Mehdi MOHAMMADI

Department of Mathematics，Payame Noor University，Tehran，Iran E-mail:mehdi.mohammadi5677@gmail.com

G.Zamani ESKANDANI

Department of Pure Mathematics，F(xiàn)aculty of Mathematical Sciences，University of Tabriz，Tabriz，Iran E-mail:zamani@tabrizu.ac.ir

Abstract In this paper，we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces.We prove strong convergence of the sequence generated by the extragradient method to a solution of the quasi-equilibrium problem.

Key words demiclosed，extragradient algorithm;quasi-equilibrium problem;quasi φ-nonexpansive mapping;strong convergence

1 Introduction

Let E be a real Banach space and C?E be a nonempty closed and convex，and K:C→2Cbe a multivalued mapping such that for all x∈C，K (x) is a nonempty closed and convex subset of C.Suppose that f:E×E→R is a bifunction.The quasi-equilibrium problem (QEP (f，K)) is to find x*∈K (x*) such that

The set of solutions of QEP (f，K) is denoted by QS (f，K).Also，the set of all fixed points of K is denoted by Fix (K).The associated Minty quasi-equilibrium problem is to find x*∈K (x*) such that f (y，x*)≤0 for all y∈K (x*).When K (x)=C for all x∈C，the quasi-equilibrium problem QEP (f，K) becomes a classical equilibrium problem EP (f，C)，also the associated Minty quasi-equilibrium problem becomes a classical Minty equilibrium problem (see[3，31]).

Equilibrium problems extend and unify many problems in optimization，variational inequalities，fi xed point theory，complementarity problems，Nash equilibria and many other problems in nonlinear analysis (see[12，15，16]).

Lately many researchers have studied approximation of solutions of an equilibrium problem (see，e.g.[15，16，18，20，22，24，27，30]and references therein).Equilibrium problems with monotone and pseudo-monotone bifunctions have been studied extensively in Hilbert，Banach as well as in topological vector spaces by many authors (e.g.[4，6，7，11，12，16，20，21，23]).Recently the extragradient method for equilibrium problems and vector equilibrium problems was studied in[13，14，19].Also，in[8]，a hybrid extragradient method was used for studying a system of unrelated mixed equilibrium problems.

Recently，quasi-equilibrium problems were studied in[3，9，31].In[10]，the authors studied the proximal point method for solving quasi-equilibrium problems in Banach spaces without assuming neither pseudo-monotonicity nor any weak continuity assumption of the bifunction in its arguments.Also，in[9]，the authors studied the extragradient method with linesearch for solving quasi-equilibrium problems in Banach spaces，assuming neither any monotonicity assumption on the bifunction nor any weak continuity assumption of the bifunction in its arguments.In both papers，they also showed the boundedness of the generated sequences implies that the solution set of the quasi-equilibrium problem is nonempty.

Motivated and inspired by the results in[9，10]，we study an extragradient method without linesearch for solving quasi-equilibrium problems in Banach spaces by assuming Lipschitz-type continuity assumption on the bifunction.Then we prove strong convergence of the generated sequence to a solution of the quasi-equilibrium problem.

This paper is organized as follows.In Section 2，we introduce some preliminary material related to the geometry of Banach spaces.In Section 3，we prove strong convergence of the sequence generated by the extragradient algorithm to a solution of quasi-equilibrium problems.

2 Preliminaries

Let E be a real Banach space with norm‖·‖and E*denote the dual of E.We denote the value of v∈E*at x∈E by〈x，v〉.When{xn}is a sequence in E，we denote strong convergence of{xn}to x∈E by xn→x and weak convergence by xn?x.The duality mapping J from E intois defined by:

Jx={v∈E*:〈x，v〉=‖x‖2=‖v‖2}

for x∈E.It is well known that when E is smooth the duality operator J is single valued.In the sequel，we recall the definitions of strictly convex，uniformly convex，smooth and uniformly smooth Banach spaces.A Banach space E is said to be strictly convex if＜1 for all x，y∈E with‖x‖=‖y‖=1 and xy.It is also said to be uniformly convex if for each ε∈(0，2]，there exists δ＞0 such that for all x，y∈E with‖x‖=‖y‖=1 and‖x-y‖≥ε，then＜1-δ.It is known that a uniformly convex Banach space is reflexive and strictly convex.A Banach space E is said to be smooth if the limit

exists for all x，y∈U={z∈E:‖z‖=1}.It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for x，y∈U.

Let E be a smooth Banach space.We use the following function studied by Alber[1]，Kamimura and Takahashi[17]and Reich[26]:

for all x，y∈E.Note that the function φ is a special Bregman distance (for more details，see[25]).It is obvious from the definition of φ that

for all x，y∈E.Notice that the duality mapping is the identity operator in Hilbert spaces.Therefore，if E is a Hilbert space，then φ(x，y)=‖x-y‖2.

Proposition 2.1([17]) Let E be a uniformly convex and smooth Banach space and let{xn}and{yn}be two sequences of E.If=0 and either{xn}or{yn}is bounded，then=0.

Proposition 2.2([17]) Let E be a reflexive，strictly convex and smooth Banach space，and let C be a nonempty closed convex subset of E and x∈E.Then there exists a unique element∈C such that

Regarding Proposition 2.2，we denote the unique element∈C by ΠC(x)，where the mapping ΠCis called the generalized projection from E onto C.It is obvious that in Hilbert spaces，ΠCis coincident with the metric projection from E onto C.In the sequel，we need the following proposition.

Proposition 2.3([17]) Let E be a smooth Banach space，C be a convex subset of E，x∈E and∈C.Then

if and only if

or equivalently，

For more details and properties of the geometry of Banach spaces，see[2].

Proposition 2.4([17]) If E is uniformly smooth，then J is uniformly norm-to-norm continuous on each bounded subset of E.

Throughout this paper we assume that E is a real Banach space which is uniformly convex and uniformly smooth unless otherwise specified.In the following definitions，suppose that C?E is nonempty closed and convex.

Definition 2.5([10]) The mapping T:C→C is called quasi φ-nonexpansive whenever Fix (T)? and φ(p，Tx)≤φ(p，x) for all (p，x)∈Fix (T)×C.

Although we will give an appropriate example of quasi φ-nonexpansive operators，but for more examples and properties of nonexpansive operators in Banach spaces，see[5，26].

Definition 2.6([10]) Suppose that K:C→2Cis a multivalued mapping such that for every x∈C，K (x) is nonempty closed and convex.K is called quasi φ-nonexpansive whenever the mapping T (·)=ΠK (·)(·) is quasi φ-nonexpansive where Π is the generalized projection.

Definition 2.7We say that K:C→2Cis lower semicontinuous at each∈C，whenever we have{xk}?C and xk→，then for any，there is a sequence{yk}with yk∈K (xk) for every k，such that yk→as k→∞.

Definition 2.8([10]) We say that K:C→2Cis demiclosed，whenever we have xk?andd (xk，K (xk))=0，then∈Fix (K).

It is well known that if T is a quasi φ-nonexpansive mapping，then Fix (T) is convex.Also，if T is demiclosed then Fix (T) is closed (see Remark 3.1 of[10]).

We introduce some conditions on the bifunction f and the multivalued mapping K which are needed in the convergence analysis.

B1f (·，·):E×E→R is continuous.

B2f is φ-Lipschitz-type continuous，i.e.，there exist two positive constants c1and c2such that

f (x，y)+f (y，z)≥f (x，z)-c1φ(y，x)-c2φ(z，y)，?x，y，z∈E.

B3f (x，·):E→R is convex for all x∈E.

B4K:C→2Cis quasi φ-nonexpansive，demiclosed and lower semicontinuous at each x∈C.

We present some comments on these assumptions.Regarding B1，we mention that our assumption is weaker than the corresponding assumption in[9].But we use an additional assumption for solving quasi-equilibrium problems which is denoted by B2.Note that B2 is an essential assumption to study the extragradient method without linesearch for solving quasiequilibrium problems.In addtion，B2 implies that f (x，x)≥0 for all x∈E.

We give now an elementary example of a bifunction f that satisfies B1-B3 with Lipschitz constants c1=c2=.This elementary example shows that our assumptions on the problem are appropriate.

Example 2.9Let f (x，y)=x2-xy.It is clear that f satisfies B1 and B3，hence we check B2.Take x，y，z∈R，then we have

We also give an example of a multivalued mapping K which implies that the assumption B4 is suitable (see Example 4.1 of[10]).

Example 2.10Let E=?p=for 1＜p＜∞，and let C=，and K (·):C→2Cbe defined by K (x)=for each x∈C，where B (x，) denotes the closed ball of radiuscentered at x.It can be shown that K (·):C→2Cis a multivalued mapping with nonempty closed and convex values，which is quasi φ-nonexpansive，lower semicontinuous and demiclosed.Hence B4 is satisfied.

In order to well definedness and boundedness of the generated sequences by the extragradient method in this paper，we assume that the solution set of the quasi-equilibrium problem is nonempty，in fact，we assume that the set

DS (f，K):={x∈K (x):f (y，x)≤0，?y∈C}，

is nonempty and then we prove our convergence results.

3 Strong Convergence of the Extragradient Algorithm

In this section，we study the strong convergence of the sequence generated by the extragradient (Korpelevich’s) method for solving quasi-equilibrium problems in Banach spaces.In order to find the relationship between Korpelevich’s extragradient method and the following algorithm，see[13].We will assume that E is a uniformly smooth and uniformly convex Banach space，C?E is nonempty closed and convex.Let K:C→2Cbe a multivalued quasi φ-nonexpansive mapping，and let f:E×E→R be a bifunction，and the assumptions B1-B4 are satisfied.We also assume that DS (f，K)?.

Algorithm 3.1(1) InitializationTake x0∈C and 0＜α≤λk≤β＜and γk∈for some positive real numbers α，β and ε，and for all k.

(2) Iterative stepGiven xn，define

If n=0，set C0=C∩H0.Otherwise，let

Determine the next approximation xn+1as

where

In order to prove the strong convergence of the sequences generated by Algorithm 3.1，we need the following lemmas.

Lemma 3.2If DS (f，K)?，then the sequences{yn}，{zn}，{un}and{vn}generated by Algorithm 3.1 are well defined.

ProofFor all x∈C，K (x) is a nonempty closed and convex subset of C.Therefore it is obvious that the sequence{yn}is well defined.

Now we prove that the sequences{zn}and{un}are well defined.It is enough to show that the sequence{zn}is well defined.Consider the function

Since γk∈，we have Dn?Hn.Let x*∈DS (f，K).Therefore x*∈Fix (K).Since yn=，and ΠK (·)(·) is a quasi φ-nonexpansive mapping，we have

for all x*∈DS (f，K).Therefore DS (f，K)?Hnfor all n.Hence DS (f，K)?Cnfor all n.It is easy to see that Cnis a nonempty closed and convex subset of C.Therefore the sequence{vn}is well defined. □

Note that in Lemma 3.4，we will prove that the sequence{xn}generated by Algorithm 3.1 is well defined.But，to simplify the proof of the lemma，we first present the following lemma.

Lemma 3.3If DS (f，K)? and the sequences{xn}，{yn}，{zn}，{un}and{vn}are generated by Algorithm 1，then

where x*∈DS (f，K).

ProofBy (3.3)，we get

Therefore，

Similarly to this argument，since znsolve the minimization problem in (3.2)，we have:

Let x*∈DS (f，K).Therefore we have f (zn，x*)≤0.Next set y=x*in (3.12) and y=unin (3.13)，we obtain respectively

On the other hand，since f is φ-Lipschitz-type continuous，we have:

Note that by (3.14)，(3.15) and (3.16)，we obtain

From (3.17) and the definition of φ，it is easy to see

By hypothesis，we have x*∈Fix (K).Now，since yn=，and ΠK (·)(·) is a quasi φ-nonexpansive mapping，we have

Since DS (f，K)?Cnfor all n，hence x*∈Cnfor all n.On the other hand，we have vn=by (3.6).Therefore Proposition 2.3 implies that

This shows that

Now (3.18)，(3.19) and (3.21) imply that

Lemma 3.4If DS (f，K)?，then DS (f，K)?Mn∩Nn.Therefore the sequence{xn}generated by Algorithm 3.1 is well defined.

ProofFirst we show that DS (f，K)?Mn∩Nnfor all k≥0.Let

Since γn∈，we have Fn?Mn.Let x*∈DS (f，K)，then by a proof similar to the proof of Lemma 3.3，we can prove that

for all x*∈DS (f，K).Therefore DS (f，K)?Fnfor all n≥0，which implies that DS (f，K)?Mnfor all n≥0.Next，by induction，we show that DS (f，K)?Mn∩Nn，for all n≥0.Indeed，we have DS (f，K)?M0∩N0，because N0=E.Assume that DS (f，K)?Mn∩Nnfor some n≥0.Since xn+1=，we have by Proposition 2.3 that

〈z-xn+1，Jx0-Jxn+1〉≤0，?z∈Mn∩Nn.

Since DS (f，K)?Mn∩Nn，

Now，since〈z-xn+1，Jx0-Jxn+1〉≤0，?z∈DS (f，K)，hence the definition of Nn+1implies that DS (f，K)?Nn+1，and so DS (f，K)?Mn∩Nnfor all n≥0.Finally，since DS (f，K) is nonempty，hence Mn∩Nnis nonempty，therefore xn+1is well defined. □

Lemma 3.5If DS (f，K)?，then the sequences{xn}，{yn}and{vn}are bounded and

ProofFrom the definition of Nn，we have xn=.For each u∈DS (f，K)，since DS (f，K)?Nnandis the generalized projection onto Nn，we have〈u-xn，Jx0-Jxn〉≤0 by Proposition 2.3，this implies that

Thus，the sequence{xn}is bounded by (2.3).

Next the definition of xn+1implies that xn+1∈Nn.Therefore we have〈xn+1-xn，Jx0-Jxn〉≤0 by Proposition 2.3 which implies that

with deg p=deg q=N，N∈N+.If

Hence φ(xn，x0)≤φ(xn+1，x0).So，the sequence{φ(xn，x0)}is non-decreasing.On the other hand，since{xn}is bounded，thereforeexists.We also have

φ(xn+1，xn)≤φ(xn+1，x0)-φ(xn，x0).

Taking to the limit in the above inequality as n→∞，one gets

Now，by Proposition 2.1，we have

Let x*∈DS (f，K).Since yn=and ΠK (·)(·) is a quasi φ-nonexpansive mapping，i.e.，K (·) is a quasi φ-nonexpansive mapping，we have

Now the boundedness of{xn}implies that{yn}is bounded by using (3.25) and (2.3).Also，Lemma 3.3 shows that

where an argument similar to the above implies that{vn}is bounded.

Since xn+1∈Mn，from the definition of Mn，we have

Therefore，by the Cauchy-Schwarz inequality，we have

Now since{vn}and{xn}are bounded，=0 and γn≥ε＞0，we have=0.Therefore Proposition 2.1 implies that

On the other hand，since vn∈Hn，from the definition of Hn，we have

Again，by the Cauchy-Schwarz inequality，we have

Since{xn}and{yn}are bounded，=0 and γn≥ε＞0，we have=0.Hence Proposition 2.1 shows that

Lemma 3.6If DS (f，K)? and the sequences{yn}，{zn}，{un}and{vn}are generated by Algorithm 3.1，then

ProofBy Lemma 3.3，we have

where x*∈DS (f，K).Note that (3.32) implies that

Uniform smoothness of E implies uniform norm-to-norm continuity of J on each bounded set of E.Therefore，from Lemma 3.5，we get

We also have 0＜α≤λk≤β＜for some positive real numbers α，β by hypothesis.Therefore (3.33) implies that

Now (2.3) implies that

Proposition 3.7Let f be a bifunction，K (·) be a multivalued quasi φ-nonexpansive mapping and the assumptions B1-B4 are satisfied.If there exists a subsequenceof{xn}such that?p，then p∈C∞∩Fix (K)，where C∞=.

ProofBy Lemma 3.5 we have=0.This implies that

Then since K is demiclosed，therefore we have p∈Fix (K).Now we prove that p∈C∞.Note that=0 by Lemma 3.5，hence we have?p.Since C∞=，it is sufficient to prove that p∈Ckfor all integers k to obtain that p∈C∞.Note that the sequence{Cn}is nonincreasing，now let N be a fixed integer，hence there is j＞N such that for all k≥j we have

Consequently，the set CNis closed and p∈CN.Since N is arbitrary，we obtain that

Proposition 3.8Assume that f is a bifunction，K (·) is a multivalued quasi φ-nonexpansive mapping and the assumptions B1-B4 are satisfied.If DS (f，K)?，then the sequence{xn}generated by Algorithm 3.1 converges strongly to an element of C∞∩Fix (K).

ProofSince{xn}is bounded by lemma 3.5，hence there exists a subsequenceof{xn}such that?p as n→∞.Then Proposition 3.7 shows that p∈C∞∩Fix (K) and hence C∞∩Fix (K)?.It is clear that C∞∩Fix (K) is closed and convex，hence we can definewhere Π is the generalized projection map onto C∞∩Fix (K).From the definition of Nn，we have xn=.Since C∞∩Fix (K)?Nnandis the generalized projection map onto Nn，hence for∈C∞∩Fix (K)?Nn，we have≤0 by Proposition 2.3.This implies that φ(xn，x0)≤.Therefore we have

Theorem 3.9Suppose that f is a bifunction，K (·) is a multivalued quasi φ-nonexpansive mapping and the assumptions B1-B4 are satisfied.If DS (f，K)? then the sequence{xn}generated by Algorithm 3.1 converges strongly to a solution of the quasi-equilibrium problem QEP (f，K).

ProofBy the proof of Lemma 3.6，we have

Since f (zn，zn)≥0 by B2，replacing y with znin (3.12)，we have

In the sequel，by (3.15) and (3.16)，we have

This implies that

Taking limit from (3.39) and (3.41)，using (3.37) and (3.38)，we have

Note that by (3.12)，we have

On the other hand，since xn→by Proposition 3.8.Hence，by Lemmas 3.5 and 3.6，we have zn→.Now，take any y∈，since K is lower semicontinuous at∈C，therefore there is a sequence{ξn}such that ξn∈K (xn) and ξn→y.Replacing y with ξnin (3.43)，and then taking limit when n→∞，we have