Let *E* be a real Banach space, *E** the dual space of *E* and *C* a nonempty closed convex subset of *E*. Let *f* be a bifunction from *C* × *C* to ℝ, where ℝ denotes the set of real numbers.

In this paper, we consider the following equilibrium problem. Find *p* ∈ *C* such that

We denote *EP*(*f*) the solution set of the equilibrium problem (1.1). That is,

Given a mapping *Q* : *C* → *E**, let

Then *p* ∈ *EP*(*f*) if and only if *p* is a solution of the following variational inequality problem. Find *p* such that

Numerous problems in physics, optimization and economics reduce to find a solution of (1.1) (see [1–4]). Let *T* : *C* → *C* be a mapping.

The mapping *T* is said to be asymptotically regular on *C* if for any bounded subset *K* of *C*,

The mapping *T* is said to be closed if for any sequence {*x*_{
n
} } ⊂ *C* such that

and

then *Tx*_{0} = *y*_{0}.

A point *x* ∈ *C* is a fixed point of *T* provided *Tx* = *x*. In this paper, we denote *F*(*T*) the fixed point set of *T* and denote → and ⇀ the strong convergence and weak convergence, respectively.

Recall that the mapping *T* is said to be nonexpansive if

*T* is said to be quasi-nonexpansive if *F*(*T*) ≠ Ø and

*T* is said to be asymptotically nonexpansive if there exists a sequence {*k*_{
n
} } ⊂ [1, ∞) with *k*_{
n
} → 1 as *n* → ∞ such that

*T* is said to be asymptotically quasi-nonexpansive if *F*(*T*) ≠ Ø and there exists a sequence {*k*_{
n
} } ⊂ [1, ∞) with *k*_{
n
} → 1 as *n* → ∞ such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] in 1972. They proved that if *C* is nonempty bounded closed and convex then every asymptotically nonexpansive self-mapping *T* on *C* has a fixed point in uniformly convex Banach spaces. Further, the fixed point set of *T* is closed and convex.

Recently, many authors considered the problem of finding a common element in the set of fixed points of a nonexpansive mapping and in the set of solutions of the equilibrium problem (1.1) based on iterative methods in the framework of real Hilbert spaces; see, for instance [4, 6–14] and the references therein. However, there are few results presented in Banach spaces.

In this paper, we will consider the problem in a Banach space. Before proceeding further, we give some definitions and propositions in Banach spaces.

Let *E* be a Banach space with the dual *E**. We denote by *J* the normalized duality mapping from *E* to 2^{E*}defined by

where 〈•,•〉 denotes the generalized duality pairing.

A Banach space *E* is said to be strictly convex if for all *x*, *y* ∈ *E* with ||*x*|| = ||*y*|| = 1 and *x* ≠ *y*. It is said to be uniformly convex if lim_{n→∞}||*x*_{
n
}- *y*_{
n
}|| = 0 for any two sequences {*x*_{
n
}} and {*y*_{
n
}} in *E* such that ||*x*_{
n
}|| = ||*y*_{
n
}|| = 1 and

Let *U*_{
E
} = {*x* ∈ *E* : ||*x*|| = 1} be the unit sphere of *E*. Then the Banach space *E* is said to be smooth provided

exists for each *x*, *y* ∈ *U*_{
E
} . It is said to be uniformly smooth if the limit (1.3) is attained uniformly for *x*, *y* ∈ *U*_{
E
} . It is well known that if *E* is uniformly smooth, then *J* is uniformly norm-to-norm continuous on each bounded subset of *E*. It is also well known that if *E* is uniformly smooth if and only if *E** is uniformly convex.

Recall that a Banach space *E* has the Kadec-Klee property [15–17], if for any sequence {*x*_{
n
} } ⊂ *E* and *x* ∈ *E* with *x*_{
n
} ⇀ *x* and ||*x*_{
n
} || → ||*x*||, then ||*x*_{
n
} - *x*|| → 0 as *n* → ∞. It is well known that if *E* is a uniformly convex Banach space, then *E* has the Kadec-Klee property.

As we all know that if *C* is a nonempty closed convex subset of a Hilbert space *H* and *P*_{
C
} : *H* → *C* is the metric projection of *H* onto *C*, then *P*_{
C
} is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [18] recently introduced a generalized projection operator Π _{
C
} in a Banach space *E* which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that *E* is a smooth Banach space. Consider the functional defined by

Observe that, in a Hilbert space *H*, (1.4) is reduced to *ϕ*(*x*, *y*) = ||*x-y*||^{2} , *x*, *y* ∈ *H*. The generalized projection Π _{
C
} : *E* → *C* is a mapping that assigns to an arbitrary point *x* ∈ *E* the minimum point of the functional *ϕ*(*x*, *y*), that is, , where is the solution to the minimization problem

The existence and uniqueness of the operator Π _{
C
} follows from the properties of the functional *ϕ*(*x*, *y*) and strict monotonicity of the mapping *J* (see, for example, [15, 17–19]). We know that Π _{
C
} = *P*_{
C
} in Hilbert spaces. It is obvious from the definition of function *ϕ* that

**Remark 1.1**. Let *E* be a reflexive, strictly convex and smooth Banach space. Then for *x*, *y* ∈ *E*, *ϕ*(*x*, *y*) = 0 if and only if *x* = *y*. It is sufficient to show that if *ϕ*(*x*, *y*) = 0 then *x* = *y*. From (1.5), we have ||*x*|| = ||*y*||. This implies that 〈*x*, *Jy*〉 = ||*x*||^{2} = ||*Jy*||^{2}. From the definition of *J*, we have *Jx* = *Jy*. Therefore, we have *x* = *y* (see [15, 17]).

Let *C* be a nonempty closed convex subset of *E* and *T* a mapping from *C* into itself. A point *p* in *C* is said to be an asymptotic fixed point of *T*[20] if *C* contains a sequence {*x*_{
n
} } which converges weakly to *p* such that

The set of asymptotic fixed points of *T* will be denoted by .

A mapping *T* from *C* into itself is said to be relatively nonexpansive [21–23] if and

for all *x* ∈ *C* and *p* ∈ *F*(*T*).

The mapping *T* is said to be relatively asymptotically nonexpansive [24] if and there exists a sequence {*k*_{
n
} } ⊂ [1, ∞) with *k*_{
n
} → 1 as *n* → ∞ such that

for all *x* ∈ *C*, *p* ∈ *F*(*T*) and *n* ≥ 1. The asymptotic behavior of a relatively nonexpansive mapping was studied in [21–23].

The mapping *T* is said to be *ϕ*-nonexpansive if

for all *x*, *y* ∈ *C*.

The mapping *T* is said to be quasi-*ϕ*-nonexpansive [25–27] if *F*(*T*) ≠ ∅ and

for all *x* ∈ *C* and *p* ∈ *F*(*T*).

The mapping *T* is said to be asymptotically *ϕ*-nonexpansive if there exists a sequence {*k*_{
n
} } ⊂ [1, ∞) with *k*_{
n
} → 1 as *n* → ∞ such that

for all *x*, *y* ∈ *C*.

The mapping *T* is said to be asymptotically quasi-*ϕ*-nonexpansive [27, 28] if *F*(*T*) ≠ ∅ and there exists a sequence {*k*_{
n
} } ⊂ [0, ∞) with *k*_{
n
} → 1 as *n* → ∞ such that

for all *x* ∈ *C*, *p* ∈ *F*(*T*) and *n* ≥ 1.

**Remark 1.2**. The class of (asymptotically) quasi-*ϕ*-nonexpansive mappings is more general than the class of relatively (asymptotically) nonexpansive mappings which requires the restriction: . In the framework of Hilbert spaces, (asymptotically) quasi-*ϕ*-nonexpansive mappings is reduced to (asymptotically) quasi-nonexpansive mappings (cf. [29–32]).

We assume that *f* satisfies the following conditions for studying the equilibrium problem (1.1).

(A1): *f*(*x*, *x*) = 0∀*x* ∈ *C*;

(A2): *f* is monotone, i.e., *f*(*x*, *y*) + *f*(*y*, *x*) ≤ 0∀*x*, *y* ∈ *C*;

(A3): lim sup_{t↓0}*f* (*tz* + (1 - *t*)*x*, *y*) ≤ *f*(*x*, *y*)∀*x*, *y*, *z* ∈ *C*;

(A4): for each *x* ∈ *C*, *y* α *f*(*x*, *y*) is convex and weakly lower semi-continuous.

Recently, Takahashi and Zembayshi [33] considered the problem of finding a common element in the fixed point set of a relatively nonexpansive mapping and in the solution set of the equilibrium problem (1.1) (cf. [32]).

**Theorem TZ**. ([33]) *Let E be a uniformly smooth and uniformly convex Banach space and let C be a nonempty closed convex subset of E. Let f be a bifunction from C* × *C to* ℝ *satisfying* (*A* 1)*-*(*A* 4) *and let T be a relatively nonexpansive mapping from C into itself such that F*(*T*) ∩ *EP*(*f*) ≠ Ø. *Let* {*x*_{
n
} } *be a sequence generated by*

*for every n* ≥ 0, *where J is the duality mapping on E*, {*α*_{
n
} } ⊂ [0, 1] *satisfies*

*and* {*r*_{
n
}} ⊂ [*a*, ∞) *for some a* > 0. *Then* {*x*_{
n
}} *converges strongly to* ∏_{F(T)∩EP(f)}*x*, *where* ∏_{F(T)∩EP(f)}*is the generalized projection of E onto F* (*T*) ∩ *EP* (*f* ).

Very recently, Qin et al. [25] further improved Theorem TZ by considering shrinking projection methods which were introduced in [34] for quasi-*ϕ*-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.

**Theorem QCK**. [25]*Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Ban ach space E. Let f be a bifunction from C* × *C to* ℝ *satisfying* (*A* 1)-(*A* 4) *and let T* : *C* → *C be a closed quasi-ϕ-nonexpansive mappings such that*. *Let* {*x*_{
n
}} *be a sequence generated in the following manner:*

*where J is the duality mapping on E and* {*α*_{
n
} } *is a sequence in* [0, 1] *satisfying*

*and* {*r*_{
n
} } ⊂ [*a*, ∞) *for some a* > 0. *Then* {*x*_{
n
} } *converges strongly to*.

In this paper, we considered the problem of finding a common element in the fixed point set of an asymptotically quasi-*ϕ*-nonexpansive mapping which is an another generalization of asymptotically nonexpansive mappings in Hilbert spaces and in the solution set of the equilibrium problem (1.1). The results presented this paper mainly improve the corresponding results announced in [33].

In order to prove our main results, we need the following lemmas.

**Lemma 1.3**. [18]*Let C be a nonempty closed convex subset of a smooth Banach space E and x* ∈ *E. Then x*_{0} = ∏_{
C
}*x if and only if*

**Lemma 1.4**. [18]*Let E be a reflexive, strictly convex and smooth Banach space, C a nonempty closed convex subset of E and x* ∈ *E*. *Then*

**Lemma 1.5**. *Let E be a strictly convex and smooth Banach space, C a nonempty closed convex subset of E and T* : *C* → *C a quasi-* *ϕ* *-nonexpansive mapping. Then F*(*T*) *is a closed convex subset of C*.

*Proof*. Let {*p*_{
n
} } be a sequence in *F*(*T* ) with *p*_{
n
} → *p* as *n* → ∞. Then we have to prove that *p* ∈ *F*(*T*) for the closedness of *F*(*T*). From the definition of *T*, we have

which implies that *ϕ*(*p*_{
n
} , *Tp*) → 0 as *n* → ∞. Note that

Letting *n* → ∞ in the above equality, we see that *ϕ*(*p*, *Tp*) = 0. This shows that *p* = *Tp*.

Next, we show that *F*(*T*) is convex. To end this, for arbitrary *p*_{1}, *p*_{2} ∈ *F* (*T*), *t* ∈ (0, 1), putting *p*_{3} = *tp*_{1} + (1 - *t*)*p*_{2}, we prove that *Tp*_{3} = *p*_{3}. Indeed, from the definition of *ϕ*, we see that

This implies that *p*_{3} ∈ *F* (*T* ). This completes the proof.

Now we will improve the above Lemma 1.6 as follows.

**Lemma 1.6**. *Let E be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property, C a nonempty closed convex subset of E and T* : *C* → *C a closed and asymptotically quasi-* *ϕ* *-nonexpansive mapping. Then F*(*T*) *is a closed convex subset of C*.

*Proof*. It is easy to check that the closedness of *F*(*T*) can be deduced from the closedness of *T*. We mainly show that *F*(*T*) is convex. To end this, for arbitrary *p*_{1}, *p*_{2} ∈ *F*(*T*), *t* ∈ (0, 1), putting *p*_{3} = *tp*_{1} + (1 - *t*)*p*_{2}, we prove that *Tp*_{3} = *p*_{3}.

Indeed, from the definition of *ϕ*, we see that

This implies that

From (1.5), we see that

It follows that

This shows that the sequence {*J*(*T*^{n}*p*_{3})}is bounded. Note that *E** is reflexive; we may, without loss of generality, assume that *J*(*T*^{n}*p*_{3}) ⇀ *e** ∈ *E**. In view of the reflexivity of *E*, we have *J*(*E*) = *E**. This shows that there exists an element *e* ∈ *E* such that *Je* = *e**. It follows that

Taking lim inf_{n→ ∞}on the both sides of above equality, we obtain that

This implies that *p*_{3} = *e*, that is, *Jp*_{3} = *e**. It follows that *J*(*T*^{n}*p*_{3}) ⇀ *Jp*_{3} ∈ *E**.

In view of the Kadec-Klee property of *E** and (1.9), we have

Note that *J*^{-1} : *E** → *E* is demi-continuous, we see that *T*^{n} *p*_{3} ⇀ *p*_{3}. By virtue of the Kadec-Klee property of *E* and (1.8), we have *T*^{n}*p*_{3} → *p*_{3} as *n* → ∞. Hence

as *n* → ∞. In view of the closedness of *T*, we can obtain that *p*_{3} ∈ *F* (*T*). This shows that *F*(*T*) is convex. This completes of proof

**Lemma 1.7**. [35, 36]*Let E be a smooth and uniformly convex Banach space and let r >* 0. *Then there exists a strictly increasing, continuous and convex function g* : [0, 2*r*] → *R such that g*(0) = 0 *and*

*for all x*, *y* ∈ *B*_{
r
} = {*x* ∈ *E* : ||*x*|| ≤ *r*} *and t* ∈ [0, 1].

**Lemma 1.8**. *Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let f be a bifunction from C* × *C to* ℝ *satisfying* (*A* 1)*-*(*A* 4). *Let r >* 0 *and x* ∈ *E. Then we have the followings*.

(a): ([1]) *There exists z* ∈ *C such that*

(b): *(Refs*. [25, 33]*) Define a mapping T*_{
r
} : *E* → *C by*

*Then the following conclusions hold:*

(1): *S*_{
r
} *is single-valued;*

(2): *S*_{
r
} *is a firmly nonexpansive-type mapping, i.e., for all x*, *y* ∈ *E*,

(3): *F*(*S*_{
r
} ) = *EP*)(*f*);

(4): *S*_{
r
} *is quasi-* *ϕ* *-nonexpansive;*

(5): *ϕ*(*q*, *S*_{
r
}*x*) + *ϕ*(*S*_{
r
}*x, x*) ≤ *ϕ* (*q*, *x*), ∀*q* ∈ *F*(*S*_{
r
} );

(6): *EP*(*f*) *is closed and convex*.