1. Introduction and Preliminaries

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

(1.1)

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

(1.2)

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 Tx0 = y0.

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 2E*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 limn→∞||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

(1.3)

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

(1.4)

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

(1.5)

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 supt↓0f (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

(1.6)

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:

(1.7)

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 x0 = ∏ 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 p1, p2 ∈ F (T), t ∈ (0, 1), putting p3 = tp1 + (1 - t)p2, we prove that Tp3 = p3. Indeed, from the definition of ϕ, we see that

This implies that p3 ∈ 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 p1, p2 ∈ F(T), t ∈ (0, 1), putting p3 = tp1 + (1 - t)p2, we prove that Tp3 = p3.

Indeed, from the definition of ϕ, we see that

This implies that

From (1.5), we see that

(1.8)

It follows that

(1.9)

This shows that the sequence {J(Tnp3)}is bounded. Note that E* is reflexive; we may, without loss of generality, assume that J(Tnp3) ⇀ 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 infn→ ∞on the both sides of above equality, we obtain that

This implies that p3 = e, that is, Jp3 = e*. It follows that J(Tnp3) ⇀ Jp3 ∈ 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 Tn p3 ⇀ p3. By virtue of the Kadec-Klee property of E and (1.8), we have Tnp3 → p3 as n → ∞. Hence

as n → ∞. In view of the closedness of T, we can obtain that p3 ∈ 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, 2r] → 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.

2. Main Results

Theorem 2.1. Let E be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property and C a nonempty closed convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A 1)-(A 4) and T : C → C a closed and asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C andis nonempty and bounded. Let {x n } be a sequence generated in the following manner:

wherefor each n ≥ 1, {α n } is a real sequence in [0, 1] such that lim infn→ ∞α n (1 - α n ) > 0, {r n } is a real sequence in [a, ∞), where a is some positive real number and J is the duality mapping on E. Then the sequence {x n } converges strongly to, whereis the generalized projection from E onto.

Proof. First, we show that C n is closed and convex by induction on n ≥ 1. It is obvious that C1 = C is closed and convex. Suppose that C m is closed and convex for some integer m. For z ∈ C m , we see that ϕ(z, u m ) ≤ ϕ(z, x m ) + (k m -1)M m is equivalent to

It is easy to see that Cm+1is closed and convex. This proves that C n is closed and convex for each n ≥ 1. This in turn shows that x0 is well defined. Putting , we from Lemma 1.8 see that is quasi-ϕ-nonexpansive.

Now, we are in a position to prove that for each n ≥ 1. Indeed, is obvious. Suppose that for some positive integer m. Then, , we have

(2.1)

which shows that w ∈ Cm+1. This implies that for each n ≥ 1.

On the other hand, it follows from Lemma 1.4 that

for each and for each n ≥ 1. This shows that the sequence ϕ(x n , x0) is bounded. From (1.5), we see that the sequence {x n } is also bounded. Since the space is reflexive, we may, without loss of generality, assume that x n ⇀ p. Not that C n is closed and convex for each n ≥ 1. It is easy to see that p ∈ C n for each n ≥ 1. Note that

It follows that

This implies that

Hence, we have ||x n || → ||p|| as n → ∞. In view of the Kadec-Klee property of E, we obtain that x n → p as n → ∞.

Next, we show that p ∈ F(T). By the construction of C n , we have that Cn+1⊂ C n and . It follows that

Letting n → ∞, we obtain that ϕ(xn+1, x n ) → 0. In view of xn+1∈ Cn+1, we have

It follows that

(2.2)

From (1.5), we see that

(2.3)

It follows that

(2.4)

This implies that {Ju n } is bounded. Note that E is reflexive and E* is also reflexive. We may assume that Ju n → x* ∈ E*. In view of the reflexivity of E, we see that J(E) = E*. This shows that there exists an x ∈ E such that Jx = x*. It follows that

Taking lim infn→∞the both sides of above equality yields that

That is, p = x, which in turn implies that x* = Jp. It follows that Ju n → Jp ∈ E*. From (2.4) and E* has the Kadec-Klee property, we obtain that

Note that J-1 : E* → E is demi-continuous. It follows that u n → p. From (2.3) and E has the Kadec-Klee property, we obtain that

(2.5)

Note that

It follows that

(2.6)

Since J is uniformly norm-to-norm continuous on any bounded sets, we have

(2.7)

Let r = supn≥0{||x n ||, ||Tnx n ||}. Since E is uniformly smooth, we know that E* is uniformly convex. In view of Lemma 1.7, we see that

It follows that

On the other hand, we have

It follows from (2.6) and (2.7) that

(2.8)

In view of limn→∞(k n -1) M n = 0 and (2.8) and the assumption lim infn→∞α n (1 - α n ) > 0, we see that

It follows from the property of g that

(2.9)

Since x n → p as n →∞ and J : E → E* is demi-continuous, we obtain that Jx n → Jp ∈ E*. Note that

This implies that ||Jx n || → ||Jp|| as n → ∞. Since E* has the Kadec-Klee property, we see that

(2.10)

Note that

From (2.9) and (2.10), we obtain at

(2.11)

Note that J-1 : E* → E is demi-continuous. It follows that Tnx n → p. On the other hand, we have

In view of (2.11), we obtain that ||Tnx n || → ||p|| as n → ∞. Since E has the Kadec-Klee property, we obtain that

(2.12)

Note that

It follows from the asymptotic regularity of T and (2.12) that

That is, TTnx n - p → 0 as n → ∞: It follows from the closedness of T that Tp = p:

Next, we show that p ∈ EF(f): From (2.1), we have

(2.13)

In view of and Lemma 1.8, we obtain

(2.14)

It follows from (2.8) that

From (1.5), we see that ||u n || - ||y n || → 0 as n → ∞. In view of u n → p as n → ∞, we have

(2.15)

It follows that

(2.16)

Since E* is reflexive, we may assume that Jy n → q*∈ E*: In view of J(E) = E*, we see that there exists q ∈ E such that Jq = q*. It follows that

Taking lim infn→∞the both sides of above equality yields that

That is, p = q, which in turn implies that q* = Jp. It follows that Jy n → Jp ∈ E*. From (2.16) and E* has the Kadec-Klee property, we obtain that

Note that J-1 : E* → E is demi-continuous. It follows that y n → p. From (2.15) and E has the Kadec-Klee property, we obtain that

(2.17)

Note that

It follows from (2.5) and (2.17) that

(2.18)

Since J is uniformly norm-to-norm continuous on any bounded sets, we have

From the assumption r n ≥ a, we see that

(2.19)

In view of , we see that

It follows from the condition (A 2) that

By taking the limit as n → ∞ in the above inequality, we from conditions (A 4) and (2.19) obtain that

For 0 < t < 1 and y ∈ C, define y t = ty + (1 - t)p. It follows that y t ∈ C, which yields that f(y t , p) ≤ 0. It follows from conditions (A 1) and (A 4) that

That is,

Letting t ↓ 0, from condition (A 3), we obtain that f(p, y) ≥ 0 ∀y ∈ C: This implies that p ∈ EP(f). This shows that .

Finally, we prove that . From , we see that

Since for each n ≥ 1, we have

(2.20)

Letting n → ∞ in (2.20), we see that

In view of Lemma 1.3, we can obtain that . This completes the proof.

Remark 2.2. Theorem 2.1 improves Theorem QCK in the following aspects:

  1. (a)

    From a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property;

  2. (b)

    From a quasi-ϕ-nonexpansive mapping to an asymptotically quasi-ϕ-non-expansive mapping.

From the definition of quasi-ϕ-nonexpansive mappings, we see that every quasi-ϕ-nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive with the constant sequence {1}. From the proof of Theorem 2.1, we have the following results immediately.

Corollary 2.3. Let E be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property and C a nonempty closed convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A 1)-(A 4) and T : C → C a closed and quasi- ϕ -nonexpansive mapping. Assume thatis nonempty.

Let {x n } be a sequence generated in the following manner:

where {α n } is a real sequence in [0, 1] such that lim infn→∞α n (1 - α n ) > 0, {r n } is a real sequence in [a, ∞), where a is some positive real number and J is the duality mapping on E. Then the sequence {x n } converges strongly to, whereis the generalized projection from E onto.

Remark 2.4. Corollary 2.3 improves Theorem TZ in the following aspects.

  1. (a)

    For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property (note that every uniformly convex Banach space has the Kadec-Klee property).

  2. (b)

    For the mappings, we extend the mapping from a relatively nonexpansive mapping to a quasi-ϕ-nonexpansive mapping (we remove the restriction , where denotes the asymptotic fixed point set).

  3. (c)

    For the algorithms, we remove the set "W n " in Theorem TZ.