschitz continuous such that g…x; y† ˆ g…y; x† 8x; y 2 H and for each given x 2 H , the function h…y; u† ˆ hx u; g…y; u†i is 0-DQCV in y. Let N : H  H ! H be a-strongly monotone with respect to E and b-Lipschitz continuous in the first argument. Let N …
;
† be n-Lipschitz continuous in the second argument. Let / : H  H ! R be such that for each fixed z 2 H , /…
; z† is a lower semicontinuous g-subdifferentiable proper functional on H satisfying g…H † \ dom D/…
; z† 6ˆ ;: Suppose that there exists a constant q > 0 such that for each x; y; z 2 H

D/…
;x†

…z† JqD/…
;y† …z† 6 lkx yk; …3:2†

Jq and

 s p2 k ˆ 1‡ 1 2c ‡ r ‡ lf < 1; d q 2 sa > dnj…1 k† ‡ …b2 2 n2 j2 †…s2 d2 …1 k† †; q 2 2 ‰sa dnj…1 k†Š …b2 2 n2 j2 †‰s2 d2 …1 k† Š q sa ‡ dnj…k 1† < ; s…b2 2 n2 j2 † s…b2 2 n2 j2 † 1 X kei iˆ1

ei 1 k < 1 8d 2 …0;1†: di …3:3†

Then the iterative sequences fxn g, fun g, fvn g and fzn g generated by Algorithm 3.1 converge strongly to x ; u ; v and z respectively and …x ; u ; v ; z † is a solution of problem (2.3).

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Proof. By Algorithm 3.1, we have

h

kxn‡1 xn k ˆ …1 k†xn ‡ k xn

i g…xn † ‡ JqD/…
;zn † …g…xn † qN …un ; vn †† h ‡ en …1 k†xn 1 k xn 1 g…xn 1 † ‡ JqD/…
;zn 1 † …g…xn 1 †

i

en 1 qN …un 1 ; vn 1 ††

6 …1

k†kxn xn 1 k ‡ kkxn xn 1 …g…xn † g…xn 1 ††k

D/…
;zn † ‡ k J q …g…xn † qN …un ; vn †† JqD/…
;zn 1 † …g…xn 1 †

qN …un 1 ; vn 1 †† ‡ ken en 1 k: …3:4†

By Theorem 2.2 and condition (3.2), we have

D/…
;zn † …g…xn † qN …un ; vn †† JqD/…
;zn 1 † …g…xn 1 †

J q

qN …un 1 ; vn 1 ††

6 kJqD/…
;zn † …g…xn †

qN …un ; vn †† JqD/…
;zn † …g…xn 1 † q…g…xn 1 †

qN …un 1 ; vn 1 ††k ‡ JqD/…
;zn † …g…xn 1 † qN …xn 1 ; vn 1 ††

JqD/…
;zn 1 † …g…xn 1 † qN …un 1 ; vn 1 ††

s 6 kg…xn † g…xn 1 † q…N …un ; vn † N …un 1 ; vn 1 ††k ‡ lkzn zn 1 k d s s 6 kxn xn 1 …g…xn † g…xn 1 ††k ‡ kxn xn 1 q…N …un ; vn † d d s N …un 1 ; vn ††k ‡ q kN …un 1 ; vn † N …un 1 ; vn 1 †k ‡ lkzn zn 1 k: …3:5† d Since g is c-strongly monotone and r-Lipschitz continuous, we obtain p kxn xn 1 …g…xn † g…xn 1 ††k 6 1 2c ‡ r2 kxn xn 1 k:

…3:6†

Since N …
;
† is a-strongly monotone with respect to E and b-Lipschitz continuous in the ®rst argument, and E is -Lipschitz continuous, we obtain kxn

xn ˆ kxn

1

N …un 1 ; vn ††k2

q…N …un ; vn † x n 1 k2

‡ q2 kN …un ; vn † 2

2qhN …un ; vn † N …un 1 ; vn †k 2qakxn

N …un 1 ; vn †; xn

xn 1 i

2 2

xn 1 k ‡ q2 b2 kun

6 kxn

xn 1 k

6 …1

2qa†kxn

6 …1

2qa ‡ q2 b2 2 …1 ‡ n 1 †2 †kxn

2

2

un 1 k

2

xn 1 k ‡ q2 b2 …1 ‡ n 1 † ‰H …E…xn †; E…xn 1 ††Š x n 1 k2 :

2

…3:7†

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279

Since N …
;
† is n-Lipschitz continuous in the second argument and F is jLipschitz continuous, we have kN …un 1 ; vn †

N …un 1 ; vn 1 †k 6 nkvn

vn 1 k

1

6 n…1 ‡ n †H …F …xn †; F …xn 1 †† 6 nj…1 ‡ n 1 †kxn

xn 1 k:

…3:8†

By f-Lipschitz continuity of G, we have kzn

zn 1 k 6 …1 ‡ n 1 †H …G…xn †; G…xn 1 †† 6 f…1 ‡ n 1 †kxn

xn 1 k:

…3:9†

From (3.4)±(3.9) it follows that kxn‡1 6 ‰…1

xn k

s p2 s 1 2c ‡ r ‡ k k† ‡ k 1 ‡ d d 

q 2 1 2qa ‡ q2 b2 2 …1 ‡ n 1 †

s ‡ k qnj…1 ‡ n 1 † ‡ klf…1 ‡ n 1 †Škxn xn 1 k ‡ ken d ˆ ‰kkn ‡ …1 k† ‡ ktn …q†Škxn xn 1 k ‡ ken en 1 k ˆ hn kxn where

xn 1 k ‡ ken

en 1 k

en 1 k;

…3:10†

p kn ˆ …1 ‡ s=d† 1 2c ‡ r2 ‡ lf…1 ‡ n 1 †; q 2 tn …q† ˆ …s=d†‰ 1 2qa ‡ q2 b2 2 …1 ‡ n 1 † ‡ qnj…1 ‡ n 1 †Š and hn ˆ kkn ‡ …1

k† ‡ ktn …q†:

Letting h ˆ kk ‡ …1 k† ‡ kt…q†, where k ˆ …1 ‡ sd† q t…q† ˆ …s=d†‰ 1 2qa ‡ q2 b2 2 ‡ qnjŠ, we have kn ! k; tn …q† ! t…q†; hn ! h;

p 1 2c ‡ r2 ‡ lf and

as n ! 1:

From condition (3.3) it follows that h < 1 and hence there exist n0 > 0 and h0 2 …h; 1† such that hn 6 h0 for all n P n0 : Therefore, by (3.10), we have that kxn‡1

xn k 6 h0 kxn

xn 1 k ‡ ken

en 1 k 8n P n0 :

It follows from (3.11) that for all n > n0 , kxn‡1

xn k 6 hn0

where rn ˆ ken

n0

kxn0 ‡1

xn0 k ‡

nX n0 kˆ1

hk0 1 rn

…k 1† ;

en 1 k 8n > n0 . Hence, for any m P n > n0 , we have

…3:11†

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X.P. Ding / Appl. Math. Comput. 122 (2001) 267±282

kxm

xn k 6

m 1 X

kxi‡1

xi k

iˆn

6

m 1 X iˆn

ˆ

m 1 X iˆn

hi0 n0 kxn0 ‡1 hi0 n0 kxn0 ‡1

xn0 k ‡ xn0 k ‡

" i n0 m 1 X X iˆn m 1 X

kˆ1

" h

i

iˆn

# hk0 1 ri …k 1†

i n0 X ri

…k 1† i …k 1† h kˆ1 0

# :

…3:12†

P1 i Since iˆ1 ri =d < 1 8d 2 …0; 1† and h0 < 1, it follows from (3.12) that kxm xn k ! 0; as n ! 1; and hence fxn g is a Cauchy sequence in H . Let xn ! x . By Algorithm 3.1 and Lipschitz continuity of E, F and G, we have kun‡1 kvn‡1 kzn‡1

un k6…1 ‡ …n ‡ 1† 1 †H …E…xn‡1 †;E…xn ††6…1 ‡ …n ‡ 1† 1 †kxn‡1

xn k;

1

1

xn k;

1

1

xn k:

vn k6…1 ‡ …n ‡ 1† †H …F …xn‡1 †;F …xn ††6…1 ‡ …n ‡ 1† †jkxn‡1 zn k6…1 ‡ …n ‡ 1† †H …G…xn‡1 †;G…xn ††6…1 ‡ …n ‡ 1† †fkxn‡1

It follows that fun g, fvn g and fzn g are also Cauchy sequences in H . We can assume that un ! u , vn ! v and zn ! z , respectively. Noting un 2 E…xn †, we have d…u ; E…x †† 6 ku 6 ku

un k ‡ d…un ; E…xn †† ‡ H …E…xn †; E…x †† un k ‡ kxn

x k ! 0;

as n ! 1:

Hence, we have u 2 E…x †. Similarly, we have v 2 F …x † and z 2 G…x †. It is easy to show that from xn‡1 ˆ …1 k†xn ‡ k‰xn g…xn †‡ JqD/…
;zn † …g…xn † qN …un ; vn ††Š, we can obtain 

g…x † ˆ JqD/…
;z † …g…x †

qN …u ; v ††:

By Theorem 3.1, …x ; u ; v ; z † is a solution of the generalized quasi-variationallike inclusion problem (2.3).  Remark 3.3. For an appropriate choice of the mappings E; F ; G; N ; g; g and /, Theorem 3.2 includes many known results of generalized quasi-variational inclusions as special cases, see [1±6]. We emphasize that for each z 2 H , the functional /…
; z† may be nonconvex in Theorem 3.2. Theorem 3.3. Let E; F ; G : H ! CB…H † be -Lipschitz continuous, j-Lipschitz continuous and f-Lipschitz continuous, respectively. Let S; T ; g : H ! H be bLipschitz continuous, n-Lipschitz continuous and r-Lipschitz continuous, respectively. Let T be a-strongly monotone with respect to E and g be c-strongly monotone. Let K : H ! 2H be such that for each x 2 H , K…x† is nonempty closed convex set in H and the projection operator PK…x† of H onto K…x† satisfies

X.P. Ding / Appl. Math. Comput. 122 (2001) 267±282

kPK…x† …z†

PK…y† …z†k 6 lkx

yk

281

8x; y; z 2 H :

Suppose that there exists a q > 0 such that p k ˆ 2 1 2c ‡ r2 ‡ lf < 1; q a > …1 k†nj ‡ …2 b2 n2 j2 †…2k k 2 †; q 2 ‰a …1 k†njŠ …2 b2 n2 j2 †…2k k 2 † q a ‡ nj…k 1† 6 : 2 b2 n2 j2 2 b2 n2 j2 Then the iterative sequences fxn g, fun g, fvn g and fzn g generated by Algorithm 3.2 converge strongly to x ; u ; v and z , respectively, and …x ; u ; v ; z † is a solution of the completely generalized strongly nonlinear implicit quasi-variational inclusion problem (2.7). Proof. Let g…x; y† ˆ x y and /…x; y† ˆ IK…y† …x† for all x; y 2 H , where IK…y† …x† is the indicator function of K…y†. Then g is 1-strongly monotone and 1-Lipschitz cintinuous. Let N …u; v† ˆ S…u† ‡ T …v† for all u; v 2 H . Then, it is easy to check that N …
;
† is b-Lipschitz continuous in the ®rst argument and n-Lipschitz continuous in the second argument and a-strongly monotone with respect to E in the ®rst argument. The conclusion follows from Theorem 3.2.  Remark 3.4. Theorem 3.3 improves Theorems 4.1 and 4.2 of Huang [11] and, in turn, includes many known results of (generalized) variational and quasivariational inequalities as special cases [8±22]. References [1] A. Hassouni, A. Mouda®, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 (3) (1994) 706±712. [2] S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl. 201 (3) (1996) 609±630. [3] H.J. Huang, Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett. 9 (3) (1996) 25±29. [4] K.R. Kazmi, Mann and Ishikawa perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl. 209 (2) (1997) 572±584. [5] X.P. Ding, Perturbed proximal point algorithm for generalized quasivariational inclusions, J. Math. Anal. Appl. 210 (1) (1997) 88±101. [6] X.P. Ding, Proximal point algorithm with errors for generalized strongly nonlinear quasivariational inclusions, Appl. Math. Mech. 19 (7) (1998) 597±602. [7] X.J. Zhou, G. Chen, Digonal convexity conditions for problems in convex analysis and quasivariational inequalities, J. Math. Anal. Appl. 132 (1988) 213±225. [8] M.A. Noor, Nonconvex functions and variational inequalities, J. Optim. Theory Appl. 87 (3) (1995) 615±630. [9] M.A. Noor, K.I. Noor, T.M. Rassia, Invitation to variational inequalities, in: Analysis, Geometry and Groups: A Riemann Legacy Volume, Hadronic, FL, 1993, pp. 373±448.

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