Iterative algorithm for solving mixed quasi variational like inequalities with skew symmetric terms in Banach spaces

QUASI-VARIATIONAL-LIKE INEQUALITIES WITH

SKEW-SYMMETRIC TERMS IN BANACH SPACES

Received 1 April 2006; Accepted 28 May 2006

We develop an iterative algorithm for computing the approximate solutions of mixed quasi-variational-like inequality problems with skew-symmetric terms in the setting of reflexive Banach spaces. We use Fan-KKM lemma and concept ofη-cocoercivity of a com-position mapping to prove the existence and convergence of approximate solutions to the exact solution of mixed quasi-variational-like inequalities with skew-symmetric terms. Furthermore, we derive the posteriori error estimates for approximate solutions under quite mild conditions.

Copyright © 2006 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that variational inequality theory, as a very effective and powerful tool, not only has stimulated new results dealing with partial differential equations, but also has been applied to a large variety of problems arising in mechanics, contact problems in elasticity, optimization and control problems, management science, operations research, general equilibrium problems in economics and transportation, unilateral problems, ob-stacle problems, and so forth. Because of its wide applications, the classical variational inequality has been studied and generalized in various directions previously by many authors. Among these generalizations, mixed variational-like inequality problem is of in-terest and importance. One of the most important and inin-teresting problems in the theory of variational inequality is the development of an efficient and implementable algorithm for solving variational inequality and its generalizations. These methods include projec-tion method and its variant forms, linear approximaprojec-tion, descent and Newton’s methods, and method based on the auxiliary principle technique. The method based on auxil-iary principle technique was first suggested by Glowinski et al. [6] for solving variational inequalities in 1981. Subsequently, it has been used to solve a number of generalizations

of classical variational inequalities; see, for example, [1,4,8,14–16] and the references therein.

LetH be a real Hilbert space whose inner product and norm are denoted by·,·, and · , respectively. LetDbe a nonempty convex subset ofH. LetT,A:D→H, and η:D×D→Hbe mappings, and letf :D→Rbe a real-valued function. Recently, Ansari and Yao [1] (see also [10]) considered and studied the mixed variational-like inequality problem (MVLIP), which is to findu∗∈Dsuch that

Tu∗_−Au∗_,ηv,u∗_+f(v)−fu∗_{≥0, ∀v∈D. (1.1)}

WhenA≡0, this problem reduces to the following problem considered by Dien [3], Noor [11] and Siddiqi et al. [12]: findu∗∈Dsuch that

_Tu∗

,ηv,u∗+f(v)−fu∗_{≥0, ∀v∈D. (1.2)}

In [1], Ansari and Yao introduced the concepts ofη-cocoercivity,η-strong monotonic-ity, andη-strong convexity of a mapping, which generalize the definitions of cocoercivity [13,17], strong monotonicity [9], and strong convexity [9], respectively. It is easy to see thatη-cocoercivity is an intermediate concept that lies betweenη-strong monotonicity andη-monotonicity. They applied the auxiliary variational inequality technique to sug-gest an iterative algorithm for finding the approximate solutions of MVLIP and proved that these approximate solutions converge to the exact solution of MVLIP.

Motivated and inspired by the work in [1], we consider and study a class of mixed quasi-variational-like inequality problems in the setting of Banach spaces.

LetDbe a nonempty convex subset of a real Banach spaceB, letB∗be the topological dual space ofB, and letu,vbe the duality pairing betweenu∈B∗_{andv∈B. LetT,} A:D→B∗_,N:B∗_×B∗_→B∗_{, andη:D×D→Bbe mappings andw}∗_∈B∗_{. Letϕ:B×} B→R{+∞}be a real bifunction. The mixed quasi-variational-like inequality problem (MQVLIP) is to findu∈Dsuch that

N(Tu,Au)−w∗,η(v,u)+ϕ(v,u)−ϕ(u,u)≥0, ∀v∈D. (1.3)

This problem was first introduced and studied by Ding and Yao [4]. They applied the auxiliary variational inequality technique to suggest an iterative algorithm for computing the approximate solutions of MQVLIP (1.3), and provided the convergence criteria of approximate solutions to the exact solution of MQVLIP (1.3).

their ones because of the appearance of skew-symmetric termϕ(·,·) in MQVLIP (1.3); (iv) the technique of our proof is very different from the one of their proof since reflexive Banach space is more general than Hilbert space; (v) we provide the posteriori error es-timates for approximate solutions under quite mild conditions. Hence the results of this paper also extend and generalize the results of Zhu and Marcotte [17].

2. Preliminaries

In this section we will recall the following definitions and some known results.

Definition 2.1. LetDbe a nonempty subset of a Banach spaceBwith the dual spaceB∗. LetT:D→B∗_{andη:D×D→Bbe two mappings. Then,}

(i)Tis called η-monotone, if

Tu−Tv,η(u,v)≥0, ∀u,v∈D; (2.1)

(ii)Tis called η-strongly monotone, if there exists a constantβ >0, such that

Tu−Tv,η(u,v)≥βu−v2, ∀u,v∈D; (2.2)

(iii)Tis called η-cocoercive, if there exists a constantξ >0, such that

Tu−Tv,η(u,v)≥ξTu−Tv2_{, ∀u,v∈D; (2.3)}

(iv)Tis called η-relaxed monotone, if there exists a constantξ >0, such that

Tu−Tv,η(u,v)≥βu−v2_{, ∀u,v∈D; (2.4)}

(v)Tis calledLipschitz continuous, if there exists a constantL >0, such that

Tu−Tv ≤Lu−v, ∀u,v∈D; (2.5)

(vi)ηis called Lipschitz continuous, if there exists a constantδ >0, such that

_{η(u,v)≤δu−v, ∀u,v∈D. (2.6)}

We illustrate hereafter the relationships amongη-monotonicity,η-strong monotonic-ity,η-cocoercivity, and Lipschitz continuity as follows:

(i)η-strong monotonicity=⇒η-monotonicity⇐=η-cocoercivity; (ii)

Tisη-strong monotone

Tis Lipschitz continuous=⇒Tisη-cocoercive; (2.7)

(iii)

Tisη-cocoercive

Definition 2.2. The bifunctionϕ:B×B→R{+∞}is said to be skew-symmetricif

ϕ(u,u)−ϕ(u,v)−ϕ(v,u) +ϕ(v,v)≥0, ∀u,v∈B. (2.9)

The skew-symmetric bifunctions have properties which can be considered analogs of monotonicity of gradient and nonnegativity of a second derivative for convex functions. For properties and applications of skew-symmetric bifunction, reader may consult [2].

Definition 2.3. LetD be a convex subset of a Banach space Band letK:D→R be a Fr´echet differentiable function.Kis said to be

(i) η-convex[7], if

K(v)−K(u)≥K_{(u),η(v,u), ∀u,v∈D; (2.10)}

(ii) η-strongly convex[11], if there exists a constantμ >0, such that

K(v)−K(u)−K_{(u),η(v,u)≥}μ

2u−v

2_{, ∀u,v∈D. (2.11)}

In particular, ifη(v,u)=v−ufor allv,u∈D, thenKis said to be strongly convex.

Proposition2.4. LetKbe a differentiableη-strongly convex functional with constantμ >0

on a convex subsetDofB, and letη:D×D→Bbe a mapping such thatη(u,v) +η(v,u)=0, for allu,v∈D. ThenKisη-strongly monotone with constantμ >0.

Proof. SinceKisη-strongly convex, we deduce that for eachu,v∈D

K(v)−K(u)−K_{(u),η(v,u)≥}μ

2u−v 2_,

K(u)−K(v)−K_{(v),η(u,v)≥}μ

2v−u 2_.

(2.12)

Adding these two inequalities and using the condition thatη(u,v) +η(v,u)=0, we obtain

K_{(v)−K(u),η(v,u)≥μv−u}2_{. (2.13)} A functionF:D→Ris called weakly sequentially continuous atx0∈D, ifF(xk)→ F(x0) (n→ ∞) for each sequence{xk} ⊂Dconverging weakly tox0.F is called weakly sequentially continuous onD, if it is weakly sequentially continuous at each point ofD.

Lemma2.5. Letη(v,·) :D→BandKbe sequentially continuous from the weak topology

to the weak topology and from the weak topology to the strong topology, respectively, wherev is any fixed point inD. Then the functiong:D→R, defined asg(u)= K_{(u),η(v,u)for}

Proof. Letube any given point inD, and let{un} ⊂Dbe any sequence converging weakly tou. Then,K_{(un)−K(u) →0 (n→ ∞), and{η(v,un)}converges weakly toη(v,u) for}

each fixedv∈D. Observe that _g

un−g(u)=K_{un−K(u),ηv,un+K(u),ηv,un−η(v,u)}

≤K_{un−K(u)ηv,un+K(u),ηv,un−η(v,u)}

−→0 asn−→ ∞.

(2.14)

Therefore, the conclusion immediately follows.

For eachD⊂B, we denote by co(D) the convex hull ofD. A point-to-set mapping G:D→2Bis called a KKMmappingif, for every finite subset{u1,u2,...,uk}ofD,

co u1,u2,...,uk⊂ k

i=1

Gxi. (2.15)

In the next section, we will use the following result.

Lemma2.6 (Fan-KKM [5]). LetDbe an arbitrary nonempty subset in a topological vector

spaceE, and letG:D→2Ebe a KKM mapping. IfG(u)is closed for allu∈Dand is compact for at least oneu∈D, thenu∈DG(u)= ∅.

3. Iterative algorithm and its convergence

In this section, following the approach of Ansari and Yao [1], we will apply the auxiliary variational inequality technique to suggest a general algorithm for finding approximate solutions of MQVLIP (1.3) with skew-symmetric termϕ(·,·). Moreover, we will also study convergence analysis of proposed algorithm.

Algorithm 3.1. LetK:D→Rbe a given Fr´echet differentiable functional, let{ρn}∞n=0be a sequence of positive parameters, and letu0be any initial guess inD. For each given iterate un∈D, consider the following auxiliary variational inequality problem: findun+1∈D, such that

ρnNTun,Aun−ρnw∗_{+Kun+1−Kun,ηv,un+1}

+ρnϕv,un+1−ρnϕun+1,un+1≥0, ∀v∈D, (3.1)

whereK(u) is the Fr´echet derivative of a functionK:D→Ratu∈D.

Theorem3.2. LetDbe a nonempty closed convex subset of a reflexive Banach spaceBwith

the dual spaceB∗. LetT,A:D→B∗_,N:B∗_×B∗_→B∗_{, andη:D×D→Bbe mappings.}

Letw∗∈B∗ _{andϕ:B×B→R{+∞}be skew-symmetric and weakly continuous, such}

the weak topology to the strong topology. Suppose also that

(i)the mappingu→N(Tu,Au)isη-cocoercive with constantυ >0; (ii)ηis Lipschitz continuous with constantδ >0, such that

(a)η(u,v)=η(u,z) +η(z,v)for eachu,v,z∈D, (b)η(·,·)is affine in the first variable,

(c)for each fixedv∈D,u→η(v,u)is sequentially continuous from the weak topol-ogy to the weak topoltopol-ogy;

(iii)for each fixedn≥0andz∈D,

u∈D:ρnN(Tz,Az)−ρnw∗_{+K(u)−K(z),η(v,u)+ρnϕ(v,u)−ρnϕ(u,u)≥0}

(3.2)

is bounded for at least onev∈D.

Then, there is a unique solutionun+1∈Dto problem (3.1) for each given iterateun. If

ρn+1≤ρn, c1< ρn<2_δμ ₁

υ+c2

, n≥0,for somec1,c2>0, (3.3)

then

(I){un}is bounded; (II) limn→∞un+1−un =0;

(III){un}converges weakly to a solution of MQVLIP (1.3) provided that the mappingu→ N(Tu,Au)is sequentially continuous from the weak topology to the strong topology.

Proof. Existence of solutions of problem (3.1).For the sake of simplicity, we rewrite (3.1) as follows. Findu∈Dsuch that

ρnNTun,Aun−ρnw∗+K(u)−Kun,η(v,u)+ρnϕ(v,u)−ρnϕ(u,u)≥0, ∀v∈D. (3.4)

For each fixedn≥0 and eachv∈D, we define

G(v)= u∈D:ρnNTun,Aun−ρnw∗_{+K(u)−Kun,η(v,u)}

+ρnϕ(v,u)−ρnϕ(u,u)≥0. (3.5)

Note that, sincev∈G(v),G(v) is nonempty for eachv∈D. Now, we claim thatGis a KKM mapping. Indeed, suppose that there exists a finite subset{v1,v2,...,vk}ofDand thatαi≥0, for alli=1, 2,...,kwithki=1αi=1 such that u=

k

i=1αivi∈G(vi), for all i=1, 2,...,k, that is,

_ρnNTun

,Aun−ρnw∗_{+K(u)−Kun,ηvi,u}

Sinceϕ(·,v) is proper convex for eachv∈B, by virtue of assumptions (a) and (b) in (ii), we have

0=ρnNTun,Aun−ρnw∗_{+K(u)−Kun,η(u,u)+ρnϕ(u,u)−ρnϕ(u,u)}

=

ρnNTun,Aun−ρnw∗+K(u)−Kun,η _k

i=1 αivi,u

+ρnϕ _k

i=1 αivi,u

−ρnϕ(u,u)

≤

ρnNTun,Aun−ρnw∗_+K(u)−Kun,k i=1

αiηvi,u

+ρn k

i=1

αiϕvi,u−ρnϕ(u,u)

= k

i=1

αiρnNTun,Aun−ρnw∗_{+K(u)−Kun,ηvi,u+ρnϕvi,u−ρnϕ(u,u)<0,}

(3.7)

which yields a contradiction. Therefore,Gis a KKM mapping.

SinceKis sequentially continuous from the weak topology to the strong topology, and ϕ(·,·) is weakly continuous onB×B, so it follows from condition (ii)(c) andLemma 2.5

thatG(v) is a weakly closed subset ofDfor eachv∈D. Moreover, from condition (iii) we know that G(v) is weakly compact for at least one point v∈D. Hence, by using

Lemma 2.6we havev∈DG(v)= ∅, which clearly implies that there exists at least one solution to problem (3.1).

Uniqueness of solutions of problem (3.1). Letuandube two solutions of problem (3.1). Then,

ρnNTun,Aun−ρnw∗_{+K(u)−Kun,η(v,u)+ρnϕ(v,u)−ρnϕ(u,u)≥0, (3.8)}

_ρnNTun

,Aun−ρnw∗_{+K(u)−Kun,η(v,u)+ρnϕ(v,u)−ρnϕ(u,u)≥0, (3.9)}

for allv∈D.

Note thatη(u,v)=η(u,z) +η(z,v) for allu,v,z∈Dimpliesη(u,v)= −η(v,u) for all u,v∈Dand thatϕ(·,·) is skew-symmetric. Takingv=uin (3.8) andv=uin (3.9), and adding these inequalities, we get

−K_{(u)−K(u),η(u,u)=K(u),η(u,u)+K(u),η(u,u)}

=ρnNTun,Aun−ρnw∗_{+K(u)−Kun,η(u,u)}

+ρnNTun,Aun−ρnw∗_{+K(u)−Kun,η(u,u)}

≥ρnϕ(u,u)−ϕ(u,u)−ϕ(u,u) +ϕ(u,u)≥0,

which hence implies that

_K

(u)−K_{(u),η(u,u)≤0. (3.11)}

SinceKisη-strongly monotone with constantμ >0 byProposition 2.4, from (3.11) we obtain

μu−u2_{≤K(u)−K(u),η(u,u)≤0, (3.12)}

and sou=u. This shows that the solution of problem (3.1) is unique.

Letu∗∈Dbe any fixed solution of MQVLIP (1.3). Sinceun+1is the unique solution to problem (3.1), we get

K_{un+1−Kun,ηv,un+1+ρnNTun,Aun−w}∗_,ηv,un+1

+ρnϕv,un+1−ρnϕun+1,un+1≥0, ∀v∈D. (3.13)

We consider a functionΛdefined by

Λ(u,ρ)=Φ(u) +Ω(u,ρ), (3.14)

whereΩ(u,ρ)=ρ[N(Tu∗,Au∗)−w∗_,η(u,u∗_)+ϕ(u,u∗_)−ϕ(u∗_,u∗_{)] and}

Φ(u)=Ku∗_{−K(u)−K(u),ηu}∗_{,u. (3.15)}

Fromη-strong convexity ofK, we obtain

Φun≥ _μ

2

un−u∗2

≥0. (3.16)

SinceΩ(un,ρn) is nonnegative, we have

Λun,ρn≥ _μ

2

un−u∗2

≥0. (3.17)

Let us study the variation ofΛfor one stage ofAlgorithm 3.1:

Γn+1

n =Λun+1,ρn+1−Λun,ρn. (3.18)

Then we haveΓn+1n =s1+s2+s3, where

s1=Kun−Kun+1−K_{un,ηun,un+1,}

s2=K_{un−Kun+1,ηu}∗_,un+1,

s3=ρn+1NTu∗_,Au∗_−w∗_,ηun+1,u∗_+ϕun+1,u∗_−ϕu∗_,u∗

−ρnNTu∗_,Au∗_−w∗_,ηun,u∗_+ϕun,u∗_−ϕu∗_,u∗_.

By usingη-strong convexity ofK, we haves1≤ −(μ/2)un+1−un2_{. Also, sinceρn+1≤ρn} andu∗is a solution of MQVLIP (1.3), we immediately derive

s3≤ρnNTu∗_,Au∗_−w∗_,ηun+1,u∗_+ϕun+1,u∗_−ϕu∗_,u∗_{. (3.20)}

Now, puttingu=u∗_{in (3.13), we get}

s2≤ρnNTun,Aun−w∗,ηu∗,un+1+ϕu∗,un+1−ϕun+1,un+1. (3.21)

Thus, in terms of skew-symmetry ofϕwe deduce from (3.20) and (3.21) that

s2+s3≤ρnNTu∗_,Au∗_−w∗_,ηun+1,u∗_+ϕun+1,u∗_−ϕu∗_,u∗

+ρnNTun,Aun−w∗_,ηu∗_,un+1+ϕu∗_{,un+1−ϕun+1,un+1}

=ρnNTu∗_,Au∗_−w∗_,ηun+1,u∗_{+NTun,Aun−w}∗_,ηu∗_,un+1

−ρnϕu∗_,u∗_−ϕun+1,u∗_−ϕu∗_{,un+1+ϕun+1,un+1}

≤ −ρnNTu∗_,Au∗_{−NTun,Aun,ηu}∗_,un+1

= −ρnNTu∗_,Au∗_{−NTun,Aun,ηu}∗_,un

−ρnNTu∗_,Au∗_{−NTun,Aun,ηun,un+1}

≤ −ρnυNTu∗_,Au∗_−NTun,Aun2

(usingη-cocoercivity)

+ρnδNTu∗,Au∗−NTun,Aunun+1−un (using Lipschitz continuity), (3.22)

which hence implies that

Γn+1

n =s1+s2+s3 ≤ −μ

2un+1−un 2

−υρnNTun,Aun−NTu∗,Au∗2

+δρnNTun,Aun−NTu∗,Au∗un+1−un.

(3.23)

Then, by using the inequality

ρnNTun,Aun−NTu∗_,Au∗_un+1−un

≤ _ρ2

n 2ω

NTun,Aun−NTu∗_,Au∗2 +

_ω 2

un+1−un2 ,

whereωis a positive number chosen so thatω < μ/δ, we get

Γn+1 n ≤ −

_μ 2−

δω 2

un+1−un2

−ρnυ−δρn 2ω

NTun,Aun−NTu∗_,Au∗2 . (3.25)

Forc1< ρn<2ω/δ(1/υ+c2), wherec1>0 andc2>0, we have

ρnυ−δρn 2ω

> c1υ− δ 2ω·

2ω δ1/υ+c2

= c1c2υ

1/υ+c2, (3.26)

and hence

Γn+1 n ≤ −

_μ 2−

δω 2

un+1−un2 −

_c1c2υ (1/υ+c2)

NTun,Aun−NTu∗_,Au∗2 . (3.27)

Thus, forω < μ/δ,Γn+1n is negative unlessun+1=un andN(Tun,Aun)=N(Tu∗,Au∗). Then, according to (3.13),unis a solution of MQVLIP (1.3).

Note that the sequence{Λ(un,ρn)}is strictly decreasing. But since it is positive, it must converge and the difference between two consecutive terms tends to zero, that is,Γn+1n →0 asn→ ∞. Therefore,un+1−unandN(Tun,Aun)−N(Tu∗,Au∗)converge to zero. Moreover, since the sequence{Λ(un,ρn)}converges, it is bounded, and so is{un}also according to (3.17).

Letu be a weak cluster point of the sequence{un}, and let{uni}be a subsequence converging weakly tou. By using (3.13), sinceKis Lipschitz continuous with constant κ >0 andρn> c1, it is known that for eachv∈D,

_NTun

,Aun−w∗_{,ηv,un+1+ϕv,un+1−ϕun+1,un+1}

≥ − ₁

ρn

K_{un+1−Kun,ηv,un+1}

≥ −_c1δK_{un+1−Kunv−un+1.}

(3.28)

which hence implies that

NTuni,Auni

−w∗_,ηv,un

i+1

+ϕv,uni+1

−ϕuni+1,uni+1

≥ − _δ

c1

K_un

i+1

−K_un

iv−uni+1.

(3.29)

since the mappingu→N(Tu,Au) is sequentially continuous from the weak topology to the strong topology, we infer that for eachv∈D,

_N

Tuni,Auni

−w∗_,ηv,un

i+1

−N(Tu,Au)−w∗_,η(v,u)

≤NTuni,Auni

−N(Tu,Au),ηv,uni+1

+N(Tu,Au)−w∗_,ηv,un

i+1−η(v,u) ≤NTuni,Auni

−N(Tu,Au)ηv,uni+1

+N(Tu,Au)−w∗_,ηv,un

i+1

−η(v,u)−→0 asn−→ ∞.

(3.30)

Consequently, lettingi→ ∞and using the weak continuity ofϕ(·,·), we conclude from (3.29) that

N(Tu,Au)−w∗_{,η(v,u)+ϕ(v,u)−ϕ(u,u)≥0, ∀v∈D. (3.31)}

This shows thatuis a solution of MQVLIP (1.3). Furthermore,N(Tu,Au)=N(Tu∗,Au∗) since limn→∞N(Tuni,Auni)−N(Tu,Au) =0 and limn→∞N(Tun,Aun)−N(Tu∗, Au∗_{) =0.}

Finally, we claim that{un}converges weakly to a solution of MQVLIP (1.3). Indeed, it suffices to show that{un}has the unique weak cluster point. Letuandube two weak cluster points of{un}. Then, both weak cluster points can be used as the aboveu∗ _to

define the Lyapunov functionΛ. This yields two possible Lyapunov functions, denoted byΛandΛ, respectively. It was proven thatΛ(un,ρn) has a limit that may depend on the solutionu∗used to defineΛ; then, the corresponding limits will be denoted bylandl, re-spectively. Consider the subsequences{ni}and{mj}such that{uni}and{umj}converge weakly touandu, respectively. Then it is easy to see thatN(Tu,Au)=N(Tu,Au). Hence we get

Λuni,ρni

=K(u)−Kuni

−K_un

i

,ηu,uni

+ρni _N

(Tu,Au)−w∗_,ηun

i,u

+ϕuni,u

−ϕ(u,u)

=K(u)−Kuni

−K_un

i

,ηu,uni

+ρni

N(Tu,Au)−w∗_,ηun

i,u

+ϕuni,u

−ϕ(u,u)

+K(u)−K(u)−K_un

i

,η(u,u)

+ρni

N(Tu,Au)−w∗,η(u,u)+ϕuni,u

−ϕ(u,u)−ϕuni,u+ϕ(u,u)

=Λuni,ρni

+K(u)−K(u)−K_un

i

,η(u,u)

+ρni

N(Tu,Au)−w∗_{,η(u,u)+ϕun}

i,u

Now letρ∗=limn→∞ρn. SinceΛ(uni,ρni)→landΛ(uni,ρni)→l, sinceKis sequentially continuous from the weak topology to the strong topology, and sinceϕ(·,·) is weakly continuous, so, lettingn→ ∞and using theη-strong convexity ofK, we conclude from (3.32) that

l=l+K(u)−K(u)−K_(u),η(u,u)

+ρ∗N(Tu,Au)−w∗,η(u,u)+ϕ(u,u)−ϕ(u,u)

≥l+ _μ

2

u−u2_.

(3.33)

By interchanging the role ofuanduand of the subsequences{ni}and{mj}, the same calculations yield

l≥l+ _μ

2

u−u2_{. (3.34)}

Then, 0≤(μ/2)u−u2_{≤l−land 0≤(μ/2)u−} 2_{≤l−l. This implies thatu=u.}

Corollary3.3. LetDbe a nonempty closed convex subset of a reflexive Banach spaceBwith

the dual spaceB∗. LetT,A:D→B∗_,N:B∗_×B∗_→B∗_{, andη:D×D→Bbe mappings.}

Letw∗∈B∗ _{andϕ:B×B→R{+∞}be skew-symmetric and weakly continuous, such}

that for eachv∈B,int(domϕ(·,v))∩D= ∅andϕ(·,v)is proper convex. Suppose that K:D→Risη-strongly convex with constantμ >0, andKis sequentially continuous from the weak topology to the strong topology. Suppose also that

(i)the mappingu→N(Tu,Au)isη-strongly monotone with constantβ >0and Lips-chitz continuous with constantL >0;

(ii)ηis Lipschitz continuous with constantδ >0, such that (a)η(u,v)=η(u,z) +η(z,v)for eachu,v,z∈D, (b)η(·,·)is affine in the first variable,

(c)for each fixedv∈D,u→η(v,u)is sequentially continuous from the weak topol-ogy to the weak topoltopol-ogy;

(iii)for each fixedn≥0andz∈D,

u∈D:ρnN(Tz,Az)−ρnw∗_{+K(u)−K(z),η(v,u)+ρnϕ(v,u)−ρnϕ(u,u)≥0}

(3.35)

is bounded for at least onev∈D.

Then, there is a unique solutionun+1∈Dto problem (3.1) for each given iterateun. If

c1< ρn<_L2δ2₂βμ

then{un}converges strongly to a solutionu∗∈Dof MQVLIP (1.3). Moreover, ifKis Lipschitz continuous with constantκ >0, then we have the posteriori error estimates:

_un+1−u∗_≤κδ βρn+

Lδ β

un+1−un. (3.37)

Proof. We consider the variation of the functionΦin (3.15) for one stage ofAlgorithm 3.1,

Δn+1

n =Φun+1−Φun, (3.38)

wheres1=K(un)−K(un+1)− K_{(un),η(un,un+1), and}

s2=K_{un−Kun+1,ηu}∗_{,un+1. (3.39)}

By usingη-strong convexity ofKand the fact thatη(u,v)= −η(v,u), for allu,v∈D, we have

s1≤ − _μ

2

un−un+12_.

(3.40)

Also, from (3.21) it follows that

s2≤ρnNTun,Aun−w∗_,ηu∗_,un+1+ϕu∗_{,un+1−ϕun+1,un+1}

=ρnNTun+1,Aun+1−NTu∗_,Au∗_,ηu∗_,un+1

+ρnNTun,Aun−NTun+1,Aun+1,ηu∗,un+1

+ρnNTu∗,Au∗−w∗_,ηu∗_,un+1+ϕu∗_{,un+1−ϕun+1,un+1.}

(3.41)

By usingη-strong monotonicity of mappingu→N(Tu,Au), we have

ρnNTun+1,Aun+1−NTu∗_,Au∗_,ηu∗_{,un+1≤ −βρnun+1−u}∗2

. (3.42)

By using Lipschitz continuity of mappingsu→N(Tu,Au) andη, we get

ρnNTun,Aun−NTun+1,Aun+1,ηu∗,un+1

≤ρnLδun−un+1u∗_−un+1

≤ _μ

2

un+1−un2 +

_ρ2 nL2δ2

2μ

un+1−u∗2_.

By using skew-symmetry of ϕand the fact that u∗ is a solution of MQVLIP (1.3), we obtain

ρnNTu∗_,Au∗_−w∗_,ηu∗_,un+1+ϕu∗_{,un+1−ϕun+1,un+1}

= −ρnNTu∗_,Au∗_−w∗_,ηun+1,u∗_+ϕun+1,u∗_−ϕu∗_,u∗

−ρnϕun+1,un+1−ϕu∗_{,un+1−ϕun+1,u}∗_+ϕu∗_,u∗

≤ρn·0 +ρn·0=0.

(3.44)

Utilizing the estimates given in (3.42)–(3.44), we conclude from (3.41) that

s2≤ −βρnun+1−u∗2 +

_μ 2

un+1−un2 +

ρ2

nL2δ2 2μ

_un+1−u∗2

. (3.45)

This, together with (3.40), yields that

Δn+1

n =s1+s2≤ρ2n

−_ρnβ +L 2_δ2 2μ

un+1−u∗2

. (3.46)

Observe that (3.36) implies that

ρ2 n

−_ρnβ +L 2_δ2 2μ

≤ρ2

n

−βL2δ2+c2 2βμ +

L2_δ2 2μ

= −ρ2nc2 2μ ≤ −

c2_1c2

2μ . (3.47)

Now substituting the estimate (3.47) in (3.46), we derive

Δn+1 n ≤

−c12c2

2μ

un+1−u∗2

. (3.48)

This shows thatΔn+1n is negative unlessun+1=u∗. The sequence{Φ(un)}is strictly de-creasing. But since it is positive, it must converge and the difference between two con-secutive terms tends to zero, that is,Δn+1_n →0 asn→ ∞. Therefore, it follows from (3.48) that{un}converges strongly tou∗.

Now, by using (3.13) withv=u∗_{, we infer that}

K_{un+1−Kun,ηu}∗_{,un+1+ρnNTun,Aun−w}∗_,ηu∗_,un+1

+ρnϕu∗,un+1−ρnϕun+1,un+1≥0. (3.49)

(1.3), we have that

κδun+1−unun+1−u∗_{+ρnLδun+1−unun+1−u}∗

≥K_{un+1−Kun,ηu}∗_,un+1

+ρnNTun,Aun−NTun+1,Aun+1,ηu∗,un+1

≥ρnNTun+1,Aun+1−NTu∗_,Au∗_,ηun+1,u∗

+ρnNTu∗,Au∗−w∗_,ηun+1,u∗_+ϕun+1,u∗_−ϕu∗_,u∗

+ρnϕun+1,un+1−ϕu∗,un+1−ϕun+1,u∗+ϕu∗,u∗ ≥ρnNTun+1,Aun+1−NTu∗,Au∗,ηun+1,u∗

≥βρnun+1−u∗2 ,

(3.50)

which hence implies that

κδun+1−unun+1−u∗_{+ρnLδun+1−unun+1−u}∗_{≥βρnun+1−u}∗2_. (3.51)

We obtain inequality (3.37) after division byun+1−u∗_{, which we assume nonzero;}

otherwise, the result is trivial.

Acknowledgments

The first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher-Education Institutions of MOE, China, and the Dawn Program Foundation in Shanghai. The second author is grateful to the Depart-ment of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia, for providing excellent research facilities. The third author was partially supported by a grant from the National Science Council.

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Lu-Chuan Ceng: Department of Mathematics, Shanghai Normal University, Shanghai 200234, China E-mail address:[email protected]

Qamrul Hasan Ansari: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Current address: Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, P.O. Box 1169, Dhahran 31261, KSA

E-mail address:[email protected]

Jen-Chih Yao: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan