Math. Nachr. 253, 45 – 54 (2003) / DOI 10.1002/mana.200310044

Generalized multivalued nonlinear quasivariational inclusions

Zeqing Liu∗1 and Shin Min Kang∗∗ 2

1 Department of Mathematics, Liaoning Normal University, P. O. Box 200, Dalian, Liaoning 116029, People’s Republic ofChina

2 Department of Mathematics, Gyeongsang National University, Chinju 660 – 701, Korea

Received 11 October 2001, revised 6 June 2002, accepted 24 June 2002Published online 24 April 2003

Key words Generalized multivalued nonlinear quasivariational inclusion, strongly monotone mappingMSC (2000) 47J20, 49J40

In this paper, we introduce and study a few classes of generalized multivalued nonlinear quasivariational in-clusions and generalized nonlinear quasivariational inequalities, which include many classes of variational in-equalities, quasivariational inequalities and variational inclusions as special cases. Using the resolvent operatortechnique for maximal monotone mapping, we construct some new iterative algorithms for finding the approxi-mate solutions of these classes of quasivariational inclusions and quasivariational inequalities. We establish theexistence of solutions for this generalized nonlinear quasivariational inclusions involving both relaxed Lipschitzand strongly monotone and generalized pseudocontractive mappings and obtain the convergence of iterative se-quences generated by the algorithms. Under certain conditions, we derive the existence of a unique solution forthe generalized nonlinear quasivariational inequalities and obtain the convergence and stability results of theNoor type perturbed iterative algorithm. The results proved in this paper represent significant refinements andimprovements of the previously known results in this area.

1 Introduction

Variational inequalities arise in various models for a lot of mathematical, physical, regional, engineering, andother problems. It is well–known that the theory of variational inequalities provides the most general, natural,simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinearproblems, see, for example, [2], [3], [6] – [10] and the references therein. In recent years, variational inequalitieshave been extended and generalized in various directions for their own sake and for their applications. Quasivari-ational inequalities and variational inclusions are very important generalizations of variational inequalities.

In 1996, Noor [6] and Huang [2] introduced and studied the generalized multivalued strongly nonlinear qua-sivariational inequalities for compact valued mappings and the set–valued nonlinear generalized variational in-clusions for closed bounded valued mappings, respectively, they constructed a few algorithms for finding theapproximate solutions of their quasivariational inequalities and variational inclusions and established the conver-gence of iterative sequences generated by these algorithms. Afterwards, a few researchers [3], [7] have extendedand generalized the resules due to Noor [6] and Huang [2] in various different aspects. Recently, Verma [8] – [10]has considered the solvability based on iterative algorithms for a few classes of variational inequalities involvingrelaxed Lipschitz mappings and generalized pseudo–contractions.

Inspired and motivated by the recent research work going on in this field, we introduce and study a fewclasses of generalized multivalued nonlinear quasivariational inclusions and generalized nonlinear quasivaria-tional inequalities. These classes are the most general and include lots of variational inequalities, quasivariationalinequalities and variational inclusions as special cases. Using the resolvent operator technique for maximal mono-tone mapping, we establish the equivalence between the fixed point problems and the generalized multivaluednonlinear quasivariational inclusions (resp. the generalized nonlinear quasivariational inequalities) and suggest

∗ e–mail: zeqingliu@sina.com.cn∗∗ Corresponding author: e–mail: smkang@nongae.gsnu.ac.kr, Phone: 82 55 751 5966, Fax: 82 55 755 1917

c© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0025-584X/03/25305-0045 $ 17.50+.50/0

46 Liu and Kang: Multivalued nonlinear quasivariational inclusions

and analyze a few iterative algorithms for solving these classes of quasivariational inclusions and quasivariationalinequalities. The convergence analyses are also discussed. The results proved in this paper extend, improve andunify the corresponding results in [2], [3], [6] – [10].

2 Preliminaries

Let H be a Hilbert space endowed with a norm ‖ · ‖ and an inner product 〈·, ·〉, respectively, 2H and CB(H)denote the families of all the nonempty subsets and all the nonempty closed bounded subsets of H , respectively,H(·, ·) denote the Hausdorff metric on CB(H). Suppose that M : H ×H → 2H is a multivalued mapping suchthat for each fixed t ∈ H, M(·, t) : H → 2H is a maximal monotone mapping and g(H) ∩ dom(M(·, t)) �= ∅.Let I denote the identity mapping on H .

Given mappings A, B, C, D : H → 2H , g, a, b, c, d : H → H, N : H × H × H → H and f ∈ H , weconsider the following problem:

Find u ∈ H, x ∈ Au, y ∈ Bu, z ∈ Cu, w ∈ Du such that gu ∈ dom (M(·, dw)) and

f ∈ gu − N(ax, by, cz) + M(gu, dw) , (2.1)

which is called the generalized multivalued nonlinear quasivariational inclusion.Problem (2.1) has many important and significant applications in fluid flow through porous media, oceanogra-

phy, elasticity, structural analysis, optimization, and operations research. See for example [6] and the referencestherein.

Special Cases(a) If a = b = c = d = I , and A, B, C and D are singlevalued mappings, then problem (2.1) collapses to

finding u ∈ H such that gu ∈ dom(M(·, Du)) and

f ∈ gu − N(Au, Bu, Cu) + M(gu, Du) , (2.2)

which is called the generalized nonlinear quasivariational inequality.(b) If f = 0, a = b = c = d = C = I , and N(x, y, z) = gz −N(x, y) for all x, y, z ∈ H , then problem (2.1)

is equivalent to finding u ∈ H, x ∈ Au, y ∈ Bu, z ∈ Du such that gu ∈ dom(M(·, z)) and

0 ∈ N(x, y) + M(gu, z) , (2.3)

which is known as the generalized nonlinear set–valued mixed quasivariational inequality, and studied by Huanget al. [3].

Remark 2.1 For appropriate and suitable choices of the mappings g, a, b, c, d, A, B, C, D, N, M and theelement f, a number of known classes of variational and quasivariational inequalities, studied previously by afew authors in [2], [3], [6] – [10] can be obtained as special cases of problems (2.1) or (2.2).

Let H be a Hilbert space and G : H → 2H be a maximal monotone mapping. For any fixed ρ > 0, themapping JG

ρ : H → H defined by

JGρ (x) = (I + ρG)−1(x) for all x ∈ H

is said to be the resolvent operator of G. It is known that the resolvent operator JGρ is singlevalued and nonex-

pansive.

Definition 2.2 ([1]) Let T be a selfmapping of H, x0 ∈ H and let xn+1 = f(T, xn) define an iterationprocedure which yields a sequence of points {xn}n≥0 in H . Suppose that {x ∈ H : Tx = x} �= ∅ and {xn}n≥0

converges to a fixed point u of T . Let {yn}n≥0 ⊂ H and εn = ‖yn+1 − f(T, yn)‖. If limn→∞ εn = 0 impliesthat limn→∞ yn = u, then the iteration procedure defined by xn+1 = f(T, xn) is said to be T –stable or stablewith respect to T .

Definition 2.3 A mapping g : H → H is said to be strongly monotone and Lipschitz continuous if there existconstants α > 0, β > 0 such that

〈gx − gy, x − y〉 ≥ α ‖x − y‖2 and ‖gx − gy‖ ≤ β ‖x − y‖ for all x , y ∈ H ,

respectively.

Math. Nachr. 253 (2003) 47

Definition 2.4 A mapping N : H × H × H → H is said to be Lipschitz continuous with respect to the firstargument if there exists a constant s > 0 such that

‖N(x, u, v) − N(y, u, v)‖ ≤ s ‖x − y‖ for all x , y , u , v ∈ H .

In a similar way, we can define Lipschitz continuity of the mapping N(·, ·, ·) with respect to the second orthird argument.

Definition 2.5 A multivalued mapping B : H → CB(H) is said to be strongly monotone with respect to themapping b : H → H and the second argument of N : H ×H ×H → H if there exists a constant t > 0 such that

〈N(p, bx, q) − N(p, by, q), u − v〉 ≥ t ‖u − v‖2 for all u , v , p , q ∈ H ,

x ∈ Bu , y ∈ Bv .

Definition 2.6 A multivalued mapping A : H → CB(H) is said to be relaxed Lipschitz with respect to themapping a : H → H and the first argument of N : H × H × H → H if there exists a constant t > 0 such that

〈N(ax, p, q) − N(ay, p, q), u − v〉 ≤ −t ‖u − v‖2 for all u , v , p , q ∈ H ,

x ∈ Au , y ∈ Av .

Definition 2.7 A multivalued mapping C : H → CB(H) is said to be generalized pseudocontractive withrespect to the mapping c : H → H and the third argument of N : H × H × H → H if there exists a constantt > 0 such that

〈N(p, q, cx) − N(p, q, cy), u − v〉 ≤ t ‖u − v‖2 for all u , v , p , q ∈ H ,

x ∈ Cu , y ∈ Cv .

Definition 2.8 A multivalued mapping A : H → CB(H) is said to be H–Lipschitz continuous if there existsa constant t > 0 such that

H(Ax, Ay) ≤ t ‖x − y‖ for all x , y ∈ H .

Lemma 2.9 ([4]) Let {αn}n≥0, {βn}n≥0 and {δn}n≥0 be nonnegative sequences satisfying

αn+1 ≤ (1 − γn)αn + βnγn + δn for all n ≥ 0 ,

where {γn}n≥0 ⊂ [0, 1],∑∞

n=0 γn = ∞, limn→∞ βn = 0 and∑∞

n=0 δn < ∞. Then

limn→∞αn = 0 .

3 Existence and convergence results

In this section, we establish a few existence results of solutions for problem (2.1) involving relaxed Lipschitzand strongly monotone and generalized pseudocontractive mappings and prove several convergence theorems ofiterative schemes generated by Algorithm 3.3.

Lemma 3.1 Let ρ and t be positive parameters. Then the following conditions are equivalent:(i) problem (2.1) has a solution u ∈ H, x ∈ Au, y ∈ Bu, z ∈ Cu, w ∈ Du with gu ∈ dom(M(·, dw));

(ii) there exist u ∈ H, x ∈ Au, y ∈ Bu, z ∈ Cu, w ∈ Du satisfying

gu = JM(·,dw)ρ ((1 − ρ)gu + ρN(ax, by, cz) + ρf)) , (3.1)

where JM(·,dw)ρ denotes the resolvent operator of M(·, dw);

(iii) the multivalued mapping F : H → 2H defined by

Fq =⋃

x∈Aq,y∈Bq,z∈Cq,w∈Dq

[(1 − t)q+t

(q − gq+JM(·,dw)

ρ ((1 − ρ)gq+ρN(ax, by, cz)+ρf))]

for all q ∈ H

(3.2)

has a fixed point u ∈ H.

48 Liu and Kang: Multivalued nonlinear quasivariational inclusions

P r o o f. Note that (3.1) holds if and only if

(1 − ρ)gu + ρN(ax, by, cz) + ρf ∈ gu + ρM(gu, dw) ,

which is equivalent to

f ∈ gu − N(ax, by, cz) + M(gu, dw) .

On the other hand, F has a fixed point u ∈ H if and only if there exist x ∈ Au, y ∈ B(u), z ∈ Cu, w ∈ Dusuch that

u = (1 − t)u + t(u − gu + JM(·,dw)

ρ ((1 − ρ)gu + ρN(ax, by, cz) + ρf)),

which is equivalent to (3.1). This completes the proof.

Remark 3.2 Lemma 3.1 extends Lemma 2.1 in [2], Lemma 3.1 in [3], [6], [7] and Theorem 2.1 in [10].

Based on Lemma 3.1 and Nadler’s result, we suggest the following algorithm for problem (2.1).

Algorithm 3.3 Let g, a, b, c, d : H → H, A, B, C, D : H → CB(H), N : H × H × H → H and f ∈ H .Given u0 ∈ H , x0 ∈ Au0, y0 ∈ Bu0, z0 ∈ Cu0 and w0 ∈ Du0, compute un+1 by the iterative scheme

un+1 = (1 − t)un + t(un − gun + JM(·,dwn)

ρ ((1 − ρ)gun + ρN(axn, byn, czn) + ρf)), (3.3)

xn ∈ Aun , ‖xn − xn+1‖ ≤ (1 + (n + 1)−1

)H(Aun, Aun+1) ,

yn ∈ Bun , ‖yn − yn+1‖ ≤ (1 + (n + 1)−1

)H(Bun, Bun+1) ,

zn ∈ Cun , ‖zn − zn+1‖ ≤ (1 + (n + 1)−1

)H(Cun, Cun+1) ,

wn ∈ Dun , ‖wn − wn+1‖ ≤ (1 + (n + 1)−1

)H(Dun, Dun+1)

(3.4)

for all n ≥ 0, where t and ρ are positive parameters with t ≤ 1.

Now we construct the Noor type perturbed iterative algorithm as follows:

Algorithm 3.4 Let g, A, B, C, D : H → H, N : H × H × H → H and f ∈ H . Given u0 ∈ H , computeun+1 by the iterative scheme

un+1 = (1 − an)un

+ an

(vn − gvn + JMn(·,Dvn)

ρ [(1 − ρ)gvn + ρN(Avn, Bvn, Cvn) + ρf ])

+ anpn ,

vn = (1 − bn)un

+ bn

(wn − gwn + JMn(·,Dwn)

ρ [(1 − ρ)gwn + ρN(Awn, Bwn, Cwn) + ρf ])

+ bnqn ,

wn = (1 − cn)un

+ cn

(un − gun + JMn(·,Dun)

ρ [(1 − ρ)gun + ρN(Aun, Bun, Cun) + ρf ])

+ cnrn

(3.5)

for all n ≥ 0, where {pn}n≥0, {qn}n≥0 and {rn}n≥0 are sequences of the elements in H introduced to take intoaccount a possible inexact computation, {Mn}n≥0 is a sequence of maximal monotone mappings approximatingM on H × H , and {an}n≥0, {bn}n≥0 and {cn}n≥0 are real sequences satisfying the following conditions:

0 ≤ an , bn , cn ≤ 1 for all n ≥ 0 ; (3.6)∞∑

n=0

an = ∞ . (3.7)

Math. Nachr. 253 (2003) 49

In case cn = 0 and Mn = M for all n ≥ 0, then Algorithm 3.4 reduces to the Ishikawa type perturbediterative algorithm. In case bn = cn = 0 and Mn = M for all n ≥ 0, then Algorithm 3.4 reduces to the Manntype perturbed iterative algorithm.

Remark 3.5 Algorithms 3.3 and 3.4 include Algorithm 2.1 in [2], Algorithms 3.1 and 3.2 in [3], [7], Algo-rithms 4.1 – 4.3 in [6] and Algorithm 3.1 in [8], [9] as special cases.

Theorem 3.6 Assume that f ∈ H and there exists a constant µ > 0 satifying∥∥JM(·,x)

ρ (z) − JM(·,y)ρ (z)

∥∥ ≤ µ ‖x − y‖ for all x , y , z ∈ H , ρ > 0 . (3.8)

Let g, a, b, c, d : H → H be Lipschitz continuous with constants l, α, β, γ, δ, respectively, and g be stronglymonotone with constant h. Let N : H×H×H → H be Lipschitz continuous with constants ξ, η, ζ with respectto the first, second and third arguments, respectively. Assume that A, B, C, D : H → CB(H) are H–Lipschitzcontinuous with constants p, q, r, s, respectively, A is relaxed Lipschitz with constant σ with respect to a andthe first argument of N , B is strongly monotone with constant ν with respect to b and the second argument of N ,and C is generalized pseudocontractive with constant m with respect to c and the third argument of N . Let

k = 2√

1 − 2h + l2 + µδs ,

i = 1 + 2σ + ξ2α2p2 ,

j =√

1 − 2ν + η2β2q2 +√

1 + 2m + ζ2γ2r2 −√

1 − 2h + l2 ≥ 0 ,

P = i − j2 , Q = 1 + σ − (1 − k)j , R = 2k − k2 .

(3.9)

Suppose that there exists a constant ρ ∈ (0, 1] satisfying

k + ρj < 1 , (3.10)

and one of the following conditions:

P > 0 , |Q| >√

RP ,∣∣ρ − QP−1

∣∣ < P−1√

Q − RP ; (3.11)

P = 0 , Q > 0 , ρ > 2−1Q−1R ; (3.12)

P < 0 ,∣∣ρ − QP−1

∣∣ > −P−1√

Q2 − RP . (3.13)

Then problem (2.1) has a solution u ∈ H, x ∈ Au, y ∈ Bu, z ∈ Cu, w ∈ Du with gu ∈ dom(M(·, dw)) andthe sequences {un}n≥0, {xn}n≥0, {yn}n≥0, {zn}n≥0 and {wn}n≥0 defined in Algorithm 3.3 converge stronglyto u, x, y, z, w, respectively.

P r o o f. Put En = (1−ρ)gun +ρN(axn, byn, czn)+ρf and E = (1−ρ)gu+ρN(ax, by, cz) +ρf . Noticethat g is Lipschitz continuous and strongly monotone. This yields that

‖un − un−1 − (gun − gun−1)‖ ≤√

1 − 2h + l2 ‖un − un−1‖ . (3.14)

Since A is relaxed Lipschitz with respect to a and the first argument of N, B is strongly monotone with respect tob and the second argument of N , and C is generalized pseudocontractive with respect to C and the third argumentof N , it follows that

‖(1 − ρ)(un − un−1) + ρ(N(axn, byn, czn) − N(axn−1, byn, czn))‖2

≤ [(1 − ρ)2 − 2ρ(1 − ρ)σ + ρ2ξ2α2p2

(1 + n−1

)2] ‖un − un−1‖2 ,(3.15)

‖N(axn−1, byn, czn) − N(axn−1, byn−1, czn) − (un − un−1)‖2

≤ (1 − 2ν + η2β2q2

(1 + n−1

)2) ‖un − un−1‖2 ,(3.16)

and

‖N(axn−1, byn−1, czn) − N(axn−1, byn−1, czn−1) + un − un−1‖2

≤ (1 + 2m + ζ2γ2r2

(1 + n−1

)2)‖un − un−1‖2 .(3.17)

50 Liu and Kang: Multivalued nonlinear quasivariational inclusions

Since JM(·,y)ρ is nonexpansive, by (3.3), (3.4), (3.8), (3.9), (3.14) – (3.17) and the assumptions of Theorem 3.6

we deduce that

‖un+1 − un‖ ≤ (1 − t) ‖un − un−1‖ + t ‖un − un−1 − (gun − gun−1)‖+ t

∥∥JM(·,dwn)ρ (En) − JM(·,dwn)

ρ (En−1)∥∥

+ t∥∥JM(·,dwn)

ρ (En−1) − JM(·,dwn−1)ρ (En−1)

∥∥

≤ (1 − t + t

√1 − 2h + l2 + tµδs

(1 + n−1

)) ‖un − un−1‖+ t(1 − ρ) ‖gun − gun−1 − (un − un−1)‖ (3.18)

+ t ‖(1 − ρ)(un − un−1) + ρ(N(axn, byn, czn) − N(axn−1, byn, czn))‖+ tρ ‖N(axn−1, byn, czn) − N(axn−1, byn−1, czn) − (un − un−1)‖+ tρ ‖N(axn−1, byn−1, czn) − N(axn−1, byn−1, czn−1) + un − un−1‖

≤ (1 − (1 − θn)t) ‖un − un−1‖ ,

where

θn = (2 − ρ)√

1 − 2h + l2 + µδs(1 + n−1

)

+√

(1 − ρ)2 − 2ρ(1 − ρ)σ + ρ2ξ2α2p2(1 + n−1)2

+ ρ√

1 − 2ν + η2β2q2(1 + n−1)2 + tρ√

1 + 2m + ζ2γ2r2(1 + n−1)2

−→ θ = k +√

(1 − ρ)2 − 2ρ(1 − ρ)σ + ρ2ξ2α2p2 + ρj

as n → ∞. From (3.9) and (3.10), we get that

θ < 1 ⇐⇒ (i − j2

)ρ2 − 2ρ(1 + σ − (1 − k)j) < −(

2k − k2). (3.19)

Since one of (3.11) – (3.13) is satisfied, by (3.19) we conclude easily that θ < 1. Put L = 12 (1 + θ). Then there

exists a positive integer T such that θn < L < 1 for all n ≥ T. It follows from (3.18) that

‖un+1 − un‖ ≤ (1 − (1 − L)t) ‖un − un−1‖ for all n ≥ T , (3.20)

which implies that {un}n≥0 is a Cauchy sequence. (3.4) and (3.20) yield that {xn}n≥0, {yn}n≥0, {zn}n≥0 and{wn}n≥0 are Cauchy sequences. Consequently there exist u, x, y, z, w ∈ H satisfying un → u, xn → x,yn → y, zn → z, wn → w as n → ∞. Observe that

d(x, Au) ≤ ‖x − xn‖ + H(Aun, Au) ≤ ‖x − xn‖ + p ‖un − u‖ −→ 0

as n → ∞. That is, x ∈ Au. Similarly, we have y ∈ Bu, z ∈ Cu and w ∈ Du. By virtue of (3.8) and thenonexpansivity of J

M(·,y)ρ , we know that

∥∥JM(·,dwn)ρ (En) − JM(·,dw)

ρ (E)∥∥

≤ µδ ‖wn − w‖ + (1 − ρ)l ‖un − u‖ + ρ(ξα ‖xn − x‖ + ηβ ‖yn − y‖ + ζγ ‖zn − z‖) ,

which implies that

limn→∞JM(·,dwn)

ρ (En) = JM(·,dw)ρ (E) .

It follows from (3.3) that

u = (1 − t)u + t(u − gu + JM(·,dw)

ρ ((1 − ρ)gu + ρN(ax, by, cz) + ρf)).

From the above equation and Lemma 3.1 we obtain that u ∈ H, x ∈ Au, y ∈ Bu, z ∈ Cu and w ∈ Du withgu ∈ dom(W (·, dw)) are a solution of the generalized multivalued nonlinear quasivariational inclusion (2.1).This completes the proof.

Math. Nachr. 253 (2003) 51

Several proofs similar to that of Theorem 3.6 give the following results and are thus omitted.

Theorem 3.7 Let M, f, g, a, b, c, d, N, A, B, C, D, R and k be as in Theorem 3.6. Suppose that ηβq ≥ν, i = ξαp, P = i2 − j2, Q = σ − (1 − k)j and j = l +

√1 − 2ν + η2β2q2 +

√1 + 2m + ζ2γ2r2. If there

exist a constant ρ > 0 satisfying (3.10) and one of (3.11) – (3.13), then the conclusions of Theorem 3.6 hold.

Theorem 3.8 Let M, f, g, a, b, c, d, N, A, C, D, P, R and k be as in Theorem 3.6. Let B : H → CB(H)be H–Lipschitz continuous with constant q, and Q = 1+σ−m−(1−k)j, i = 1+2(σ−m)+(ξαp+ζγr)2 andj = ηβq − √

1 − 2h + l2 ≥ 0. If there exists a constant ρ ∈ (0, 1] satisfying (3.10) and one of (3.11) – (3.13),then the conclusions of Theorem 3.6 hold.

Theorem 3.9 Let M, f, g, a, b, c, d, N, A, B, C, D, R and k be as in Theorem 3.8. Let i = ξαp, j =l + ηβq, P = i2 − j2 and Q = σ − m − (1 − k)j. If there exist a constant ρ > 0 satisfying (3.10) and one of(3.11) – (3.13), then the conclusions of Theorem 3.6 hold.

Remark 3.10 Theorems 3.6 – 3.9 extend, improve and unify Theorem 3.1 in [2], [7] – [9], Theorem 4.1 in [3],[6] and Theorem 2.2 in [10].

4 Stability results

In this section, we show some existence results of a unique solution for problem (2.2) dealing with relaxedLipschitz mappings, strongly monotone mappings and generalized pseudocontractive mappings, and derive a fewconvergence and stability results of Algorithm 3.4.

Theorem 4.1 Let M, f, g, A, B, C, D, P, Q, R and N be as in Theorem 3.6. Let {Mn}n≥0 be a sequenceof multivalued mappings from H × H into 2H such that for each fixed y ∈ H and n ≥ 0, Mn(·, y) is a maximalmonotone mapping on H , and

∥∥JMn(·,x)ρ (z) − JMn(·,y)

ρ (z)∥∥ ≤ µ ‖x − y‖ for all x , y , z ∈ H , n ≥ 0 , ρ > 0 ; (4.1)

limn→∞

∥∥JMn(·,x)ρ (y) − JM(·,x)

ρ (y)∥∥ = 0 for all x , y ∈ H , ρ > 0 . (4.2)

Let A be relaxed Lipschitz with constant σ with respect to I and the first argument of N, B be strongly monotonewith constant ν with respect to I and the second argument of N , and C be generalized pseudocontractive withconstant m with respect to I and the third argument of N . Let {zn}n≥0 be any sequence in H and define{εn}n≥0 ⊂ [0, +∞) by

εn =∥∥zn+1 −

{(1 − an)zn

+ an

[yn − gyn + JMn(·,Dyn)

ρ ((1 − ρ)gyn + ρN(Ayn, Byn, Cyn) + ρf)]+ anpn

}∥∥ ,

yn = (1 − bn)zn

+ bn

[xn − gxn + JMn(·,Dxn)

ρ

((1 − ρ)gxn + ρN(Axn, Bxn, Cxn) + ρf

)]+ bnqn ,

xn = (1 − an)zn

+ cn

[zn − gzn + JMn(·,Dzn)

ρ

((1 − ρ)gzn + ρN(Azn, Bzn, Czn) + ρf

)]+ cnrn

for all n ≥ 0, and

k = 2√

1 − 2h + l2 + µs , i = 1 + 2σ + ξ2p2 ,

j =√

1 − 2ν + η2q2 +√

1 + 2m + ζ2r2 −√

1 − 2h + l2 ≥ 0 ;(4.3)

limn→∞ ‖pn‖ = 0 , (4.4)

and one of the following conditions holds:

the sequences {qn}n≥0 and {rn}n≥0 are bounded and limn→∞ bn = 0 ; (4.5)

the sequence {rn}n≥0 is bounded and limn→∞ ‖qn‖ = lim

n→∞ cn = 0 ; (4.6)

limn→∞ ‖qn‖ = lim

n→∞ ‖rn‖ = 0 . (4.7)

52 Liu and Kang: Multivalued nonlinear quasivariational inclusions

If there exists a constant ρ ∈ (0, 1] satisfying (3.10) and one of (3.11) – (3.13), then problem (2.2) hasa unique solution x ∈ H with gx ∈ dom(W (·, Dx)) and the sequence {un}n≥0 defined in Algorithm 3.4converges strongly to x. Moreover, if there exists a constant a such that

an ≥ a > 0 for all n ≥ 0 , (4.8)

then limn→∞ zn = x if and only if limn→∞ εn = 0.

P r o o f. Let x, y be arbitrary elements in H, Ex = (1−ρ)gx+ρN(Ax, Bx, Cx)+ρf and F be the mappingdefined in Lemma 3.1. Following the method of proof in Theorem 3.6 and using the assumptions in Theorem4.1, we conclude that

‖Fx − Fy‖ ≤ (1 − t) ‖x − y‖ + t ‖x − y − (gx − gy)‖+ t

∥∥JM(·,Dx)ρ (Ex) − JM(·,Dx)

ρ (Ey)∥∥

+ t∥∥JM(·,Dx)

ρ (Ey) − JM(·,Dy)ρ (Ey) ‖

≤ (1 − (1 − θ)t) ‖x − y‖ ,

(4.9)

where

θ = k +√

(1 − ρ)2 − 2ρ(1 − ρ)σ + ρ2ξ2p2 + ρj . (4.10)

It is easy to verify that (3.10) and one of (3.11) – (3.13) imply that θ ∈ (0, 1). Note that t ∈ (0, 1]. It follows from(4.9) that F is a Banach contraction mapping. Therefore F has a unique fixed point x ∈ H, which is a uniquesolution of problem (2.2) by Lemma 3.1. By (3.5) and (3.6) we know that

‖un+1 − x‖ ≤ (1 − an) ‖un − x‖+ an

∥∥vn − x − (gvn − gx) + JMn(·,Dvn)ρ (Evn) − JM(·,Dx)

ρ (Ex)∥∥

+ an‖pn‖≤ (1 − an) ‖un − x‖ + θan ‖vn − x‖ + anhn + an ‖pn‖ ,

(4.11)

where

hn =∥∥JMn(·,Dx)

ρ (Ex) − JM(·,Dx)ρ (Ex)

∥∥ −→ 0 as n → ∞ . (4.12)

Similarly, we have

‖vn − x‖ ≤ (1 − bn) ‖un − x‖ + θbn ‖wn − x‖ + bnhn + bn ‖qn‖ , (4.13)

and

‖wn − x‖ ≤ (1 − (1 − θ)cn) ‖un − x‖ + cnhn + cn ‖rn‖ . (4.14)

Substituting (4.13) and (4.14) into (4.11), we get that

‖un+1 − x‖ ≤ (1 − (1 − θ)an) ‖un − x‖+ an(bn(cn ‖rn‖ + hn + ‖qn‖ ) + hn + ‖pn‖ ) .

(4.15)

It follows from Lemma 2.9, (3.6), (3.7), (4.4), (4.10), (4.12), (4.15) and one of (4.5) – (4.7) that limn→∞ un = x.As in the proof of (4.15), by (4.8) we infer that

∥∥(1 − an)zn + an

[yn − gyn + JMn(·,Dyn)

ρ (E(yn))]+ anpn − x

∥∥

≤ (1 − (1 − θ)a) ‖zn − x‖ + bn(cn ‖rn‖ + hn + ‖qn‖ ) + hn + ‖pn‖ .(4.16)

Math. Nachr. 253 (2003) 53

Suppose that limn→∞ zn = x. According to (4.4), (4.12), (4.16) and one of (4.5) – (4.7), we conclude that

ε ≤ ‖zn+1 − x‖ +∥∥(1 − an)zn + an

[yn − gyn + J

Mn(·,Dyn)ρ (E(yn))

]+ anpn − x

∥∥≤ ∥∥zn+1 − x‖ + (1 − (1 − θ)a) ‖zn − x‖ + bn(cn ‖rn‖ + hn + ‖qn‖ ) + hn + ‖pn‖−→ 0

as n → ∞. That is, limn→∞ εn = 0.Conversely, suppose that limn→∞ εn = 0. By virtue of (4.16), we obtain that

‖zn+1 − x‖ ≤ (1 − (1 − θ)a) ‖zn − x‖ + bn(cn ‖rn‖ + hn + ‖qn‖ ) + hn + ‖pn‖ + εn . (4.17)

It follows from Lemma 2.9, (4.4), (4.17) and one of (4.5) – (4.7) that limn→∞ αn = 0. That is, limn→∞ zn = x.This completes the proof.

A few proofs similar to that of Theorems 3.6 – 3.9 and 4.1 give the following results and are thus omitted.

Theorem 4.2 Let M, f, g, A, B, C, D, R, k, {Mn}n≥0, {εn}n≥0, {xn}n≥0, {yn}n≥0, {zn}n≥0 and N beas in Theorem 4.1, (4.4) and one of (4.5) – (4.7) hold. Suppose that ηq ≥ ν, i = ξq, P = i2−j2, Q= σ−(1−k)jand j = l +

√1 − 2ν + η2q2 +

√1 + 2m + ζ2r2. Suppose that there exists a constant ρ > 0 satisfying (3.10)

and one of (3.11) – (3.13). Then the conclusions Theorem 4.1 hold.

Theorem 4.3 Let M, f, g, A, C, D, P, R, k, {Mn}n≥0, {εn}n≥0, {xn}n≥0, {yn}n≥0, {zn}n≥0 and N beas in Theorem 4.1, (4.4) and one of (4.5) – (4.7) hold. Let B : H → CB(H) be H–Lipschitz continuous withconstant q, and Q = 1+σ−m−(1−k)j, i = 1+2(σ−m)+(ξp+ζr)2 and j = ηq−√

1 − 2h + l2 ≥ 0. Supposethat there exists a constant ρ ∈ (0, 1] satisfying (3.10) and one of (3.11) – (3.13). Then Then the conclusionsTheorem 4.1 hold.

Theorem 4.4 Let M, f, g, A, B, C, D, R, k, {Mn}n≥0, {εn}n≥0, {xn}n≥0, {yn}n≥0, {zn}n≥0 and N beas in Theorem 4.3, (4.4) and one of (4.5) – (4.7) hold. Let P = i2 − j2, Q = σ − m − (1 − k)j, i = ξp + ζrand j = l + ηq. Suppose that there exists a constant ρ > 0 satisfying (3.10) and one of (3.11) – (3.13). Then theconclusions Theorem 4.1 hold.

Remark 4.5 Theorems 4.1 – 4.4 extend Theorem 5.1 of Huang – Bai – Cho – Kang [3] in the following ways:(a) the Ishikawa type perturbed iterative algorithm in [3] is replaced by the more general Noor type perturbed

iterative algorithm.(b) the generalized nonlinear mixed quasivariational inequality involving strongly monotone mappings in [3] is

replaced by the more general generalized nonlinear quasivariational inequality (2.2) involving strongly monotonemappings, relaxed Lipschitz mappigs and generalized pseudocontractive mappings.

(c) conditions (3.11) – (3.13) are weaker than condition (5.4) in [3].(d) the sequences {‖pn‖}n≥0, {‖qn‖}n≥0 and {‖rn‖}n≥0 may not converge to zero simultaneously.

Acknowledgements This work was supported by Korea Research Foundation Grant (KRF–2001–005–D00002).

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