Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces

Nonlinear Analysis 69 (2008) 3259–3272www.elsevier.com/locate/na

Existence and iterative approximations of solutions for mixedquasi-variational-like inequalities in Banach spaces

Zeqing Liua,∗, Zhengsheng Chena, Shin Min Kangb, Jeong Sheok Umec

a Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaoning 116029, People’s Republic of Chinab Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, Republic of Korea

c Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

Received 12 June 2007; accepted 13 September 2007

Abstract

In this paper, we introduce and investigate a new class of mixed quasi-variational-like inequalities in reflexive Banach spaces.By applying a minimax inequality due to Ding–Tan and a lemma due to Chang, we establish some existence and uniqueness resultsof solution for the mixed quasi-variational-like inequality. Next, by using a KKM theorem due to Fan and an auxiliary principletechnique due to Cohen, we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated bythe iterative algorithms. Our results extend, improve and unify several known results in the literature.c© 2007 Elsevier Ltd. All rights reserved.

MSC: 47J20; 49J40

Keywords: Mixed quasi-variational-like inequality; Iterative algorithm; KKM theorem; Auxiliary principle technique; Strongly monotone mapping;Relaxed cocoercive mapping; Cocoercive mapping; Partially relaxed monotone mapping

1. Introduction

It is well-known that variational inequality theory provides the most natural, direct, simple, unified and efficientframework for a general treatment of a wide class of unrelated linear and nonlinear problems arising in elasticity,oceanography, economics, transportation, operations research, structural analysis, and engineering sciences. Fordetails, we refer the reader to [1–32] and the references therein. Cohen [4] established the existence of solutionsfor a class of variational inequalities which is equivalent to a class of auxiliary minimization problems. Ansari andYao [2], Chang [3], Yao [27,28] obtained the existence of solutions for several kinds of variational inequalities andvariational-like inequalities in Hilbert spaces and mixed nonlinear variational-like inequalities. Liu, Ume and Kang[22] studied a class of general strongly nonlinear quasi-variational inequalities in Hilbert spaces. Recently, Deng[9] and Liu, Ume and Kang [24] established the existence and uniqueness results of solutions for two classes of

∗ Corresponding author.E-mail addresses: [email protected] (Z. Liu), [email protected] (Z. Chen), [email protected] (S.M. Kang),

[email protected] (J.S. Ume).

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.09.015

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http://dx.doi.org/10.1016/j.na.2007.09.015

3260 Z. Liu et al. / Nonlinear Analysis 69 (2008) 3259–3272

mixed nonlinear variational-like inequalities and generalized nonlinear variational-like inequalities in reflexive Banachspaces, respectively.

Motivated and inspired by the results in [2–32], we introduce a new class of mixed quasi-variational-likeinequalities in reflexive Banach spaces. By applying a minimax inequality due to Ding and Tan [5] and a lemma due toChang [3], we give the existence and uniqueness theorems of solution for the mixed quasi-variational-like inequalityin reflective Banach spaces under weaker assumptions. For finding the approximate solutions of the mixed quasi-variational-like inequality, we suggest two iterative algorithms, one of which is introduced by the auxiliary principletechnique due to Cohen [4]. Under certain conditions we investigate the convergence criteria of iterative sequencespresented in the algorithms. Our results improve and generalize many known results in this field [2–10,21,22,26–28].

2. Preliminaries

Let D be a nonempty convex subset of a Banach space B, B∗ be a topological dual space of B, and 〈u, v〉 be thepairing between u ∈ B∗ and v ∈ B. Let T, A,G : D → B∗, N : B∗

× B∗× B∗

→ B∗, η : D × D → B be mappings.Assume that ϕ : B × B → (−∞,+∞] is a weakly continuous function, and a : B × B → (−∞,+∞] is a coercivecontinuous bilinear form, that is, there exist c > 0 and d > 0 such that

(A1) a(v, v) ≥ c‖v‖2,∀v ∈ D;

(A2) a(u, v) ≤ d‖u‖‖v‖,∀u, v ∈ D.

For a given f ∗∈ B∗, we consider the following mixed quasi-variational-like inequality problem (MQVLIP):

Find u ∈ D such that

〈N (T u, Au,Gu)− f ∗, η(v, u)〉 + ϕ(v, u)− ϕ(u, u)+ a(u, v − u) ≥ 0, ∀v ∈ D. (1.1)

Special cases:

(A) If N (T u, Au,Gu) = N (T u, Au), a(u, v) = 0 for all u, v ∈ D, then MQVLIP (1.1) reduces to the followingproblem: find u ∈ D such that

〈N (T u, Au)− f ∗, η(v, u)〉 + ϕ(v, u)− ϕ(u, u) ≥ 0, ∀v ∈ D,

which was studied by Ding and Yao [9] and Liu, Ume and Kang [24] in Banach spaces and Liu, Sun, Shim andKang [20] in Hilbert spaces, respectively.

(B) If N (T u, Au,Gu) = N (T u, Au), a(u, v) = 0, ϕ(v, u) = f (v) for all u, v ∈ D, where f : B → (−∞,+∞],then the MQVLIP (1.1) reduces to the following mixed variational-like inequality problem: find u ∈ D such that

〈N (T u, Au)− f ∗, η(v, u)〉 + f (v)− f (u) ≥ 0, ∀v ∈ D,

which was investigated and introduced by Ding [7,8] in Banach spaces, and by Ansari and Yao [2] in Hilbertspaces, respectively.

(C) If N (T u, Au,Gu) = T u − Au, a(u, v) = 0, ϕ(v, u) = f (v), and η(u, v) = gu − gv for all u, v ∈ D, whereg : D → B, then the MQVLIP (1.1) reduces to the following problem: find u ∈ D such that

〈T u − Au, gu − gv〉 ≥ f (u)− f (v), ∀v ∈ D,

which was studied by Yao [27].(D) If N (T u, Au,Gu) = T u, a(u, v) = 0, ϕ(v, u) = f (v), η(u, v) = u −v for all u, v ∈ B, then the MQVLIP (1.1)

collapses to finding u ∈ D such that

〈T u, v − u〉 + f (v)− f (u) ≥ 0, ∀v ∈ D,

which was introduced and studied by Cohen [4].

In brief, for appropriate and suitable choices of the mappings N , T, A,G, η and the functionals ϕ and a, one canobtain a number of known variational inequalities and variational-like inequalities in [2,4,6–10,22–24,27] as specialcases of the MQVLIP (1.1). Thus the MQVLIP (1.1) is a more general and unifying one.

Definition 2.1. Let D be a nonempty subset of a Banach B with the dual B∗. Let T, A,G : D → B∗, N :

B∗× B∗

× B∗→ B∗, and η : D × D → B be mappings.

Z. Liu et al. / Nonlinear Analysis 69 (2008) 3259–3272 3261

(1) N is said to be Lipschitz continuous in the first argument if there exists a constant σ > 0 satisfying

‖N (u, x, y)− N (v, x, y)‖ ≤ σ‖u − v‖, ∀u, v, x, y ∈ B∗;

(2) N is said to be η-cocoercive with respect to T in the first argument if there exists a constant α > 0 satisfying

〈N (T u, x, y)− N (T v, x, y), η(u, v)〉 ≥ α‖N (T u, x, y)− N (T v, x, y)‖2, ∀u, v ∈ D, x, y ∈ B∗;

(3) N is said to be η-strongly monotone with respect to T in the second argument if there exists a constant β > 0satisfying

〈N (x, T u, y)− N (x, T v, y), η(u, v)〉 ≥ β‖u − v‖2, ∀u, v ∈ D, x, y ∈ B∗;

(4) N is said to be η-strongly monotone with respect to T, A and G if there exists a constant ν > 0 satisfying

〈N (Au, T u,Gu)− N (Av, T v,Gv), η(u, v)〉 ≥ ν‖u − v‖2, ∀u, v ∈ D;

(5) N is said to be η-partially relaxed monotone with respect to T in the third argument if there exists a constantγ > 0 satisfying

〈N (x, y, T u)− N (x, y, T v), η(w, v)〉 ≥ −γ ‖w − u‖2, ∀u, v, w ∈ D, x, y ∈ B∗

;

(6) N is said to be η-relaxed cocoercive with respect to T in the second argument if there exists a constant t > 0satisfying

〈N (x, T u, y)− N (x, T v, y), η(u, v)〉 ≥ −t‖N (x, T u, y)− N (x, T v, y)‖2, ∀u, v ∈ D, x, y ∈ B∗;

(7) N is said to be hemicontinuous with respect to T and A in the first and second arguments if for any x, y ∈ D andz ∈ B∗, the mapping t → 〈N (T (t x + (1 − t)y), A(t x + (1 − t)y), z), η(x, y)〉 is continuous on [0, 1];

(8) T is said to be Lipschitz continuous if there exists a constant r > 0 satisfying

‖T u − T v‖ ≤ r‖u − v‖, ∀u, v ∈ D;

(9) η is said to be Lipschitz continuous if there exists a constant δ > 0 satisfying

‖η(u, v)‖ ≤ δ‖u − v‖, ∀u, v ∈ D.

Similarly, we can define the Lipschitz continuity of N in the second and third arguments, respectively.

Definition 2.2 ([32]). Let ψ : D × D → (−∞,+∞] be a functional. ψ is said to be 0-diagonally concave in the firstargument for any finite set {v1, . . . , vn} ⊆ D and any u =

∑ni=1 λivi with λi ≥ 0 and

∑ni=1 λi = 1,

n∑i=1

λiψ(vi , u) ≤ 0.

Definition 2.3. Let D be a nonempty convex subset of a Banach space B, and T, A,G : D → B∗, N :

B∗× B∗

× B∗→ B∗ and η : D × D → B be mappings and f ∗

∈ B∗. N , T, A,G, f ∗ and η are said to havethe 0-diagonally concave relation on D if the function ψ : D × D → (−∞,+∞] defined by

ψ(v, u) = 〈N (T u, Au,Gu)− f ∗, η(u, v)〉, ∀u, v ∈ D,

is 0-diagonally concave in the first argument, that is, for any finite set {v1, . . . , vm} ⊆ D and for any u =∑n

i=1 λiviwith λi ≥ 0 and

∑ni=1 λi = 1,

n∑i=1

λiψ(vi , u) ≤ 0.

Definition 2.4 ([1]). A bifunctional ϕ : B × B → (−∞,+∞] is said to be skew-symmetric if

ϕ(u, u)− ϕ(u, v)− ϕ(v, u)+ ϕ(v, v) ≥ 0, ∀u, v ∈ B.


Definition 2.5 ([2,12]). Let D be a nonempty convex subset of a Banach space B and h : D → R be a Frechetdifferentiable function. h is said to be

(1) η-convex if

h(v)− h(u) ≥ 〈h′(u), η(v, u)〉, ∀u, v ∈ D;

(2) η-strongly convex if there exists a constant α > 0 satisfying

h(v)− h(u)− 〈h′(u), η(v, u)〉 ≥α

2‖u − v‖2, ∀u, v ∈ D.

Lemma 2.1 ([2]). Let η : D × D → B be continuous from the weak topology to the weak topology. Let h : D → Rbe Frechet differentiable such that h′ is continuous from the weak topology to the strong topology, respectively. Thenthe mapping g : D → R defined by g(x) = 〈h′(x), η(y, x)〉 for each fixed y ∈ D is also continuous from the weaktopology to the strong topology.

For D ⊆ B, we define by conv(D) the convex hull of B. The set-valued mapping P : D → 2B is said to be aKKM mapping, if for any finite subset {x1, . . . , xk} of D,

conv({x1, . . . , xk}) ⊆

k⋃i=1

P(xi ).

Lemma 2.2 ([11]). Let D be an arbitrary nonempty set in a topological vector space B, and let P : D → 2B be aKKM mapping. If P(x) is closed for each x ∈ B and is compact for at least one x ∈ B, then

⋂x∈B P(x) 6= ∅.

Lemma 2.3 ([5]). Let D be a nonempty convex subset of a topology vector space B and let ψ : D×D → (−∞,+∞]

be such that

(C1) for each v ∈ D, ψ(v, ·) is lower semicontinuous on each nonempty compact subset of D;(C2) for each nonempty finite subset {v1, . . . , vn} ⊆ D and for each u =

∑ni=1 λivi with λi ≥ 0,

∑ni=1 λi = 1,

min1≤i≤n ψ(vi , u) ≤ 0;(C3) there exists a nonempty compact convex subset D0 of D and a nonempty compact subset K of D such that for

each u ∈ D \ K , there is v ∈ conv(D0⋃

{u}) with ψ(v, u) > 0.

Then there exists u ∈ K , such that ψ(v, u) ≤ 0 for all v ∈ D.

Lemma 2.4 ([3]). Let X be a nonempty closed convex subset of a Hausdorff linear topological space E, and letφ,ψ : X × X → R be mappings satisfying the following conditions:

(C4) ψ(x, y) ≤ φ(x, y) for all x, y ∈ X, and ψ(x, x) ≥ 0 for all x ∈ X;(C5) for each x ∈ X, φ(x, ·) is upper semicontinuous in the second argument;(C6) for each y ∈ X, the set {x ∈ X : ψ(x, y) < 0} is convex;(C7) there exists a nonempty compact set K ⊆ X and x0 ∈ K such that ψ(x0, y) < 0 for all y ∈ X \ K .

Then there exists y ∈ K such that φ(x, y) ≥ 0 for all x ∈ X.

3. Existence and uniqueness of solution

In this section, we study the existence and uniqueness of solution for the MQVLIP (1.1).

Theorem 3.1. Let D be a nonempty closed convex subset of a reflexive Banach space B with dual space B∗. LetT, A,G : D → B∗, N : B∗

× B∗× B∗

→ B∗, and η : D × D → B be mappings. Let f ∗∈ B∗ and

ϕ : B × B → (−∞,+∞] be a weakly continuous bifunctional satisfying int{v ∈ D : ϕ(v, v) < +∞} 6= ∅,and a : B × B → R be a coercive continuous bilinear form with (A1) and (A2). Suppose that

(i) the mapping u 7→ N (T u, Au,Gu) is continuous from the weak topology on B to the strong topology on B∗ andis η-strongly monotone with respect to T, A and G with constant α > d;

(ii) η is Lipschitz continuous with constant λ ≥ 0, satisfying η(u, v) = −η(v, u) for all u, v ∈ D, and for each fixedv ∈ D, η(·, v) is continuous from the weak topology to the weak topology;


(iii) the mappings N , T, A,G, f ∗ and η have the 0-diagonally concave relation on D;(iv) ϕ is skew-symmetric on B × B and for each u ∈ B, ϕ(·, u) is proper convex on B.

Then the MQVLIP (1.1) has a unique solution u ∈ D.

Proof. Define a functional ψ : D × D → (−∞,+∞] by

ψ(v, u) = 〈N (T u, Au,Gu)− f ∗, η(u, v)〉 + ϕ(u, u)− ϕ(v, u)− a(u, v − u), ∀u, v ∈ D.

Now we assert that for each fixed v ∈ D, the function u 7→ 〈N (T u, Au,Gu)− f ∗, η(u, v)〉 is weakly continuouson any weakly compact subset D1 of D. In fact, for any weakly convergent net {uα} ⊆ D1 with uα ⇀ u ∈ D1, itfollows from (i) and (ii) that

‖N (T uα, Auα,Guα)− N (T u, Au,Gu)‖ → 0 and η(uα, v) ⇀ η(u, v).

In view of the weak continuity of η(·, v) for each v ∈ D and the weak compactness of D1, we infer easily that foreach v ∈ D, the set {η(u, v) : u ∈ D1} is also weakly compact on B and hence {η(uα, v)} ⊆ {η(u, v) : u ∈ D1} isbounded. Consequently, we deduce that for each fixed v ∈ D,

|〈N (T uα, Auα,Guα)− f ∗, η(uα, v)〉 − 〈N (T u, Au,Gu)− f ∗, η(u, v)〉|

≤ |〈N (T uα, Auα,Guα)− N (T u, Au,Gu), η(uα, v)〉| + |〈N (T u, Au,Gu)− f ∗, η(uα, v)− η(u, v)〉|

≤ ‖N (T uα, Auα,Guα)− N (T u, Au,Gu)‖‖η(uα, v)‖ + |〈N (T u, Au,Gu)− f ∗, η(uα, v)− η(u, v)〉|

→ 0.

Hence for each fixed v ∈ D, the function

u 7→ ψ(v, u) = 〈N (T u, Au,Gu)− f ∗, η(u, v)〉 + ϕ(u, u)− ϕ(v, u)− a(u, v − u)

is weakly continuous on each weakly compact subset D1 ⊆ D.Next we claim that ψ satisfies the condition (ii) of Lemma 2.3. Otherwise, there exist {v1, . . . , vm} ⊂ D and

u =∑m

i=1 λivi for some λi ≥ 0 with∑m

i=1 λi = 1 such that ψ(vi , u) > 0 for all 1 ≤ i ≤ m, that is,

〈N (T u, Au,Gu)− f ∗, η(u, vi )〉 + ϕ(u, u)− ϕ(vi , u)− a(u, vi − u) > 0, 1 ≤ i ≤ m,

which implies thatm∑

i=1

λi 〈N (T u, Au,Gu)− f ∗, η(u, vi )〉 + ϕ(u, u)−

m∑i=1

λiϕ(vi , u)−

m∑i=1

λi a(u, vi − u) > 0.

Since for each u ∈ B, v → ϕ(v, u) is proper convex and a(u, 0) = 0, it follows thatm∑

i=1

λi 〈N (T u, Au,Gu)− f ∗, η(u, vi )〉 > 0,

which contradicts the fact that the mappings N , T, A,G, f ∗ and η have the 0-diagonally concave relation on D.Therefore the condition (ii) of Lemma 2.3 holds. Note that for each v ∈ B, ϕ(·, v) is proper convex weakly lowersemicontinuous functional and int{v ∈ D : ϕ(v, v) < +∞} 6= ∅. Take v∗

∈ int{v ∈ D : ϕ(v, v) < +∞}. ByProposition I.2.6 of Pascali and Sburlan [25], ϕ(·, v∗) is subdifferentiable at v∗. Hence we obtain that

ϕ(u, v∗)− ϕ(v∗, v∗) ≥ 〈r, u − v∗〉, ∀r ∈ ∂ϕ(·, v∗), u ∈ B.

Since ϕ is skew-symmetric, it follows that

ϕ(u, u)− ϕ(v∗, u) ≥ ϕ(u, v∗)− ϕ(v∗, v∗) ≥ 〈r, u − v∗〉, ∀r ∈ ∂ϕ(·, v∗), u ∈ B.

By the η-strong monotonicity of the mapping u 7→ N (T u, Au,Gu) and η(u, v) = −η(v, u) for all u, v ∈ D, we getthat

ψ(v∗, u) = 〈N (T u, Au,Gu)− f ∗, η(u, v∗)〉 + ϕ(u, u)− ϕ(v∗, u)− a(u, v∗− u)

≥ 〈N (T v∗, Av∗,Gv∗)− N (T u, Au,Gu), η(v∗, u)〉 − 〈N (T v∗, Av∗,Gv∗)− f ∗, η(v∗, u)〉

− 〈r, u − v∗〉 + a(v∗

− u, v∗− u)− a(v∗, v∗

− u)


≥ α‖u − v∗‖

2− λ‖N (T v∗, Av∗,Gv∗)− f ∗

‖‖u − v∗‖ − ‖r‖‖u − v∗

‖

+ c‖u − v∗‖

2− d‖v∗

‖‖u − v∗‖

≥ ‖u − v∗‖[(α + c)‖u − v∗

‖ − λ‖N (T v∗, Av∗,Gv∗)− f ∗‖ − ‖r‖ − d‖v∗

‖].

Let R =1α+c [λ‖N (T v∗, Av∗,Gv∗) − f ∗

‖ + ‖r‖ + d‖v∗‖] and K = {u ∈ D : ‖u − v∗

‖ ≤ R}. Thus K andD0 = {v∗

} are both weakly compact convex subsets of D. For each u ∈ D \ K , there exists v∗∈ conv(D0

⋃{u})

such that ψ(v∗, u) > 0 and the condition (iii) of Lemma 2.3 is satisfied. It follows from Lemma 2.1 that there existsu ∈ D such that ψ(v, u) ≤ 0 for all v ∈ D,

〈N (T u, Au,Gu)− f ∗, η(v, u)〉 − ϕ(u, u)+ ϕ(v, u)+ a(u, v − u) ≥ 0, ∀v ∈ D.

Therefore, u is a solution of the MQVLIP (1.1).Finally we prove the uniqueness of solution for the MQVLIP (1.1). Suppose that u1 and u2 are any two solutions

of the MQVLIP (1.1). It follows that

〈N (T u1, Au1,Gu1)− f ∗, η(v, u1)〉 − ϕ(u1, u1)+ ϕ(v, u1)− a(u1, v − u1) ≥ 0, ∀v ∈ D, (3.1)

〈N (T u2, Au2,Gu2)− f ∗, η(v, u2)〉 − ϕ(u2, u2)+ ϕ(v, u2)− a(u2, v − u2) ≥ 0, ∀v ∈ D. (3.2)

Notice that η(u, v) = −η(v, u) for all u, v ∈ D, (C1) holds and ϕ is skew-symmetric. Taking v = u2 in (3.1) andv = u1 in (3.2), we obtain that

α‖u1 − u2‖2

≤ 〈N (T u1, Au1,Gu1)− N (T u2, Au2,Gu2), η(u1, u2)〉 ≤ d‖u1 − u2‖2,

which implies that u1 = u2 by α > d . This completes the proof. �

Remark 3.1. If N (x, y, z) = N (x, y) and a(x, y) = 0 for all x, y, z ∈ B∗, then Theorem 3.1 reduces to Theorems3.1 and 3.2 of Ding and Yao [9] and Liu, Ume and Kang [24] in Banach spaces, respectively. If N (x, y, z) = x − yfor all x, y, z ∈ B∗, ϕ(v, u) = f (v), η(u, v) = gu − gv for all u, v ∈ B, where g : D → B, f : B → (−∞,+∞],then Theorem 3.1 improves and generalizes the corresponding results in [7,8,10,20,27,28].

Theorem 3.2. Let D, B∗, B, a and f ∗ be as in Theorem 3.1. Let T, A,G : D → B∗, N : B∗× B∗

× B∗→ B∗, and

η : D × D → B be mappings. Suppose that

(i) For any x, y, z ∈ B∗ and v ∈ D, the mapping u 7→ 〈N (x, y, z), η(u, v)〉 is convex and lower semicontinuous onD, and N is hemicontinuous with respect to T and A in the first and second arguments;

(ii) N is η-strongly monotone with constant ξ > 0 with respect to T in the first argument, and η-relaxed cocoercivewith constant ζ > 0 with respect to A in the second argument, and N is Lipschitz continuous in the second andthird arguments with constants s > 0 and t > 0, respectively, and A is Lipschitz continuous with constant r > 0;

(iii) η is Lipschitz continuous with constant λ > 0 satisfying η(u, v) = −η(v, u) for all u, v ∈ D;(iv) ϕ : B × B → (−∞,+∞] satisfies that

(a) ϕ is linear in the first argument;(b) ϕ is convex in the second argument;(c) ϕ(u, v)− ϕ(u, w) ≤ ϕ(u, v − w),∀u, v, w ∈ B;(d) ϕ(u, v) ≤ β‖u‖‖v‖,∀u, v ∈ B, where β > 0 is a constant.

If G : D → B∗ is completely continuous and ξ > r2s2ζ + β + d, then the MQVLIP (1.1) has a solution u ∈ D.

Proof. First of all we prove that for each fixed u ∈ D, there exists a unique point w ∈ D such that

〈N (T w, Aw,Gu)− f ∗, η(v, w)〉 + ϕ(w, v)− ϕ(w, w)+ a(w, v − w) ≥ 0, ∀v ∈ D. (3.3)

Let u ∈ D. Define the functionals φ and ψ : D × D → (−∞,+∞] by

φ(v,w) = 〈N (T v, Av,Gu)− f ∗, η(v,w)〉 + ϕ(w, v)− ϕ(w,w)+ a(v, v − w),

ψ(v,w) = 〈N (Tw, Aw,Gu)− f ∗, η(v,w)〉 + ϕ(w, v)− ϕ(w,w)+ a(w, v − w)(3.4)

for all v,w ∈ D.


Now we check that the functionals φ and ψ satisfy all the conditions of Lemma 2.4 in the weak topology. Note that(iii) leads to η(x, x) = 0 for all x ∈ D. It follows from (3.4) that for all v,w ∈ D, φ(v, v) ≥ 0 and

φ(v,w)− ψ(v,w) = 〈N (T v, Av,Gu)− N (Tw, Av,Gu), η(v,w)〉

+ 〈N (Tw, Av,Gu)− N (Tw, Aw,Gu), η(v,w)〉 + a(v − w, v − w)

≥ (c + ξ)‖v − w‖2− ζ‖N (Tw, Av,Gu)− N (Tw, Aw,Gu)‖2

≥ (c + ξ − r2s2ζ )‖v − w‖2

≥ 0,

which imply that φ and ψ satisfy the condition (C4) of Lemma 2.4. It follows from (a), (b), (c) and (d) that for anyu, v, w ∈ B

ϕ(u, v)− ϕ(u, w) ≤ ϕ(u, v − w) ≤ β‖u‖‖v − w‖

and

−(ϕ(u, v)− ϕ(u, w)) ≤ ϕ(−u, v)− ϕ(−u, w) ≤ β‖ − u‖‖v − w‖,

which give that

|ϕ(u, v)− ϕ(u, w)| ≤ β‖u‖‖v − w‖, ∀u, v, w ∈ B,

that is, for each fixed u ∈ B, ϕ(u, ·) is continuous on the second argument on B. It follows from (i) and (3.4) that foreach v ∈ D, φ(v, ·) is upper semicontinuous in the second argument and the set {v ∈ D : ψ(v,w) < 0} is convex foreach w ∈ D. Therefore the conditions (C5) and (C6) of Lemma 2.4 hold. Take v0 ∈ int D. Put

R0 =1

ξ − r2s2ζ − β − d(λ‖N (T v0, Av0,Gu)‖ + ‖ f ∗

‖ + β‖v0‖ + d‖v0‖)

and

K = {w ∈ D : ‖w − v0‖ ≤ R0}.

Obviously, K is a weakly compact subset of D and for any w ∈ D \ K ,

ψ(v0, w) = 〈N (Tw, Aw,Gu)− f ∗, η(v0, w)〉 + ϕ(w, v0)− ϕ(w,w)+ a(w, v0 − w)

≤ 〈N (Tw, Aw,Gu)− N (T v0, Aw,Gu), η(v0, w)〉 − 〈N (T v0, Aw,Gu)

− N (T v0, Av0,Gu), η(w, v0)〉 + 〈N (T v0, Av0,Gu), η(v0, w)〉

+ 〈 f ∗, η(v0, w)〉 + ϕ(w − v0, v0 − w)+ ϕ(v0, v0 − w)+ a(w − v0, v0 − w)+ a(v0, v0 − w)

≤ −ξ‖w − v0‖2+ r2s2ζ‖w − v0‖

2+ λ(‖N (T v0, Av0,Gu)‖ + ‖ f ∗

‖)‖w − v0‖ + β‖w − v0‖2

+β‖v0‖‖w − v0‖ + d‖w − v0‖2+ d‖v0‖‖w − v0‖

≤ −‖w − v0‖[(ξ − r2s2ζ − β − d)‖w − v0‖ − λ(‖N (T v0, Av0,Gu)‖ + ‖ f ∗‖)

−β‖v0‖ − d‖v0‖] < 0,

which yields that the condition (C7) of Lemma 2.4 holds. It follows from Lemma 2.4 that there exists a w ∈ D suchthat φ(v, w) ≥ 0 for all v ∈ D, that is,

〈N (T v, Av,Gu)− f ∗, η(v, w)〉 + ϕ(w, v)− ϕ(w, w)+ a(v, v − w) ≥ 0, ∀v ∈ D. (3.5)

Let t be in [0, 1] and v be in D. Replacing v by vt = tv + (1 − t)w in (3.5), we conclude that

〈N (T vt , Avt ,Gu)− f ∗, η(vt , w)〉 + ϕ(w, vt )− ϕ(w, w)+ a(vt , vt − w) ≥ 0, ∀v ∈ D. (3.6)

Notice that a is bilinear, ϕ is convex in the second argument, and ∀x, y, z ∈ B∗ and v ∈ D, u 7→ 〈N (x, y, z), η(u, v)〉is convex in D. By (3.6), we deduce that

t〈N (T vt , Avt ,Gu)− f ∗, η(v, w)〉 + t[ϕ(w, v)− ϕ(w, w)+ a(vt , v − w)] ≥ 0, ∀v ∈ D,

which implies that

〈N (T vt , Avt ,Gu)− f ∗, η(v, w)〉 + ϕ(w, v)− ϕ(w, w)+ a(vt , v − w) ≥ 0, ∀v ∈ D. (3.7)


Letting t → 0 in (3.7), by (i) we obtain that

〈N (T w, Aw,Gu)− f ∗, η(v, w)〉 + ϕ(w, v)− ϕ(w, w)+ a(w, v − w) ≥ 0, ∀v ∈ D.

That is, w is the solution of (3.3). Now we show the uniqueness of solution of (3.3). Suppose that (3.3) possesses twosolutions w1, w2 ∈ D. It follows that

〈N (Tw1, Aw1,Gu)− f ∗, η(w2, w1)〉 + ϕ(w1, w2)− ϕ(w1, w1)+ a(w1, w2 − w1) ≥ 0,

〈N (Tw2, Aw2,Gu)− f ∗, η(w1, w2)〉 + ϕ(w2, w1)− ϕ(w2, w2)+ a(w2, w1 − w2) ≥ 0.

Adding these inequalities, we infer that

〈N (Tw1, Aw1,Gu)− f ∗, η(w2, w1)〉 + 〈N (Tw2, Aw2,Gu)− f ∗, η(w1, w2)〉

+ϕ(w2 − w1, w1 − w2)− a(w1 − w2, w1 − w2) ≥ 0,

which means that

0 ≤ (c + d)‖w1 − w2‖2

≤ (c + ξ − r2s2ζ − β)‖w1 − w2‖2

≤ 0

by ξ > r2s2ζ + β + d . Consequently, w1 = w2. That is, w is the unique solution of (3.3). Hence there exists amapping F : D → D satisfying F(u) = w, where w is the unique solution of (3.3) for each u ∈ D.

Next we show that the mapping F is completely continuous. Let w1, w2 be arbitrary elements in D. By (3.3), weinfer that

〈N (T (Fw1), A(Fw1),Gw1)− f ∗, η(Fw2, Fw1)〉 + ϕ(Fw1, Fw2)− ϕ(Fw1, Fw1)

+ a(Fw1, Fw2 − Fw1) ≥ 0 (3.8)

〈N (T (Fw2), A(Fw2),Gw2)− f ∗, η(Fw1, Fw2)〉 + ϕ(Fw2, Fw1)− ϕ(Fw2, Fw2)

+ a(Fw2, Fw1 − Fw2) ≥ 0. (3.9)

Adding (3.8) and (3.9), we get that

c‖Fw1 − Fw2‖2

≤ a(Fw1 − Fw2, Fw1 − Fw2)

≤ −〈N (T (Fw1), A(Fw1),Gw1)− N (T (Fw2), A(Fw1),Gw1), η(Fw1, Fw2)〉

− 〈N (T (Fw2), A(Fw2),Gw1)− N (T (Fw2), A(Fw1),Gw1), η(Fw2, Fw1)〉

− 〈N (T (Fw2), A(Fw2),Gw2)− N (T (Fw2), A(Fw2),Gw1), η(Fw2, Fw1)〉

+ϕ(Fw1 − Fw2, Fw2 − Fw1)

≤ −ξ‖Fw1 − Fw2‖2+ r2s2ζ‖Fw1 − Fw2‖

2+ tλ‖Gw1 − Gw2‖‖Fw1 − Fw2‖

+β‖Fw1 − Fw2‖2,

that is,

‖Fw1 − Fw2‖ ≤λt

c + ξ − r2s2ζ − β‖Gw1 − Gw2‖, ∀w1, w2 ∈ D. (3.10)

Because G is completely continuous, it follows from (3.10) that F : D → D is completely continuous. By theSchauder fixed-point theorem, F has a fixed point u ∈ D, which is a solution of the MQVILP (1.1). This completesthe proof. �

Remark 3.2. Let B be a Hilbert space. If N (x, y, z) = x + y for all x, y, z ∈ B∗ and η(u, v) = u − v for allu, v ∈ D, then Theorem 3.2 collapses to Theorem 2 in Chang [3]. If N (x, y, z) = M(x, y) + z for all x, y, z ∈ B∗

and ϕ(v, u) = f (v) for all u, v ∈ D, where f : B → (−∞,+∞], then Theorem 3.2 reduces to Theorem 3.1 in Liu,Sun, Shim and Kang [21].

Theorem 3.3. Let D, B∗, B, a, f ∗, N , ϕ, η, T and A be as in Theorem 3.2. Suppose that G : D → B∗ is Lipschitzcontinuous with constant l > 0. If

0 <λlt

c + ξ − r2s2ζ − β< 1, (3.11)

then the MQVILP (1.1) has a unique solution u ∈ D.


Proof. Let u1 and u2 be arbitrary elements in D. As in the proof of Theorem 3.2, by (3.10) we obtain that

‖Fu1 − Fu2‖ ≤λt

c + ξ − r2s2ζ − β‖Gu1 − Gu2‖ ≤

λlt

c + ξ − r2s2ζ − β‖u1 − u2‖,

it follows from (3.11) that G : D → B∗ is a contraction mapping and hence it has a unique fixed point u ∈ D, whichis the unique solution of the MQVILP (1.1). This completes the proof. �

4. Iterative algorithms and convergence

In this section, we introduce two iterative algorithms for the MQVLIP (1.1) and investigate the convergence of thealgorithms.

Algorithm 4.1. Let D be a nonempty bounded closed convex subset of a reflexive Banach space B. Let ρ > 0 be aconstant. For any x0 ∈ D, compute {xn}n≥0 ⊆ D by solving the following auxiliary variational inequality : for eachn ≥ 0, find x ∈ D such that

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(x)− h′(xn), η(v, x)〉 + ρϕ(v, x)− ρϕ(x, x)+ ρa(x, v − x) ≥ 0,

∀v ∈ D, (4.1)

where h′ is the Frechet derivative of the functional h : D → R. Let xn+1 denote the solution of the auxiliary variationalinequality (4.1). That is,

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(xn+1)− h′(xn), η(v, xn+1)〉 + ρϕ(v, xn+1)

−ρϕ(xn+1, xn+1)+ ρa(xn+1, v − xn+1) ≥ 0, ∀v ∈ D. (4.2)

For X ⊆ D, Xw

denotes the weak closure of X .

Theorem 4.1. Let D be a nonempty bounded closed convex subset of a reflexive Banach space B with the dual spaceB∗. Let T, A,G : D → B∗, N : B∗

× B∗× B∗

→ B∗, and η : D × D → B be mappings. Let f ∗∈ B∗,

ϕ : B × B → (−∞,+∞] be skew-symmetric on B × B and for each v ∈ B, ϕ(·, v) be proper convex and weaklycontinuous in the first argument on B. Assume that (A1) and (A2) hold. Suppose that

(i) the mapping u 7→ N (T u, Au,Gu) is continuous from the weak topology on B to the strong topology on B∗;(ii) η is Lipschitz continuous with constant λ > 0 such that

(a) η(x, y) = η(x, z)+ η(z, y) for all x, y, z ∈ D,(b) for each fixed u, w ∈ D, 〈N (T u, Au,Gu), η(w, ·)〉 is concave,(c) for each fixed y ∈ D, η(y, ·) is sequentially continuous from the weak topology to the weak topology;

(iii) N is η-cocoercive in the first argument with respect to T with constant σ > 0, η-strongly monotone with respectto A in the second argument with constant δ > 0 and η-partially relaxed monotone with respect to G in the thirdargument with constant γ > 0, and N is Lipschitz continuous in the second and third arguments with constantss > 0 and t > 0, respectively, A and G are Lipschitz continuous with constants r > 0 and l > 0, respectively,where δ > λlt;

(iv) h : D → R is η-strongly convex with constant µ > 0, and its derivative h′ is sequentially continuous from theweak topology to the strong topology.

Then the following conclusions hold:

(1) The MQVLIP (1.1) has a unique solution x∗∈ D;

(2) For each x0 ∈ D and n ≥ 0, the auxiliary variational inequality (4.1) possesses a unique solution xn+1 ∈ Dsatisfying (4.2). Furthermore, if ρ satisfies the following condition:

0 < ρ ≤2µδσ

r2λ2s2σ + 4δγ σ + λ2δ, (4.3)

then the sequence {xn}n≥0 generated by the Algorithm 4.1 converges strongly to the unique solution x∗∈ D of the

MQVLIP (1.1).


Proof. (1) Since N is η-cocoercive in the first argument with respect to T , and η-strongly monotone with respect toA and G is Lipschitz continuous, it follows that

〈N (T u, Au,Gu)− N (T v, Av,Gv), η(u, v)〉

= 〈N (T u, Au,Gu)− N (T v, Au,Gu), η(u, v)〉 + 〈N (T v, Au,Gu)− N (T v, Av,Gu), η(u, v)〉

+ 〈N (T v, Av,Gu)− N (T v, Av,Gv), η(u, v)〉

≥ σ‖N (T u, Au,Gu)− N (T v, Au,Gu)‖2+ δ‖u − v‖2

− λtl‖u − v‖2

≥ (δ − λtl)‖u − v‖2, ∀u, v ∈ D.

Thus N is η-strongly monotone with constant α = δ−λtl > 0 with respect to T, A and G. That is, the condition (i) ofTheorem 3.1 is satisfied. It follows from (b) that the condition (iii) of Theorem 3.1 is satisfied. Notice that condition(a) implies that η(u, u) = 0 and η(u, v) = −η(v, u) for all u, v ∈ D. It follows from Theorem 3.1 that the MQVLIP(1.1) has a unique solution x∗

∈ D.(2) Let x0 ∈ D and n be a fixed nonnegative integer. Next we show the existence of solution of the auxiliary

variational inequality (4.1). Define a multi-valued mapping P : D → 2D by

P(v) = {x ∈ D : 〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(x)− h′(xn), η(v, x)〉 + ρϕ(v, x)

− ρϕ(x, x)+ ρa(x, v − x) ≥ 0}, ∀v ∈ D.

Clearly, v ∈ P(v) 6= ∅ for any v ∈ D. Now we claim that P is a KKM mapping. Otherwise, there is a finite subset{v1, . . . , vk} of D and αi ≥ 0, i ∈ {1, 2, . . . , k} with

∑ki=1 αi = 1 such that

x =

k∑i=1

αivi∈P(vi ), ∀i ∈ {1, 2, . . . , k}.

It follows that

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(x)− h′(xn), η(vi , x)〉 + ρϕ(vi , x)− ρϕ(x, x)+ ρa(x, vi − x) < 0,

∀i ∈ {1, 2, . . . , k},

which implies that

0 ≤

k∑i=1

αi 〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(x)− h′(xn), η(vi , x)〉

+ ρ

k∑i=1

αi [ϕ(vi , x)− ϕ(x, x)+ a(x, vi − x)] < 0

by the convexity of ϕ(·, v) and (b). This is a contradiction. Hence P : D → 2D is a KKM mapping.Since P(v)

wis a weakly closed subset of the bounded set D ⊆ B, it is weakly compact. It follows from Lemma 2.2

that ⋂v∈D

P(v)w

6= ∅.

Let x ∈⋂v∈D P(v)

w. Consequently, for each v ∈ D there exists a sequence {ym}m≥1 ⊆ P(v) such that ym ⇀ x ∈ D.

It follows that

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(ym)− h′(xn), η(v, ym)〉

≥ −ρ[ϕ(v, ym)− ϕ(ym, ym)+ a(ym, v − ym)], ∀v ∈ D,m ≥ 1

and hence

limm→∞

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(ym)− h′(xn), η(v, ym)〉

≥ limm→∞

{−ρ[ϕ(v, ym)− ϕ(ym, ym)+ a(ym, v − ym)]}, ∀v ∈ D.


Note that ϕ(·, v) is proper convex and weakly continuous in the weak topology for each v ∈ D. By Lemma 2.1, wegain that

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(x)− h′(xn), η(v, x)〉 ≥ −ρ[ϕ(v, x)− ϕ(x, x)+ a(x, v − x)], ∀v ∈ D.

That is, the auxiliary variational inequality (4.1) possesses a solution x ∈ D.Now we prove the uniqueness of solution for the auxiliary variational inequality (4.1). Let y ∈ D be also a solution

of the auxiliary variational inequality (4.1). It follows that

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(x)− h′(xn), η(v, x)〉 + ρϕ(v, x)− ρϕ(x, x)+ ρa(x, v − x) ≥ 0,

∀v ∈ D (4.4)

and

〈ρN (T xn, Axn,Gxn)− ρ f ∗+ h′(y)− h′(xn), η(v, y)〉 + ρϕ(v, y)− ρϕ(y, y)+ ρa(y, v − y) ≥ 0,

∀v ∈ D. (4.5)

Taking v = y in (4.4), v = x in (4.5) and adding these inequalities, we obtain that

ρ〈N (T xn, Axn,Gxn)− f ∗, η(y, x)〉 + 〈h′(x)− h′(xn), η(y, x)〉 + ρ〈N (T xn, Axn,Gxn)− f ∗, η(x, y)〉

+ 〈h′(y)− h′(xn), η(x, y)〉 + ρ[ϕ(y, x)− ϕ(x, x)+ ϕ(x, y)− ϕ(y, y)] − ρa(x − y, x − y) ≥ 0,

which implies that

〈h′(x), η(y, x)〉 ≥ −〈h′(y), η(x, y)〉.

By the η-strong convexity of h, we infer that

h(y)− h(x)− (µ/2)‖x − y‖2

≥ −h(x)+ h(y)+ (µ/2)‖x − y‖2,

that is,

µ‖x − y‖2

≤ 0,

which gives that x = y. Thus the solution x of the auxiliary variational inequality (4.1) is unique. Put xn+1 = x . It isclear that xn+1 satisfies (4.2).

Finally we prove the convergence of the sequence {xn}n≥0 generated by Algorithm 4.1 under condition (4.3).Define a functional Λ : D → (−∞,+∞) by

Λ(v) = h(x∗)− h(v)− 〈h′(v), η(x∗, v)〉, v ∈ D.

Let n be a nonnegative integer. By the η-strong convexity of h, (a) and (4.1) with v = x∗, x = xn+1, we conclude that

Λ(xn)− Λ(xn+1) = h(xn+1)− h(xn)− 〈h′(xn), η(x∗, xn)〉 + 〈h′(xn+1), η(x

∗, xn+1)〉

= h(xn+1)− h(xn)− 〈h′(xn), η(x∗, xn+1)〉 − 〈h′(xn), η(xn+1, xn)〉

+ 〈h′(xn+1), η(x∗, xn+1)〉

= h(xn+1)− h(xn)− 〈h′(xn), η(xn+1, xn)〉 + 〈h′(xn+1)− h′(xn), η(x∗, xn+1)〉

≥ µ/2‖xn+1 − xn‖2− ρ〈N (T xn, Axn,Gxn)− f ∗, η(x∗, xn+1)〉

− ρ[ϕ(x∗, xn+1)− ϕ(xn+1, xn+1)+ a(xn+1, x∗− xn+1)]

≥ µ/2‖xn+1 − xn‖2− ρ〈N (T xn, Axn,Gxn)− f ∗, η(x∗, xn+1)〉

+ ρϕ(xn+1, x∗)− ρϕ(x∗, x∗)− ρa(xn+1, x∗− xn+1). (4.6)

Setting u = x∗ and v = xn+1 in (1.1) and combining it with (4.6), we infer that

Λ(xn)− Λ(xn+1) ≥ µ/2‖xn+1 − xn‖2+ ρ〈N (T xn, Axn,Gxn)− f ∗, η(xn+1, x∗)〉

− ρ〈N (T x∗, Ax∗,Gx∗)− f ∗, η(xn+1, x∗)〉 + ρc‖xn+1 − x∗‖

2

= µ/2‖xn+1 − xn‖2+ Q + ρc‖xn+1 − x∗

‖2, (4.7)


where,

Q = ρ〈N (T xn, Axn,Gxn)− N (T x∗, Ax∗,Gx∗), η(xn+1, x∗)〉

= ρ〈N (T xn, Axn,Gxn)− N (T x∗, Ax∗,Gxn), η(xn+1, x∗)〉

+ ρ〈N (T x∗, Ax∗,Gxn)− N (T x∗, Ax∗,Gx∗), η(xn+1, x∗)〉

= ρ〈N (T xn, Axn,Gxn)− N (T x∗, Ax∗,Gxn), η(xn+1, xn)〉 + ρ〈N (T xn, Axn,Gxn)

− N (T x∗, Ax∗,Gxn), η(xn, x∗)〉 + ρ〈N (T x∗, Ax∗,Gxn)− N (T x∗, Ax∗,Gx∗), η(xn+1, x∗)〉

= ρ〈N (T xn, Axn,Gxn)− N (T x∗, Axn,Gxn), η(xn+1, xn)〉 + ρ〈N (T x∗, Axn,Gxn)

− N (T x∗, Ax∗,Gxn), η(xn+1, xn)〉 + ρ〈N (T xn, Axn,Gxn)− N (T x∗, Axn,Gxn), η(xn, x∗)〉

+ ρ〈N (T x∗, Axn,Gxn)− N (T x∗, Ax∗,Gxn), η(xn, x∗)〉 + ρ〈N (T x∗, Ax∗,Gxn)

− N (T x∗, Ax∗,Gx∗), η(xn+1, x∗)〉

≥ −ρ‖N (T xn, Axn,Gxn)− N (T x∗, Axn,Gxn)‖‖η(xn+1, xn)‖ − ρ‖N (T x∗, Axn,Gxn)

− N (T x∗, Ax∗,Gxn)‖‖η(xn+1, xn)‖ + ρσ‖N (T xn, Axn,Gxn)− N (T x∗, Axn,Gxn)‖2

+ ρδ‖xn − x∗‖

2− ργ ‖xn − xn+1‖

2

≥ −ρλ‖N (T xn, Axn,Gxn)− N (T x∗, Axn,Gxn)‖‖xn − xn+1‖ − ρλsr‖xn − x∗‖‖xn − xn+1‖

+ ρσ‖N (T xn, Axn,Gxn)− N (T x∗, Axn,Gxn)‖2+ ρδ‖xn − x∗

‖2− ργ ‖xn − xn+1‖

2

≥ −ρλ2

4σ‖xn − xn+1‖

2− ρλsr‖xn − x∗

‖‖xn − xn+1‖ + ρδ‖xn − x∗‖

2− ργ ‖xn − xn+1‖

2

= ρ

(−γ −

λ2

4σ

)‖xn − xn+1‖


‖‖xn − xn+1‖ + ρδ‖xn − x∗‖

2. (4.8)

Thus (4.7) and (4.8) ensure that

Λ(xn)− Λ(xn+1) ≥

[µ

2− ρ

(γ +

λ2

4σ

)]‖xn − xn+1‖


‖‖xn − xn+1‖

+ ρδ‖xn − x∗‖

2+ ρc‖xn+1 − x∗

‖2

≥

(ρδ −

ρ2s2λ2r2

2µ− 4ργ −ρλ2

σ

)‖xn − x∗

‖2+ ρc‖xn+1 − x∗

‖2

≥ ρc‖xn+1 − x∗‖

2. (4.9)

It follows from (4.3) and (4.9) that the sequence {Λ(xn)}n≥0 is decreasing and nonnegative by the η-strong convexityof h. Therefore the sequence {Λ(xn)}n≥0 converges to some number. It follows from (4.9) that

limn→∞

‖xn+1 − x∗‖

2= 0,

that is, the sequences {xn}n≥0 converges strongly to x∗, which is the solution of the MVQLIP (1.1). This completesthe proof. �

Remark 4.1. Algorithm 4.1 in Ding and Yao [9] and Algorithm 4.2 in Liu, Ume and Kang [24] are special cases ofAlgorithm 4.1 in this paper, which also extends, improves and unifies the corresponding algorithms of Ansari and Yao[2], Ding [7,8], Fang and Huang [10] and Yao [27,28].

Based on the proofs of Theorems 3.2 and 3.3, we suggest a new algorithm for the MVQLIP (1.1).

Algorithm 4.2. For any u0 ∈ D, compute {un}n≥0 and {wn}n≥0 ⊆ D by the following iterative schemes:

〈N (Twn, Awn,Gun)− f ∗, η(v,wn)〉 + ϕ(wn, v)− ϕ(wn, wn)+ a(wn, v − wn) ≥ 0 (4.10)

and

〈N (T un+1, Aun+1,Gwn)− f ∗, η(v, un+1)〉 + ϕ(un+1, v)− ϕ(un+1, un+1)+ a(un+1, v − un+1) ≥ 0, (4.11)

for all v ∈ D and n ≥ 0.


Theorem 4.2. Let the conditions of Theorem 3.3 be satisfied. Then the sequences {un}n≥0 and {wn}n≥0 defined byAlgorithm 4.2 converge strongly to the unique solution u ∈ D of the MVQLIP (1.1), and has the following errorestimate:

‖un − u‖ ≤k2n

1 − k2 ‖u0 − u‖, ∀n ≥ 1, (4.12)

where k =λtl

c+ξ−r2s2ζ−β.

Proof. It follows from Theorem 3.3 that the MVQLIP (1.1) has a unique solution u ∈ D satisfying

〈N (T u, Au,Gu)− f ∗, η(v, u)〉 + ϕ(u, v)− ϕ(u, u)+ a(u, v − u) ≥ 0. (4.13)

Taking v = u in (4.10), v = wn in (4.13), and adding these inequalities, we obtain that

c‖wn − u‖2

≤ a(wn − u, wn − u)

≤ −〈N (Twn, Awn,Gun)− N (T u, Awn,Gun), η(wn, u)〉 − 〈N (T u, Awn,Gun)

− N (T u, Au,Gun), η(wn, u)〉 − 〈N (T u, Au,Gun)

− N (T u, Au,Gu), η(wn, u)〉 + ϕ(u − wn, wn − u)

≤ −ξ‖wn − u‖2+ r2s2ζ‖wn − u‖

2+ tλl‖un − u‖‖wn − u‖ + β‖wn − u‖

2, ∀n ≥ 0,

which means that

‖wn − u‖ ≤ k‖un − u‖, ∀n ≥ 0. (4.14)

From (4.11) and (4.13), we deduce similarly that

‖un+1 − u‖ ≤ k‖wn − u‖, ∀n ≥ 0. (4.15)

It follows from (4.14) and (4.15) that

‖un+1 − u‖ ≤ k2‖un − u‖ ≤ · · · ≤ k2(n+1)

‖u0 − u‖, ∀n ≥ 0. (4.16)

In view of (3.11), (4.14) and (4.16), we gain that {un}n≥0 and {wn}n≥0 converge strongly to u ∈ D. Therefore for anygiven m ≥ n ≥ 1, we infer that

‖un − un+m‖ ≤

m−1∑i=0

‖un+i − un+i+1‖ ≤

m−1∑i=0

k2(n+i)‖u0 − u‖ ≤

k2n

1 − k2 ‖u0 − u‖. (4.17)

Letting m → ∞ in (4.17), we get that (4.12) holds. It follows from (4.12) and (4.14) that

limn→∞

un = limn→∞

wn = u.

This completes the proof. �

Acknowledgements

This work was supported by the Science Research Foundation of Educational Department of Liaoning Province(20060467) and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic ResearchPromotion Fund) (KRF-2006-312-C00026).

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Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces